FAKULTÄT FÜR
ELEKTROTECHNIK,
INFORMATIK UND
MATHEMATIK
Development and Real-time Implementation
of Digital Signal Processing Algorithms
for Coherent Optical Receivers
Zur Erlangung des akademischen Grades
DOKTORINGENIEUR (Dr.-Ing.)
der Fakultät für Elektrotechnik, Informatik und Mathematik
der Universität Paderborn
vorgelegte Dissertation
von
Dipl.-Ing. Timo Pfau
Stuttgart
Referent: Prof. Dr.-Ing. Reinhold Noé
Korreferent: Prof. Dr.-Ing. Ulrich Rückert
Tag der mündlichen Prüfung: 05.03.2009
Paderborn, den 13.03.2009
Diss. EIM-E/251
Abstract
The continuous increase of the worldwide data traffic demands new concepts for data
transmission in the optical fiber-based backbone networks. One promising way to increase
the capacity of the existing fiber infrastructure is to use multilevel modulation formats in
combination with polarization-multiplexing and coherent detection. Though elaborate
transmitters and receivers are required to transmit multiple bits per symbol, but this also
enables a very efficient utilization of the available bandwidth. The development of
coherent optical receivers thereby profits from advancements in integrated circuit
technologies that allow the digital realization of the required signal processing.
In this dissertation all necessary algorithms for the signal processing in a coherent digital
receiver are presented. The main focus thereby lies on the algorithms for polarization
control and carrier recovery. A digital polarization control is required to realize a
polarization-multiplexed transmission system without optical polarization control. Both a
non-data-aided and a decision-directed polarization control algorithm are presented. For
the latter an extension is proposed to enable also the compensation of intersymbol
interference.
The most time-critical task in coherent receivers for optical transmission systems is it to
recover the carrier phase from the received symbols. Due to the large linewidth of the
distributed feedback (DFB) lasers employed in commercial systems a high phase noise
tolerance is required. Several algorithms have been proposed to solve this problem This
dissertation compares the different approaches at the example of the quadrature phase shift
keying (QPSK) modulation format. Additionally a novel feed-forward carrier recovery for
arbitrary quadrature amplitude modulation (QAM) constellations is proposed. Together
with the other carrier recovery schemes it is analyzed for QPSK, but additionally also for
higher-level square QAM.
Finally the results of the real-time implementation of a polarization-multiplexed
synchronous optical QPSK transmission system are presented, which was developed in the
framework of the synQPSK project funded by the European Commission. The algorithms
implemented in the coherent receiver and their parameters are optimized based on the
simulation results of this thesis. Both the single-polarization QPSK transmission system
and the polarization-multiplexed QPSK transmission system presented in this dissertation
are the worldwide first that were realized with a real-time coherent digital receiver and
standard DFB lasers.
Zusammenfassung
Der kontinuierliche Anstieg des weltweiten Datenverkehrs erfordert neue
Datenübertragungskonzepte für die auf optischen Glasfasern basierenden Backbone-Netze.
Eine vielversprechende Möglichkeit, die Kapazität der bestehenden Glasfaser-Infrastruktur
zu erhöhen, ist der Einsatz von mehrstufigen Modulationsverfahren in Kombination mit
Polarisationsmultiplex und kohärentem Empfang. Zwar werden aufwendige Sender und
Empfänger benötigt, um mehrere Bit pro Symbol zu übertragen, aber das ermöglicht auch
eine sehr effiziente Nutzung der verfügbaren Bandbreite. Die Entwicklung kohärenter
optischer Empfänger profitiert dabei von den Fortschritten in der integrierten Schaltung-
stechnik, die eine digitale Realisierung der erforderlichen Signalverarbeitung ermöglicht.
In dieser Dissertation werden alle zur Signalverarbeitung in einem kohärenten digitalen
Empfänger benötigten Algorithmen vorgestellt. Der Schwerpunkt liegt dabei auf den
Algorithmen zur Polarisationsregelung und Trägerrückgewinnung. Eine digitale
Polarisationsregelung wird benötigt, um ein Übertragungssystem mit
Polarisationsmultiplex ohne optische Polarisationsregelung zu realisieren. Sowohl ein
datenunabhängiger und ein entscheidungsgesteuerter Polarisationsregel-Algorithmus
werden vorgestellt. Für letzteren wird eine Erweiterung vorgeschlagen, die zusätzlich die
Kompensation von Intersymbolstörungen ermöglicht.
Die zeitkritischste Aufgabe für den kohärenten Empfänger eines optischen
Übertragungssystems ist die Rückgewinnung der Trägerphase aus den empfangenen
Symbolen. Aufgrund der hohen Linienbreite der in kommerziellen Systemen eingesetzten
DFB-Laser wird eine hohe Phasenrauschtoleranz benötigt. Mehrere Algorithmen wurden
zur Lösung dieses Problems vorgeschlagen. Diese Dissertation vergleicht die
verschiedenen Ansätze am Beispiel der Quadratur-Phasenumtastung (QPSK). Zusätzlich
wird eine neuartige vorwärtsgekoppelte Trägerrückgewinnung für Quadratur-
Amplitudenmodulation (QAM) mit beliebigen Konstellationen vorgeschlagen. Zusammen
mit den anderen Verfahren zur Trägerrück-gewinnung wird sie für QPSK, aber auch für
höherstufige quadratische QAM analysiert.
Schließlich werden die Ergebnisse einer Echtzeit-Implementierung eines synchronen
optischen Übertragungssystems mit Polarisationsmultiplex vorgestellt, das im Rahmen des
EU-geförderten synQPSK-Projekts entwickelt wurde. Die in dem kohärenten Empfänger
implementierten Algorithmen und ihre zugehörigen Parameter wurden mithilfe der
Simulationsergebnisse dieser Arbeit optimiert. Sowohl das QPSK Übertragungssystem mit
einfacher Polarisation als auch das QPSK Übertragungssystem mit Polarisationsmultiplex
sind weltweit die ersten, die mit einem kohärenten digitalen Echtzeit-Empfänger und
Standard-DFB-Lasern realisiert wurden.
i
Publications
Articles
T. Pfau, S. Hoffmann, R. Noé, „Hardware-efficient Coherent Digital Receiver Concept
with Feed-forward Carrier Recovery for M-QAM Constellations”, IEEE J.
Lightwave Technol., accepted for publication
S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, R. Noé, „Multiplier-
free Realtime Phase Tracking for Coherent QPSK Receivers”, IEEE Photon.
Technol. Lett., Vol. 21, No. 3, Feb. 1, 2009, pp. 137-139
M. El-Darawy, T. Pfau, S. Hoffmann, R. Peveling, C. Wördehoff, B. Koch, M.
Porrmann, O. Adamczyk, R. Noé, “Fast Adaptive Polarization and PDL Tracking in
a Real-time FPGA-based Coherent PolDM-QPSK Receiver”, IEEE Photon.
Technol. Lett., Vol. 20, No. 21, Nov. 1, 2008, pp. 1796-1798
S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wördehoff, R. Peveling, M.
Porrmann, R. Noé, “Frequency and Phase Estimation for Coherent QPSK
Transmission with Unlocked DFB Lasers”, IEEE Photon. Technol. Lett., Vol. 20,
No. 18, Sept. 15, 2008, pp. 1569-1571
T. Pfau, S. Hoffmann, O. Adamczyk, R. Peveling, V. Herath, M. Porrmann, R. Noé,
“Coherent optical communication: Towards real-time systems at 40 Gbit/s and
beyond”, Optics Express, Vol. 16, No. 2, Jan. 21, 2008
T. Pfau, R. Peveling, J. Hauden, N. Grossard, H. Porte, Y. Achiam, S. Hoffmann, S.
Ibrahim, O. Adamczyk, S. Bhandare, D. Sandel, M. Porrmann, R. Noé, “Coherent
Digital Polarization Diversity Receiver for Real-Time Polarization-Multiplexed
QPSK Transmission at 2.8 Gbit/s”, IEEE Photon. Technol. Lett., Vol. 19, No. 24,
Dec. 15, 2007, pp. 1988-1990
T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, S. Bhandare, O. Adamczyk, M.
Porrmann, R. Noé, Y. Achiam, “Synchronous QPSK transmission at 1.6 Gbit/s with
standard DFB lasers and real-time digital receiver”, Electron. Lett., Vol. 42, No. 20,
Sept. 28, 2006, pp. 1175-1176
T. Pfau, S. Hoffmann, R. Peveling, S. Bhandare, S. K. Ibrahim, O. Adamczyk, M.
Porrmann, R. Noé, Y. Achiam, “First Real-Time Data Recovery for Synchronous
QPSK Transmission with Standard DFB Lasers”, IEEE Photon. Technol. Lett.,
Vol. 18, No. 18, Sept. 15, 2006, pp. 1907-1909
ii
Conference papers
T. Pfau, R. Peveling, V. Herath, S. Hoffmann, C. Wördehoff, O. Adamczyk, M.
Porrmann, R. Noé, “Towards Real-Time Implementation of Coherent Optical
Communications”, Proc. OFC/NFOEC’09, OThJ4 (invited), March 22-26, 2009,
San Diego, CA, USA
V. Herath, R. Peveling, T. Pfau, O. Adamczyk, S. Hoffmann, C. Wördehoff, M.
Porrmann, R. Noé, “Chipset for a Coherent Polarization-Multiplexed QPSK
Receiver”, Proc. OFC/NFOEC’09, OThE2, March 22-26, 2009, San Diego, CA,
USA
M. El-Darawy, T. Pfau, C. Wördehoff, B. Koch, S. Hoffmann, R. Peveling, M.
Porrmann, R. Noé, “Real-time 40 krad/s Polarization Tracking with 6 dB PDL in
Digital Synchronous Polarization-Multiplexed QPSK Receiver”, Proc. ECOC‘08,
We3.E.4, Sept. 21-25, 2008, Brussels, Belgium
T. Pfau, M. El-Darawy, C. Wördehoff, R. Peveling, S. Hoffmann, B. Koch, O.
Adamczyk, M. Porrmann, R. Noé, “32-krad/s Polarization and 3-dB PDL Tracking
in a Real-time Digital Coherent Polarization-Multiplexed QPSK Receiver”, Proc.
IEEE/LEOS Summer Topicals 2008, MC2.4, July 21-23, 2008, Acapulco, Mexico
R. Noé, S. Hoffmann, T. Pfau, O. Adamczyk, V. Herath, R. Peveling, M. Porrmann,
“Real-time Digital Polarization and Carrier Recovery in a Polarization Multiplexed
Synchronous Optical QPSK Transmission”, Proc. IEEE/LEOS Summer Topicals
2008, MC2.1 (invited), July 21-23, 2008, Acapulco, Mexico
S. Hoffmann, T. Pfau, O. Adamczyk, C. Wördehoff, R. Peveling, M. Porrmann, R.
Noé, S. Bhandare, “Frequency Estimation and Compensation for Coherent QPSK
Transmission with DFB Lasers”, Proc. COTA‘08, CWB4, July 13-16, 2008, Boston,
MA, USA
T. Pfau, R. Noé, “Real-time Digital Coherent QPSK Transmission: Algorithms and
Technologies”, Proc. HDOC-WS‘08, pp. 83-88, June 25-26, 2008, Tokyo, Japan
T. Pfau, C. Wördehoff, R. Peveling, S. K. Ibrahim, S. Hoffmann, O. Adamczyk, S.
Bhandare, M. Porrmann, R. Noé, A. Koslovsky, Y. Achiam, D. Schlieder, N.
Grossard, J. Hauden, H. Porte, “Ultra-fast Adaptive Digital Polarization Control in a
Real-time Coherent Polarization-Multiplexed QPSK Receiver”, Proc.
OFC/NFOEC‘08, OTuM3, Feb. 24-28, 2008, San Diego, CA, USA
T. Pfau, R. Peveling, F. Samson, J. Romoth, S. Hoffmann, S. Bhandare, S. Ibrahim, D.
Sandel, O. Adamczyk, M. Porrmann, R. Noé, J. Hauden, N. Grossard, H. Porte, D.
Schlieder, A. Koslovsky, Y. Benarush, Y. Achiam, “Polarization-Multiplexed 2.8
Gbit/s Synchronous QPSK Transmission with Real-Time Digital Polarization
Tracking”, Proc. ECOC‘07, 8.3.3, Sept. 16-20, 2007, Berlin, Germany
T. Pfau, R. Peveling, S. Hoffmann, S. Bhandare, S. Ibrahim, D. Sandel, O. Adamczyk,
M. Porrmann, R. Noé, Y. Achiam, D. Schlieder, A. Koslovsky, Y. Benarush, J.
Hauden, N. Grossard, H. Porte, “PDL-Tolerant Real-Time Polarization-Multiplexed
QPSK Transmission with Digital Coherent Polarization Diversity Receiver”, Proc.
IEEE/LEOS Summer Topicals’07, Ma3.3, July 23-25, 2007, Portland, OR, USA
iii
T. Pfau, O. Adamczyk, V. Herath, R. Peveling, S. Hoffmann, M. Porrmann, R. Noé,
“Real-time Optical Synchronous QPSK Transmission with DFB Lasers”, Proc.
IEEE/LEOS Summer Topicals’07, Ma3.2 (invited), July 23-25, 2007, Portland, OR,
USA
R. Noé, T. Pfau, O. Adamczyk, R. Peveling, V. Herath, S. Hoffmann, M. Porrmann, S.
Ibrahim, S. Bhandare, “Real-time Digital Carrier & Data Recovery for a
Synchronous Optical Quadrature Phase Shift Keying Transmission System”, Proc.
IMS‘07, TH2E-01 (invited), June 3-8, 2007, Honolulu, HI, USA
S. Hoffmann, T. Pfau, R. Peveling, S. Bhandare, O. Adamczyk, M. Porrmann, R. Noé,
“PLL-free coherent optical QPSK transmission with real-time digital phase
estimation using DFB lasers”, Proc. ITG-Workshop Modellierung photonischer
Komponenten und Systeme, Feb. 12-13, 2007, Munich, Germany
R. Noé, T. Pfau, Y. Achiam, F.-J. Tegude, H. Porte, “Integrated Components for
Optical QPSK Transmission”, Proc. FiO‘06, FMF4, October 8-12, 2006, Rochester,
NY, USA
R. Noé, T. Pfau, “Synchronous Demodulation of Optical Phase Shift Keying in
Coherent Systems with DFB Lasers”, Proc. FiO‘06, FMF3 (invited), October 8-12,
2006, Rochester, NY, USA
T. Pfau, S. Hoffmann, R. Peveling, S. Bhandare, O. Adamczyk, M. Porrmann, R. Noé,
Y. Achiam, “1.6 Gbit/s Real-Time Synchronous QPSK Transmission with Standard
DFB Lasers”, Proc. ECOC‘06, Mo4.2.6, Sept. 24-28, 2006, Cannes, France
S. Hoffmann, T. Pfau, R. Peveling, S. Bhandare, O. Adamczyk, M. Porrmann, R. Noé,
“Synchrone 1,6 Gbit/s-QPSK-Datenübertragung in Echtzeit mit DFB-Lasern“, Proc.
ITG-Workshop Modellierung photonischer Komponenten und Systeme, July 17-18,
2006, Nürnberg, Germany
S. Hoffmann, T. Pfau, O. Adamczyk, R. Peveling, M. Porrmann, R. Noé, “Hardware-
Efficient and Phase Noise Tolerant Digital Synchronous QPSK Receiver Concept”,
Proc. OAA/COTA‘06, CThC6, June 25-30, 2006, Whistler, Canada
T. Pfau, S. Hoffmann, R. Peveling, S. Bhandare, S. K. Ibrahim, O. Adamczyk, M.
Porrmann, R. Noé, Y. Achiam, “Real-time Synchronous QPSK Transmission with
Standard DFB Lasers and Digital I&Q Receiver”, Proc. OAA/COTA‘06, CThC5,
June 25-30, 2006, Whistler, Canada
I
Table of contents
1 INTRODUCTION ................................................................................................................... 1
1.1 THE EUROPEAN SYNQPSK PROJECT ..................................................................................... 2
1.2 OUTLINE OF THE THESIS ........................................................................................................ 4
2 FUNDAMENTALS ................................................................................................................. 5
2.1 M-ARY QUADRATURE AMPLITUDE MODULATION ................................................................. 5
2.1.1 QAM constellations with equidistant-phases ............................................................... 5
2.1.2 Square QAM constellations .......................................................................................... 7
2.1.3 Differential encoding and decoding ............................................................................. 9
2.2 COHERENT OPTICAL QAM TRANSMISSION SYSTEM ............................................................ 11
2.2.1 Optical QAM transmitter ............................................................................................ 11
2.2.2 Polarization-multiplexed QAM transmitter ................................................................ 12
2.2.3 Optical transmission link impairments ....................................................................... 13
2.2.4 Coherent optical QAM receiver with digital signal processing ................................. 17
3 DIGITAL SIGNAL PROCESSING ALGORITHMS FOR COHERENT OPTICAL
RECEIVERS .......................................................................................................................... 23
3.1 CONSTRAINTS FOR ALGORITHMS IN DIGITAL RECEIVERS FOR COHERENT OPTICAL
COMMUNICATION ................................................................................................................ 23
3.1.1 Feasibility of parallel processing ............................................................................... 24
3.1.2 Hardware efficiency ................................................................................................... 25
3.1.3 Tolerance against feedback delays ............................................................................. 26
3.2 CLOCK RECOVERY ............................................................................................................... 29
3.3 POLARIZATION CONTROL & EQUALIZATION ....................................................................... 29
3.3.1 Non-data-aided polarization control .......................................................................... 30
3.3.2 Decision-directed polarization control ...................................................................... 31
3.3.3 Decision-directed ISI compensation
........................................................................... 32
3.4 FEED-FORWARD CARRIER RECOVERY ................................................................................. 34
3.4.1 Viterbi & Viterbi algorithm ........................................................................................ 35
3.4.2 Weighted Viterbi & Viterbi algorithm ........................................................................ 36
3.4.3 Barycenter algorithm.................................................................................................. 37
3.4.4 Feed-forward carrier recovery for arbitrary QAM constellations ............................. 41
3.4.5 Hardware effort .......................................................................................................... 45
3.5 DATA RECOVERY ................................................................................................................. 46
3.5.1 Data recovery for QAM constellations with equidistant-phases ................................ 46
3.5.2 Data recovery for square QAM constellations ........................................................... 47
3.6 INTERMEDIATE FREQUENCY CONTROL ................................................................................ 48
3.6.1 External LO frequency control ................................................................................... 48
3.6.2 Internal intermediate frequency compensation .......................................................... 48
4 SIMULATION RESULTS .................................................................................................... 49
4.1 QPSK CARRIER RECOVERY ................................................................................................. 49
4.1.1 QPSK carrier phase estimator efficiency and mean squared error ........................... 50
4.1.2 QPSK phase noise tolerance ...................................................................................... 54
II
4.1.3 QPSK analog-to-digital converter resolution ............................................................ 60
4.1.4 QPSK phase resolution............................................................................................... 61
4.2 QAM CARRIER RECOVERY .................................................................................................. 63
4.2.1 Square QAM phase angle resolution
.......................................................................... 63
4.2.2 Square QAM phase estimator efficiency .................................................................... 64
4.2.3 Square QAM phase noise tolerance ........................................................................... 70
4.2.4 Square QAM analog-to-digital converter resolution ................................................. 73
4.2.5 Square QAM internal resolutions ............................................................................... 74
4.3 POLARIZATION CONTROL AND PMD COMPENSATION ......................................................... 75
4.3.1 Comparison of polarization control algorithms
......................................................... 75
4.3.2 Verification of the ISI compensation algorithm ......................................................... 78
5 IMPLEMENTATION OF A SYNCHRONOUS OPTICAL QPSK TRANSMISSION
SYSTEM WITH REAL-TIME COHERENT DIGITAL RECEIVER ............................ 87
5.1 SINGLE-POLARIZATION SYNCHRONOUS QPSK TRANSMISSION WITH REAL-TIME FPGA-
BASED COHERENT RECEIVER ............................................................................................... 87
5.1.1 Single-polarization synchronous QPSK transmission setup ...................................... 87
5.1.2 Self-homodyne experiment results at 800 Mb/s
.......................................................... 91
5.1.3 Intradyne experiment results at 800 Mb/s .................................................................. 92
5.1.4 Intradyne experiment results at 1.6 Gb/s .................................................................... 93
5.1.5 System optimizations & comparison of 90° hybrid with 3x3 coupler ......................... 94
5.1.6 Comparison of experimental with simulation results ................................................. 96
5.2 POLARIZATION-MULTIPLEXED SYNCHRONOUS QPSK TRANSMISSION WITH REAL-TIME
FPGA-BASED COHERENT RECEIVER .................................................................................... 97
5.2.1 Polarization-multiplexed QPSK transmission setup .................................................. 98
5.2.2 Influence of different carrier recovery filter widths ................................................. 104
5.2.3 Polarization tracking capability ............................................................................... 105
5.2.4 Polarization tracking capability with optimized VHDL code ................................... 108
5.2.5 Influence of PDL on the receiver sensitivity ............................................................. 109
5.3 POLARIZATION-MULTIPLEXED SYNCHRONOUS QPSK TRANSMISSION WITH REAL-TIME
ASIC BASED COHERENT RECEIVER ................................................................................... 110
5.3.1 Transmission with and without polarization crosstalk ............................................. 111
5.3.2 Influence of different carrier recovery filter widths ................................................. 112
5.3.3 Single-polarization vs. polarization-multiplexed QPSK transmission ..................... 113
6 DISCUSSION ....................................................................................................................... 115
7 SUMMARY .......................................................................................................................... 117
8 OUTLOOK .......................................................................................................................... 119
9 BIBLIOGRAPHY ................................................................................................................ 120
10 LIST OF FIGURES & TABLES ........................................................................................ 125
III
Glossary
Latin symbols
Variable Unit Description
a
~
V Electrical drive signal of upper MZM
b
~ V Electrical drive signal of lower MZM
sΔ rad Gaussian distributed random variable for continuous phase
noise
ϕ
ˆ rad Estimated carrier phase
fΔHz Sum laser linewidth
dB3
fΔ Hz Full width at half maximum
DFB
fΔHz DFB laser linewidth
ECL
fΔ Hz ECL linewidth
B Number of test carrier phase angles
bi i-th input bit sequence into the transmitter
bmin Index of minimum squared distance sum
B
r
Hz Reference bandwidth
c Constellation point in the complex plane
c m/s Light velocity
c
t
s Control time constant
c
k
Transmitted complex symbol
DCD s/m2 Chromatic dispersion parameter
di Distance of test sample to closest constellation point in
i-th block
DPMD ms Polarization mode dispersion parameter
e(NCR) Estimator efficiency for filter half width NCR
Ea V/m Output electrical field of the upper MZM
Eb V/m Output electrical field of the lower MZM
eCL
K
Clock phase error signal
EC
W
V/m Electrical field of the transmitter laser
Ei V/m Electrical field of i-th optical 90° hybrid output
E
l
V/m Input electrical field into the lower MZM
E
L
O V/m Electrical field of local oscillator signal
E
RX
V/m Input electrical field of the optical receiver
E
S
J Energy per symbol
ET
X
V/m Output electrical field of the optical transmitter
Eu V/m Input electrical field into the upper MZM
IV
F Differential coding penalty
fc Hz Carrier frequency
g Control gain
I
I
A Differential output current of inphase photodiodes
I
Q
A Differential output current of quadrature photodiodes
J Fiber Jones matrix
k Discrete time index
K V/A Transimpedance amplifier transfer ratio
l Number of pipeline stages
Lfibe
r
m Fiber length
LPMDE PMD emulator filter length
M QAM modulation level
(number of constellation points)
m Number of parallel modules
M Polarization control matrix
Mi Dispersion compensation matrix of i-th tap
n Refractive index
n Complex Gaussian noise variable
N0 W/Hz Noise power spectral density
na Amplitude number
NCR Carrier recovery filter half width
nd Differential half-plane/quadrant/sector number
ni Inphase number
n
j
Jump number
NPMDC PMD compensator filter half width
n
q
Quadrature number
n
t
Transmitter half-plane/quadrant/sector number
P W Optical power
p Number of sectors for equidistant-phase constellations
Pin W Fiber input power
P
L
O W Local oscillator power
P
N
W Optical noise power
Pou
t
W Fiber output power
P
S
W Optical signal power
Q Correlation matrix for polarization control
Qi Correlation matrix for i-th tap for dispersion compensation
R A/W Photodiode responsivity
Rb b/s Bit rate
R
S
baud Symbol/Baud rate
si Squared distance sum in i-th block
t s Time
C Optical 3 dB coupler transfer matrix
V
T Polarization control error matrix for CMA
Tb s Bit duration
T
S
s Symbol duration
u Power parameter for Viterbi & Viterbi carrier recovery
U Carrier recovery filter input
U
I
V Transimpedance amplifier output signal (inphase)
U
Q
V Transimpedance amplifier output signal (quadrature)
V Carrier recovery filter output
v
g
m/s Group velocity
vi i-th Wiener filter coefficient
W Number of averaged correlation matrices
X Discrete signal after carrier recovery
x FIR/IIR filter input signal
Y Discrete signal after polarization control and dispersion
compensation
y FIR/IIR filter output signal
Z Discrete signal after analog-to-digital converter
zi i-th output signal of 3x3 coupler
Greek symbols
Variable Unit Description
2
n
σ
Gaussian noise variance
ϑ
rad Modulation free symbol phase
φ
rad (S)MLPA filter cell output
γ
rad Symmetrie angle of constellation diagram
α Fiber attenuation coefficient
αi i-th FIR tap coefficient
αPDL PDL coefficient
β Propagation constant
βi i-th IIR tap coefficient
δ, ε rad Phase offset parameters of Jones matrix
Δ Processing delay
ΔτDGD s Differential group delay
Δψ rad Gaussian distributed random variable for discrete phase
noise
θ, ζrad (S)MLPA filter cell inputs
λ m Wavelength
υ rad Polarization cross-talk parameter of Jones matrix
φCL
K
rad Clock phase
VI
φi rad Test carrier phase of i-th block
χi Correlation factor between 0-th and i-th dispersion
compensation filter input
ψ
I
F rad Carrier phase
ψ
L
O rad Local oscillator phase
ψ
S
rad Signal phase
ω
I
F Hz Angular carrier frequency
ω
L
O Hz Angular local oscillator frequency
ω
S
Hz Angular signal frequency
Acronyms and Abbreviations
Abbreviation Description
100GbE 100 Gigabit Ethernet
ADC Analog-to-Digital Converter
ASE Amplified Spontaneous Emission
ASIC Application-specific Integrated Circuit
ASK Amplitude Shift Keying
AWG Arrayed-waveguide Grating
AWGN Additive White Gaussian Noise
BER Bit Error Rate
BERT Bit Error Rate Tester
Bit Binary digit
BPF Bandpass Filter
BPSK Binary Phase Shift Keying
CD Chromatic Dispersion
CMA Constant Modulus Algorithm
CMOS Complementary Metal–Oxide–Semiconductor
CORDIC Coordinate Rotation Digital Computer
CRLB Cramér-Rao Lower Bound
CW Continuous Wave
DAC Digital-to-Analog Converter
DBPSK Differential Binary Phase Shift Keying
DCF Dispersion Compensating Fiber
DD Decision-Directed
DEMUX Demultiplexer
DFB Distributed Feedback
DGD Differential Group Delay
DQPSK Differential Quadrature Phase Shift Keying
DSPU Digital Signal Processing Unit
VII
DWDM Dense Wavelength Division Multiplexing
ECL External-cavity Laser
ECOC European Conference on Optical Communication
EDFA Erbium-Doped Fiber Amplifier
FF Flip-Flop
FFT Fast Fourier Transform
FIR Finite Impulse Response
FP6 6th Framework Programme
FPGA Field-Programmable Gate Array
FWHM Full Width at Half Maximum
GVD Group Velocity Dispersion
HWP Half-Wave Plate
IEEE Institute of Electrical and Electronics Engineers
IF Intermediate Frequency
IFFT Inverse Fast Fourier Transform
IIR Infinite Impulse Response
ISI Intersymbol Interference
ITU-T International Telecommunication Union - Telecommunication
Standardization Sector
LO Local Oscillator
LUT Loop-Up Table
MGT Multi-Gigabit Transceiver
MLPA Maximum Likelihood Phase Approximation
M-QAM M-ary Quadrature Amplitude Modulation
MSE Mean Squared Error
MUX Multiplexer
MZM Mach-Zehnder-Modulator
NDA Non-Data-Aided
NFOEC National Fiber Optic Engineers Conference
OFC Optical Fiber Communication Conference and Exposition
OFDM Orthogonal Frequency Division Multiplexing
ONT Optische Nachtichtentechnik und Hochfrequenztechnik
(Optical Communication and High Frequency Engineering)
OOK On-Off-Keying
OSNR Optical Signal-to-Noise Ration
PBC Polarization beam combiner
PBS Polarization beam splitter
PC Personal Computer
PDG Polarization-dependent Gain
PDL Polarization-dependent Loss
PLL Phase-locked Loop
PMD Polarization Mode Dispersion
VIII
PMDC Polarization Mode Dispersion Compensator
PMDE Polarization Mode Dispersion Emulator
PM-QPSK Polarization-Multiplexed Quadrature Phase Shift Keying
PRBS Pseudo-Random Binary Sequence
PSK Phase Shift Keying
QAM Quadrature Amplitude Modulation
QPSK Quadrature Phase Shift Keying
QWP Quarter-Wave Plate
RTL Register Transfer Level
SCT Schaltungstechnik (System and Circuit Technology)
SMF Single-Mode fiber
SMLPA Selective Maximum Likelihood Phase Approximation
SNR Signal to Noise Ratio
SOP State of Polarization
SPM Self-Phase Modulation
V&V Viterbi & Viterbi
VCO Voltage-Controlled Oscillator
VHDL VHSIC Hardware Description Language
VHSIC Very High Speed Integrated Circuit
VOA Variable Optical Attenuator
WDM Wavelength Division Multiplexing
XPM Cross-Phase Modulation
1 Introduction
1
1 Introduction
Coherent optical receivers that use either homodyne or heterodyne detection have
significant advantages over traditional optical direct detection receivers because they
linearly down-convert the optical signal to electrical signals. Therefore the receiver
sensitivity is shot-noise limited, if the local oscillator (LO) power is sufficiently high.
In the 1980s this property of high receiver sensitivity directed a lot of research towards the
development and implementation of coherent long-distance optical transmission systems
without repeaters [1; 2; 3; 4]. But the invention of the erbium-doped fiber amplifier
(EDFA) and its fast deployment in commercial transmission systems dramatically reduced
the interest in coherent technologies [5; 6].
In EDFA-based systems amplified spontaneous emission (ASE) rather than shot noise
determines the signal-to-noise ratio (SNR), which made the shot-noise limited receiver
sensitivity of coherent receivers less significant. Additional technical difficulties inherent
in coherent receivers also prevented further investigations. The disadvantage of heterodyne
receivers is that an intermediate frequency (IF) higher than the symbol rate is required.
Thus the receiver bandwidth must be more than twice as large as for baseband and direct
detection receivers. The homodyne receiver operates at the baseband, but requires a stable
locking of the transmitter and local oscillator frequency and phase. With standard
distributed feedback (DFB) lasers stable locking using a phase-locked loop (PLL) could
not be demonstrated [7]. Coherent receivers with analog feed-forward carrier recovery
showed sufficient phase noise tolerance [8], but could not prevail over the less complex
direct detection receivers.
In contrast the EDFA technology revolutionized research in optical communication in the
1990s. Thanks to the large bandwidth of EDFAs wavelength division multiplexing (WDM)
techniques became possible and dramatically increased the transmission capacity of optical
fibers.
In recent years research about coherent receivers experienced a revival [9]. Due to the
ever-increasing bandwidth demand researchers are looking for ways to exploit the optical
bandwidth more efficiently by using coherent transmission with multilevel modulation
formats. The development thereby profited from the fact that over the past years the
bandwidth and clock frequencies for digital signal processing circuits increased faster than
the symbol rate for optical communication. Therefore the electrical signals in a coherent
receiver can now be processed in a digital signal processing unit (DSPU). By means of
feed-forward carrier recovery the inphase and quadrature component of the complex
amplitude of the optical carrier is recovered digitally and in a stable manner [10; 11].
1 Introduction
2
Moreover, all linear optical distortions (polarization transformations, polarization mode
dispersion, chromatic dispersion) can theoretically be equalized without any losses [12;
13].
The main research focus was laid on the investigation of synchronous optical quadrature
phase shift keying (QPSK) combined with polarization division multiplex. Compared to
standard on-off-keying (OOK) the line rate is 4 times lower, the needed number of photons
per bit less than half as high, the tolerance to chromatic dispersion about 5 times better, the
tolerance to polarization mode dispersion about 3 times better, and the tolerance against
fiber nonlinearities, in particular cross phase modulation, is excellent [14]. Therefore it is
an extremely attractive modulation format for metropolitan-area and long-haul fiber
communication. Distinct advantages exist also over other modulation formats, such as
duobinary modulation, differential binary phase shift keying (DBPSK) or differential
quadrature phase shift keying (DQPSK) [9; 15].
The first transmission experiments with coherent digital receivers were realized using
digital storage oscilloscopes and offline signal processing in a personal computer (PC)
[16]. The reason was that some key components to realize a real-time coherent digital
receiver did not exist yet. For this reason in 2004 the University of Paderborn, Photline
Technologies, CeLight Israel and the Innovative Processing AG started the synQPSK-
project, which aimed at the development these key components.
But not only QPSK attracts the attention of the research community, also higher level
quadrature amplitude modulation (QAM) with the main focus on square QAM
constellations is interesting as it allows to increase the spectral efficiency even beyond the
one of polarization-multiplexed QPSK [17; 18]. Although high-level QAM is more
susceptible to noise, which makes it less attractive for long-haul applications, but its
ultimate spectral efficiency makes QAM very interesting for metropolitan and regional
area networks, especially for next-generation networks beyond 100 Gb/s.
But as for coherent QPSK transmission the key components were missing in 2004, today
even the main key algorithm for coherent QAM transmission with high-level constellations
is not available: A feed-forward carrier recovery algorithm with a sufficiently high phase
noise tolerance that allows the employment of standard DFB lasers.
1.1 The European synQPSK project
The synQPSK project, funded by the European Commission within the 6th Framework
Programme (FP6) under the contract 004631, was started on July 1, 2004. The research
consortium was coordinated by the University of Paderborn from Germany with the
working groups ONT (Optical Communication and High Frequency Engineering) and SCT
(System and Circuit Technology). The additional partners were Photline Technologies
1 Introduction
3
from France, CeLight Israel and the Innovative Processing AG from Germany, which latter
was replaced after the first project year by the University of Duisburg-Essen, Germany.
The overall project goal was to develop all necessary components that could not be found
on the market for a synchronous optical QPSK transmission system combined with
polarization division multiplex, and to validate them in a 10 Gbaud, 40 Gb/s “synQPSK”
testbed.
The identified key components were LiNbO3 QPSK modulators required in the transmitter,
integrated coherent receiver frontends consisting of LiNbO3 optical 90° hybrids co-
packaged with InP balanced photoreceivers, and SiGe/CMOS integrated electronic circuits
for analog-to-digital conversion and digital signal processing. Figure 1.1 shows the layout
of the synQPSK project and links the consortium partners to their respective development
task.
TXlaser
Frontendwith common package
WDMtransmission
QPSKmodulators,
polarization multiplex
SiGe/CMOS
integrated
electrical
circuits
LOlaser
Optical90°
hybrids,
polarization
diversity
Balanced
photo‐
receivers
UPb University of Paderborn
PHT Photline Technologies
CIL CeLight Israel LTd.
IPAG/
UDE
Innovative Processing AG/
University of Duisburg-Essen
Figure 1.1: Simplified system schematic for the synQPSK project with partners’ contributions highlighted
Within the University of Paderborn the development tasks were distributed as follows: The
working group “Optical Communication and High Frequency Engineering” (ONT) headed
by Prof. Dr.-Ing. Reinhold Noé was responsible for algorithm development, system
simulations, development of high-speed analog-to-digital converters in SiGe technology
and the design of full-custom demultiplexers in CMOS. Additionally the working group
was responsible for the synQPSK testbed, i.e. the initial operation and validation of all
components and the assembly of a fully functional coherent polarization-multiplexed
QPSK transmission system.
The working group “System and Circuit Technology” (SCT) of Prof. Dr.-Ing. Ulrich
Rückert was responsible for the hardware implementation of the algorithms provided by
ONT, the integration of full custom demultiplexers in the DSPU standard cell design and
the backend development of the CMOS application-specific integrated circuits (ASIC).
Most of the work that is related to synchronous QPSK transmission and presented in this
dissertation was conducted in the framework of the synQPSK project.
1 Introduction
4
1.2 Outline of the thesis
At first the theoretical description of a fiber-optic transmission system with coherent
receiver and digital signal processing is presented. Starting from the specification of two
main classes of constellations for M-ary quadrature amplitude modulation (M-QAM), i.e.
QAM constellations with equidistant-phases and square QAM constellations, and their
generation in an optical transmitter is described. Then the main distortions that occur while
the optical signal is traveling though the fiber are summarized, and finally the coherent
detection of the signal in an optical polarization diversity receiver with subsequent analog-
to-digital conversion is explained.
Before going into detail in chapter 3 about the algorithms required in a digital signal
processing unit (DSPU) of a coherent optical receiver, chapter 3.1 summarizes the
constraints for these algorithms to be suitable for real-time implementation. Then the
algorithms for clock recovery, polarization control, dispersion compensation, carrier
recovery and intermediate frequency control are described. Two of the core elements of
this thesis are presented in this chapter: The dispersion compensation algorithm as well as
the carrier recovery for arbitrary QAM constellations were developed within this
dissertation.
In chapter 4 the simulation results for polarization control, dispersion compensation and
carrier recovery are presented. The purpose of the simulations is to demonstrate the
applicability and performance of the newly proposed algorithms, and for the QPSK carrier
recovery to compare the performance against state-of-the-art techniques. Additionally the
simulations were required to determine the key parameters for a hardware implementation
of a real-time synchronous QPSK receiver in the framework of the synQPSK project.
The setup for this hardware implementation and the measurement results derived from it
are finally outlined in chapter 5. The structure of this chapter follows the implementation
sequence of the system, from single-polarization QPSK transmission to polarization-
multiplexed QPSK transmission, both based on a field-programmable gate array (FPGA)
for digital signal processing, to the final polarization-multiplexed synchronous QPSK setup
with specifically developed SiGe and CMOS application-specific integrated circuits
(ASIC). A discussion of the achieved results followed by a summary and an outlook close
the thesis.
2 Fundamentals
5
2 Fundamentals
In digital communication systems information is sent from a source through a transmission
channel to a remote sink. In fiber-optic communication the source is represented by the
optical transmitter. According to the applied modulation format it maps the transmitted
sequence of binary digits (bit) with the bit rate bb TR 1
=
to symbols with the symbol or
baud rate SS TR 1=. Tb and TS are the bit and symbol duration, respectively. The ratio
Sb RR specifies the number of bits per symbol and is a measure for the spectral efficiency
of the modulation format. The symbols are impressed on a carrier signal that can be sent
through the optical fiber to the optical receiver. The receiver then recovers the symbols
from the received signal and reconstructs the bit sequence. Although there are different
types of optical receivers, this dissertation only considers coherent optical receivers. In the
following these different components of a fiber-optic transmission system are described in
more detail.
2.1 M-ary quadrature amplitude modulation
In quadrature amplitude modulation (QAM) data is transported by modulating the
amplitude of two carriers, which have the same frequency fc but are 90° out of phase. They
can therefore be called quadrature carriers – hence the name of the scheme [19]. A
convenient way to represent digital QAM schemes is the constellation diagram. Inphase
and quadrature modulation are represented as real and imaginary parts of a complex
number. The number of symbols M in the constellation diagram defines the order of a
digital QAM format, which can therefore be named M-ary QAM or M-QAM.
But to specify the order of a QAM constellation is not sufficient to uniquely qualify a M-
QAM format, because the M symbols can be arbitrarily distributed over the complex plane.
Thus also the shape of the QAM constellation must be considered. In this thesis I will
concentrate on the two most important kinds of shapes for QAM constellations, which are
mostly used in commercial transmission systems: Equidistant-phase constellations and
square QAM constellations.
2.1.1 QAM constellations with equidistant-phases
A QAM constellation scheme with equidistant-phases is also referred to as phase shift
keying (PSK), if the amplitude is constant, or combined amplitude and phase shift keying
(ASK-PSK), if also the amplitude is modulated. The most commonly used modulation
schemes of this QAM sub-class are binary phase shift keying (BPSK) with a spectral
2 Fundamentals
6
efficiency of 1 bit/symbol and quadrature phase shift keying (QPSK) with a spectral
efficiency of 2 bit/symbol. The constellation diagrams of the two schemes with the
corresponding Gray-coded bit assignments are depicted in Figure 2.1. The colored areas
represent the tolerable corruption by noise while the correspondent symbol is still detected
correctly at the receiver.
01
01 00
11 10
Re
Im
Re
Im
n
t
=0n
t
=1
n
t
=0n
t
=1
n
t
=3n
t
=2
Figure 2.1: BPSK (left) and QPSK (right) constellation diagrams
The symbols positions in the complex plane for BPSK are given by the formula
{
}
{
}
1 ,0exp
BPSK
∈
⋅
=tt nnjc
π
(2.1)
and for QPSK by
{}
3 ,2 ,1 ,0
42
exp2
QPSK ∈
⎭
⎬
⎫
⎩
⎨
⎧+= tt nnjc
ππ
. (2.2)
In case of BPSK nt can be regarded as a half-plane number, in case of QPSK as a quadrant
number. The bit-to-symbol assignment is calculated by converting the binary value of nt to
Gray-code [20].
Equation (2.1) and (2.2) imply that for QPSK twice the signal power is required compared
to BPSK to achieve the same distance between adjacent constellation points. Thus for the
same signal power the distance between adjacent symbols is reduced. This shows that
increasing the order of QAM allows the transmission of more bits per symbol, but at the
price of a less reliable detection at the receiver.
But also higher level modulation formats are possible. Figure 2.2 shows a ASK-8-PSK
constellation diagram for a spectral efficiency of 4 bit/symbol. The symbol positions in the
complex plane are given by
{
}
{}
7 ,... ,1 ,0
2 ,1
4
exp
PSK8ASK ∈
∈
⎭
⎬
⎫
⎩
⎨
⎧
=
−−
t
a
ta n
n
njnc
π
. (2.3)
2 Fundamentals
7
The bit-to-symbol assignment depends now on nt, which can be considered as a segment
number and determines the first three bits of a symbol. The amplitude number na
determines if the symbol is lying on the inner or outer circle represented by the last bit.
n
t
=5
0000
0010
0001
0011
0111
0100
0101
1101 1100
1110
1111
1000
1011
1001
n
t
=0
n
t
=1
n
t
=2
n
t
=3
n
t
=4
n
t
=6
n
t
=7
0110
1010
Re
Im
Figure 2.2: ASK-8-PSK constellation diagram
In Figure 2.2 the disadvantage of QAM constellation scheme with equidistant-phases for
higher-order constellations becomes obvious. The distances to adjacent symbols are
smaller for symbols on the inner circle than for the symbols on the outer circle. In systems
where phase noise is dominant, this does not matter, but for systems where additive white
Gaussian noise (AWGN) dominates, square QAM constellations are more tolerant against
noise than equidistant-phase constellations [21].
2.1.2 Square QAM constellations
In square QAM constellations the symbols are placed on a square grid with equal vertical
and horizontal spacing. Due to the uniform distribution square QAM constellations are less
susceptible to AWGN than QAM constellation scheme with equidistant-phases. Figure 2.3
shows different square QAM constellation diagrams ranging from 4-QAM, which is
equivalent to QPSK and has a spectral efficiency of 2 bit/symbol, to 256-QAM with a
spectral efficiency of 8 bit/symbol.
2 Fundamentals
8
Re
Im
4-QAM
64-QAM
256-QAM
128-QAM
32-QAM
16-QAM
Figure 2.3: Square QAM constellation diagrams
As QPSK is the simplest square QAM, it is straightforward to describe the positions of the
symbols for M-QAM by extending equation (2.2) by two new variables ni and nq, which
describe the additional amplitude modulation along the real (inphase) and imaginary
(quadrature) axis, respectively. If
(
)
M
2
log is an even number, then the number of
amplitude levels on the real and imaginary axis is M and the positions of constellation
points are given by
{}
[]
{}
[]
{
}
{}
.
12 ,... ,1 ,0
12 ,... ,1 ,0
Imsgn2Resgn2 QPSKQPSKQPSKQAM −∈
−∈
⋅++=
−Mn
Mn
ncjnccc
q
i
qiM (2.4)
As for QPSK nt can be considered as a quadrant number represented by two bits, and ni
and nq each represent half of the remaining bits. It is sufficient to separately Gray-encode
nt, ni and nq. The resulting constellation will also be Gray-encoded.
If
()
M
2
log is an odd number, then the constellation diagram is not an ideal square as can
be seen in Figure 2.3. But it can be easily constructed by extending the constellation
diagram of the 2M square QAM by adding 8M additional amplitude levels on the
real and imaginary axes. In this case the constellation points with simultaneous
8Mni≥ and 8Mnq≥ are unused.
2 Fundamentals
9
The bit-to-symbol assignment for square QAM constellations is exemplified for 16-QAM
in Figure 2.4. nt is represented by the first two bits, ni corresponds to the 3rd bit, nq to the 4th
bit. The colored areas show the tolerable corruption by noise while the corresponding
symbol is still detected correctly at the receiver.
0000
0001
0010
0011
0111
0100
0110
1110 1100
11011111
1000
1011
1010
n
t
=0
0101
1001
Re
Im
n
t
=1
n
t
=2 n
t
=3
n
q
=0
n
q
=1
n
i
=1
n
i
=0n
i
=0
n
i
=1
n
q
=1
n
q
=0
Figure 2.4: Square 16-QAM constellation diagram and bit-to-symbol assignment
By comparing Figure 2.4 to Figure 2.2 it becomes obvious why square QAM constellations
are preferable in AWGN-dominated transmission systems.
2.1.3 Differential encoding and decoding
A problem in QAM detection at the receiver is that the constellations are rotationally
symmetric by the angle t
n
π
2 . Due to this nt-fold phase ambiguity the absolute phase
rotation of the constellation introduced by the transmission channel cannot be recovered by
the receiver. To overcome this problem differential encoding at the transmitter and
corresponding differential decoding at the receiver can be applied [21]. Differential
encoding means that the information is contained in the phase difference between two
consecutive symbols rather than in the absolute phase. The drawback is that if one bit is
detected wrongly the differential decoding causes two consecutive bits to be wrong.
2 Fundamentals
10
Therefore it is desirable to apply differential encoding only to as few bits as possible. This
is referred to as partial differential encoding.
As the phase ambiguities of all QAM constellations presented in the sections 2.1.1 and
2.1.2 only depend on nt, it is sufficient to solely differentially encode nt
(
)
{
}
(
)
1maxmod
,1,,
+
+
=−tktkdkd nnnn , (2.5)
where k is the discrete time index and
{
}
1max
+
t
n is the possible number of values of nt.
Thus the range of values of nd and nt is the same. Differential decoding at the receiver
undoes the differential encoding by calculating
(
)
{
}
(
)
1maxmod
ˆˆˆ 1,,,
+
−
=−tkdkdkt nnnn . (2.6)
Figure 2.5 depicts the partial differential encoding process for a square 16-QAM
constellation. The encircled symbol pairs mark deviations from ideal Gray coding due to
the differential encoding process.
Re
Im
00
11
00
11
11
00
11
00
00
11 01
10
01
10
10
01
10
01
01
10
Figure 2.5: Partial differential encoding for a square 16-QAM constellation
Thus two effects degrade the performance of a transmission system if differential encoding
is applied: A symbol error causes at least two bit errors due to the comparisons used in the
differential decoding process described by equation (2.6), and additional errors may occur
due to the deviation from ideal Gray coding. Both effects are considered in the differential
coding penalty F defined as the bit error probability ratio of the differentially coded system
to the non-differentially coded system [21]. In [22] it is shown that for square QAM
constellations this coding penalty is given by
2 Fundamentals
11
(
)
()
12
log
12
−
+= M
M
F. (2.7)
Because the relative number of differentially encoded bits decreases as the total number of
bits per symbol increases, the differential coding penalty drops from 2 for QPSK to nearly
1 for high-order QAM formats (Table 2.1).
Table 2.1: Differential coding penalty for different square QAM constellations
Constellation Bits per symbol Differential coding penalty F
4-QAM 2 2.00 (3.0 dB)
16-QAM 4 1.67 (2.2 dB)
64-QAM 6 1.43 (1.5 dB)
256-QAM 8 1.27 (1.0 dB)
1024-QAM 10 1.16 (0.6 dB)
An alternative to the differential encoding/decoding process is it to use framing
information to resolve the phase ambiguity of the constellation diagram at the receiver
[23]. But in the simulations as well as in the experiments presented in this dissertation no
framing information is transmitted. Therefore differential encoding has to be applied.
2.2 Coherent optical QAM transmission system
2.2.1 Optical QAM transmitter
There are many possible implementations for an optical QAM transmitter. In this section I
present a transmitter architecture that is most commonly used and is compatible to
arbitrary QAM constellations. In literature it is often referred to as IQ-modulator or nested
Mach-Zehnder-modulator [24]. Figure 2.6 shows the schematic of such a transmitter.
…
k
b,0
k
b,1
{}
kM
b,1log2−
k
b,2
differential
encoding
CW laser
(
)
tES
Signal laser
DAC
DAC
QAM
constellation
mapping
3π/2
MZM
MZM
3 dB coupler 3 dB coupler
{}
k
cRe
{}
k
cIm
optical signals
electrical signals
(
)
ta
~
(
)
tb
~
(
)
tEu
(
)
tEl
()
tEa
()
tEb
(
)
tETX
Figure 2.6: Optical QAM transmitter structure
2 Fundamentals
12
The electrical field
()
(
)
(
)
ttj
S
CW
SS
ePtE
ψω
+
=2 generated by a continuous wave (CW) laser
is split by a directional coupler with the transfer matrix
⎥
⎦
⎤
⎢
⎣
⎡
=1
1
2
1
j
j
C (2.8)
into an upper path
()
(
)
SStj
S
uePtE
ψω
+
=2 and a lower path
()
(
)
SS tj
S
lePjtE
ψω
+
=2.
()
S
CW PE 221 2= is the power of the CW laser,
π
ω
2
S is the optical carrier frequency.
()
tEu in the upper path is modulated in a Mach-Zehnder modulator (MZM) by the
electrical driving signal
()
ta
~. In the lower path the electrical signal
()
tb
~
modulates
(
)
tEl
in a MZM. The continuous signals
(
)
ta
~ and
(
)
tb
~
correspond to the discrete samples
{}
k
cRe and
{}
k
cIm , respectively. The modulated optical signals in the upper and lower
paths can be written as
()
(
)
(
)
(
)
() ()
()()
ttj
S
b
ttj
S
a
SS
SS
ePtbtE
ePtatE
ψω
ψω
+
+
=
=
2
~
2
~
.
(2.9)
The additional phase shift of 23
π
in the lower path as depicted in Figure 2.6 is already
considered in the equation for
(
)
tEb. After combination in the following cross coupler we
obtain the optical signal
() () ()
[]
(
)
(
)
(
)
(
)
ttj
S
tj
S
TX
SSSS ePtcePtjbtatE
ψωψω
++ =+= . (2.10)
()
tETX is the output signal of the transmitter with the optical power PS. At the time instants
kTS with ...2 ,1 ,0
±
±=k and T being the symbol duration,
(
)
S
kTc corresponds to the
discrete symbol ck in the constellation diagram.
2.2.2 Polarization-multiplexed QAM transmitter
To generate a polarization-multiplexed transmission signal the electrical field from the CW
laser must be split by a polarization beam splitter (PBS) into two branches:
()
()
()()
ttj
S
S
yCW
xCW SS
e
P
P
tE
tE
ψω
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
2
2
2
1
,
,
(2.11)
Then the signals are fed into two parallel QAM transmitter as described above. After
modulation the signals are recombined in a polarization beam combiner (PBC) to form the
polarization-multiplexed transmission signal
2 Fundamentals
13
()
()
(
)
()
()()
ttj
S
y
x
yTX
xTX SS
eP
tc
tc
tE
tE
ψω
+
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
2
1
,
,. (2.12)
Figure 2.7 shows the structure of a polarization-multiplexed QAM transmitter.
CW laser
⎥
⎦
⎤
⎢
⎣
⎡
yCW
xCW
E
E
,
,
yS
E,
xS
E,
signal laser
PBS
Single polarization QAM transmitter
Single polarization QAM transmitter
PBC
xTX
E,
yTX
E,
⎥
⎦
⎤
⎢
⎣
⎡
yTX
xTX
E
E
,
,
Figure 2.7: Polarization-multiplexed QAM transmitter
2.2.3 Optical transmission link impairments
This section introduces the main optical transmission link impairments that alter the signal
while it travels through the fiber.
2.2.3.1 Attenuation
The attenuation caused by optical fibers limits the performance of fiber-optic
communication systems by reducing the average power that reaches the receiver [24].
Since optical receivers need a certain minimum amount of power to recover the signal
accurately, the transmission distance is inherently limited.
Under quite general conditions power attenuation inside an optical fiber is governed by
P
dz
dP
α
−= , (2.13)
where P is the optical power in the fiber. The attenuation coefficient α includes material
absorption as well as other sources of power attenuation. If Pin is the power launched at the
input of a fiber of length Lfiber, the output power Pout from (2.13) is given by
{
}
fiberinout exp LPP
α
−
=. (2.14)
It is customary to express α in the units of dB/km by using the relation
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
out
in
10
fiber
kmdB log
10
P
P
L
α
(2.15)
and to refer to it as the fiber loss.
2.2.3.2 Polarization crosstalk & polarization-dependent loss
Variations in the shape of the core of a SMF cause random changes of the polarization of a
pulse travelling through the fiber [24]. Therefore the state of polarization (SOP) is arbitrary
2 Fundamentals
14
at the receiver of an optical transmission system. It is common to describe the change of
the SOP by a unitary Jones matrix [25] given by
() (){}
(
)
(
)
{
}
(
)
(){}
()
(){}
()
⎥
⎦
⎤
⎢
⎣
⎡−
=−
−
22
22
cossin
sincos
tjtj
tjtj
etet
etet
t
δε
εδ
υυ
υυ
J. (2.16)
The time-variant parameter υ(t) describes the cross-talk between the two polarization
modes, δ(t) and ε(t) denote the phase differences. The input signal to the receiver is then
given by the fiber input signal at the transmitter multiplied by the fiber Jones matrix.
()
() () ()
()
(
){}
(
)
(
)
(
)
{
}
(
)
()
(){}
()
() (){}
()
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
−
=
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
tEettEet
tEettEet
tE
tE
t
tE
tE
TX,y
tj
TX,x
tj
TX,y
tj
TX,x
tj
TX,y
TX,x
RX,y
RX,x
22
22
cossin
sincos
δε
εδ
υυ
υυ
J (2.17)
The Jones matrix J(t) is time-variant. Slow variations of J(t) are caused by temperature
drifts, but also very fast polarization change speeds with several krad/s on the Poincaré
sphere are possible. These fast fluctuations are caused by movements of the fiber, e.g. by
vibrations of DCF coils [26].
But not only the SOP changes while the signal is travelling through the fiber, the two
polarization modes can also suffer from different rates of loss due to asymmetries of the
fiber [24]. This effect is referred to as polarization-dependent loss (PDL). In an optical
transmission system it can be modeled by a lumped PDL element placed between 2 unitary
Jones matrices.
()
() () ()
(
)
()
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎥
⎦
⎤
⎢
⎣
⎡
tE
tE
tt
tE
tE
TX,y
TX,x
RX,y
RX,x
0
PDL
10
01 JJ
α
(2.18)
It is customary to express PDL in the unit of dB by using the relation
(
)
PDL10dBPDL, log20
α
α
=
. (2.19)
It is also possible that 1
PDL >
α
. In this case the effect is referred to as polarization-
dependent gain (PDG).
2.2.3.3 Chromatic dispersion
Dispersion is a major source of signal distortion in optical fiber transmission systems [24;
27]. Single-mode fibers (SMF) have the advantage that intermodal dispersion is absent
because the energy of the injected pulse is transported only by a Single-mode. However
pulse broadening does not disappear altogether due to chromatic dispersion.
Chromatic dispersion occurs because all optical signals have a finite spectral width, and
different spectral components travel with different speeds through the fiber. One cause of
this velocity difference is that the refractive index n(ω) of a SMF is frequency-dependent.
This is called material dispersion and it is the dominant source of chromatic dispersion in
2 Fundamentals
15
single-mode fibers. Another cause of dispersion is that the cross-sectional distribution of
light within the fiber also changes for different wavelengths. Shorter wavelengths are more
completely confined to the fiber core, while a larger portion of the optical power at longer
wavelengths propagates in the cladding. Since the index of the core is greater than the
index of the cladding, this difference in spatial distribution causes a change in propagation
velocity. This phenomenon is known as waveguide dispersion. Waveguide dispersion is
relatively small compared to material dispersion.
The chromatic dispersion property of an optical fiber is given by the group-velocity
dispersion parameter DCD, which is usually expressed in ps/nm/km [24]. In general, for a
signal with an angular frequency ω(β) at a propagation constant β, i.e. the electromagnetic
fields in the propagation direction z oscillate proportional to
(
)
tzj
e
ωβ
−, the dispersion
parameter DCD is defined as
ω
λ
π
ω
β
λ
π
d
dv
v
c
d
dc
Dg
g
222
2
2
CD 22 =−= . (2.20)
where
ω
π
λ
c2= is the vacuum wavelength and
β
ω
ddvg
=
is the group velocity.
2.2.3.4 Polarization mode dispersion
In realistic fibers random imperfections in the circular symmetry cause the two
polarizations within the fiber to travel at different speeds [24]. This phenomenon causes
pulse broadening and is called polarization mode dispersion. Due to the random
characteristic of the fiber imperfections the pulse broadening effect corresponds to a
random walk. Thus the differential group delay (DGD) ΔτDGD is proportional to the square
root of the fiber length Lfiber.
fiberPMDDGD LD=Δ
τ
(2.21)
The PMD parameter DPMD of the fiber is usually expressed in kmps and is a measure
for the asymmetry of the fiber. For a standard single-mode fiber (SMF) [ITU-T G.652] the
PMD parameter is kmps 1.0
PMD =D [28].
A good model to emulate PMD is to use the filter structure depicted in Figure 2.8 [29].
J
1
τ
1
J
L
τ
2
J
2
E
in,x
E
in,y
E
out,x
E
out,y
τ
L
Figure 2.8: Polarization mode dispersion emulator (PMDE)
The PMD emulator (PMDE) is given by a cascade, which consists alternately of a unitary
Jones matrix as described in equation (2.16) and a component that adds the additional
2 Fundamentals
16
delay τ to one of the two polarizations. Its total differential group delay depends on the
values of the Jones matrices. With random Jones matrices and PMDLPMDE
τ
τ
τ
τ
===
=
...
21
the expectation of DGD
τ
Δ
becomes
PMDEPMDDGD L
ττ
=Δ . (2.22)
2.2.3.5 Amplified spontaneous emission
To compensate for the fiber loss introduced in section 2.2.3.1 optical amplifiers are used to
regenerate the signal before detection at the receiver. The most common amplifier is the
erbium-doped fiber amplifier (EDFA) [5; 6]. It has a huge amplification window that can
cover both the optical C-band (1525 nm ≤ λ ≤ 1565 nm) and L-band (1570 nm ≤ λ ≤
1610 nm). The signal is amplified by being multiplexed in the doped fiber with a pump
laser signal at a wavelength of 980 nm or 1480 nm. The pump laser excites the trivalent
Erbium ions (Er+3) into a higher energy level. By interaction with a photon at the signal
wavelength the ion can decay back to a lower energy level by emitting a photon with the
same wavelength as the signal. This effect is called stimulated emission [30].
But the Erbium ions that are excited by the pump laser can also decay back to a lower
energy level spontaneously. This amplified spontaneous emission (ASE) reduces the
efficiency of the amplifier and generates noise at the receiver. The effect of ASE at the
receiver can be described by an additive white Gaussian noise (AWGN) process
()
(
)
(
)
tntEtE
+
=
TXRX , (2.23)
where n(t) is a complex Gaussian random variable with zero mean and variance 2
n
σ
.
An important measure for optical transmission systems is the optical signal to noise ratio
(OSNR), which is defined as the ratio of the signal power PS to the noise power PN
corrupting the signal [24].
r
S
N
S
BN
P
P
P
0
OSNR
ρ
== (2.24)
The average noise power is given by the noise power spectral density N0 within the
reference bandwidth Br. For noise in both polarizations 2
=
ρ
, and for noise only in a
single-polarization 1=
ρ
. Because of the wide dynamic range of optical signals, the
OSNR is usually expressed in a logarithmic decibel scale.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
r
S
N
S
BN
P
P
P
0
1010dB log10log10OSNR
ρ
(2.25)
2 Fundamentals
17
In simulations a normalized signal to noise ratio is used. It is defined as the ratio of the
energy per symbol ES to the noise power spectral density N0. It is related to the OSNR by
the equation
S
r
N
SS
R
B
P
P
N
E
ρ
⋅=
0
, (2.26)
where RS is the symbol rate of the system.
2.2.4 Coherent optical QAM receiver with digital signal processing
2.2.4.1 Polarization diversity coherent optical receiver frontend
Figure 2.9 shows a polarization diversity coherent optical receiver frontend [24; 10]. It
consists of a local oscillator lasers, two polarization beam splitters (PBS), two optical 90°
hybrids and four differential photodiode pairs with transimpedance amplifiers. It can be
considered as an optical down-converter with optical-to-electric conversion. By
superimposing the received optical signal with the local oscillator the frequency band of
the signal is down-converted into the baseband (homodyne detection) or to an intermediate
band with a center frequency at least twice as large as the signal bandwidth (heterodyne
detection).
CW laser
⎥
⎦
⎤
⎢
⎣
⎡
yRX
xRX
E
E
,
,
⎥
⎦
⎤
⎢
⎣
⎡
yLO
xLO
E
E
,
,
xRX
E,
yRX
E,
yLO
E,
xLO
E,
90
o
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
E
xI
I,
xQ
I,
yI
I,
yQ
I,
optical signals
electrical signals
local oscillator
optical 90ohybrids
PBS
PBS
xI
U,
xQ
U,
yI
U,
yQ
U,
90
o
Figure 2.9: Polarization diversity coherent receiver frontend
2 Fundamentals
18
The receiver input signal is given by the transmitted signal (2.10) multiplied by a Jones
matrix J (2.17) and corrupted by additive white Gaussian noise.
()
() () ()
()
(
)
()
(){}
()
() (){}
()
()
(){}
()
() (){}
()
()
()
()
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
−
=
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
tn
tn
tEettEet
tEettEet
tn
tn
tE
tE
t
tE
tE
y
x
TX,y
tj
TX,x
tj
TX,y
tj
TX,x
tj
y
x
TX,y
TX,x
RX,y
RX,x
~
~
cossin
sincos
~
~
22
22
δε
εδ
υυ
υυ
J
(2.27)
()
tn x
~
and
()
tn y
~
are complex Gaussian random variables with zero mean and the variance
2
n
σ
. Chromatic dispersion and polarization-mode dispersion are not considered yet. The
received signal and the local oscillator laser signal
()
()
()()
ttj
LO
LO
yLO
xLO LOLO
e
P
P
tE
tE
ψω
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
2
2
2
1
,
,
(2.28)
with the power PLO and the optical frequency
π
ω
2
LO are split into their two polarization
components by two PBS. Then the signals are fed into two optical 90° hybrids. There the
signals ERX,x and ERX,y are superimposed with the four quadrature states in the complex-
field space associated with the local oscillator signals ELO,x and ELO,y, respectively. Thus
the output signals of the two hybrids are given by
() () ()
() () ()
,
2
2
1
,
2
1
2
1
,,4,3
,,2,1
⎟
⎠
⎞
⎜
⎝
⎛±=
⎟
⎠
⎞
⎜
⎝
⎛±=
tE
j
tEtE
tEtEtE
xLOxRX
xLOxRX
(2.29)
() () ()
() () ()
.
2
2
1
,
2
1
2
1
,,8,7
,,6,5
⎟
⎠
⎞
⎜
⎝
⎛±=
⎟
⎠
⎞
⎜
⎝
⎛±=
tE
j
tEtE
tEtEtE
yLOyRX
yLOyRX
(2.30)
After detection of the outputs of the optical 90° hybrid in differential photoreceivers with
the responsivity R and current to voltage conversion in the transimpedance amplifiers with
the transfer ratio K, the output voltages of the coherent receiver frontend become
() () () ()
()
() ()
{
}
() () () ()
()
() ()
{}
,Im
22
,Re
22
,,
2
4
2
3
,,
,,
2
2
2
1
,,
tEtE
KR
tEtE
KR
tIKtU
tEtE
KR
tEtE
KR
tIKtU
xLOxRX
xQxQ
xLOxRX
xIxI
+
+
=−==
=−==
(2.31)
2 Fundamentals
19
() () () ()
()
() ()
{
}
() () () ()
()
() ()
{}
.Im
22
,Re
22
,,
2
8
2
7
,,
,,
2
6
2
5
,,
tEtE
KR
tEtE
KR
tIKtU
tEtE
KR
tEtE
KR
tIKtU
yLOyRX
yQyQ
yLOyRX
yIyI
+
+
=−==
=−==
(2.32)
The two output signals of one polarization can be considered as one complex signal. Using
(2.12) and (2.17) they are given by
() () () () ()
(){}
()
() (){}
()
()
()
()()
()
,
2
sincos
2
2
22
,,
,,
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡+−=
=+=
−+−
−
+
tne
PP
tcettcet
KR
tEtE
KR
tjUtUtU
x
tj
LOS
y
tj
x
tj
xLOxRX
xQxIx
SLOSLO
ψψωω
εδ
υυ
(2.33)
() () () () ()
(){}
()
() (){}
()
()
()
()()
()
,
2
sincos
2
2
22
,,
,,
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡++=
=+=
−+−
−
+
tne
PP
tcettcet
KR
tEtE
KR
tjUtUtU
y
tj
LOS
x
tj
y
tj
yLOyRX
yQyIy
SLOSLO
ψψωω
εδ υυ
(2.34)
where LOSIF
ω
ω
ω
−= and LOSIF
ϕ
ϕ
ϕ
−
=
are the frequency and phase differences
between the signal and local oscillator lasers.
(
)
tn x and
(
)
tn y are different complex
Gaussian random variables, but have the same variance 2
n
σ
.
It can be seen, that Ix(t) and Iy(t) contain the transmitted data symbols cx(t) and cy(t).
However the two polarization channels are mixed and additionally the received
constellation is rotating with the intermediate frequency
π
ω
2
IF . Therefore a polarization
control and an IF carrier recovery are necessary in order to recover the transmitted data.
2.2.4.2 Analog-to-digital conversion and digital signal processing
To be able to process the received data in the digital domain, the four output voltages of
the coherent receiver frontend are sampled by analog-to-digital converters (ADC). The
sampling rate of the ADCs should be at least as high as the symbol rate of the system. In
practical systems either TS-spaced sampling or TS/2-spaced sampling is implemented.
For TS-spaced sampling the discrete output signal of the ADCs is given by
2 Fundamentals
20
(){}
()
()
(
)
{
}
(
)
(
)
(){}
()
() (){}
()
()
()()
(
)
()
{} {}
{} {}
()
.
cossin
sincos
:
cossin
sincos
,
,
,
2
,
2
,
2
,
2
22
22
,
,
,⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
−
=
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
−
∝
⎥
⎦
⎤
⎢
⎣
⎡
+
−
−
+
−
−
ky
kx
kTj
ky
j
kkx
j
k
ky
j
kkx
j
k
Sy
Sx
kTkTj
Sy
kTj
SSx
kTj
S
Sy
kTj
SSx
kTj
S
ky
kx
n
n
e
cece
cece
kTn
kTn
e
kTcekTkTcekT
kTcekTkTcekT
Z
Z
kIFIF
kk
kk
IFIF
ψω
δε
εδ
ψω
δε
εδ
υυ
υυ
υυ
υυ
(2.35)
For TS/2-spaced sampling every second sample corresponds to the samples obtained with
TS-spaced sampling. The other samples represent symbol transitions. This so-called
oversampling can be advantageous for the compensation of dispersive effects, where the
information of one symbol is spread over several samples. Figure 2.10 shows the setup of a
coherent digital receiver.
CW laser
90
90
optical 90 hybrids
PBS
PBS
(
)
tU xI ,
(
)
tU xQ,
(
)
tU yI ,
(
)
tU yQ,
ADC
ADC
ADC
ADC
kT
S
{
}
kx
Z,
Re
{
}
kx
Z,
Im
{
}
ky
Z,
Re
{
}
ky
Z,
Im
(
)
()
⎥
⎦
⎤
⎢
⎣
⎡
tE
tE
yRX
xRX
,
,
(
)
()
⎥
⎦
⎤
⎢
⎣
⎡
tE
tE
yLO
xLO
,
,
DSPU
k
b,0
ˆ
k
b,1
ˆ
{}
kM
b,1log2 2
ˆ−
k
b,2
ˆ
…
optical signals
electrical signals
Figure 2.10: Coherent optical receiver with analog-to-digital conversion and digital signal processing
Although it is possible to build a coherent optical receiver with analog signal processing
[8; 10], a digital implementation has several considerable advantages.
• Although a manual control of the SOP has successfully been employed in the
analog electrical domain [31], an automatic tracking of the SOP has not been
demonstrated yet. Therefore an optical polarization control would be required in a
commercial system[32]. In contrast in a digital coherent receiver an electronic
polarization control with automatic tracking of the SOP is feasible.
• Digital signal processing allows the use of very sophisticated algorithms, e.g. for
PMD/CD compensation or mitigation of nonlinear effects. In analog circuits only
some simple operations can be realized.
2.2.4.3 Phase noise in a coherent digital receiver
Phase noise originating from both the transmitter and local oscillator laser is a major
source of distortion in coherent optical receivers. It is caused by random fluctuations of the
2 Fundamentals
21
instantaneous frequency of the lasers due to their finite linewidth. Typically the linewidth
of a laser is specified as the full width at half maximum (FWHM) dB3
fΔ of its optical
power spectrum [33].
The phase fluctuations are described by the so called Wiener-Lévy process [34], which
describes a random walk phase modulation. In a coherent digital receiver with discrete
variables this random walk is given by
kkk
ψ
ψ
ψ
Δ
+
=
−1. (2.36)
The zero mean Gaussian random variable k
ψ
Δ
is referred to as the step-size of the random
walk. The fluctuation speed of the process is set by the variance
S
Tf dB3
22Δ=
Δ
πσ
(2.37)
of k
ψ
Δ. Figure 2.11 shows examples for the random walk of k
ψ
for different linewidth-
times-symbol-duration products S
Tf dB3
Δ. Figure 2.12 depicts the corresponding Lorentzian
carrier power spectra given by
()
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛Δ+Δ
Δ
=ΔΔ 2
dB3
2
dB3
dB3
2
1
2
1
;
SS
S
SS
TffT
Tf
TffTP
π
. (2.38)
Figure 2.11: Examples of the phase noise process ψk for different values of Δf3dBTS
02000 4000 6000 8000 10000
-25
-20
-15
-10
-5
0
5
Discrete time index: k
ψk
[rad]
Δf3dBTS=10-4
Δf3dBTS=10-3
Δf3dBTS=10-2
2 Fundamentals
22
Figure 2.12: Lorentzian carrier power spectra for different values of Δf3dBTS
-0.04 -0.03 -0.02 -0.01 00.01 0.02 0.03 0.04
-50
-40
-30
-20
-10
0
10
Normalized frequency
Δ
fT
S
Power [dB]
Δf
3dB
T
S
=10
-4
Δf
3dB
T
S
=10
-3
Δf
3dB
T
S
=10
-2
3 Digital signal processing algorithms for coherent optical receivers
23
3 Digital signal processing algorithms for
coherent optical receivers
This chapter presents the algorithms that are necessary to recover the data in a coherent
optical receiver with digital signal processing (Figure 3.1). But as the high data rates in
optical communication of 43 Gb/s, 112 Gb/s or even above generate stringent constraints
for these algorithms section 3.1 first summarizes these constraints. They are not taken from
a book but derived from my personal experience gained during my research. The
subsequent algorithms for clock recovery, polarization control, ISI compensation, carrier &
data recovery and intermediate frequency control all fulfil these requirements.
LO
ADC
ADC
ADC
ADC
DSPU
Digital Signal Processing Unit
Local
oscillator
90 hybrid
90 hybrid
I1
Q1
I2
Q2
TX
single chip
or
modular system
Figure 3.1: Coherent optical receiver structure
3.1 Constraints for algorithms in digital receivers for coherent
optical communication
This section summarizes and exemplifies the main real-time constraints for coherent
optical receiver algorithms. Three main constraints are identified [4; 12]:
• Feasibility of parallel processing
• Possibility of a hardware-efficient implementation
• Tolerance against feedback delays
The justifications for these constraints are presented in the following sections.
3 Digital signal processing algorithms for coherent optical receivers
24
3.1.1 Feasibility of parallel processing
Algorithms that support multi-Gb/s operations must allow parallel processing as shown in
Figure 3.2 [12]. The DSPU cannot operate directly at the sampling clock frequency of the
analog-to-digital converter, which is in general 10 GHz or higher, but requires
demultiplexing to process the data in m parallel modules at clock frequencies below
1 GHz. This allows automated generation of the layout, which is indispensable due to the
complexity of the system. A comparison between the sampling clock frequency and the
divided clock shows that at least m = 16 parallel modules are required. Algorithms for real-
time applications should therefore theoretically allow parallel processing with an unlimited
number of demultiplexed channels. The requirement for this is that (intermediate) results
within one module do not depend on results calculated at the same time in other parallel
modules.
… … ……
1:m
demux
ADC
1:m
demux
ADC
1:m
demux
ADC
1:m
demux
ADC
memory or flip-flops
processing block
…………
memory or flip-flops
memory or flip-flops
processing block
memory or flip-flops
I1 Q1 I2 Q2
memory or
flip-flops
memory or
flip-flops
processing
block
T-spaced sampling:
f
IN
= 1/T
S
T/2-spaced sampling:
f
IN
= 2/T
S
f
DIV
= f
IN
/m
feedback
path
m:1
mux
m:1
mux
m:1
mux
m:1
mux
f
OUT
= 1/T
S
Figure 3.2: Interfacing between ADCs and DSPU and internal structure of the DSPU
A good example to explain the feasibility of parallel processing is the comparison of two
filter structures: Finite (FIR) and infinite (IIR) impulse response filters. Figure 3.3 depicts
their structures in both serial and parallel systems. It can be seen that it is easily possible to
parallelize an FIR filter. Although the output signal
22110 −−
+
+
=kkkk xxxy
α
α
α
(3.1)
3 Digital signal processing algorithms for coherent optical receivers
25
depends on information provided by several parallel modules it does not depend on the
result of the same calculation performed in these modules. In contrast, it is a big challenge
to realize the parallel structure shown for the IIR filter, because the result
2211 −−
+
+
=kkkk yyxy
β
β
(3.2)
depends on results calculated at the same time in other parallel modules. A very low clock
frequency or a low number of parallel channels would be needed to allow all calculations
to be executed within one clock cycle. Neither of these requirements is fulfilled in coherent
digital receivers for optical transmission system.
x
k
T T
+y
k
α
0
α
1
α
2
x
k-1
x
k-2
y
k
TT
+
x
k
β
1
β
2
y
k-1
y
k-2
α
1
α
2
α
0
α
1
α
2
α
0
α
1
α
2
α
0
α
1
α
2
x
k
+
x
k-1
+
x
k-2
+
x
k-3
+
y
k
y
k-1
y
k-2
y
k-3
α
0
x
k
+
x
k-1
+
x
k-2
+
x
k-3
+
y
k
y
k-1
y
k-2
y
k-3
β
1
β
2
β
1
β
2
β
1
β
2
serial
parallel
data not
instantly available
Figure 3.3: Serial and parallel FIR and IIR filter structures
3.1.2 Hardware efficiency
Another important constraint for real-time coherent receiver algorithms is that they must
allow for a hardware-efficient implementation. This requirement also originates from the
parallel processing in the DSPU. Since most of the required algorithm blocks have to be
implemented m times within the DSPU, computationally intensive algorithms require a
huge amount of chip area and therefore increase power consumption and cost. The
algorithms considered for a chip implementation should therefore not only be evaluated by
performance, but also by hardware efficiency.
One way to increase hardware efficiency is to use signal transformations, e.g. FFT/IFFT,
log-function or transformation of in-phase and quadrature signal pairs into polar
coordinates [35]. Although the transformation itself requires additional hardware effort,
this is often beneficial because subsequent calculations are simplified. For example an FFT
3 Digital signal processing algorithms for coherent optical receivers
26
can reduce a convolution to multiplications, or the log-function or polar coordinates allow
replacing multiplications by summations.
The use of look-up tables (LUT) is another effective way to reduce the required hardware.
Coordinate transformations and nonlinear functions are the main candidates for a LUT
implementation, but it can also be beneficial to replace a multiplication by a LUT,
especially if the required precision of the result is low [35].
This directly implies the last but most important way to increase efficiency: The
optimization of the required precision. In microprocessors and computers it is common to
use standard precisions. The MATLAB® programming software for example stores integer
variables with 8, 16, 32 or 64 bit precision and floating point variables with 32 or 64 bit
precision [IEEE 754-1985]. This is useful to allow the efficient compilation of software.
However, in an application-specific hardware design each additional bit increases the
required number of gates and hence chip area, power consumption and cost. Therefore all
calculations should be optimized to just the precision that is necessary to achieve the
required accuracy.
3.1.3 Tolerance against feedback delays
The last important consideration for algorithms for real-time applications is the tolerable
feedback delay [4]. In simulation or offline processing feedback delays of 1 symbol are
easy to achieve, but this is impossible in a real-time system designed for multi-Gb/s
operation. The reasons are the parallel processing and massive pipelining, which is
required to cope with the high data rates. Pipelining means that only fractions of the whole
algorithm are processed within one clock cycle and the intermediate results are stored in
buffers, e.g. memory or flip-flops (FF) as shown in Figure 3.2. Therefore it can take easily
100 symbol durations until a received symbol has an impact on the feedback signal.
Algorithms for polarization control and CD/PMD compensation usually employ integral
controllers with time constants in the μs-range to update their tap coefficients [12]. In these
cases the additional delay due to pipelining, which is in the ns-range, can be neglected. But
the feedback delay can have a severe impact on the performance of algorithms that require
an instantaneous feedback, e.g. decision-directed carrier recovery, which is often used in
offline signal processing for higher-order QAM [36; 37].
Figure 3.4 shows the general structure of a such a decision-directed carrier & data recovery
module with an optimum feedback delay 1
=
Δ
[38]. The phase estimator uses the
estimated carrier phase 1,
ˆ−kIF
ψ
to derotate the input symbol k
Y, and the result is fed into a
decision device ([ ]D denotes the output of the decision device).
[]
{
}
[
]
D
kIF
k
D
kk jYYX 1,
exp
ˆ−
−=
′
=
ψ
)
(3.3)
3 Digital signal processing algorithms for coherent optical receivers
27
The carrier phase estimate kIF,
ˆ
ψ
is calculated with
{}
{
}()
∑
=
−− −= CR
0
,ˆ
argarg
ˆ
N
i
ikik
ikIF XYw
ψ
, 1
CR
0
=
∑
=
N
i
i
w, 0≥
i
w, (3.4)
where wi are weighting coefficients (e.g. Wiener filter coefficients, which will be described
in detail in section 3.4.2) and NCR is the FIR filter length. If only preceding symbols are
used as inputs to the FIR filter, a feedback delay of 1
=
Δ
becomes theoretically possible
[38].
decision
circuit +
-
exp{-j( )}
arg( )
arg( )
W(z)
Y
k
^
FF
X
k
^
filter function
ψ
k
1
ˆ−
−k
j
e
ψ
Figure 3.4: Decision-directed carrier recovery with Δ = 1
However, as explained above in a practical implementation it is impossible to achieve
1=Δ . In order to support 100 Gb/s or higher data throughput, practical DSP circuits use
massive parallelization and pipelining to realize a synchronous carrier & data recovery as
explained in section 3.1.1. Figure 3.5 shows the structure of such an implementation for
decision-directed carrier recovery. It can be seen that due to the parallel processing of m
consecutive samples the feedback delay between two symbols within one module is equal
to the degree of parallelism m. Even if the latest phase estimate from all parallel outputs is
fed back into each module as shown in Figure 3.5, the average delay amounts to
()
21
+
m.
Taking also the number of pipeline stages l into account (represented by the flip-flops (FF)
in Figure 3.5), which means that it takes l clock cycles until an input sample has an impact
on the feedback value, the total average feedback delay is
()
2
1
1
+
+⋅−=Δ m
ml . (3.5)
Note that the minimum number of pipeline stages is 1, because this is the minimum
number of clock cycles required in a digital feedback system (see Figure 3.4).
3 Digital signal processing algorithms for coherent optical receivers
28
W(z)
exp{-j( )}FF
Decision
circuit +
-
arg( )
arg( )
FF
FF
FFFF
Decision
circuit +
-
arg( )
arg( )
FF
FF
FFFF
Decision
circuit +
-
arg( )
arg( )
FF
FF
FFFF
decision
circuit +
-
arg( )
arg( )
FF
FF
FFFF
FF
FF
Y
k
Y
k-m+1
X
k-2m
^
X
k-3m+1
^
Ψ
k-4m
^
mk
j
e5
ˆ−
−
ψ
Figure 3.5: Decision-directed carrier recovery in a realistic receiver with parallel and pipelined signal
processing.
To determine the effect of Δ on the phase noise tolerance of the receiver, let Δ−kIF ,
ˆ
ψ
be a
perfect estimate of Δ−kIF ,
ψ
, i.e. Δ−Δ−
=
kIFkIF ,,
ˆ
ψ
ψ
and k
Y is only corrupted by phase noise.
With (3.3) k
Y′ can be written as
(
)
{
}
{
}
kIF
k
kIFkIF
kk jYjYY ,,, expexp
ψ
ψ
ψ
′
=
−
=
′Δ− . (3.6)
According to (2.36) k
ψ
′ is given by
∑Δ−=
Δ− Δ=−=
′k
ki
ikIFkIFkIF
ψψψ
ψ
,,, . (3.7)
kIF,
ψ
′ is a random Gaussian variable with zero mean and variance
()
S
Tf ⋅Δ⋅Δ=⋅Δ= Δ
πσσ
ψ
2
22 . Therefore the standard deviation of the phase noise increases
by the factor Δ.
In a practical implementation, assuming e.g. parallel processing with m = 32 and l = 5
pipeline stages, the average feedback delay is Δ = 144.5 samples. This shows that a
decision-directed carrier recovery is fairly unfeasible because the phase noise tolerance is
reduced by a factor of ~12. Consequently any feedback loop must be avoided in the carrier
recovery process, especially for higher-order QAM constellations with their inherently
smaller phase noise tolerance. In particular, the carrier cannot be recovered in a decision-
directed manner when normal DFB lasers are employed.
A possibility to avoid feedback loops is to apply a feed-forward structure. Several feed-
forward carrier recovery schemes will be presented in section 3.4.
3 Digital signal processing algorithms for coherent optical receivers
29
3.2 Clock recovery
Clock recovery at the receiver is essential for any data transmissions system, as otherwise a
recovery of the transmitted data becomes impossible. Therefore a PLL is required at the
receiver to lock the receiver clock phase RXCLK,
ϕ
to the phase of the transmitter clock
TXCLK,
ϕ
. This is implemented using a voltage-controlled oscillator (VCO) as receiver clock
source. It is controlled using a clock phase error signal. A common way to generate this
clock phase error signal eCLK,k is exploiting the correlation of consecutive samples in an
oversampled signal (e.g. TS/2-spaced sampling) [39]. By calculating
11
,CLK +− −= kkkk
kZZZZe (3.8)
the expectation of eCLK,k for a trailing receiver clock phase is 0
TXCLK,RXCLK,
CLK, <
<
ϕϕ
k
e, and it
is 0
TXCLK,RXCLK,
CLK, >
>
ϕϕ
k
e for a leading receiver clock phase. Therefore the receiver VCO
can be locked to the transmitter clock phase by integrating eCLK,k and using it as a control
signal for the VCO.
The description for the clock phase error signal generation is only described for single-
polarization transmission. An extension to two polarizations is straightforward. However,
in a practical implementation an even simpler error signal calculation using only the sign
of either the inphase or quadrature channel of one polarization is sufficient.
{}
[
]
{
}
[]
{
}
[
]
{
}
[
]
11
,CLK ResgnResgnResgnResgn +−
−
=kkkk
kZZZZe (3.9)
This significantly reduces the required hardware effort for the error signal calculation,
because only operations on bits are required rather than on the whole sample.
3.3 Polarization control & equalization
The purpose of a polarization control circuit is the compensation of cross-talk between the
two transmitted polarization channels. The cross-talk between these channels is described
by the fiber Jones matrix introduced in section 2.2.3.2. Therefore compensation requires
multiplication of the received signal with the estimated inverse of the Jones matrix Mk.
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
k
yk
xk
Z
Z
Y
Y
,
,
,
,M
(3.10)
In order to obtain Mk a non-data-aided (NDA) approach can be employed, where the input
and output data of the polarization controller is used to estimate the Jones matrix. Another
option is to use a decision-directed (DD) approach where the data before and behind the
carrier & data recovery is used for the estimation process.
3 Digital signal processing algorithms for coherent optical receivers
30
But not only polarization cross-talk corrupts the signal. Dispersive effects such as
chromatic dispersion (CD) (section 2.2.3.3) or polarization mode dispersion (section
2.2.3.4) add additional intersymbol interference, i.e. cross-talk between consecutive
symbols. In order to compensate for these effects a novel algorithm is proposed in this
thesis, which is based on the principles used for the decision-directed polarization control.
3.3.1 Non-data-aided polarization control
To demultiplex the two polarization channels in a non-data-aided approach the constant
modulus algorithm (CMA) is used. The algorithm exploits the fact that polarization cross-
talk causes the amplitude of the signal to fluctuate. Thus by minimizing these fluctuations
and forcing the signal on the unity circle a perfect separation of two polarization channels
can be achieved. The required error signal is calculated as
(
)
()
+
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
=
yk
xk
ykyk
xkxk
kZ
Z
YY
YY
,
,
,
2
,
,
2
,
1
1
T, (3.11)
and the polarization control matrix is incrementally updated by
Δ−−
+
=kkk gTMM 1. (3.12)
Figure 3.6 depicts the CMA graphically. A detailed derivation of the CMA can be found in
[40], its adaptation for polarization control in a coherent optical receiver is explained in
[41].
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
Y
Y
,
,
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
Z
Z
,
,
carrier & data
recovery
correlation
matrix update
M
k
T
k
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
c
c
,
,
ˆ
ˆ
Figure 3.6: Non-data-aided polarization control algorithm
A disadvantage of this algorithm is that in general 1−
≠kk JM , because the CMA only
minimizes the cross-talk between the two polarization channels but does not recover the
phase offset between them. To be able to achieve also a phase alignment between the two
polarization modes an additional control circuit needs to be added or as an alternative a
decision-directed polarization control algorithm can be applied.
3 Digital signal processing algorithms for coherent optical receivers
31
3.3.2 Decision-directed polarization control
To estimate the Jones matrix J in a decision-directed approach the recovered symbol
[]
T
ykxk cc ,, ˆˆ must be correlated with the output signal of the polarization control block
[12]. Using equation (2.35) and (3.10) the output signal of the polarization controller is
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
∝
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
j
yk
xk
kk
yk
xk
n
n
e
c
c
Y
YkIF
,
,
,
,
,
,,
ψ
JM . (3.13)
The correlation matrix k
Q is therefore given by
+
−⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=
yk
xk
j
yk
xk
kc
c
e
Y
Yk
,
,
ˆ
,
,
ˆ
ˆ
ϕ
Q, (3.14)
where k
ϕ
ˆ is the estimated carrier phase. The expectation k
Q of the matrix k
Q is a
perfect estimate of the matrix product MkJk.
kkk JMQ = (3.15)
Therefore by calculating
111
1
:−
Δ−Δ−
−
Δ−
−
Δ−Δ−
−
Δ− === kkkkkkk JMMJMQM , (3.16)
where Δ is the processing delay of the system introduced in section 3.1.3, the polarization
can be controlled electronically and penalty-free if 0→
−
=
Δ
Δ−kkk JJJ . Figure 3.7
visualizes the structure of the algorithm.
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
Y
Y
,
,
carrier & data
recovery
correlationmatrix update
Mk
Qk
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
c
c
,
,
ˆ
ˆ
k
ϕ
ˆ
⎥
⎦
⎤
⎢
⎣
⎡
yk
xk
Z
Z
,
,
Figure 3.7: Decision-directed polarization control algorithm
The implementation of equation (3.16) poses several challenges. First to calculate the
inverse of a matrix in a digital circuit is very complex. But the calculation can be
simplified by using the Taylor series [42]
()( )
∑
∞
=
−−−=
0
1
i
ii 1Q1Q . (3.17)
3 Digital signal processing algorithms for coherent optical receivers
32
If 1Q →, which is the goal of the polarization control, the Taylor series is dominated by
its first order element and Mk can be calculated by
()
(
)
Δ−Δ−Δ−Δ−Δ− −+=−+= kkkkkk MQ1MMQ11M :. (3.18)
The second problem is that in general kk QQ ≠ applies. Therefore the update algorithm
for Mk must be extended:
()
()
()
Δ−−−
−
=
Δ−−Δ−−− ⎟
⎠
⎞
⎜
⎝
⎛−+= ∑1
1
0
1
:Wk
W
w
wkWkk W
gMQ1MM . (3.19)
If 1=g then the expectation k
Q is calculated by averaging over W correlations.
Therefore Mk can be updated every W clock cycles. If 1
=
W then Mk is updated
incrementally every clock cycle using directly the correlation result Δ−k
Q multiplied with
the control gain g. The control time constant ct is given by
St T
g
W
c=. (3.20)
This means that within the time c a control error decays to the 1/e-fold error. The ratio W/g
should be in the range of [102, 104], depending on whether an accurate or a fast
polarization control should be realized.
In principle, the polarization control could also recover the carrier, but the time constant c
is much too large to be able to track the phase noise caused by standard DFB lasers.
Therefore an additional feed-forward carrier recovery must be applied.
3.3.3 Decision-directed ISI compensation
The decision-directed polarization control algorithm presented in section 3.3.2 can be
extended to additionally allow for intersymbol interference (ISI) compensation caused e.g.
by PMD. Therefore equation (3.10) is changed to the following FIR filter structure:
∑
−= −
−⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡PMDC
PMDC ,
,
,
,
,N
Ni yik
xik
ki
yk
xk
Z
Z
Y
YM. (3.21)
NPMDC is the FIR filter half width, i.e. 2NPMDC+1 input vectors are weighted by the
compensation matrices Mi and summed up.
To update the filter matrices the correlations between the filter output vectors of the NPMDC
preceding and subsequent symbols are correlated with the current recovered symbol
similar to equation (3.14).
3 Digital signal processing algorithms for coherent optical receivers
33
PMDC
,
,
ˆ
,
,
,
,
, ..., ,1
ˆ
ˆ
2
1Nn
c
c
e
Y
Y
Y
Y
yk
xk
j
yk
xk
n
ynk
xnk
nkn
nk =
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⎥
⎦
⎤
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛+=
+
−
−
−−
ϕ
χχ
Q
(3.22)
The parameter χn compensates for inherent correlations between neighboring symbols and
depends on the applied oversampling. For TS/2-spaced sampling χ±1 = ½, all other χn,
|n| ≠ 1, are zero.
The expectations kn,
Q, 0≠n of the correlation results are perfect estimates of the ISI
between the symbols with the indices k and k-n. More precise the expectation kn,
Q is a
perfect estimate of the matrix that describes how the symbol
[
]
T
ykxk XX ,, is coupled into
the symbol
[]
T
ynkxnk XX ,, −− . Polarization crosstalk within the center symbol can be
compensated by using equation (3.18). Therefore if
[
]
T
ykxk YY ,, were decoded a second
time ISI and polarization crosstalk could be significantly reduced by calculating
()
∑
=+
+
−
−
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−+
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡PMDC
N
nynk
xnk
kn
ynk
xnk
kn
yk
xk
k
yk
xk
yk
xk
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
1,
,
,
,
,
,
,
,
,0
,
,
,
,1
~
~
QQQ . (3.23)
By setting equation (3.21) into equation (3.23) and neglecting all input samples
[]
T
yikxik ZZ ,, −− with PMDC
Ni > one obtains
()
∑
−= −
−⎥
⎦
⎤
⎢
⎣
⎡
+=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡PMDC
N
Ni yik
xik
kiki
yk
xk
Z
Z
Y
Y
PMDC ,
,
,,
,
,~
~
~
MM , (3.24)
where the ki,
~
M are given by
()
()
()
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
>−−−
=−−−
<−−−
=
∑∑
∑∑
∑∑
−
−=
+
−
=
+
−
−==
−
−−=
+
=
+
0
0
0
~
1
,,
1
,,,,0
1
,,
1
,,,0,0
1
,,
1
,,,,0
,
PMDC
PMDC
PMDC
PMDC
PMDC
PMDC
i
i
i
Nn
kinkn
iN
n
kinknkik
Nn
knkn
N
n
knknkk
iNn
kinkn
N
n
kinknkik
ki
MQMQMQ1
MQMQMQ1
MQMQMQ1
M. (3.25)
Taking into account that in general knkn ,, QQ ≠ holds in a practical system, the
compensation matrices can be updated by
PMDC,,, ..., ,1
~Ni
W
g
kikiki =
′
+= Δ−Δ− MMM (3.26)
with
3 Digital signal processing algorithms for coherent optical receivers
34
()
()
()
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
>
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛−
=
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛−
<
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛−
=
′
∑∑∑∑∑
∑∑∑∑∑
∑∑∑∑∑
−
−=
+
−
=
−
−
=
+
−
=
−−
−
=
−
−
−=
−
=
−
=
−
=
−−
−
=
−
−
−−=
+
−
=
−
=
+
−
=
−−
−
=
−
0
0
0
~
1
,
1
0
,
1
,
1
0
,,
1
0
,0
1
,
1
0
,
1
,
1
0
,,0
1
0
,0
1
,
1
0
,
1
,
1
0
,,
1
0
,0
,
PMDC
PMDC
PMDC
PMDC
PMDC
PMDC
i
i
i
Nn
kin
W
w
wkn
iN
n
kin
W
w
wknki
W
w
wk
Nn
kn
W
w
wkn
N
n
kn
W
w
wknk
W
w
wk
iNn
kin
W
w
wkn
N
n
kin
W
w
wknki
W
w
wk
ki
MQMQMQ1
MQMQMQ1
MQMQMQ1
M
(3.27)
The derived update algorithm allows to efficiently compensate for ISI caused by CD or
PMD. But updating the compensation matrices is computationally very expensive because
()
113 PMDCPMDC
+
+NN complex matrix multiplications are required. In order to reduce the
complexity of the updating process it is possible to consider only the dominant summand
in the calculation for each ki,
~
M. To identify this summand the energy distribution of the
spread symbol can be investigated. Most of the energy is contained in the center symbol
and it decays towards both directions. Thus each compensation matrix ki,
~
M is dominated
by the summand kki ,0, MQ−. If the other summands are neglected equation (3.25) reduces
to
()
⎪
⎩
⎪
⎨
⎧
=−
≠−
=−
0
0
~
,0,0
,0,
,i
i
kk
kki
ki MQ1
MQ
M. (3.28)
Still all ki,−
Q, 0
≠
i are forced towards zero. Hence no performance degradation has to
be expected compared to the updating process given in equation (3.25). Another way to
derive equation (3.28) is also described in [43].
3.4 Feed-forward carrier recovery
After polarization control and intermediate frequency compensation, which will be
described in section 3.6, the input signal into the carrier recovery block is given by
()
⎥
⎦
⎤
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
∝
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
ky
kx
j
ky
kx
j
ky
j
kx
ky
kx
n
n
e
c
c
eY
eY
Y
YkIF
ky
kx
,
,
,
,
,
,
,
,,
,
,
ψ
ϕ
ϕ
(3.29)
with
{
}
{}
⎥
⎦
⎤
⎢
⎣
⎡
′
++
′
++
=
⎥
⎦
⎤
⎢
⎣
⎡
kykIFky
kxkIFkx
ky
kx
nc
nc
,,,
,,,
,
,
arg
arg
ψ
ψ
ϕ
ϕ
. (3.30)
The input signal Yk is sampled at the symbol rate, and perfect intermediate frequency
compensation and equalization are assumed.
3 Digital signal processing algorithms for coherent optical receivers
35
As the polarizations are perfectly separated and the carrier recovery is identical for both
polarizations in the following the algorithms are described for a single-polarization system
and the indices denoting the polarization channel are omitted.
3.4.1 Viterbi & Viterbi algorithm
The Viterbi & Viterbi algorithm allows feed-forward carrier recovery for constellations
with equidistant symbol phases [44]. If the phase offset between two adjacent symbols is
p
π
2 the modulation can be eliminated by raising the input signal Yk to the pth power. In
the case of QPSK p = 4.
{
}
4 ,2 ,0∈= ueYU k
jp
u
k
k
ϕ
(3.31)
Due to
{
}
π
=
p
k
carg the result is independent of modulation-induced phase changes [9].
The factor u allows to choose whether Uk should depend on the amplitude k
Y or not. For
u = p equation (3.31) equates to p
k
kYU =, for 0
=
u it can be rewritten as k
jp
keU
ϕ
=.
The resulting frequency multiplied carrier components are then filtered to remove
distortions caused by noise.
∑
−=
−
+
=CR
CR
12
1
CR
N
Nn
nkk U
N
V (3.32)1
NCR is referred to as filter half width, i.e. 2NCR+1 values are summed up to calculate Vk.
0
CR =N is equivalent to asynchronous demodulation of the signal.
The filter also alters the phase angle: kIF
jp
keV ,
ˆ
ψ
−∝ . In the noise-free case
kIFkIF pp ,,
ˆ
ψ
ψ
=. To finally recover the carrier phase the phase of Vk must be divided by a
factor of p.
()
kkIF V
parg
1
ˆ,=
ψ
(3.33)
Figure 3.8 shows the schematic of the Viterbi & Viterbi carrier recovery algorithm.
1 For a joined carrier recovery for both polarizations equation (3.32) must be extended to
()
()
∑
−=
−− +
+
=
CR
CR
,,
CR 122
1N
Nn
ynkxnkk UU
N
V
3 Digital signal processing algorithms for coherent optical receivers
36
……
U
k
U
k-N
CR
U
k+N
CR
| · |
u
arg{·}
p
exp{j(·)}
Y
k
∑
+12
1
CR
NkIF ,
ˆ
ψ
arg{·}
1/p
Figure 3.8: Viterbi & Viterbi feed-forward carrier recovery
3.4.2 Weighted Viterbi & Viterbi algorithm
Looking again at the statistical properties of IF
ψ
given by equation (2.36) it can be seen
that most information about kIF ,
ψ
is contained in the symbol k itself and that it is
symmetrically decaying for symbols with increasing distance to k. As in the filter function
(3.32) of the original Viterbi & Viterbi algorithm all symbols are weighted with the same
weight
()
121 CR +N, it is only optimal for 0
2=
Δ
σ
. It can be optimized by using a Wiener
filter with variable weights that reflect the information content about kIF ,
ψ
in the weighted
symbol [34].
∑
−=
−
=CR
CR
N
Nn
nknk UV
ν
(3.34)2
The optimal values for the Wiener coefficients n
ν
depend on the ratio of laser-induced
phase noise to the angular portion of AWGN 22
n′
Δ
σσ
and are given by [38]
n
n
n
κ
σ
σ
κ
κ
ν
2
2
2
1′
Δ
−
= (3.35)
with
1
2
1
1
2
1
1
2
2
2
2
2
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+=
′
Δ
′
Δ
nn
σ
σ
σ
σ
κ
. (3.36)
The sum of the Wiener coefficients is 1=
∑
∞
−∞=nn
ν
. Figure 3.9 shows the modified Viterbi
& Viterbi algorithm containing the additional weighting factors.
2 For a joined carrier recovery for both polarizations equation (3.34) must be extended to
()
∑
−=
−− +=
CR
CR
,,
N
Nn
ynkxnk
n
kUUV
ν
3 Digital signal processing algorithms for coherent optical receivers
37
……
U
k
U
k-N
CR
| · |
u
arg{·}
p
exp{j(·)}
Y
k
kIF ,
ˆ
ψ
arg{·}
1/p
v
0
v
-1
v
1
v
-N
CR
v
-N
CR
v
0
v
-1
v
1
v
-N
CR
v
-N
CR
U
k-1
U
k+1
U
k+N
CR
∑
Figure 3.9: Weighted Viterbi & Viterbi feed-forward carrier recovery
The value 22
n′
Δ
σσ
can be estimated in the receiver by comparing the noise variances in the
tangential and radial directions.
(
)
1
1
ˆ
2
2
,
2
2
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
−
=
′
Δ
k
k
kIFk
n
Y
Y
ψϕ
σ
σ
(3.37)
As 22
n′
Δ
σσ
is in general not time-variant its estimation can be done by software with a
slow update rate. Another option is it to omit the estimation of 22
n′
Δ
σσ
and set it to a
constant value.
A further increase in hardware efficiency can be achieved if the Wiener coefficients n
ν
are
rounded to multiples of a−
2,
{
}
... ,2 ,1 ,0∈a. In a digital circuit this allows to realize the
calculation of the products nk
nU−
ν
by using simple shift-and-add operations instead of
multipliers.
3.4.3 Barycenter algorithm
The Viterbi & Viterbi algorithm recovers the carrier phase using operations on complex
numbers. However the amplitudes of these numbers contain only limited information about
the carrier phase, if any. Therefore it is straightforward to estimate the carrier angle IF
ψ
in
the angular domain using only
{
}
k
kYarg
=
ϕ
. This algorithm is referred to as barycenter
algorithm [45] or maximum likelihood phase approximation (MLPA) [46].
In a first step modulation-induced phase changes are removed. If the constellation is
rotationally symmetric by the angle p
π
2, a modulation-free symbol phase is obtained by
calculating
3 Digital signal processing algorithms for coherent optical receivers
38
p
kk
π
ϕϑ
2
mod
=. (3.38)
This operation is similar to Viterbi & Viterbi carrier recovery with u = 0 and NCR = 0,
because it equates to
()
{
}
p
k
kYp arg1=
ϑ
with
[
)
p
k
π
ϑ
2,0
∈
.
To remove distortions caused by the angular portion of AWGN the k
ϑ
need to be filtered.
But in contrast to filtering in the complex plane simple averaging over several consecutive
phase angles is not possible due to the limited co-domain of k
ϑ
. To overcome this problem
the statistical properties of IF
ψ
can be exploited. Therefore the filter is constructed in a
cellular approach, in which each cell performs the following function [47]:
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
<−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+
+
<−≤−
+
≥−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−
+
=
ppp
ppp
ppp
kk
kk
kk
kk
kk
kk
k
π
ζθ
ππ
ζθ
π
ζθ
ππ
ζθ
π
ζθ
ππ
ζθ
φ
2
mod
2
2
mod
2
2
mod
2
(3.39)
θk and ζk are the input angles of the filter cell. Similar to the differential decoding process
described in section 3.4.1, always the shortest possible physical path between θk and ζk is
chosen for calculation of the mean angle k
φ
. Larger filter structures can be designed by
combining several filter cells in a tree structure. Depending on the arrangement of the filter
cells several filter structures are possible for the same filter half width NCR. Table 3.1
shows the structures that are examined in this thesis.
3 Digital signal processing algorithms for coherent optical receivers
39
Table 3.1: Barycenter carrier recovery filter structures
NCR Filter structure3
1
k
ϑ
1−k
ϑ
1+k
ϑ
kIF,
ˆ
ψ
2
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
kIF ,
ˆ
ψ
3
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
kIF ,
ˆ
ψ
4
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
kIF ,
ˆ
ψ
5
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
kIF ,
ˆ
ψ
3 For a joined carrier recovery for both polarizations the outputs from two filters xkIF ,,
ˆ
ψ
and ykIF ,,
ˆ
ψ
can be
combined in an additional filter cell to determine the joined carrier phase kIF,
ˆ
ψ
.
3 Digital signal processing algorithms for coherent optical receivers
40
6
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
6−k
ϑ
6+k
ϑ
kIF ,
ˆ
ψ
7
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
6−k
ϑ
6+k
ϑ
7−k
ϑ
7+k
ϑ
kIF ,
ˆ
ψ
9
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
6−k
ϑ
6+k
ϑ
7−k
ϑ
7+k
ϑ
8+k
ϑ
9+k
ϑ
8−k
ϑ
9−k
ϑ
kIF ,
ˆ
ψ
11
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
6−k
ϑ
6+k
ϑ
7−k
ϑ
7+k
ϑ
8+k
ϑ
9+k
ϑ
8−k
ϑ
9−k
ϑ
10−k
ϑ
11−k
ϑ
10+k
ϑ
11+k
ϑ
kIF ,
ˆ
ψ
13
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
6−k
ϑ
6+k
ϑ
7−k
ϑ
7+k
ϑ
8+k
ϑ
9+k
ϑ
8−k
ϑ
9−k
ϑ
10−k
ϑ
11−k
ϑ
10+k
ϑ
11+k
ϑ
12−k
ϑ
13−k
ϑ
12+k
ϑ
13+k
ϑ
kIF ,
ˆ
ψ
15
k
ϑ
1−k
ϑ
1+k
ϑ
2−k
ϑ
2+k
ϑ
3−k
ϑ
3+k
ϑ
4+k
ϑ
4−k
ϑ
5−k
ϑ
5+k
ϑ
6−k
ϑ
6+k
ϑ
7−k
ϑ
7+k
ϑ
8+k
ϑ
9+k
ϑ
8−k
ϑ
9−k
ϑ
10−k
ϑ
11−k
ϑ
10+k
ϑ
11+k
ϑ
12−k
ϑ
13−k
ϑ
12+k
ϑ
13+k
ϑ
14−k
ϑ
15−k
ϑ
14+k
ϑ
15+k
ϑ
kIF ,
ˆ
ψ
A closer look at the structures depicted in Table 3.1 unveils that the barycenter algorithm
also incorporates a weighting of the input coefficients. For example in the filter with
NCR = 1 the center input is fed twice into the filter and is therefore also weighted twice
compared to the other inputs. For NCR = 13 the center input is weighted four times and its
neighbour inputs twice as strong as the other inputs.
Using the final filter output, which represents the estimated carrier phase kIF ,
ψ
, the
transmitted data can be recovered using equation (3.46).
3 Digital signal processing algorithms for coherent optical receivers
41
Problematic for the performance of the filter is the case if for one of the filter cells
p
kk
πζθ
≈− . In this case two results have roughly the same probability. Hence the
output of the filter cell is highly unreliable and might severely falsify the overall filter
output. This can be avoided by adding reliability information to each intermediate result.
Then a subsequent filter cell can decide on the basis of this reliability information, if it is
advantageous to discard an unreliable input and just feed-through the other one, or in case
of two reliable inputs to perform the averaging function according to equation (3.39). This
improved barycenter algorithm is referred to as selective maximum likelihood phase
approximation (SMLPA) [46].
3.4.4 Feed-forward carrier recovery for arbitrary QAM constellations
The described feed-forward carrier recovery concepts until now are only able to efficiently
recover the carrier phase for QAM constellations with equidistant-phases. But their
performance for square or other possible QAM constellations is very poor [48]. They either
require averaging over a large number of symbols (NCR > 100) or to use only dedicated
symbols for carrier recovery that fulfill the equidistant-phase constraint [49]. Both
significantly reduces the phase noise tolerance of the algorithms.
3.4.4.1 Square QAM constellations
In this dissertation a novel feed-forward carrier recovery concept is proposed that is able to
recover the carrier phase from arbitrary QAM constellations. Figure 3.10 shows a block
diagram of the proposed carrier recovery for square QAM constellations. As these
constellations are the most important for practical systems the derivation of the concept is
first presented for square QAM constellations. Afterwards the concept will be generalized
to arbitrary constellations.
3 Digital signal processing algorithms for coherent optical receivers
42
……
decision
circuit | |
2
∑
+
-
……
exp{jφ
b
}
|d
k,b
|
2
|d
k-N
CR
,b
|
2
|d
k+N
CR
,b
|
2
Y
k
s
k,b
Y
k
block B-1:
test phase φ
B-1
block 1:
test phase φ
1
block 0:
test phase φ
0
min( )
X
k,b
^
MUX
s
k,0
s
k,1
s
k,B-1
X
k,0
X
k,1
X
k,B-1
X
k
^
^
^
^
…
d
k,b
Figure 3.10: Feed-forward carrier recovery for square QAM constellations
The input signal Zk of the coherent receiver is sampled at the symbol rate, and perfect clock
recovery and equalization are assumed. To recover the carrier phase in a pure feed-forward
approach the received signal Zk is rotated by B test carrier phase angles b
ϕ
with
2
π
ϕ
⋅= B
b
b,
{
}
1,...,1,0
−
∈
Bb . (3.40)
Then all rotated symbols are fed into a decision circuit, which will be described in section
3.5.2, and the squared distance 2
,bk
d to the closest constellation point is calculated in the
complex plane:
{} {}
⎣⎦
{}
2
,
2
2
,
ˆ
exp
expexp
bkbk
D
bkbkbk
XjZ
jZjZd
−=
−=
ϕ
ϕϕ
(3.41)
In order to remove noise distortions, the distances of 12 CR
+
N consecutive test symbols
rotated by the same carrier phase angle b
ϕ
are summed up:
3 Digital signal processing algorithms for coherent optical receivers
43
∑
−=
−
=CR
CR
2
,,
N
Nn
bnkbk ds
(3.42)4
The optimum value of NCR depends on the laser linewidth-times-symbol-duration product
and will be evaluated by simulation in section 4.2.2.
After filtering the optimum phase angle is determined by searching the minimum sum of
distance values k
bk
smin,
,. As the decoding was already executed in (3.41), the recovered
output symbol k
X
ˆ can be selected by a switch controlled by the index bmin,k of the
minimum distance sum:
k
bkk XX min,
,
ˆ
:
ˆ= (3.43)
3.4.4.2 Arbitrary QAM constellations
The proposed feed-forward carrier recovery concept can also be applied to arbitrary QAM
constellations. If the constellation diagram is rotationally symmetric by the angle
γ
, then
b
ϕ
must be selected as
γϕ
⋅= B
b
b,
{
}
1,...,1,0
−
∈
Bb . (3.44)
For square QAM constellations 2
π
γ
=
as used in equation (3.40). Without rotational
symmetry
π
γ
2= must be used.
Due to the k-fold ambiguity of the recovered phase with
γ
π
2
=
k,
{}
⎡⎤
k
2
log bits should
be differentially encoded/decoded, where
⎡
⎤
x is the smallest integer larger than or equal to
x.
3.4.4.3 Hardware-efficient implementation
Due to the extremely high degree of parallelization (almost each functional block is in total
required B·m times, where m is the number of parallel modules due to demultiplexing
introduced in section 3.1.1), a highly efficient implementation of the algorithm is required
to make it feasible for implementation in a real-time receiver. Therefore this subsection
lists some recommendations for a hardware-efficient implementation.
4 For a joined carrier recovery for both polarizations equation (3.42) must be extended to
∑
−=
−− ⎟
⎠
⎞
⎜
⎝
⎛+=
CR
CR
2
,,
2
,,,
N
Nn
ybnkxbnkbk dds
3 Digital signal processing algorithms for coherent optical receivers
44
3.4.4.3.1 Efficient calculation of vector rotations
The rotation of a symbol in the complex plane normally requires a complex multiplication,
consisting of four real-valued multiplications with subsequent summation. This would lead
to a large number of multiplications to be executed, while achieving a sufficient resolution
B for the carrier phase values b
ϕ
. The hardware effort would therefore become prohibitive.
Applying the CORDIC (coordinate rotation digital computer) algorithm can dramatically
reduce the necessary hardware effort to calculate the B rotated test symbols [50; 51]. This
algorithm can compute vector rotations simply by summation and shift operations. As for
the calculation of the B rotated copies of the input vector intermediate results can be reused
for different rotation angles, only
{
}
∑
=
+
B
b
b
2
log
1
1
2 shift-and-add operations are required to
generate the B test symbols. For example to generate B = 32 rotated copies of Zk the
CORDIC algorithm requires only 124 shift-and-add operations instead of 124 real valued
multiplications and 62 adders.
3.4.4.3.2 Calculation of the distance to the closest constellation point
To determine the closest constellation point bk
X,
ˆ the rotated symbols are fed into a
decision circuit. The squared distance 2
,bk
d calculated with equation (3.41) can be
rewritten as
[]
()
[]
()
{}
[]
[]
()
{}
[]
[]
()
.
ˆ
ImexpIm
ˆ
ReexpRe
ImRe
2
,
2
,
2
,
2
,
2
,
bkbkbkbk
bkbkbk
XjZXjZ
ddd
−+−=
+=
ϕϕ
(3.45)
Implementing this formula literally into hardware would result into two multipliers and
three adders/subtractors. But a closer examination of (3.41) and (3.45) reveals that the
results of the subtractions are relatively small because the distance to the closest
constellation point is calculated. Therefore the most significant bits (MSBs) of the absolute
value of the subtraction result will always be zero and can be discarded to reduce the
hardware effort. As the required accuracy for 2
,bk
d is also moderate, it can be determined
using a look-up table (LUT) or basic logic functions more efficiently than with multipliers.
3.4.4.3.3 Filter function
Highly parallelized systems allow a very resourceful implementation of the summation of
2NCR+1 consecutive values. The adders can be arranged in a binary tree structure where
intermediate results from different modules are reused in neighboring modules. This leads
to a moderate hardware effort.
3 Digital signal processing algorithms for coherent optical receivers
45
3.4.5 Hardware effort
In this section the hardware effort is estimated for the different carrier recovery algorithms
at the example of QPSK. For this purpose the required number of basic functional blocks
such as multipliers, adders/subtractors, look-up tables (LUT), comparators and switches is
analyzed. Table 3.2 lists the required blocks for each of the different algorithms. Note that
a complex multiplication requires 4 real-valued multiplications and 3 adders. A complex
addition or subtraction requires 2 real-valued adders/subtractors.
Table 3.2: Required hardware components for different carrier recovery algorithms
Algorithm Multiplier LUT Adder/
Subtractor Comparator Switch
Reuse of
intermediate
results
weighted unweighted
V&V, u=0 0 3 4NCR 0 0
9
V&V, u=2 0 5 4NCR 0 0
9
V&V, u=4 10 1 4NCR+4 0 0
9
V&V, u=0 4NCR 3 4NCR 0 0
8
V&V, u=2 4NCR 5 4NCR 0 0
8
V&V, u=4 4NCR+10 1 4NCR+4 0 0
8
MLPA 0 1 4 NCR 2NCR NCR 9
SMLPA 0 1 4 NCR 4NCR 2NCR 9
QAM 0 B
{
}
CR
log
1
122
2BN
B
b
b+
∑=
+B B 9
The hardware effort for the functional blocks in Table 3.2 reduces from left to right. Thus
the most hardware-efficient algorithm is the MLPA. The total component count may be
larger than for the unweighted Viterbi & Viterbi (V&V) algorithm with u = 0, but the 2
additional look-up tables are more costly than some comparators and switches.
Table 3.2 also reveals the big disadvantage of the weighted V&V algorithm. Not only
requires the algorithm a large number of multipliers, additionally due to the weighting
almost no intermediate results from the summation process can be reused in other parallel
modules. Thus the hardware effort is significantly larger than for the other algorithms
except the QAM carrier recovery.
The estimated hardware effort for the QAM carrier recovery is roughly B times larger than
for the Viterbi & Viterbi or the (S)MLPA algorithm. This may be viewed as a large
increase in hardware consumption since B is in the range of 16 to 64. But taking into
account that the algorithm is designed for high-order QAM constellations, and that the
hardware effort for polarization control or chromatic dispersion (CD) and polarization
mode dispersion (PMD) compensation is also much higher than the hardware effort for
3 Digital signal processing algorithms for coherent optical receivers
46
QPSK carrier recovery, the implementation of the QAM carrier recovery algorithm can
still be achieved with reasonable effort.
3.5 Data recovery
When the carrier phase estimate kIF,
ˆ
ψ
is available the transmitted data can be recovered.
The data recovery process is slightly different for QAM constellations with equidistant-
phases and square QAM constellations.
3.5.1 Data recovery for QAM constellations with equidistant-phases
For QAM constellations with equidistant-phases the amplitude and phase information can
be recovered independently.
The amplitude information na can be recovered without carrier recovery. It can be
extracted from the magnitude of the received signal by simple threshold decision. The
optimum thresholds can be determined from the average power of the signal and depend on
the number of amplitude levels in the constellation and the number of constellation points
per amplitude level.
The recovered phase information represented by the estimated sector number kd
n,
ˆ depends
on the recovered carrier. It can be determined with kIF ,
ˆ
ψ
by
()
⎥
⎦
⎥
⎢
⎣
⎢+−= 2
1
2
ˆ
ˆ,,
π
ψϕ
p
nkIFkkd , (3.46)
where
⎣⎦
x is the biggest integer ≤ x. The differential sector number kd
n,
ˆ introduced in
section 2.1.3 is used at this point because kIF,
ˆ
ψ
has a p-fold ambiguity and thus the
absolute phase cannot be unambiguously recovered and differential encoding/decoding is
required.
To overcome the ambiguity of IF
ψ
ˆ the statistical properties of IF
ψ
can be exploited. As
described in section 2.2.4.3 phase noise can be modeled as a Wiener-Lévy process using
equation (2.36). Hence the most likely value for kIF,
ˆ
ψ
is the one closest to 1,
ˆ−kIF
ψ
.
However choosing the correct value from the interval
[
)
π
2,0 requires the availability of
1,
ˆ−kIF
ψ
. This violates the constraint “Feasibility of parallel processing” described in section
3.1.1. A solution to this problem is proposed in [52]. The idea is to choose kIF,
ˆ
ψ
always
from the interval
[
)
p
π
2,0 and to extend the formula (2.6) for differential decoding to
3 Digital signal processing algorithms for coherent optical receivers
47
(
)
{
}
(
)
1maxmod
ˆˆˆ ,1,,,
+
+
−= −tkjkdkdkt nnnnn . (3.47)
kj
n, is referred to as jump number because it indicates when the physical course of kIF ,
ˆ
ψ
crosses the boarders of the interval
[
)
p
π
2,0 and the actual course of kIF,
ˆ
ψ
makes a jump
p
kIFkIF
πψψ
>− −1,, ˆˆ . Therefore kj
n, is calculated by
⎪
⎩
⎪
⎨
⎧
>−−
≤−
−<−
=
−
−
−
p
p
p
n
kIFkIF
kIFkIF
kIFkIF
kj
πψψ
πψψ
πψψ
1,,
1,,
1,,
,
ˆˆ
1
ˆˆ
0
ˆˆ
1
. (3.48)
3.5.2 Data recovery for square QAM constellations
For the data recovery from square QAM constellations the evaluation of amplitude and
phase information cannot be separated like for equidistant-phase constellations. Thus no
data can be recovered without previous carrier recovery.
At first the differentially encoded quadrant number nd,k is recovered from the square QAM
constellation. The decoding process is the same as for the phase information in QAM
constellations with equidistant-phases and p = 4:
()
⎥
⎦
⎥
⎢
⎣
⎢+−= 2
12
ˆ
ˆ,,
π
ψϕ
kIFkkd
n (3.49)
Using this information the recovered symbol can be rotated into the first quadrant of the
complex plane by
⎟
⎠
⎞
⎜
⎝
⎛+−
=
′2
ˆ
ˆ,,
π
ψ
kdkIF nj
kk eYY . (3.50)
The inphase and quadrature numbers ni,k and nq,k can be recovered from Re{Yk’} and
Im{Yk’}, respectively, using two threshold deciders. The threshold levels can be
determined from the average signal power and depend on the number of constellation
points.
Finally the quadrant number kd
n,
ˆ has to be differentially decoded to obtain kt
n,
ˆ. The
differential decoding process is the same as for QAM constellations with equidistant-
phases given by equation (3.47) and
{
}
3max
=
t
n.
(
)
4mod
ˆˆˆ ,1,,, kjkdkdkt nnnn
+
−
=− (3.51)
Only the calculation of the jump number nj,k has to be adapted to the carrier recovery
process and is calculated with
3 Digital signal processing algorithms for coherent optical receivers
48
⎪
⎩
⎪
⎨
⎧
>−−
≤−
−<−
=
−
−
−
21
20
21
1min,min,
1min,min,
1min,min,
,
Bbb
Bbb
Bbb
n
kk
kk
kk
kj . (3.52)
3.6 Intermediate frequency control
In a coherent optical receiver with feed-forward carrier recovery a locking of the signal and
local oscillator laser frequencies is not required. However the frequencies of the lasers
should be sufficiently close to not degrade the carrier recovery process. Therefore either an
external LO frequency control or internal intermediate frequency compensation is required.
3.6.1 External LO frequency control
One option to control the intermediate frequency (IF) between the signal and local
oscillator is to directly control the LO laser frequency, e.g. by altering a portion of the LO
bias current. In this thesis the method will be referred to external frequency control. As
control error signal the jump numbers nj,k defined in equation (3.48) can be used. If the
course of the recovered carrier phase continuously deceases the LO frequency is too low. If
it continuously increases the LO frequency is too high. These states are indicated by the
fact that 1
,−=
kj
n occurs more often that 1
,
=
kj
n or vice versa. Thus by feeding the nj,k
into an external integral controller the LO laser frequency can be locked to the signal laser
frequency.
3.6.2 Internal intermediate frequency compensation
Instead of directly controlling the LO laser frequency it is also possible to compensate for
the IF in the DSPU. Like for the external LO frequency control the jump numbers nj,k can
be used as control error signal. But instead of outputting the nj,k from the DSPU they are
integrated internally.
∑
−∞=
=k
i
ijIFkIF ng ,,
ˆ
ω
(3.53)
kIF ,
ˆ
ω
is the estimated product T
IF
ω
. As only a coarse compensation of the IF is required,
the control gain gIF can be small. The IF is compensated by altering the phase of the
received symbols.
()
1,, ˆˆ
,
,
,
,
,
,
,ˆ
exp −Δ−Δ− +−
Δ−
−∞= ⎥
⎦
⎤
⎢
⎣
⎡
=
⎭
⎬
⎫
⎩
⎨
⎧−
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
′
′∑kIFkIF
j
yk
xk
k
i
iIF
yk
xk
yk
xk e
Z
Z
j
Z
Z
Z
Z
ωω
ω
(3.54)
Δ is the processing delay of the system.
4 Simulation Results
49
4 Simulation results
The performance of the polarization control, ISI compensation and carrier recovery
algorithms described in chapter 3 is investigated in extensive simulations. The goals are to
define the optimum system parameters such as control gain or filter width and to determine
the tolerances against various distortions such as AWGN, polarization cross-talk, PMD,
phase noise and quantization effects. Especially the simulation results for a QPSK
transmission system are required for the implementation of a real-time polarization-
multiplexed synchronous QPSK system. All simulations were programmed and executed
with MATLAB®.
4.1 QPSK carrier recovery
The feed-forward carrier recovery algorithms for a QAM constellation with equidistant-
phases described in section 3.4 are compared in Monte-Carlo simulations of a QPSK
transmission system. The QPSK constellation is chosen as a reference because a
polarization-multiplexed QPSK system is the target system of the synQPSK project to be
realized in a real-time laboratory experiment. The following simulation results were used
to determine the main system parameters. Another more general reason for the detailed
investigation of the QPSK format is its superior noise immunity that makes it most
attractive for long-haul fiber transmission systems. Each data point is based on the
simulation of 2,000,000 symbols. The results are compared against the theoretically
optimal receiver sensitivity [19] given by
[]
BERQ 1
0
−
=
N
ES. (4.1)
ES/N0 is the normalized optical signal to noise ratio introduced in section 2.2.3.5, and BER
is the target bit error rate. Note that this equation does not take the coding penalty due to
differential encoding/decoding into account. Thus the differential coding penalty F = 2 for
QPSK (Table 2.1) contributes to the sensitivity penalties determined by the simulations.
As another reference also the feed-forward carrier recovery for arbitrary QAM
constellations described in section 3.4.4 is analyzed for QPSK. The hardware-effort for this
algorithm is much higher than for the other considered algorithms, thus for a practical
QPSK implementation the QAM carrier recovery will be of low interest. But QPSK is the
only modulation format, where the QAM algorithm performance can be compared against
efficient state-of-the-art techniques. For higher-order QAM formats only poorly conceived
techniques exist which have either a low practicality or a low phase noise tolerance.
4 Simulation Results
50
4.1.1 QPSK carrier phase estimator efficiency and mean squared error
In the following the mean squared error (MSE) and the efficiency e(NCR) of the different
carrier phase estimators are analyzed. The MSE allows quantifying how much the
estimated carrier phase
ψ
ˆ differs from the true value of
ψ
and is given by
()
(
)
2
ˆˆ
MSE
ψψψ
−= . (4.2)
To determine the efficiency of an estimator, its MSE is compared against the Cramér-Rao
lower bound (CRLB), which specifies the lowest possible mean squared error achievable
by an unbiased estimator [53]. To calculate the CRLB the contribution of AWGN to phase
noise is approximated to be Gaussian with the variance
(
)
1
0
2−
NES [54]. With this
approximation, which has a high accuracy for high OSNRs, the CRLB depending on the
filter half width NCR is given by
()
S
E
N
N
N212
1
CRLB 0
CR
CR ⋅
+
=. (4.3)
Then the efficiency e(NCR) of the carrier phase estimators is defined as
() ()
() ()()
1
ˆ
122
ˆ
MSE
CRLB
2
CR
0CR
CR ≤
−+
==
ψψ
ψ
NE
NN
Ne
S
. (4.4)
As the CRLB is the lowest possible mean squared error, the maximum estimator efficiency
is 1.
4.1.1.1 Carrier phase estimator efficiency for Δf·TS = 0
Figure 4.1 shows the efficiency e(NCR) in the absence of laser-induced phase noise
(Δf·TS = 0) and two different OSNRs for all carrier recovery algorithms that were presented
in section 3.4.
Normally for NCR = 0 one would expect that e(0) = 100%, because the actual symbol phase
is used as carrier phase, i.e. no estimation of the carrier is performed. The reason for
e(0) < 100% is that the approximation of the AWGN-induced phase noise variance is too
optimistic, especially for lower OSNRs.
4 Simulation Results
51
Figure 4.1: QPSK carrier phase estimator efficiency for OSNR = 10 dB (left) and OSNR = 16 dB (right)
The comparison of the different carrier phase estimators shows that the Viterbi & Viterbi
(V&V) algorithm with u = 0 yields the best performance reaching almost 100% efficiency
for OSNR = 16 dB. Increasing u to 2 or 4 degrades the efficiency. This becomes evident if
one considers that the amplitude contains no information about the carrier phase IF
ψ
. As
the amplitude of a QPSK signal is constant, amplitude fluctuations are only caused by
noise, whose influence can be reduced by considering only the phase of the received
symbols for carrier recovery. Note that a differentiation between the Viterbi & Viterbi
algorithms with and without weighted averaging is not necessary for Δf·TS = 0, because all
wi have the same value and thus both filters are identical.
Comparing the MLPA algorithm against the other approaches, for OSNR = 10 its
performance is inferior. But it can be significantly improved by adding reliability
information to each intermediate result, referred to as the SMLPA algorithm. The effect
that the two algorithms do not generate a smooth curve is caused by the filter structures
that vary for each value of NCR (compare Table 3.1). The reason that for OSNR = 16 the
two algorithms yield exactly the same performance is due to the fact that for high OSNRs
virtually all intermediate results are reliable and therefore the output of both algorithms is
the same.
The 4-QAM carrier recovery achieves an efficiency comparable to the V&V algorithm
with u = 0, admittedly only for OSNR = 10 and short filter lengths. And for OSNR = 16 dB
the efficiency even degrades dramatically for larger filters. This result is not surprising if
one recalls that the QAM carrier recovery algorithm has an inherent phase quantization,
which is set to log2{B} = 5 in this simulation. This limits the achievable mean squared
0 5 10 15
50
55
60
65
70
75
80
85
90
95
100
Filter half width N
Efficiency [%]
ONSR = 10 dB
0 5 10 15
50
55
60
65
70
75
80
85
90
95
100
Filter half width N
Efficiency [%]
V&V, u=0
V&V, u=2
V&V, u=4
MLPA
SMLPA
4-QAM
ONSR = 16 dB
4 Simulation Results
52
error of the estimation process and thus degrades the efficiency, especially for low OSNRs
and large filters, where the CRLB is low.
4.1.1.2 Carrier phase estimator mean squared error for Δf·TS > 0
In the presence of laser-induced phase noise (Δf·TS > 0) the analysis of the estimator
efficiency yields only minor information, because the CRLB is only valid for the
estimation of constant values. But this requirement is not any more fulfilled if laser phase
noise is considered next to AWGN. Therefore in this section the mean squared errors of the
estimator outputs are depicted. The CRLB is plotted as a reference curve.
Figure 4.2 shows the mean squared error of the different carrier recovery algorithms for
OSNR = 10 dB and OSNR = 16 dB. As the linewidth of DFB lasers employed in
commercial optical transmission systems is usually in the range of 1 MHz to 10 MHz and
the symbol rates range from 10 Gbaud to 40 Gbaud, the carrier recovery should be able to
tolerate linewidth-times-symbol-duration products as high as 3
10−
=⋅Δ S
Tf . Therefore the
considered linewidth-times-symbol-duration products are Δf·TS = 10-4 (top left),
Δf·TS = 4·10-4 (top right) and Δf·TS = 10-3 (bottom).
Note that the Viterbi & Viterbi algorithm with u = 2 is not considered any more for the
subsequent simulations, because its performance is inferior to the corresponding algorithm
with u = 0 while the required computational effort for it is higher.
4 Simulation Results
53
Figure 4.2: Carrier phase estimator mean squared error for different values of Δf·TS
The results for the MSE are similar to the ones obtained for the estimator efficiency for
Δf·TS = 0. The best performance is also achieved using the Viterbi & Viterbi algorithm
with u = 0 – but only if weighted averaging is applied. Especially for higher phase noise
values the SMLPA algorithm often outperforms the Viterbi & Viterbi algorithm without
weighted filter inputs. This can be explained by the fact that depending on the filter
structure the SMLPA filter also weights its input values. This is advantageous in the
presence of phase noise. As the QAM carrier phase estimator does not have this advantage
it is not surprising that its MSE is very similar to the one of the unweighted Viterbi &
Viterbi algorithm.
Figure 4.2 also unveils that most algorithms have an optimum filter width that varies for
different values of Δf·TS. The optimum values for NCR are between 2 and 6 for Δf·TS = 10-3
and increase for lower linewidth-times-symbol-duration products. An exception to this is
0 5 10 15
10
-3
10
-2
10
-1
Filter half width N
MSE [rad
2
]
Δ
f
⋅
T
S
= 10
-4
CRLB
V&V, u=0
V&V, u=4
MLPA
SMLPA
4-QAM
OSNR = 16 dB
OSNR = 10 dB
weighted averaging
0 5 10 15
10
-3
10
-2
10
-1
Filter half width N
MSE [rad
2
]
Δ
f
⋅
T
S
= 4
⋅
10
-4
CRLB
V&V, u=0
V&V, u=4
MLPA
SMLPA
4-QAM
OSNR = 10 dB
weighted averaging
OSNR = 16 dB
0 5 10 15
10
-3
10
-2
10
-1
Filter half width N
MSE [rad2]
Δ
f
⋅
T
S
= 10
-3
CRLB
V&V, u=0
V&V, u=4
MLPA
SMLPA
4-QAM
OSNR = 10 dB
OSNR = 16 dB
weighted
averaging
4 Simulation Results
54
the Viterbi & Viterbi carrier recovery with weighted averaging. As its weighting
coefficients are adaptively optimized to the ratio of laser phase noise to AWGN-induced
phase noise the value for the optimum filter half width is
∞
→
CR
N. However although
the MSE for the Viterbi & Viterbi algorithm does not degrade, but it only improves
marginally if NCR becomes larger than the optimum filter half widths of the other
approaches. The reason is that the outer weights of the filter tend to zero.
4.1.2 QPSK phase noise tolerance
In order to evaluate the effect of the different algorithm efficiencies and MSEs on the
receiver sensitivity of a coherent optical receiver, the BER for the different carrier recovery
algorithms against the OSNR is investigated. The same values are used for the linewidth-
times-symbol-duration products as in section 4.1.1. NCR = 4 and NCR = 9 are the considered
filter half widths.
4.1.2.1 Viterbi & Viterbi algorithm performance
Figure 4.3 shows the BER against the OSNR for the original Viterbi & Viterbi feed-
forward carrier recovery algorithm described in section 3.4.1 for
{
}
4 ,0
∈
u.
Figure 4.3: OSNR vs. BER for Viterbi & Viterbi carrier recovery
and different linewidth-times-symbol-duration products Δf·TS.
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
u=0, N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
Δf⋅T
S
=0
Δf⋅T
S
=1⋅10
-4
Δf⋅T
S
=4⋅10
-4
Δf⋅T
S
=1⋅10
-3
theor. opt.
u=0, N=9
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
u=4, N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
u=4, N=9
4 Simulation Results
55
The comparison of the receiver sensitivity for u = 0 and u = 4 confirms the results from
section 4.1.1 that the neglect of the symbol amplitude (u = 0) is beneficial for the carrier
recovery performance. Looking at the different values of the filter half width NCR, then no
general decision can be made for an optimum value. For low phase noise a large filter is
beneficial to reduce the negative effects of AWGN. But for high values of Δf·TS the
symbols at the edges of the filter carry only little information about the actual carrier phase
to be recovered. Therefore for the unweighted Viterbi & Viterbi algorithm a narrower filter
becomes advantageous.
As expected from the results in section 4.1.1 the simulation results for the weighted Viterbi
& Viterbi algorithm depicted in Figure 4.4 do not contain such a performance degradation
for high values of Δf·TS and large filters. Especially for u = 0 and NCR = 9 the benefit of the
weighted filtering becomes obvious. Due to the adaptation of the weighting coefficients to
the ratio of laser phase noise to the angular portion of AWGN ( 22
n′
Δ
σσ
) no penalty is
observed compared to the results for NCR = 4, even for Δf·TS = 10-3.
Figure 4.4: OSNR vs. BER for weighted Viterbi & Viterbi carrier recovery
and different linewidth-times-symbol-duration products.
Figure 4.5 compares the sensitivity penalties at BER = 10-3 both for the unweighted (left)
and weighted (right) Viterbi & Viterbi carrier recovery algorithms. For Δf·TS < 10-4 the
sensitivity improves only marginally. The reason for the residual sensitivity penalty for
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
u=0, N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
Δf⋅T
S
=0
Δf⋅T
S
=1⋅10
-4
Δf⋅T
S
=4⋅10
-4
Δf⋅T
S
=1⋅10
-3
theor. opt.
u=0, N=9
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
u=4, N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
u=4, N=9
4 Simulation Results
56
Δf·TS → 0 is the differential coding penalty explained in section 2.1.3 and an additional
implementation penalty depending on the efficiency of the carrier recovery process.
Figure 4.5: Sensitivity penalties at BER = 10-3 against Δf·TS for unweighted (a)
and weighted (b) Viterbi & Viterbi carrier recovery
4 Simulation Results
57
4.1.2.2 (S)MLPA algorithm performance
In this subsection the phase noise tolerance of the (S)MLPA algorithm presented in section
3.4.3 is evaluated. Both the original MLPA algorithm, also referred to as barycenter
algorithm, and the SMLPA algorithm which uses additional reliability information about
intermediate results are considered. Figure 4.6 depicts the corresponding BERs against
OSNR.
Figure 4.6: OSNR vs. BER for (S)MLPA carrier recovery
and different linewidth-times-symbol-duration products
The results show that for the MLPA algorithm a larger filter gives only little benefits. For
low OSNRs the improvement due to the larger filter is marginal, and for high OSNRs the
sensitivity even degrades for Δf·TS = 0. This is due to the fact that the carrier phase angles
are averaged pairwise. As for each averaging step one of three possible solutions has to be
selected, the error-susceptibility increases for larger filter widths, even for low linewidth-
times-symbol-duration products.
This drawback is overcome by adding reliability information to each intermediate result.
The simulation results for SMLPA in the bottom row of Figure 4.6 confirm a sensitivity
improvement, especially for low OSNRs and larger filters. Only for NCR = 9 and
Δf·TS = 10-3 the phase noise tolerance is slightly degraded. This can also be seen in Figure
4.7, which exemplifies the different sensitivity penalties at BER = 10-3.
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
MLPA, N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
Δf⋅T
S
=0
Δf⋅T
S
=1⋅10
-4
Δf⋅T
S
=4⋅10
-4
Δf⋅T
S
=1⋅10
-3
theor. opt.
MLPA, N=9
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
SMLPA, N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
SMLPA, N=9
4 Simulation Results
58
Figure 4.7: Sensitivity penalties at BER = 10-3 against Δf·TS for (S)MLPA carrier recovery
4.1.2.3 4-QAM carrier recovery performance
Figure 4.8 shows the simulation results for 4-QAM carrier recovery with a carrier phase
resolution of log2{B} = 5. The performance for low linewidth-times-symbol-duration
products Δf·TS ≤ 4·10-4 is comparable to the other approaches. But for Δf·TS = 10-3 the
disadvantage of constant weights for the filter inputs becomes visible, similar to the results
for the unweighted Viterbi & Viterbi algorithm in section 4.1.2.1. The sensitivity
disproportionately reduces as phase noise increases, especially for larger filters. This effect
becomes even more obvious in Figure 4.9, which shows the sensitivity penalties for
4-QAM carrier recovery at a BER of 10-3.
Figure 4.8: OSNR vs. BER for 4-QAM carrier recovery and different
linewidth-times-symbol-duration products.
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
N=4
910 11 12 13 14
10
-6
10
-5
10
-4
10
-3
10
-2
OSNR [dB]
BER
Δf⋅T
S
=0
Δf⋅T
S
=1⋅10
-4
Δf⋅T
S
=4⋅10
-4
Δf⋅T
S
=1⋅10
-3
theor. opt.
N=9
4 Simulation Results
59
Figure 4.9: Sensitivity penalties at BER = 10-3 against Δf·TS for 4-QAM carrier recovery
4.1.2.4 Summary
A comparison between all considered QPSK carrier recovery concepts unveils that the best
performance, i.e. the highest receiver sensitivity is achieved by applying the weighted
Viterbi & Viterbi algorithm with u = 0 and NCR = 9. Its minimum penalty compared to the
theoretical optimum is 0.7 dB and it has a high tolerance against phase noise. The
barycenter algorithm extended by reliability information (SMLPA) achieves roughly the
same phase noise tolerance, but its minimum sensitivity penalty exceeds the penalty of the
weighted Viterbi & Viterbi algorithm by 0.1 dB. Finally the 4-QAM carrier recovery
achieves the same receiver sensitivity as the barycenter algorithm, even though it contains
an inherent quantization of the phase, which is not considered yet for the other algorithms.
But due to its unweighted filter inputs its phase noise tolerance is inferior.
All in all the differences in receiver sensitivity due to the different carrier recovery
algorithms are small. Therefore it is difficult to give a final recommendation, which
algorithm should be employed in a real system implementation. If the strategic decision is
to develop a receiver with ultimate performance, then the Viterbi & Viterbi algorithm with
adaptive weighted averaging should be applied. However, the SMLPA algorithm offers a
high potential to reduce the number of required logic gates in a hardware implementation
at only the minor cost of slightly reduced receiver sensitivity.
4.1.2.5 Common carrier in a polarization-multiplexed QPSK receiver
In a polarization-multiplexed QPSK receiver a common carrier can be recovered from the
data of both polarizations. Figure 4.10 shows a comparison of the sensitivity penalties for
the Viterbi & Viterbi algorithm without weighting, the SMLPA algorithm and the QAM
carrier recovery for a single-polarization system with NCR = 9 and a polarization-
4 Simulation Results
60
multiplexed system with NCR = 4. Thus almost the same amount of 19 and 18 symbols,
respectively, is used to recover the carrier.
Figure 4.10: Phase noise tolerance for different carrier recovery algorithms using either the data from a
single-polarization or from both polarizations
For Δf·TS = 0 the two systems using the same carrier recovery algorithm also obtain the
same receiver sensitivity. As the estimation process is based on the same number of
symbols this result is not surprising. But the systems with joined carrier recovery tolerate
roughly twice as much phase noise as the respective algorithms using the data from only
one polarization. The reason for this better tolerance is that the algorithms with joined
carrier recovery require only half the filter width to recover the carrier phase from the same
amount of data. Considering that the symbols that are temporally closer to the symbol to be
decoded carry more information about its carrier phase than symbols that are temporally
further away, the higher accuracy of the joined carrier recovery process becomes evident.
4.1.3 QPSK analog-to-digital converter resolution
In this section the influence of the analog-to-digital converter (ADC) resolution on the
receiver sensitivity is analyzed. As the sensitivity penalty only marginally depends on the
employed carrier recovery concept, the considered algorithms are reduced to the Viterbi &
Viterbi (V&V) algorithm with u = 0, the SMLPA and the 4-QAM carrier recovery
algorithms. As the simulations are conducted with Δf·TS = 0 a differentiation between
V&V algorithms without and with weighted averaging is not necessary. The filter half
width is set to NCR = 9.
Figure 4.11 depicts the sensitivity penalty of the receiver due to the quantization of the
input samples by the ADCs. It is remarkable that the sensitivity penalty for the QAM
carrier recovery only slightly degrades for ADC resolutions down to 4 bit and is even
below the penalty of the ideal receiver. This can be explained by the fact that the QAM
4 Simulation Results
61
carrier recovery does not calculate the carrier phase as a function of the input samples, but
selects the carrier phase angle that allows the most reliable decoding. Therefore the
dependence on the accuracy of the input samples is reduced. For the other carrier recovery
algorithms (V&V and SMLPA), that determine the carrier phase directly from the input
samples, sensitivity starts degrading for ADC resolutions below 5 bit.
Figure 4.11: Sensitivity penalty vs. analog-to-digital converter resolution
for different QPSK carrier recovery algorithms
4.1.4 QPSK phase resolution
An important internal resolution for the different QPSK carrier recoveries is the required
precision of the symbol phase. All algorithms require the calculation of the phase from the
input symbol. But the received symbol is given by its real and imaginary part. Hence a
transformation into the angular domain is required. As the calculation of the argument of a
complex number is very complex to realize literally in hardware, usually the results for all
possible inputs are stored in a look-up table (LUT). Therefore the required precision of
these results is an important parameter for the digital implementation of the algorithms.
Figure 4.12 shows the sensitivity penalty due to the phase quantization. Even down to a 3
bit resolution of the symbol phase the data can still be recovered from the received symbol
with only low performance degradation. However to avoid any penalty a 5 or 6 bit
resolution should be chosen.
4 Simulation Results
62
Figure 4.12: Sensitivity penalty vs. phase resolution for different QPSK carrier recovery algorithms
4 Simulation Results
63
4.2 QAM carrier recovery
In this section the performance of the QAM carrier recovery algorithm proposed in section
3.4.4 is evaluated. The considered constellations are 4-QAM (QPSK), 16-QAM, 64-QAM
and 256-QAM. The simulations are limited to square constellations because they are easy
to generate [17; 55] and have a high noise immunity in AWGN-dominated transmission
systems [21]. The filter width is always set to NCR = 9, and each data point is based on the
simulation of 200,000 symbols. The results are compared against the theoretically
achievable sensitivity calculated with the following formula [19]:
{}
()
2
1
2
1
0
BER11
1
1
2
log
Q
3
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡−−
⎟
⎠
⎞
⎜
⎝
⎛−
−
=
−
−
M
M
M
N
ES
(4.5)
ES/N0 is the normalized optical signal to noise ratio (OSNR), M is the number of
constellation points and BER is the target bit error rate.
4.2.1 Square QAM phase angle resolution
A crucial quantity for the proposed algorithm is the required number B of the test phase
values b
ϕ
. If the required resolution is too large, the realization in hardware becomes
unfeasible. Figure 4.13 shows the sensitivity penalty at the bit error rate 10-3 for 4-QAM
and 16-QAM. The proposed algorithm and a receiver with ideal carrier recovery were
simulated with different resolutions for the carrier phase. Ideal carrier recovery means that
the receiver knows the exact carrier phase (which is only realizable in simulation) and
therefore the sensitivity penalty is only caused by differential quadrant encoding and
quantization effects.
4-QAM attains a minimum penalty of 0.5 dB for the ideal receiver and 0.7 dB for the
proposed algorithm. The penalty difference of 0.2 dB is thus the implementation-induced
penalty. For 16-QAM the minimum penalties decreases (0.4 dB for the ideal receiver,
0.6 dB for the proposed algorithm), because only 2 out of 4 transmitted bits are
differentially encoded. For all 4 receivers it can be seen that almost no additional penalty is
induced due to the quantization of the carrier phase, provided that
{}
5log2≥B. Therefore
in all following simulations for 4-QAM and 16-QAM B is set to 32.
4 Simulation Results
64
Figure 4.13: Sensitivity penalty for different numbers of test phase values φb for 4-QAM and 16-QAM
Figure 4.14 shows the same simulations for 64-QAM and 256-QAM. The minimum
penalty for 64-QAM is 0.3 dB with ideal carrier recovery and 0.5 dB using the proposed
algorithm. For 256-QAM the respective values are 0.35 dB and 0.55 dB. For both
constellations the penalty due to the quantization of the carrier phase is tolerable only if
{}
6log2≥B. The number of test phase values for 64-QAM and 256-QAM is therefore
chosen to be B = 64 in all subsequent simulations.
Figure 4.14: Sensitivity penalty for different numbers of test phase values φb for 64-QAM and 256-QAM
4.2.2 Square QAM phase estimator efficiency
In the following the efficiency of the QAM phase estimator is analyzed. It is given by the
ratio of the CRLB to the mean square error of the estimator. The formula is presented in
equation (4.3) of section 4.1.1. As the CRLB is independent of the estimator structure its
calculation does not need to be modified for higher-order QAM constellations.
4 Simulation Results
65
Figure 4.15 shows for 4-QAM and 16-QAM the mean squared error of the phase estimator
together with the theoretical optimum expressed by the CRLB (top row) and the resulting
estimator efficiency e(NCR) (bottom row) for the proposed carrier recovery concept. Figure
4.16 depicts the same information for 64-QAM and 256-QAM.
Figure 4.15: Phase estimator mean squared error and efficiency e(NCR) vs. filter half width NCR
for square 4-QAM (left) and square 16-QAM (right) constellations with log2{B} = 5
(The legends are valid for both figures of a column)
4 Simulation Results
66
Figure 4.16: Phase estimator mean squared error and efficiency e(NCR) vs. filter half width NCR
for square 64-QAM (left) and square 256-QAM (right) constellations with log2{B} = 6
(The legends are valid for both figures of a column)
For all considered constellations in the absence of phase noise the mean squared error
()
2
ˆ
ψψ
− continuously decreases for larger values of NCR. In contrast if phase noise is
present a global minimum emerges that depends on the linewidth-times-symbol-duration
product and the OSNR. It can be seen that NCR = 9, which was selected for simulations,
induces a mean squared error that is always close to this minimum, especially for lower
OSNRs. In principle by optimizing NCR for each parameter set {OSNR, ∆f·TS, M} the
performance of the receiver could have been improved. But as this is not practical in real
systems this optimization was omitted in the simulations.
4 Simulation Results
67
A result comparison among the different QAM constellations makes it apparent that the
maximum achievable efficiency reduces from ~85% for 4-QAM to ~60% for 16-QAM and
64-QAM to finally around 30% for 256-QAM. This reduction is mainly caused by the
quantization of the carrier phase that limits the minimum achievable mean squared error of
the estimator. For
{}
5log2=B (4-QAM, 16-QAM) and
{
}
6log2=B (64-QAM, 256-
QAM) the minimum mean squared errors are 2·10-4 rad2 and 5·10-5 rad2, respectively. For
4-QAM the CRLB is well above the quantization limit, thus quantization effects can
mostly be neglected. For 16-QAM and 64-QAM the CRLB is in the range of the
quantization limit. Hence the estimator efficiency is already reduced, especially for low
OSNR. If the resolution of the carrier phase for 16-QAM is increased to 6 bit, the estimator
efficiency improves to ~80% (Figure 4.17). Finally for 256-QAM the CRLB is mostly
below the 6 bit quantization limit, which of course severely degrades the estimator
efficiency.
Figure 4.17: Phase estimator mean squared error (left) and efficiency (right) vs. filter half width NCR
for a square 16-QAM constellation and log2{B} = 6
(The legends are valid for both figures)
Increasing the carrier phase resolution could improve the estimator efficiency, but at the
price of an increased hardware effort.
Figure 4.18 to Figure 4.21 show the considered square QAM constellations at the receiver
before and after carrier recovery. They verify that the selected carrier phase resolutions are
sufficient to reliably recover the investigated constellations.
510 15 20
10
-4
10
-3
10
-2
Filter half width N
Mean squared error [rad
2
]
510 15 20
0
10
20
30
40
50
60
70
80
90
100
Filter half width N
Efficiency [%]
OSNR = 16 dB
OSNR = 22 dB
CRLB
Δf⋅T
S
=0
Δf⋅T
S
=3.5⋅10
-5
Δf⋅T
S
=1.4⋅10
-4
4 Simulation Results
68
Figure 4.18: 4-QAM constellation diagram at the receiver before and after carrier recovery
for ∆f·TS = 4·10-4 (log2{B} = 5, NCR = 9)
Figure 4.19: 16-QAM constellation diagram at the receiver before and after carrier recovery
for ∆f·TS = 1.4·10-4 (log2{B} = 5, NCR = 9)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
rx data w/o carrier recovery (OSNR = 10 dB)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
rx data with carrier recovery (OSNR = 10 dB)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
rx data w/o carrier recovery (OSNR = 16 dB)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
rx data with carrier recovery (OSNR = 16 dB)
-5 0 5
-5
0
5
rx data w/o carrier recovery (OSNR = 16 dB)
-5 0 5
-5
0
5
rx data with carrier recovery (OSNR = 16 dB)
-5 0 5
-5
0
5
rx data w/o carrier recovery (OSNR = 22 dB)
-5 0 5
-5
0
5
rx data with carrier recovery (OSNR = 22 dB)
4 Simulation Results
69
Figure 4.20: 64-QAM constellation diagram at the receiver before and after carrier recovery
for ∆f·TS = 4·10-5 (log2{B} = 6, NCR = 9)
Figure 4.21: 256-QAM constellation diagram at the receiver before and after carrier recovery
for ∆f·TS = 8·10-6 (log2{B} = 6, NCR = 9)
-5 0 5
-8
-6
-4
-2
0
2
4
6
8
rx data w/o carrier recovery (OSNR = 22 dB)
-5 0 5
-8
-6
-4
-2
0
2
4
6
8
rx data with carrier recovery (OSNR = 22 dB)
-5 0 5
-5
0
5
rx data w/o carrier recovery (OSNR = 28 dB)
-5 0 5
-5
0
5
rx data with carrier recovery (OSNR = 28 dB)
-15 -10 -5 0 5 10 15
-15
-10
-5
0
5
10
15
rx data w/o carrier recovery (OSNR = 28 dB)
-15 -10 -5 0 5 10 15
-15
-10
-5
0
5
10
15
rx data with carrier recovery (OSNR = 28 dB)
-15 -10 -5 0 5 10 15
-15
-10
-5
0
5
10
15
rx data w/o carrier recovery (OSNR = 34 dB)
-15 -10 -5 0 5 10 15
-15
-10
-5
0
5
10
15
rx data with carrier recovery (OSNR = 34 dB)
4 Simulation Results
70
Another peculiarity in Figure 4.15 and Figure 4.16 is that the efficiency for the higher-
order QAM constellations tends to go to zero for short filter lengths and low OSNR. The
reason for this effect becomes obvious if one recalls the carrier recovery algorithm: The
squared distance of the received symbol to the closest constellation point is calculated for
different carrier phase angles. Looking at the developing of |dk,b|2 and sk,b over the different
b
ϕ
shown in Figure 4.22, the algorithm produces for 4-QAM one distinct minimum
already for NCR = 0. In contrast for higher-order QAM constellations several local minima
emerge. Therefore larger filters are required to identify the global minimum, especially for
lower OSNRs where the signal is strongly corrupted by noise.
Figure 4.22: Squared distance sum sb of the b-th parallel block (with test carrier phase angle φb) for different
filter half widths and different square QAM constellations
4.2.3 Square QAM phase noise tolerance
Another important property of a carrier recovery in a coherent receiver is its tolerance
against phase noise. Today’s commercial transmission systems usually employ DFB lasers,
because they are cost-efficient and have a small footprint. The linewidth of such lasers is in
0 5 10 15 20 25 30
0
2
4
6
8
10
12
14
16
18
20
b
s
b
4-QAM
N
CR
=0
N
CR
=1
N
CR
=2
N
CR
=3
N
CR
=4
N
CR
=5
N
CR
=6
N
CR
=7
N
CR
=8
N
CR
=9
0 5 10 15 20 25 30
0
2
4
6
8
10
12
14
16
18
20
b
s
b
16-QAM
N
CR
=0
N
CR
=1
N
CR
=2
N
CR
=3
N
CR
=4
N
CR
=5
N
CR
=6
N
CR
=7
N
CR
=8
N
CR
=9
010 20 30 40 50 60
0
2
4
6
8
10
12
14
16
18
20
b
s
b
64-QAM
N
CR
=0
N
CR
=1
N
CR
=2
N
CR
=3
N
CR
=4
N
CR
=5
N
CR
=6
N
CR
=7
N
CR
=8
N
CR
=9
010 20 30 40 50 60
0
2
4
6
8
10
12
14
16
18
20
b
s
b
256-QAM
N
CR
=0
N
CR
=1
N
CR
=2
N
CR
=3
N
CR
=4
N
CR
=5
N
CR
=6
N
CR
=7
N
CR
=8
N
CR
=9
4 Simulation Results
71
the range of 100 kHz < ΔfDFB < 10 MHz. Assuming a symbol rate of 10 Gbaud the
tolerable linewidth-times-symbol-duration product must be 10-5 < Δf·TS < 10-3.
Figure 4.23 shows the sensitivity penalty of the proposed carrier recovery algorithm
against the linewidth-times-symbol-duration product and compares single-polarization
carrier recovery with NCR = 9 against a joined carrier recovery in a polarization-
multiplexed receiver with NCR = 4. If no phase noise is present both approaches achieve the
same receiver sensitivity. But similar as for the QPSK carrier recovery the phase noise
tolerance of all QAM receivers can be roughly doubled by applying a joined carrier
recovery for both polarizations. Table 4.1 and Table 4.2 summarize the maximum tolerable
linewidths for a 10
Gbaud and a 100GbE system, respectively, for the different square
QAM constellations and a sensitivity penalty of 1 dB at BER = 10-3.
Figure 4.23: Receiver tolerance against phase noise for different square QAM constellations
Table 4.1: Maximum tolerable linewidth for 10 Gbaud systems with different square QAM constellations
Square
constellation
Max. tolerable Δf·TS
for 1 dB penalty
@ BER = 10-3
Max. tolerable Δf for
TS = 10 Gbaud
4-QAM 4·10-4 (1·10-3) 4 MHz (10 MHz)
16-QAM 1.4·10-4 (2.1·10-4) 1.4 MHz (2.1 MHz)
64-QAM 4·10-5 (8·10-5) 400 kHz (800 MHz)
256-QAM 8·10-6 (1.5·10-5) 80 kHz (150 kHz)
(The numbers in brackets correspond to a joined carrier recovery in a polarization-multiplexed receiver)
4 Simulation Results
72
Table 4.2: Required symbol rate and maximum tolerable linewidth to realize a 100GbE (112 Gb/s) system
with different square QAM constellations
Square
constellation Bits per symbol Required symbol rate
for 112 Gb/s
Max. tolerable Δf
for a 100GbE system
4-QAM 2 (4) 56 Gbaud (28 Gbaud) 22.4 MHz (28 MHz)
16-QAM 4 (8) 28 Gbaud (14 Gbaud) 3.9 MHz (2.9 MHz)
64-QAM 6 (12) 18.67 Gbaud (9.33 Gbaud) 750 kHz (750 kHz)
256-QAM 8 (16) 14 Gbaud (7 Gbaud) 110 kHz (105 kHz)
(The numbers in brackets correspond to a polarization-multiplexed transmission system with joined carrier
recovery for both polarization channels at the receiver)
In order to evaluate the phase noise tolerance of the algorithm also for lower BER rates,
additional long term simulations have been executed for single-polarization carrier
recovery and selected values of ∆f·TS, simulating 2,000,000 symbols per data point (Figure
4.24). Note that for BERs below 10-5 the results become inaccurate due to the low number
of errors that occurred during simulation. The theoretical optimum is calculated by
inverting equation (4.5):
{}
2
02 1
3
Q
1
1
log
2
11BER ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛−−−= N
E
M
M
M
S
(4.6)
Figure 4.24: Impact of different linewidth-times-symbol-duration products on the receiver sensitivity of
coherent QAM receivers
It can be seen that for the values of ∆f·TS causing 1 dB of penalty at a BER of 10-3, the
penalty increases for lower BERs, especially for higher-order QAM constellations. But if
4 Simulation Results
73
these values are reduced to one quarter the penalty stays almost constant even for BER
rates down to 10
-5
.
4.2.4 Square QAM analog-to-digital converter resolution
Another major obstacle to realize real-time coherent transmission systems with digital
carrier recovery is based on the required bandwidth and resolution of the analog-to-digital
converter (ADC). Figure 4.25 shows the effect of the ADC resolution on the receiver
sensitivity for the considered QAM constellations.
Figure 4.25: Receiver sensitivity penalty vs. analog-to-digital converter resolution
for different square QAM constellations
The necessary ADC resolution increases approximately by 1 bit if the number of
constellation points is multiplied by 4. This relation becomes evident by looking again at
Figure 2.3. It can be seen that increasing the number of constellations points by a factor of
4 doubles the size of the constellation diagram in real and imaginary dimensions.
Therefore, to keep the accuracy of the received samples constant, the number of ADC
quantization steps must also double. Table 4.3 summarizes the ADC requirements for a
polarization-multiplexed 100GbE transmission system. Because commercial systems will
also contain PMD and CD compensation, which necessitates oversampling, the values for
T
S
/2 sampling are also given.
4 Simulation Results
74
Table 4.3: Analog-to-digital converter requirements for a
polarization-multiplexed QAM transmission system for 100GbE
Square
constellation
ADC
bandwidth
ADC sampling rate
(T
S
/2 sampling)
ADC effective
number of bits
4-QAM 28 GHz 56 Gsample/s > 3.8
16-QAM 14 GHz 28 Gsample/s > 4.9
64-QAM 9.33 GHz 18.67 Gsample/s > 5.7
256-QAM 7 GHz 14 Gsample/s > 7.0
4.2.5 Square QAM internal resolutions
Not only the external quantization limited by the ADC resolution constrains algorithm
performance, but also internal resolutions have to be considered. Therefore an optimal
compromise must be found between performance degradation and hardware efficiency.
Especially because the proposed algorithm uses B parallel blocks per module, the hardware
efficiency of each block is crucial for the practicality of the system.
To find an efficient hardware implementation, one important goal is to avoid multipliers in
the system, since they usually utilize a lot of chip area. Figure 4.26 shows the receiver
sensitivity penalty against different resolutions of Re[d
k,b
] and Im[d
k,b
]. The results are
similar for all considered constellations and show that a resolution ≥ 4 bits is sufficient. As
for |d
k,b
|
2
the penalty for a resolution ≥ 5 bits is tolerable (Figure 4.27), the ( )
2
-operation in
equation (3.41) can be realized with a small look-up table (4 bit input, 4 bit output) or
simple logic functions.
Figure 4.26: Receiver sensitivity penalty vs. internal resolution of the distances Re[dk,b] and Im[dk,b] for
different square QAM constellations
4 Simulation Results
75
Figure 4.27: Receiver sensitivity penalty vs. internal resolution of the squared distance |dk,b|2 for different
square QAM constellations
The reason for the similar results for all simulated constellations is that the distance to the
closest constellation point is independent of the number of constellation points. This fact
was already mentioned in section 3.4.4.3.2 and shows that the hardware effort to
implement the proposed algorithm only increases moderately for higher-order QAM
constellations. The needed internal resolutions for calculation of |d
k,b
|
2
and consequently
also for the subsequent filter function are always the same.
4.3 Polarization control and PMD compensation
The polarization control algorithms presented in sections 3.3.1 and 3.3.2, and even more
the extension of the decision-directed algorithm to enable also the compensation of
dispersive effects described in section 3.3.3 require a careful determination of the
algorithm parameters to achieve an optimum performance. Additionally the simulations
verify the functionality of the novel ISI compensation algorithm and demonstrate that it
can improve the receiver sensitivity if the signal was corrupted by PMD.
T
S
/2-spaced sampling is implemented. For carrier recovery the weighted Viterbi & Viterbi
algorithm is chosen with a joined carrier recovery for both polarizations and N
CR
= 4. The
linewidth-times-symbol-duration product is set to Δf·T
S
= 10
-4
.
4.3.1 Comparison of polarization control algorithms
This section compares the polarization control algorithms presented in the sections 3.3.1
and 3.3.2. The Jones matrix governing the polarization cross-talk between the two
polarization modes is arbitrarily generated using a random number generator. Each data
point is based on the simulation of 500,000 symbols.
4 Simulation Results
76
Figure 4.28 compares the influence of the control gain g on the receiver sensitivity for the
non-data-aided constant modulus algorithm (CMA) and the decision-directed (DD)
algorithm. The higher the control gain, the faster the polarization controller can track
polarization changes.
Figure 4.28: Influence of the polarization control gain g on the receiver sensitivity
The CMA tolerates a much higher control gain than the decision-directed polarization
control. For the latter a 1 dB sensitivity penalty is already observed for g ≈ 2-7.5, whereas
for the CMA this penalty is reached only for g ≈ 2-4. This seems to indicate that the non-
data-aided algorithm allows for a 23.5 ≈ 11 times faster control of the polarization.
But looking at Figure 4.29, which shows the start-up developing of the polarization control
matrix elements, unveils that the same control gain g does not result in the same control
time constant ct. The decision-directed polarization controller converges about 2.3 times
faster than the CMA based controller with the same control gain. The reason is that for the
CMA the control update is not only weighted by the control gain, but also by the factors
2
,
1kx
Y− and 2
,
1ky
Y−, respectively. This decelerates the settlement of the controller.
4 Simulation Results
77
Non-data-aided polarization control Decision-directed polarization control
Figure 4.29: Start-up development of the matrix elements of the polarization control matrix M
To allow for a fair comparison of the polarization control algorithms, Figure 4.30 shows
the sensitivity penalties against the normalized control time constant ct/TS. Now for a
tolerable penalty of 1 dB the CMA allows only for a 5 times lower value of ct. Thus the
advantage of a faster polarization control as implicated from Figure 4.28 becomes less
significant. A system designer has to ponder, if a faster polarization control is preferable
using the CMA at the price of unaligned phases between the two polarization modes, or if
the improved phase noise tolerance due to a common carrier recovery for both
polarizations enabled by the decision-directed polarization control outweighs a slower
polarization control.
04000 8000 12000 16000
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
g = 2-8
k
Mk
04000 8000 12000 16000
-0.5
0
0.5
1
1.5
2
g = 2
-8
k
M
k
04000 8000 12000 16000
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
g = 2
-10
k
M
k
04000 8000 12000 16000
-0.5
0
0.5
1
1.5
2
g = 2
-10
k
M
k
4 Simulation Results
78
Figure 4.30: Sensitivity penalty vs. normalized polarization control time constant
for the non-data-aided and decision-directed polarization control algorithms
4.3.2 Verification of the ISI compensation algorithm
The extension of the decision-directed polarization control algorithm to allow also the
compensation of ISI caused e.g. by dispersive effects is newly proposed in this thesis. In
the following the functionality of the algorithm is verified at the example of PMD.
4.3.2.1 Impact of the control gain on the receiver sensitivity
A trade-off between control accuracy, which increases for a lower control gain g, and the
control rate, that increases with increasing control gain has also to be found for the ISI
compensation algorithm. Figure 4.31 shows the influence of different values of g on the
receiver sensitivity.
Figure 4.31: Sensitivity penalty at BER = 10-3 for different control gains of the polarization control/ISI
compensation algorithm and different values of the FIR filter half width
4 Simulation Results
79
The graph for NPMDC = 0 represents the polarization control algorithm presented in section
3.3.2. It can be implemented with a relatively low control gain. For g < 2-10 the receiver
sensitivity is virtually constant, only for g = 2-8 a small sensitivity degradation is observed.
This changes if NPMDC is increased, i.e. the algorithm is extended to allow also for ISI
compensation as described in section 3.3.3. The larger the filter width NPMDC becomes, the
higher is the required accuracy for the ISI compensation. This is apparent because
[Yx,k Yy,k]T is calculated as the sum of 2NPMDC+1 complex control matrix multiplications. If
one assumes that each control matrix is corrupted by uncorrelated AWGN with the
variance 2
C
σ
, then additional AWGN with the variance
(
)
2
PMDC 12 C
N
σ
+ is loaded to
[Yx,k Yy,k]T. Thus if the filter half width NPMDC is increased by 1 the control gain must be
reduced by 2 to compensate for the additional noise loading.
The reason that the sensitivity for NPMDC = 1 is inferior for low control gains is that the
factors χ±1 = ½ do not ideally compensate for the inherent correlations between the TS/2-
spaced samples. Thus a residual ISI remains. For larger filters this can be compensated by
the additional filter taps. For NPMDC = 1 the residual ISI degrades the receiver sensitivity.
A comparison of the continuous lines representing the simplified control matrix update
with equation (3.28) and the dashed lines representing the elaborate matrix update
according to equation (3.27) shows that the latter allows for lower control gains. This is
due to the fact that several correlation results are used to update one control matrix, which
causes a portion of the AWGN to be averaged out. Figure 4.32 exemplifies the different
receiver sensitivities for g = 2-10.
Figure 4.32: BER vs. OSNR for the simplified and original ISI compensation algorithm
with g = 2-10 and different FIR filter half widths
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
OSNR [dB]
BER
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
Simplified algorithm
Original algorithm
4 Simulation Results
80
4.3.2.2 PMD compensation performance
To be able to evaluate the performance of the ISI compensation algorithm, a PMD
emulator (PMDE) is integrated in the system simulations [29]. It implements the PMD
model described in section 2.2.3.4. The MATLAB® code for the PMDE as well as the
code to plot the DGD profiles were provided by Prof. Dr.-Ing. Reinhold Noé.
The transversal filter length is set to LPMDE = 10 and τ0 = τ1 = … = τ10 = TS/4. The average
differential group delay (DGD) generated by the PMDE corresponds to S
T
⋅
≈Δ 8.0
DGD
τ
.
Figure 4.33 shows the 8 arbitrarily generated input referred DGD profiles, which are used
to evaluate the PMD compensation performance of the algorithm [56].
4 Simulation Results
81
Figure 4.33: Input referred DGD profiles for the emulation of PMD
Figure 4.34 depicts the achieved receiver sensitivities after PMD compensation using the
original algorithm and different FIR filter half widths. The theoretical receiver sensitivity
according to equation (4.1) serves as a reference. Figure 4.35 shows the same information
for the simplified algorithm.
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
A
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
B
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
C
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
D
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
4 Simulation Results
82
Figure 4.34: Receiver sensitivity for the original ISI compensation algorithm
with different FIR filter half widths for different DGD profiles
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
E
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
F
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
G
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
H
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
A
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
B
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
C
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
D
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
4 Simulation Results
83
Figure 4.35: Receiver sensitivity for the simplified ISI compensation algorithm
with different FIR filter half widths for different DGD profiles
The simulation results verify that both algorithms efficiently compensate ISI due to PMD.
The performance of the original and simplified algorithm are thereby roughly the same. As
the simplified algorithm requires a much lower computational effort, in the following only
the simplified algorithm is considered.
As expected the receiver sensitivity increases with increasing FIR filter half width. For all
considered DGD profiles except example B the receiver sensitivity is significantly
improved and for NPMDC = 6 almost reaches the receiver sensitivity of the PMD-free case.
Figure 4.36 shows the developing of |det{Mi}| over the 2NPMDC+1 control matrices for
different values of NPMDC.
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
E
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
F
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
G
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
H
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
-6 -4 -2 0 2 4 6
10
-3
10
-2
10
-1
10
0
10
1
tap index: i
|det{M
i
}|
A
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
-6 -4 -2 0 2 4 6
10
-3
10
-2
10
-1
10
0
10
1
tap index: i
|det{M
i
}|
B
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
4 Simulation Results
84
Figure 4.36: Developing of |det{Mi}| for the simplified ISI compensation algorithm
with different FIR filter half widths for different DGD profiles
The determinant of the center matrix has the highest absolute value and it decays with
increasing distance to the center element. This confirms with the nature of dispersion,
where the energy is spread over consecutive symbols with the same characteristic. Only the
examples B and D show a deviation from this shape with |det{M±2}| > |det{M±1}| for
NPMDC > 2. It is peculiar that for example B also the receiver sensitivity significantly
degrades for NPMDC > 2. Very likely this indicates that the control locks to a local minimum
for the crosstalk between the different symbols, but does not find the global minimum. In
order to minimize the probability of this locking to a local minimum the start-up sequence
of the ISI compensator is optimized.
-6 -4 -2 0 2 4 6
10-3
10-2
10-1
100
101
tap index: i
|det{M
i
}|
C
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
-6 -4 -2 0 2 4 6
10-3
10-2
10-1
100
101
tap index: i
|det{M
i
}|
D
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
-6 -4 -2 0 2 4 6
10-3
10-2
10-1
100
101
tap index: i
|det{M
i
}|
E
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
-6 -4 -2 0 2 4 6
10-3
10-2
10-1
100
101
tap index: i
|det{M
i
}|
F
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
-6 -4 -2 0 2 4 6
10-3
10-2
10-1
100
101
tap index: i
|det{M
i
}|
G
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
-6 -4 -2 0 2 4 6
10-3
10-2
10-1
100
101
tap index: i
|det{M
i
}|
H
NPMDC=0
NPMDC=1
NPMDC=2
NPMDC=4
NPMDC=6
4 Simulation Results
85
4.3.2.3 Optimization of the start-up sequence
Figure 4.35-B/D and Figure 4.36-B/D show that it is possible that the ISI compensation
algorithm might converge to a local minimum for the crosstalk between adjacent symbols
rather than to the global minimum. However, the results also indicate that for NPMDC ≤ 2
the algorithm converged to this global minimum. Thus it seems reasonable to assume that
the probability that the algorithm converges to a local minimum increases with increasing
NPMDC. Therefore the start-up sequence of the algorithm is optimized.
The idea is that the algorithm first achieves a coarse locking to the global minimum by
updating only the closest neighbor matrices of the center control matrix, and then improves
the accuracy by successively activating the update for the control matrices for the more
distant symbols. Figure 4.37 shows the results obtained with this optimized start-up
sequence for the DGD profiles B and D.
Figure 4.37: Receiver sensitivity (top row) and developing of |det{Mi}| (bottom row) for different DGD
profiles and different FIR filter half widths for the simplified algorithm with optimized start-up sequence.
The receiver sensitivities for case B with NPMDC = 4 and NPMDC = 6 show a significant
improvement compared to the results depicted in Figure 4.34, and also the sensitivity for
case B is slightly improved. The reason is that the |det{Mi}| now continuously decay with
increasing distance to the center element. Thus the algorithm converges more likely to the
global crosstalk minimum.
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
B
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
8 9 10 11 12 13 14
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
OSNR [dB]
BER
D
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
theoretical
receiver sensitivity
-6 -4 -2 0 2 4 6
10
-3
10
-2
10
-1
10
0
10
1
tap index: i
|det{M
i
}|
B
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
-6 -4 -2 0 2 4 6
10
-3
10
-2
10
-1
10
0
10
1
tap index: i
|det{M
i
}|
D
N
PMDC
=0
N
PMDC
=1
N
PMDC
=2
N
PMDC
=4
N
PMDC
=6
4 Simulation Results
86
In the simulations the activation of the update is simply controlled by a counter, i.e. every
215 symbols another matrix update is activated until all updates are active. In a practical
system the effective FIR filter width could be adapted to the actual DGD profile by
monitoring the determinants of the control matrices. A possible rule for increasing the filter
width could be
{
}
2det 5⇒> −
l
M activate update for
{}
ll sgn+
M, (4.7)
where l is the index of the last updated control matrix. A similar rule could be applied to
reduce the filter width:
{}
{}
⇒< −
−
6
sgn 2det ll
M deactivate update for Ml and set 0M
=
:
l. (4.8)
The different switching thresholds avoid possible oscillations.
This mechanism not only solves the start-up problem, but additionally improves the
receiver sensitivity. By the deactivation of redundant FIR filter taps the noise loading of
the received signal by the ISI compensation filter is reduced.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
87
5 Implementation of a synchronous optical
QPSK transmission system with real-time
coherent digital receiver
In the framework of the European synQPSK project a real-time synchronous QPSK
transmission testbed had to be developed to verify the functionalities of the different
developed components, i.e. the QPSK modulators, optical 90° hybrids, coherent receiver
frontends and digital signal processing circuits, and to investigate their interplay. Thus the
development of the synQPSK testbed was only possible in a broad cooperation between a
multitude of researchers. The focus within this dissertation is placed on the experimental
results achieved with the testbed, as my main responsibility was the final assembly of the
testbed and the execution of the measurements. Although I also supported the DSPU
development, especially the compilation of the register transfer level (RTL) description of
the algorithms in the very high speed integrated circuit (VHSIC) hardware description
language (VHDL), these results are not addressed in this thesis. For detailed information
about the VHDL development I refer to [57; 58; 59].
5.1 Single-polarization synchronous QPSK transmission with
real-time FPGA-based coherent receiver
The first assembled testbed was mainly built from commercial components. The only
applied component developed within the synQPSK project was the optical 90° hybrid
provided by CeLight Israel [60]. The purpose of the testbed was to verify the VHDL code
developed for the CMOS chip version A, which should contain the carrier & data recovery
for a single-polarization synchronous QPSK transmission system.
5.1.1 Single-polarization synchronous QPSK transmission setup
In this subsection the assembly of the single-polarization synchronous QPSK transmission
setup is described. Figure 5.1 depticts the structure of the testbed. In the following the
different components are described in more detail.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
88
I
PRBS
ADC
signal
laser SMF
LiNbO
3
optical
90º hybrid
I
Q
digital sampling,
carrier & data recovery
automatic frequency control
LO
laser
QPSK
modulator
ECL or
DFB laser
DFB laser
Q
Xilinx
Virtex 2
FPGA
6
6
optical signals
electrical signals
ADC
400 MHz clock signal
Figure 5.1: 800 Mb/s single-polarization synchronous QPSK transmission setup
with real-time digital coherent receiver
5.1.1.1 QPSK transmitter
The output signal of a 192.5 THz DFB laser (JDSU) with a specified linewidth of
MHz 1
DFBdB,3 =Δfwas fed into a fiber-pigtailed QPSK modulator (Bookham) to generate
the QPSK optical signal. As an alternative signal laser also a tunable external-cavity laser
(ECL) with a lower linewidth of kHz 150
ECLdB,3
=
Δ
f was available. A pattern generator
was used to generate a 400 Mb/s pseudo random binary sequence (PRBS) data stream with
a switchable PRBS pattern length of 27-1 and 231-1. The 400 Mb/s data stream was split
and mutually delayed by 30 ns, which corresponds to 12 bit durations at 400 Mb/s, to
emulate two decorrelated patterns. For experimental convenience the differential precoder
described in section 2.2.1, that is normally required in a QPSK transmitter was omitted.
Two modulator drivers (TriQuint) were used to drive the QPSK modulator with the two
400 Mb/s data streams (I and Q). Thus the modulator output was an optical 2x400 Mb/s
NRZ-QPSK signal, which was then fed into the transmission fiber. Figure 5.2 shows a
photograph of the QPSK transmitter.
Figure 5.2: Single-polarization QPSK transmitter with Bookham optical QPSK modulator.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
89
5.1.1.2 Coherent optical receiver frontend
After transmission through either 2 km or 63 km of standard Single-mode fiber (SMF)
[ITU-T G.652] and a variable optical attenuator (VOA) to control the optical power the
signal was fed into the coherent optical receiver frontend. It employed an erbium-doped
fiber amplifier (EDFA) as an optical preamplifier followed by a dense wavelength division
multiplexing (DWDM) arrayed-waveguide grating (AWG) demultiplexer (DEMUX) with
Gaussian passbands and 100 GHz channel spacing. The DEMUX output signal was fed
into a second EDFA which was used for power control followed by a bandpass filter with a
width of ~20 GHz. Then the received signal was fed into a LiNbO3 optical 90° hybrid
provided by CeLight Israel, where it was superimposed with the local oscillator (LO)
signal provided from a second 192.5 THz DFB laser (JDSU) with MHz 1
DFBdB,3 =
Δ
f for
intradyne operation or from the transmitter laser for self-homodyne operation. The
polarizations of the LO and received signals were matched manually by using quarter-
wave plate arrays. The outputs of the optical 90° hybrid were detected in two differential
photodiode pairs. Their output currents were converted to voltage signals through resistive
loads and amplified in two 10 GHz bandwidth amplifiers (Picosecond).
Figure 5.3: LiNbO3 optical 90° hybrid and
associated control unit
Figure 5.4: Differential photodiode pairs and
amplifiers
5 Implementation of a synchronous optical QPSK transmission system with real-time
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90
5.1.1.3 FPGA-based digital signal processing
To digitize the received signal a MAX105EVAL evaluation board (Maxim) was used. It
was equipped with a MAX105 dual 6-bit analog-to-digital converter (ADC). It converted
the analog signals of the I and Q component to digital outputs at up to 800 MS/s with a
400 MHz, -0.5 dB analog input bandwidth. Its -3 dB analog input bandwidth is 1.5 GHz.
The dual ADC board provided LVDS digital outputs with an internal 6:12 demultiplexer
that reduced the output data rate to one half the sample clock rate. This allowed easier
interfacing with the subsequent digital signal processing unit (DSPU). Data was output in
two's complement format. In the experiment the ADCs sampled the analog input signal at
the symbol rate, i.e. with 400 MS/s.
The carrier & data recovery was implemented on a Xilinx Virtex II prototype platform
populated with a Xilinx Virtex II FPGA (XC2V2000). The board offered a sufficient
number of differential user I/Os to the FPGA, which were fast enough to accept the
200 Mb/s LVDS output data streams from the ADC. The FPGA further demultiplexed the
input data into 16 parallel channels. This reduced the clock frequency for the FPGA core to
25 MHz. The core contained the carrier and data recovery as described in section 3.4.3
with the quantizations determined in section 4.1. Additionally an external IF control was
implemented as described in section 3.6.1. The recovered data was reassembled to two
full-rate serial bit streams that were analyzed in an external bit error rate tester (BERT).
Figure 5.5 shows a photograph of the ADC and FPGA board.
Figure 5.5: Commercial ADC board (left) and FPGA board (right) for digital signal processing
The VHDL code for the FPGA was developed in cooperation between SCT and ONT. SCT
developed the core program describing the carrier & data recovery according to the
5 Implementation of a synchronous optical QPSK transmission system with real-time
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91
specifications provided by ONT [57]. In consideration of the required hardware effort for
the different algorithms (Table 3.2) and the simulation results of section 4.1 the SMLPA
algorithm with NCR = 3 was chosen for implementation. The I/O logic, clock management,
IF control and monitoring functions were developed at ONT.
5.1.2 Self-homodyne experiment results at 800 Mb/s
The first test of the system was conducted with a 27-1 PRBS as a self-homodyne
experiment at data rates of 600 Mb/s (300 Mbaud) and 800 Mb/s (400 Mbaud) using the
external-cavity laser (ECL). Figure 5.6 depicts the achieved BER. I&Q channel behavior is
very similar. At both data rates transmission was error-free during a 30 min test with
-37 dBm of received optical power. The measured receiver sensitivity at BER = 10-3 is
about −48 dBm for 600 Mb/s data rate and −49 dBm for 800 Mb/s data rate. Note that the
applicable sum linewidth-times-symbol-duration products of 10-4 and 7.5·10-5 can be
achieved with commercial DFB lasers in a 10 Gbaud system.
Figure 5.6: BER vs. preamplifier input power for synchronous QPSK transmission
with self-homodyne detection using an ECL.
Next, the ECL was replaced by the DFB laser and the transmission experiment was
repeated with data rates of 600 Mb/s, 667 Mb/s, 733 Mb/s and 800 Mb/s. Figure 5.7 shows
that due to the larger linewidth of the employed laser BER floors emerge. For simplicity
only averaged I&Q channel BERs are plotted. The minimum achieved BERs are 9.2·10-3,
6.9·10-3, 4.5·10-3 and 3.8·10-3, respectively.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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Figure 5.7: BER vs. preamplifier input power for synchronous QPSK transmission
with self-homodyne detection using a DFB laser.
5.1.3 Intradyne experiment results at 800 Mb/s
Finally, an identical second DFB laser was added and used as the LO source. Without
modulation, the I&Q beat signals detected at the coherent frontend outputs were recorded
with a digital sampling oscilloscope in x-y-mode and are displayed in Figure 5.8.
Figure 5.8: Laser beating at the output of the coherent receiver frontend
BER vs. received power is plotted in Figure 5.9 for transmission at 600 Mb/s to 800 Mb/s
data rates using 27-1 PRBS. The best measured BER result is 6.4·10-3 for 800 Mb/s, the
highest BER floor of 1.1·10-3 is yielded by the transmission at 600 Mb/s.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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Figure 5.9: BER vs. preamplifier input power for synchronous QPSK transmission
with intradyne detection using DFB lasers.
The slight increase of the BER floors compared to the self-homodyne setup is due to a
broadening of the LO laser linewidth, which is most likely caused by insufficiently filtered
reflections, as the laser only employs a single isolator.
5.1.4 Intradyne experiment results at 1.6 Gb/s
The bottle neck for the maximum data rate of the transmission system was the assembly of
the recovered data to full-rate bit streams. As the maximum supported data rate of the
FPGA I/Os was 400 Mb/s, this limited the symbol rate to 400 Mbaud. However the ADCs
supported sampling rates up to 800 MS/s, and owing to the 2 times demultiplexed outputs
of the ADC, the FPGA is also able to further process this data. Therefore the FPGA design
was optimized and the two full-rate output bit streams were replaced by lower rate outputs
that output only every fourth bit. This allowed increasing the system data rate to 1.6 Gb/s,
which was limited now by the maximum ADC sampling rate and the ADC-FPGA interface
speed.
Figure 5.10 shows the receiver sensitivity for 1.6 Gb/s transmission over distances of 2 and
63 km using PRBS with pattern lengths of 27-1 (PRBS-7) and 231-1 (PRBS-31). The best
measured BER was 2.7·10-4 with PRBS-7 transmitted over 2 km, and it was 4.4·10-4 for
PRBS-31. Both PRBS could be detected until the preamplifier input power was set below
-52 dBm. The BER floors for 63 km distance are slightly higher than for 2 km and are
3.4·10−4 for PRBS-7 and 4.0·10-4 for the PRBS-31, respectively.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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Figure 5.10: Measured BER vs. optical preamplifier input power at 1.6 Gb/s data rate.
The increase of the BER floor for longer transmission distances is most likely due to the
lack of a clock recovery circuit at the receiver and the workaround of using the transmitter
clock. This causes an increased clock jitter at the receiver for increasing transmission
distances. The sensitivity degradation for longer PRBS patterns is most likely caused by
AC coupling effects, which have a larger influence on longer patterns that contain longer
‘0’- or ‘1’-sequences.
5.1.5 System optimizations & comparison of 90° hybrid with 3x3 coupler
In order to further reduce the BER floor several system components were optimized:
• In the coherent receiver frontend the optical passband filter with a width of 20 GHz
was replaced by a filter with a lower bandwidth of 15 GHz to reduce the thermal
frontend noise.
• The filtering of the laser bias currents was improved to reduce the laser linewidth.
• Due to reliability problems in the ADC-FPGA interface the symbol rate was
reduced to 700 Mbaud.
• The receiver was extended with a clock recovery circuit as described in section 3.2.
Therefore an additional ADC was added to the system, which was clocked with an
inverted clock to enable TS/2 sampling of one of the receiver frontend output
signals (see Figure 5.12).
Next to these optimizations an alternative receiver frontend was developed using a fused
symmetric 3x3 coupler, i.e. the coupling ratio is 1:1:1. A 90° hybrid is in general realized
with integrated optics [61], with free-space micro-optics [62], or in an all-fiber approach
[63]. Its replacement by a standard component like a 3x3 coupler reduces the system cost
and is thus interesting for commercial applications. In principle also the use of an
5 Implementation of a synchronous optical QPSK transmission system with real-time
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95
asymmetric 3x3 coupler is possible, but the symmetric coupler allows for the suppression
of direct detection terms [52].
The equivalent to the 90° hybrid output signal Zk can be calculated from the symmetric 3x3
coupler output signals z1,k, z2,k, z3,k by applying the formula
{} {}
()()
kkkkk
kkk zz
j
zzzZjZZ ,3,2,3,2,1 3
3
1
3
2
ImRe −+
⎟
⎠
⎞
⎜
⎝
⎛+−=+= . (5.1)
Figure 5.11 illustrates this transformation graphically.
{}
k
ZRe
{
}
k
ZIm
k
z,1
k
z,2
k
z,3
3
,2 k
z
3
,3 k
z
−
3
,2 k
z
−3
,3 k
z
−
k
z,1
3
2
Figure 5.11: Transformation of the 3x3 coupler outputs to I&Q signals
Figure 5.12 depicts the reworked synchronous QPSK transmission setup with the 3x3
coupler employed in the coherent receiver frontend. Additionally the detailed structure for
the clock recovery is depicted. The signal conversion from the 3 input signals into I&Q
signals according to (5.1) is implemented inside the FPGA. For simplicity the factor 3 is
approximated as 1.75.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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96
6
6
6
I
Q
ADC
ADC
ADC
Xilinx Virtex II FPGA
ADC
PI
VCO
+
–
T
2
clock phase
error signal
recovered
clock
T
Signal
laser
LO
laser
PRBS
QPSK
modulator
3x3
coupler
700 MHz clock signal
80 km
SSMF
automatic frequency control
z
1
z
2
z
3
MSB
MSB
optical signals
electrical signals
Figure 5.12: Synchronous QPSK transmission setup with clock recovery and symmetrical 3x3 coupler
The averaged BER of the received signal against the preamplifier input power at the
receiver is shown in Figure 5.13. The minimum BER was 2.8·10−5 for the receiver
configuration with symmetric 3x3 coupler. For comparison the BER performance of a
receiver with a 90° hybrid is about 1 dB higher than for the 3x3 coupler and fixed
transformation function in the receiver. Also the achieved BER floor is with 1.7·10−5
slightly lower. This results from the non-ideal coupling ratios in the fused 3x3 coupler. In
contrast the 90° hybrid settings are optimal because both the coupling ratios and the phase
shift can be controlled.
Figure 5.13: Receiver sensitivity of a synchronous QPSK receiver with either a symmetric 3x3 coupler
or a 90° hybrid.
5.1.6 Comparison of experimental with simulation results
Figure 5.14 compares the measured against the simulated BER floors for the synchronous
QPSK transmission system with 90° hybrid. The different measured BER floors are caused
by different symbol rates that were varied from 200 Mbaud to 700 Mbaud. The
5 Implementation of a synchronous optical QPSK transmission system with real-time
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97
corresponding linewidth-times-symbol-duration product Δf·TS was calculated by using the
specified linewidth of the DFB lasers, i.e. Δf = 2·Δf3dB,DFB.
Figure 5.14: Simulated and measured BER floors for different linewidth-times-symbol-duration products
The experimental results are in good accordance with the simulations. The slight
deviations can be attributed to fact that the exact linewidth was not determined. Figure 5.14
shows that at 10 Gbaud operations a BER floor due to phase noise will not be detectable
any more.
5.2 Polarization-multiplexed synchronous QPSK transmission
with real-time FPGA-based coherent receiver
In order to extend the testbed to polarization-multiplexed synchronous QPSK transmission
a reconstruction of the transmitter and coherent receiver became necessary, because most
of the components were required twice. A completely new polarization-multiplexed QPSK
receiver was assembled employing two QPSK modulators provided by Photline [64]. At
the receiver side a second optical 90° hybrid (CeLight) was added. Additionally to
preparing the system for 10 Gbaud operation all photodiodes and amplifiers employed in
the receiver were replaced by components with at least 10 GHz bandwidth. Finally also the
FPGA and the associated prototyping board were replaced by a newer, larger and faster
version. The main purpose of the testbed within the synQPSK project was to verify the
VHDL code developed for the CMOS chip version B, which should next to the carrier &
data recovery also contain a polarization control unit to enable polarization-multiplexed
synchronous QPSK transmission.
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5.2.1 Polarization-multiplexed QPSK transmission setup
Figure 5.15 shows a simplified block diagram for the polarization-multiplexed
synchronous QPSK testbed at 2.8 Gb/s. The QPSK transmitter is extended with differential
precoding circuits. This makes the programming of the BERT with particularly calculated
patterns dispensable. After transmission through 80 km of standard Single-mode fiber, the
received signal power is controlled by a variable optical attenuator (VOA) and then fed
into a polarization scrambler. The amplified and filtered optical signal is split by a
polarization beam splitter (PBS) and superimposed with the local oscillator in two 90°
hybrids. Subsequent to optoelectronic and analog-to-digital conversion with
1 sample/symbol, the digitized samples are fed into the FPGA, where the digital
polarization control and the carrier & data recovery are implemented. Clock recovery and
local oscillator frequency control are implemented like for the single-polarization
experiments described in section 5.1.
6
6
6
6
DFB laser QPSK modulator
QPSK modulator
4 x 700 Mb/s
PBS
PBC
Signal laser 45
SMF
80 km
precoder
precoder
DFB laser
ADC
ADC
ADC
ADC
Xilinx
Virtex 4
FPGA
PBS
PBS
Local
oscillator 45
90 hybrid
90 hybrid
I1
Q1
I2
Q2
VOA
Polarization
scrambler
optical signals
electrical signals
Figure 5.15: 2.8 Gb/s polarization-multiplexed QPSK transmission setup with a real-time FPGA-based
synchronous coherent digital I&Q receiver
5.2.1.1 Pattern generator and QPSK precoder
Pattern generation and QPSK precoding is realized with a Xilinx RocketIO
characterization board (MK325) populated with a Virtex-II ProX FPGA (XC2VP70X).
The FPGA provides several multi-gigabit transceivers (MGT), that operate up to a serial
data rate of 10 Gb/s per channel. The block diagram of the algorithm implemented inside
the FPGA is shown in Figure 5.16.
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Xilinx Virtex II Pro-X FPGA
Pattern generator
27-1 PRBS
generator
215-1 PRBS
generator
223-1 PRBS
generator
231-1 PRBS
generator
64 bit
delay QPSK
precoder
PRBS select
MUX
MUX
MUX
„1010…“ sequence
I
Q
CLK
X
Y
Figure 5.16: Block diagram for the pattern generator and QPSK precoder implemented in an FPGA
First the FPGA generates four different pseudo-random binary sequences (PRBS) with
pattern lengths of 27-1, 215-1, 223-1 and 231-1 bit. By a selector controlled by two external
dip switches the user can choose which pattern should be transmitted. The output of the
pattern generator is split into two branches, and one branch is delayed by 64 bit for
decorrelation. Then they are fed into a QPSK precoder for differential encoding according
to equation (2.5). For parallel-to-serial conversion the built-in MGTs of the FPGA are
used.
As the pattern generator and precoder must support symbol rates up to 10 Gbaud, parallel
processing of the data has to be applied inside the FPGA. The number of parallel modules
depends on the target symbol rate and ranges from 4 parallel modules for 700 Mbaud to 80
parallel modules for a symbol rate of 10 Gbaud.
Up to a symbol rate of 5 Gbaud also the transmitter clock is generated with a MGT of the
FPGA by transmitting an alternating “1010…” sequence at twice the symbol rate of the
system. Above 5 Gbaud only a half-rate clock can be generate with the FPGA, and an
external clock multiplier has to be added to the clock output.
VHDL simulations of the program were executed with ModelSim. For VHDL synthesis,
implementation and FPGA programming the Xilinx ISETM Design Suite was used. Figure
5.17 shows the MK325 Xilinx Virtex-II ProX RocketIO characterization board, on which
the pattern generator and precoder are implemented.
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Figure 5.17: MK325 Xilinx Virtex-II ProX RocketIO characterization board
used for pattern generation and precoding
5.2.1.2 QPSK transmitter
The same 192.4 THz DFB laser (JDSU) as used for the single-polarization experiments is
fed into a polarization beam splitter, whose outputs are connected to two fiber-pigtailed
QPSK modulators (Photline) to generate polarization-multiplexed QPSK optical signals.
Two precoded data streams are provided by the pattern generator described in section
5.2.1.1. They are fed into HF amplifiers, split and delayed to emulate decorrelated patterns
for the two polarizations. The delay between the patterns for the two polarizations is
generated using different cable lengths and is set to 7 symbols.
Four modulator drivers (from TriQuint) are used to drive the QPSK modulators with the
data streams. The QPSK modulators are followed by variable attenuators to match the
powers of the two polarizations. Finally the two branches are recombined in a polarization
beam combiner. The path lengths of the two branches are matched within a sub-millimeter
scale. Figure 5.18 shows a photograph of the polarization-multiplexed QPSK transmitter.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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Figure 5.18: Polarization-multiplexed QPSK modulator.
5.2.1.3 Ultra-fast polarization scrambler
In order to be able to test the polarization tracking capabilities of the polarization control
algorithm implemented in the coherent receiver an available polarization scrambler is
integrated into the transmission channel [65]. It consists of 4 motorized fiber-optic quarter-
wave plates (QWP) followed by a variable fiber-optic PDL element. Another 4 QWPs
decorrelate the output SOP of the PDL element from the input SOP of a bulk-optic half-
wave plate (HWP), which is followed again by 4 QWPs (Figure 5.19). The 12 QWPs rotate
at different speeds ensuring that the polarization state covers all points on the Poincaré
sphere and that the position dependent loss of ~2 dB of the HWP is uncorrelated with the
variable PDL element. The rotation frequency of the motor that drives the HWP can be
adjusted between 0 to 612 Hz. With a gearbox ratio of 39:15 from the motor to the HWP (0
to 1592 Hz) and up to 8π rad rotation on the Poincaré sphere per HWP rotation, the
maximum speed of polarization changes is 40 krad/s.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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PDL HWP
Variable PDL element (0…6 dB)
Motorized half-wave
plate (bulk optic)
12 motorized
quarter-wave plates
Figure 5.19: Polarization scrambler and the corresponding fast polarization changes
displayed on the Poincaré sphere.
The polarization scrambler was tested by applying an unmodulated laser signal at its input
and measuring the SOP at its output with a polarimeter [65]. Figure 5.19 (left) shows the
SOP covering the full Poincaré sphere when only the 12 QWPs are rotating. After
additionally turning on the motor driving the HWP, the Poincaré sphere fills with points
(Figure 5.19 right) because the polarimeter sampling rate of 1 kHz is too slow to follow the
fast polarization changes.
5.2.1.4 Polarization diversity coherent optical receiver frontend
The coherent optical receiver frontend employs an EDFA as an optical preamplifier
followed by a DWDM-DEMUX with Gaussian passbands and 100 GHz channel spacing.
The DEMUX output is fed into a second EDFA which is used for power control followed
by a bandpass filter (BPF) with a width of ~15 GHz. Then the received signal is split by a
PBS and the outputs are fed into two optical 90° hybrids, were they are superimposed with
the local oscillator signal generated in a second 192.4 THz DFB laser (JDSU), which is
also split by a PBS. The outputs of the 90° hybrid are detected in four photodiode pairs
(Nortel). The differential signals between the photodiodes are generated by 4 differential
amplifiers (Micram) with a bandwidth of 40 GHz. One output signal of the amplifier is
passed through a second amplifier (Picosecond) with 10 GHz bandwidth and fed into the
ADCs. The second outputs are used for monitoring the settings of the two optical 90°
hybrids on two oscilloscopes. A schematic of the receiver frontend is shown in Figure
5.20.
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DFB laser
PBS
Local
oscillator 45
90 hybrid
Preamp.
EDFA
DEMUX
Pwr. ctrl.
EDFA
BPF
PBS
90 hybrid
HM1508
digital
oscilloscope
(in xy-mode)
HM1508
digital
oscilloscope
(in xy-mode)
Re{Z
x
}
Im{Z
x
}
Re{Z
y
}
Im{Z
y
}
optical signals
electrical signals
Figure 5.20: Polarization diversity coherent optical receiver frontend.
The optical path length difference between the two polarizations is matched within 1 mm.
The residual delay can be matched exactly by variable delay lines connected to all four
electrical outputs (I & Q of both polarizations) of the coherent receiver frontend.
5.2.1.5 FPGA-based digital signal processing
In order to evaluate the carrier & and data recovery algorithms for the CMOS chip version
B, a digital coherent receiver was built up using commercially available components
(Figure 5.21).
Figure 5.21: Commercially available analog-to-digital converters (left)
and an FPGA board for digital signal processing (center).
5 Implementation of a synchronous optical QPSK transmission system with real-time
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104
To digitize the received signals two MAX105EVAL evaluation boards (Maxim) are used.
The electronic polarization control and carrier & data recovery are implemented on a
Xilinx Virtex 4 prototype platform equipped with a Xilinx Virtex 4 FPGA (XC4VSX35).
Considering the simulation results of section 4.3 a flexible implementation for the
polarization control was chosen, allowing to set the polarization control gain either to
g = 2-4 or g = 2-6. As W = 16 correlation results are averaged before incremental update of
M, the overall system performance is similar to a system with W = 1 and g = 2-8 or g = 2-10
as simulated in section 4.3. To reduce the hardware effort only every 8th symbol is used for
polarization control update. Thus according to equation (3.20) the control time constant of
the system results to c = (16/2-4)·(8/700 Mbaud) ≈ 3 μs and
c = (16/2-6)·(8/700 Mbaud) ≈ 12 μs, respectively. A detailed description of the VHDL
program can be found in [58; 59].
Also the carrier recovery circuit is modified. The user can now select between an SMLPA
filter with either NCR = 2 or NCR = 4.
5.2.2 Influence of different carrier recovery filter widths
At first the influence of the two different carrier recovery filters is analyzed. Figure 5.22
shows the measurement results for the two polarization channels for NCR = 2 and NCR = 4,
i.e. 10 or 18 symbols are used for carrier recovery, respectively. The linewidth-times-
symbol-duration product of the system is Δf·TS ≈ 2.8·10-3.
Due to its larger bandwidth the filter with NCR = 2 allows to achieve a lower BER floor
than the filter with NCR = 4. This result is also supported by the simulation results in
section 4.1.1, which showed that for strong phase noise filters with a smaller width achieve
a lower mean squared phase estimation error. As also for low preamplifier input powers
the filter with NCR = 4 does not perform better than the filter with NCR = 2, the filter with
NCR = 2 is used for the following measurements.
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Figure 5.22: Influence of the carrier recovery filter width on the receiver sensitivity and BER floor.
A peculiarity in Figure 5.22 is that the x-polarization channel suffers from a ~1 dB penalty
compared to the y-polarization channel. This effect is caused by unequal signal powers in
the two polarization channels. The two QPSK modulators in the polarization-multiplexed
QPSK transmitter have different attenuations and the subsequent variable attenuators only
allow for a coarse power matching.
5.2.3 Polarization tracking capability
Figure 5.23 shows the achieved BER against the preamplifier input power at the coherent
receiver for different HWP rotation rates of the polarization scrambler described in section
5.2.1.3. The PDL of the scrambler is set to zero. For better readability the BERs of both
polarizations are averaged.
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Figure 5.23: Measured BER vs. optical preamplifier input power at the coherent receiver for the two different
control time constants of a) 12 μs and b) 3 μs and various polarization change speeds.
Initially the polarization scrambler is halted to measure the reference BER performance of
the system. Then the scrambler is switched on and the polarization change speeds are set to
300 rad/s, 3 krad/s, 6 krad/s and 12 krad/s on the Poincaré sphere.
In Figure 5.23 (a) the polarization control time constant is set to c = 12 μs. The minimum
bit error rate of 1.2·10-7 was degraded by the polarization scrambler to 1.2·10-6 and 1.7·10-5
for the rotation speeds of 300 rad/s and 3 krad/s, respectively. With the rotation speed of
6 krad/s the BER floor was already above 1·10-3, and for 12 krad/s the receiver was not
able to compensate any more for the polarization changes.
The results for the control time constant of 3 μs are shown in Figure 5.23 (b). At
BER = 10-4 the receiver sensitivity is degraded by 0.8 dB compared to c = 12 μs. The
minimum achievable BER of 3.1·10-7 is also slightly worse than for c = 12 μs. This
conforms with the simulation results of section 4.3, which also predicted a sensitivity
degradation for higher control gains. However, with c
= 3 μs the polarization scrambler
degrades the BER only to 4.7·10-7, 2.3·10-6 and 4.8·10-6 for the rotation speeds of 300 rad/s,
3 krad/s and 6 krad/s, respectively. Even at 12 krad/s the receiver is able to follow the
polarization changes and achieves a BER of 1.2·10-5.
Figure 5.24 shows the receiver sensitivity penalties against the polarization change speeds
for BER = 10-4. Assuming a maximum tolerable penalty of 1 dB in receiver sensitivity, the
polarization controller with c = 12 μs tolerates up to 600 rad/s. Reducing the control time
constant to c = 3 μs increases the permissible speed of the polarization changes to
3.5 krad/s.
5 Implementation of a synchronous optical QPSK transmission system with real-time
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Figure 5.24: Receiver sensitivity penalty vs. scrambling speed for a bit error rate of 10-4
In order to investigate if some rare events cause the polarization control to loose lock, the
BER is measured for a time interval of 10 minutes with a measurement update every 10 s.
The scrambling speed is chosen in such a way that it causes a 1 dB penalty in receiver
sensitivity at a BER of 1·10-4. The result is depicted in Figure 5.25. It can be seen that in
spite of fast polarization change speeds of 600 rad/s and 3.5 krad/s, respectively, the BER
remains almost constant. This clearly verifies the reliability of the system.
Figure 5.25: Long-term BER measurements with polarization change speeds
causing 1 dB loss in receiver sensitivity.
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5.2.4 Polarization tracking capability with optimized VHDL code
In order to achieve a faster polarization control the original VHDL code was optimized,
allowing to use every second symbol in the FPGA to update the polarization control [59].
This reduces the polarization control time constant by a factor of 4, i.e. it can be switched
between c = 0.75 μs and c = 3.0 μs.
First the reference BER curves had to be measured for the modified receiver. Therefore the
PDL element was set to 0 dB and the BER was measured for different optical preamplifier
input powers at polarization change speeds of 0, 10, 20, 30 and 40 krad/s. Figure 5.26 (a)
shows the results for the accurate polarization controller with c = 3.0 μs, Figure 5.26 (b)
depicts the corresponding values for the fast polarization control (c = 0.75 μs). The
accurate polarization controller achieves a 0.3 dB better sensitivity at 0 krad/s than the fast
one due to its higher control accuracy. Also the BER floor for c = 3.0 μs is with 3.8·10-7
lower than the one for c = 0.75 μs, which is 6.1·10-7. This changes dramatically if the
polarization change speed is set to 40 krad/s. While the BER floor for the fast controller
degrades only moderately to 4.3·10-6 the BER floor of the accurate controller increases to
3.1·10-3.
Figure 5.26: BER (I&Q averaged) vs. optical power at the preamplifier input for different speeds of
polarization change. (a) for c = 3.0 μs (slow control) and (b) for c = 0.75 μs (fast control).
Figure 5.27 compares the receiver sensitivities of the two polarization controllers for
different polarization change speeds at BER = 10-3. As long as the polarization change
speed is below ~7 krad/s the controller with c = 3.0 μs outperforms the one with
c = 0.75 μs. If faster polarization changes occur the controller with c = 0.75 μs is
advantageous. Assuming a tolerable sensitivity penalty of 1 dB the receiver with fast
polarization control can tolerate polarization changes speeds faster than 40 krad/s on the
Poincaré sphere, the accurate controller tolerates speeds up to 14 krad/s.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
109
Figure 5.27: Receiver sensitivity penalty vs. speed of polarization changes on the Poincaré sphere.
5.2.5 Influence of PDL on the receiver sensitivity
In order to investigate the receiver tolerance against PDL, the PDL element of the
polarization scrambler set to 6 dB. Then the BER vs. optical preamplifier input power is
measured at 0 krad/s and 40 krad/s. The results are similar for both control time constants.
Therefore, only the results for the fast polarization control with c = 0.75 μs (g = 2-4) are
shown in Figure 5.28.
Figure 5.28: BER vs. optical preamplifier input power for 0 dB and 6 dB of PDL.
The additional penalty due to PDL is uncorrelated with the speed of the polarization
changes. For 0 rad/s as well as for 40 krad/s the additional penalty due to PMD is ~1.5 dB.
Figure 5.28 also shows that distortions due to PDL have a larger impact on the receiver
sensitivity than polarization cross-talk, because the latter can be efficiently compensated.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
110
5.3 Polarization-multiplexed synchronous QPSK transmission
with real-time ASIC based coherent receiver
The main target of the University of Paderborn within the synQPSK project was the
development of a SiGe chip for analog-to-digital conversion with a sampling rate up to
10 Gsample/s [66], and of a subsequent CMOS chip for polarization control and carrier
recovery to realize a coherent synchronous polarization-multiplexed QPSK receiver that
supports 40 Gb/s operations. The VHDL code for the development of the CMOS chip was
verified in the experiments described in section 5.2.
The chips replace the commercial ADC boards and the FPGA, which limited the data rate
of the polarization-multiplexed QPSK transmission setup depicted in Figure 5.15.
Therefore four ADC chips and the CMOS chip are mounted on a common high-speed
Al2O3 ceramic board that is placed on top of custom-designed copper blocks that act as
heat sinks as well as power supply paths [67]. The whole block is placed on top of
additional heat sinks. The inputs of the ADCs are connected to the coherent receiver
frontend by SMA cables. Figure 5.29 shows the completed package for the SiGe and
CMOS chips integrated in the testbed.
Figure 5.29: SiGe ADCs and CMOS chip package integrated in the synchronous QPSK testbed.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
111
5.3.1 Transmission with and without polarization crosstalk
For the first BER measurements the data rate was set to 10 Gb/s. The polarization
scrambler was halted and the polarization was set manually in such a way that the
polarization crosstalk at the receiver was minimized. Then the BER was recorded for
different preamplifier input powers. Afterwards the input polarization at the receiver was
changed to 50% crosstalk between the two polarization channels and again the BER
performance was measured. Finally the QWPs of scrambler were turned on and generated
polarization changes with a speed of ~50 rad/s on the Poincarè sphere. Figure 5.30 shows
the achieved bit error rates for the three cases against the input power of the preamplifier.
The bit error rates of all output channels are averaged.
Figure 5.30: Bit error rate vs. preamplifier input power for synchronous polarization-multiplexed QPSK
transmission at 10 Gb/s for different polarization states at the receiver input.
For the measurements without polarization crosstalk the best performance is achieved with
a BER floor at 1.3·10-5 and a sensitivity of -44.1 dBm for a BER of 10-3. The
measurements with 50% polarization crosstalk show the worst performance with a BER
floor of 5.4·10-5 and a sensitivity of -41.4 dBm for a BER of 10-3. The results for
polarization scrambling with 50 rad/s lie in the middle with a BER floor of 3.3·10-5 and a
sensitivity of -42.6 dBm.
The measurements show that the performance of the system is limited by the switching
noise, resulting from an unsufficient filtering of power supply at the reveiver backend. The
reasons for this conclusion are the following: If the polarization crosstalk is close to zero,
the secondary diagonal elements of the polarization control matrix are close to zero, too.
Thus the switching noise in the CMOS chip is low (best case). In contrast if the
polarization crosstalk is close to 50%, all elements of the polarization control matrix toggle
and the switching noise is maximal (worst case). This correlates exactly with the BER
measurement results in Figure 5.30 and the observed power consumption of the CMOS
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
112
chip, which increases if there is crosstalk between the polarizations. The system
performance with activated scrambler consequently lies in between the best case and worst
case scenario results, as due to the scrambling of the SOP the polarization cross-talk at the
receiver varies between 0% and 50%.
5.3.2 Influence of different carrier recovery filter widths
Due to the increased data rate that is possible with the SiGe ADCs and the CMOS chip the
linewidth-times-symbol-duration product changed from Δf·TS ≈ 2.8·10-3 for the
measurements results presented in section 5.2.2 to Δf·TS ≈ 8·10-4 for the current setup.
Figure 5.31 shows the influence of the filter half width NCR on the receiver sensitivity.
The larger filter with NCR = 4 yields a better system performance with a BER floor of
3.3·10-5 and a sensitivity of -43.5 dBm for a BER of 10-3. The receiver with NCR = 2
achieves a BER floor of 9.3·10-5 and a sensitivity of -42.0 dBm. Due to the lower
linewidth-times-symbol-duration product compared to the previous measurements the
larger filter with NCR = 4 now outperforms the smaller filter with NCR = 2 by 1.5 dB in
terms of receiver sensitivity.
Figure 5.31: Influence of the filter half width NCR in the carrier recovery circuit on the BER performance of a
10 Gb/s polarization-multiplexed synchronous QPSK transmission system
But in general a lower BER floor would be expected for NCR = 2, because a lower filter
width increases the phase noise tolerance. That this cannot be observed in the
measurements shown in Figure 5.31 is due to the fact that the BER floor is not caused by
phase noise but by switching noise of the CMOS chip.
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
113
5.3.3 Single-polarization vs. polarization-multiplexed QPSK
transmission
In a final measurement the 10 Gb/s polarization-multiplexed QPSK transmission system is
compared against a 5 Gb/s single-polarization transmission system. Thus both systems
have a symbol rate of 2.5 Gbaud. Figure 5.32 shows that the polarization-multiplexed
system suffers from a ~3 dB penalty compared to the single-polarization system. This is in
accordance with theory because the polarization-multiplexed system sends twice the
number of bits per symbol. Thus for the same preamplifier input power the energy per bit
is halved compared to the single-polarization system.
Figure 5.32: Comparison of 5 Gb/s single-polarization QPSK transmission
against 10 Gb/s polarization-multiplexed QPSK transmission
The BER floors in Figure 5.32 strongly differ for the different evaluated scenarios. These
large variations are caused by the switching noise of the CMOS chip, which also differs
strongly for the different scenarios. For single-polarization transmission with manually
optimized SOP at the receiver input the digital polarization control matrix can be frozen,
i.e. the controller is disabled. This also minimizes the switching noise and a BER floor of
2·10-9 can be achieved. As soon as the SOP at the receiver input varies due to scrambling
and the digital polarization controller is switched on, the BER floor degrades to 4·10-6.
Roughly the same BER floor (1.5·10-6) is also achieved for polarization-multiplexed
transmission with manually optimized SOP. In both cases the switching noise inside the
CMOS chip is roughly the same because in both cases half of the polarization control
matrix elements is zero: In the case of single-polarization transmission the elements for the
second polarization are zero, for polarization-multiplexed transmission only the phase
offset between the two polarizations has to be controlled and thus the secondary diagonal
elements are zero. Finally if the input SOP of the polarization-multiplexed receiver is
5 Implementation of a synchronous optical QPSK transmission system with real-time
coherent digital receiver
114
scrambled and all elements of the polarization control matrix toggle the BER floor further
degrades to 3·10-5.
6 Discussion
115
6 Discussion
Until today (January 2009) most of the optical transmission experiments with coherent
detection still use offline signal processing [68; 69; 70]. Although these experiments have a
high value, especially for the evaluation of transmission impairments at high data rates
>100 Gb/s [71], special care must be taken to ensure the feasibility of the employed
algorithms for real-time implementations. The algorithmic constraints summarized in this
dissertation can help researchers to find out whether their signal processing algorithms can
operate in real-time systems [72].
The most convincing way to demonstrate the real-time feasibility of an algorithm is its
actual implementation in a real-time coherent receiver. The experimental results presented
in this thesis thereby mark milestones in the development of real-time synchronous QPSK
transmission systems as shown in Figure 6.1.
Figure 6.1: Milestones in synchronous QPSK transmission with real-time coherent digital receivers
Although researchers all over the world investigate coherent technologies for optical
communication, until today only 4 research groups published experimental results obtained
with real-time digital receivers. Among these the University of Paderborn was the first to
achieve a single-polarization QPSK transmission with digital feed-forward carrier recovery
[73] as well as a polarization-multiplexed QPSK (PM-QPSK) transmission with additional
real-time digital polarization tracking [74]. As described in the sections 5.1 and 5.2 both
systems were realized with commercially available ADCs and FPGAs for digital signal
processing. In July 2007 the Nortel group revealed the first real-time coherent PM-QPSK
[73]
[74]
[82]
[83]
[75] [79]
[80]
[67]
[81]
6 Discussion
116
receiver based on an ASIC [75]. Due to the application-specific design of the DSPU they
were the first to reach the standard data rate of 40 Gb/s. The experimental results presented
in section 5.3 using the ASIC-based receiver developed within the synQPSK project will
be published at the Optical Fiber Communication Conference and Exposition (OFC) 2009
[67].
But the presentation of a commercial polarization-multiplexed synchronous QPSK system
does not mark the end of research about coherent optical techniques. In contrast it
motivates to look further. Orthogonal frequency division multiplexing (OFDM) and
higher-level QAM formats promise spectral efficiencies far beyond the one of polarization-
multiplexed QPSK. Integrated modulators have already been developed for 16-QAM [17;
55], and the transmission of a polarization-multiplexed 128-QAM signal was demonstrated
using a pilot carrier for optical phase noise cancelling [18]. But a phase noise tolerant
carrier recovery algorithm has not been proposed yet. All published algorithms only
achieve a low laser linewidth tolerance in practical systems because of a decision-directed
feedback loop [38; 76; 77], or the use of dedicated symbols for carrier recovery [78]. The
feed-forward carrier recovery proposed in this thesis is the first one that promises a
sufficient phase noise tolerance to allow the application of DFB lasers.
7 Summary
117
7 Summary
This thesis has presented the different functional blocks and the corresponding algorithms
that are required in a digital coherent receiver for optical transmission systems: Clock
recovery, polarization control and ISI compensation, carrier recovery, data recovery and
intermediate frequency control.
One emphasis was put on the comparison of two polarization control algorithms. The non-
data-aided (NDA) approach uses the correlation of data before and after the polarization
controller to force the output data of the controller on the unity circle. This inherently
separates the two polarizations. The decision-directed (DD) approach correlates the
controller output data with the recovered symbols and forces this correlation matrix to the
unity matrix. Comparative simulations showed that the NDA approach allows an
approximately 3 times faster tracking of the SOP, but at the price of arbitrary phase
differences between the two polarization modes. In contrast the DD approach requires a
larger control time constant, but also compensates for phase offsets between the
polarization channels, thus allowing a common carrier recovery for both polarizations
which increases the phase noise tolerance.
The DD approach for polarization control has additionally been extended to enable also the
compensation of intersymbol interference (ISI). In simulations the capability of the
algorithm has been successfully demonstrated at the example if PMD compensation. If the
start-up sequence of the ISI compensator is correctly controlled, the algorithm successfully
locked to all investigated DGD profiles and efficiently compensated for ISI. The only
drawback is that for an increasing filter width the control gain has to be reduced to avoid a
degradation of the receiver sensitivity. This limits the achievable tracking speed for time-
variant impairments.
The comparison of different feed-forward carrier recovery schemes was another main topic
within this dissertation. Four different algorithms were investigated: The Viterbi & Viterbi
(V&V) algorithm, the V&V algorithm extended by adaptive weighting of the filter inputs
and the barycenter or (S)MLPA algorithm are developed for equidistant-phase
constellations. A novel feed-forward algorithm proposed in this thesis allows also for
efficient carrier recovery for arbitrary QAM constellations. All algorithms were compared
in QPSK simulations. An evaluation of the estimator efficiency in a phase noise-free
scenario and of the mean squared estimator error in simulations considering phase noise
showed that the V&V algorithm with adaptive weighting yields the best performance.
However, the SMLPA algorithm is only slightly inferior, but allows for a much more
efficient hardware implementation. Although the QAM carrier recovery performance is
7 Summary
118
similar to the other presented approaches for QPSK, its main advantage is the efficient
carrier recovery for higher-order QAM constellations. The simulation results from section
4.2 showed that with the proposed algorithm a synchronous optical 16-QAM transmission
system employing DFB lasers is already feasible today, and 64-QAM or 256-QAM system
will be realizable in the near future.
Finally the experimental results of a real-time synchronous QPSK transmission system
developed within the European synQPSK project were presented. The simulation results
presented in section 4.1 were thereby the main basis for the specifications of the digital
signal processing unit in the real-time coherent receiver. The experiments presented in the
sections 5.1 and 5.2 were the worldwide first demonstrations of a synchronous QPSK
transmission system with real-time coherent digital receiver using feed-forward carrier
recovery and of a synchronous polarization-multiplexed QPSK transmission system with
real-time coherent receiver and digital polarization tracking, respectively. After replacing
the commercial ADCs and the FPGA for signal processing in the receiver by specifically
developed high-speed ADCs and a CMOS DSPU, both also developed in the context of the
synQPSK project, the data rate could be increased to 10 Gb/s.
8 Outlook
119
8 Outlook
The algorithms and experimental results presented in this dissertation offer a broad variety
of follow-up research:
The dispersion compensation algorithm proposed in section 3.3.3 is until now only
evaluated for its PMD compensation performance. Additional simulations considering CD
as well as a comparison against other state-of-the-art dispersion compensation algorithms
are required to finally assess the feasibility of the algorithm for commercial systems.
The proposed QAM carrier recovery algorithm is evaluated in great detail in this
dissertation for 4 different square QAM constellations. However, many more constellations
are possible and might outperform the considered constellations in terms of receiver
sensitivity, transmitter and receiver complexity or tolerance against various distortions.
The research about high-level QAM formats for optical communication systems is only in
the early stages. Only little is known today about the tolerances against linear and non-
linear impairments such as PMD and CD or self-phase modulation (SPM) and cross-phase
modulation (XPM). Also the interoperability of different QAM constellations and also of
the proposed feed-forward carrier recovery with other algorithms for polarization control
or dispersion compensation needs to be assessed.
Finally based on the existing testbed for the evaluation of real-time synchronous QPSK
transmission all algorithms described in this thesis, which are not yet implemented, can be
translated to a hardware description language (e.g. VHDL) to be evaluated in real-time
transmission experiments.
9 Bibliography
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10. List of figures & tables
125
10 List of figures & tables
Figure 1.1: Simplified system schematic for the synQPSK project with partners’
contributions highlighted ....................................................................................................... 3
Figure 2.1: BPSK (left) and QPSK (right) constellation diagrams ....................................... 6
Figure 2.2: ASK-8-PSK constellation diagram ..................................................................... 7
Figure 2.3: Square QAM constellation diagrams .................................................................. 8
Figure 2.4: Square 16-QAM constellation diagram and bit-to-symbol assignment .............. 9
Figure 2.5: Partial differential encoding for a square 16-QAM constellation ..................... 10
Figure 2.6: Optical QAM transmitter structure ................................................................... 11
Figure 2.7: Polarization-multiplexed QAM transmitter ...................................................... 13
Figure 2.8: Polarization mode dispersion emulator (PMDE) .............................................. 15
Figure 2.9: Polarization diversity coherent receiver frontend ............................................. 17
Figure 2.10: Coherent optical receiver with analog-to-digital conversion and digital signal
processing ............................................................................................................................ 20
Figure 2.11: Examples of the phase noise process ψk for different values of Δf3dBTS ......... 21
Figure 2.12: Lorentzian carrier power spectra for different values of Δf3dBTS .................... 22
Figure 3.1: Coherent optical receiver structure .................................................................. 23
Figure 3.2: Interfacing between ADCs and DSPU and internal structure of the DSPU ..... 24
Figure 3.3: Serial and parallel FIR and IIR filter structures ................................................ 25
Figure 3.4: Decision-directed carrier recovery with Δ = 1 .................................................. 27
Figure 3.5: Decision-directed carrier recovery in a realistic receiver with parallel and
pipelined signal processing. ................................................................................................. 28
Figure 3.6: Non-data-aided polarization control algorithm ................................................. 30
Figure 3.7: Decision-directed polarization control algorithm ............................................. 31
Figure 3.8: Viterbi & Viterbi feed-forward carrier recovery ............................................... 36
Figure 3.9: Weighted Viterbi & Viterbi feed-forward carrier recovery .............................. 37
Figure 3.10: Feed-forward carrier recovery for square QAM constellations ..................... 42
Figure 4.1: QPSK carrier phase estimator efficiency for OSNR = 10 dB (left) and OSNR =
16 dB (right) ........................................................................................................................ 51
Figure 4.2: Carrier phase estimator mean squared error for different values of Δf·TS ........ 53
Figure 4.3: OSNR vs. BER for Viterbi & Viterbi carrier recovery and different linewidth-
times-symbol-duration products Δf·TS. ................................................................................ 54
Figure 4.4: OSNR vs. BER for weighted Viterbi & Viterbi carrier recovery and different
linewidth-times-symbol-duration products. ........................................................................ 55
Figure 4.5: Sensitivity penalties at BER = 10-3 against Δf·TS for unweighted (a) and
weighted (b) Viterbi & Viterbi carrier recovery .................................................................. 56
Figure 4.6: OSNR vs. BER for (S)MLPA carrier recovery and different linewidth-times-
symbol-duration products .................................................................................................... 57
Figure 4.7: Sensitivity penalties at BER = 10-3 against Δf·TS for (S)MLPA carrier recovery
............................................................................................................................................. 58
Figure 4.8: OSNR vs. BER for 4-QAM carrier recovery and different linewidth-times-
symbol-duration products. ................................................................................................... 58
10. List of figures & tables
126
Figure 4.9: Sensitivity penalties at BER = 10-3 against Δf·TS for 4-QAM carrier recovery 59
Figure 4.10: Phase noise tolerance for different carrier recovery algorithms using either the
data from a single-polarization or from both polarizations ................................................. 60
Figure 4.11: Sensitivity penalty vs. analog-to-digital converter resolution for different
QPSK carrier recovery algorithms ...................................................................................... 61
Figure 4.12: Sensitivity penalty vs. phase resolution for different QPSK carrier recovery
algorithms ............................................................................................................................ 62
Figure 4.13: Sensitivity penalty for different numbers of test phase values φb for 4-QAM
and 16-QAM ........................................................................................................................ 64
Figure 4.14: Sensitivity penalty for different numbers of test phase values φb for 64-QAM
and 256-QAM ...................................................................................................................... 64
Figure 4.15: Phase estimator mean squared error and efficiency e(NCR) vs. filter half width
NCR for square 4-QAM (left) and square 16-QAM (right) constellations with log2{B} = 5 65
Figure 4.16: Phase estimator mean squared error and efficiency e(NCR) vs. filter half width
NCR for square 64-QAM (left) and square 256-QAM (right) constellations with log2{B} = 6
............................................................................................................................................. 66
Figure 4.17: Phase estimator mean squared error (left) and efficiency (right) vs. filter half
width NCR for a square 16-QAM constellation and log2{B} = 6 ......................................... 67
Figure 4.18: 4-QAM constellation diagram at the receiver before and after carrier recovery
for ∆f·TS = 4·10-4 (log2{B} = 5, NCR = 9) ............................................................................. 68
Figure 4.19: 16-QAM constellation diagram at the receiver before and after carrier
recovery for ∆f·TS = 1.4·10-4 (log2{B} = 5, NCR = 9) ........................................................... 68
Figure 4.20: 64-QAM constellation diagram at the receiver before and after carrier
recovery for ∆f·TS = 4·10-5 (log2{B} = 6, NCR = 9) .............................................................. 69
Figure 4.21: 256-QAM constellation diagram at the receiver before and after carrier
recovery for ∆f·TS = 8·10-6 (log2{B} = 6, NCR = 9) .............................................................. 69
Figure 4.22: Squared distance sum sb of the b-th parallel block (with test carrier phase
angle φb) for different filter half widths and different square QAM constellations ............ 70
Figure 4.23: Receiver tolerance against phase noise for different square QAM
constellations ....................................................................................................................... 71
Figure 4.24: Impact of different linewidth-times-symbol-duration products on the receiver
sensitivity of coherent QAM receivers ................................................................................ 72
Figure 4.25: Receiver sensitivity penalty vs. analog-to-digital converter resolution for
different square QAM constellations ................................................................................... 73
Figure 4.26: Receiver sensitivity penalty vs. internal resolution of the distances Re[dk,b]
and Im[dk,b] for different square QAM constellations ......................................................... 74
Figure 4.27: Receiver sensitivity penalty vs. internal resolution of the squared distance
|dk,b|2 for different square QAM constellations .................................................................... 75
Figure 4.28: Influence of the polarization control gain g on the receiver sensitivity .......... 76
Figure 4.29: Start-up development of the matrix elements of the polarization control matrix
M ......................................................................................................................................... 77
Figure 4.30: Sensitivity penalty vs. normalized polarization control time constant for the
non-data-aided and decision-directed polarization control algorithms ............................... 78
Figure 4.31: Sensitivity penalty at BER = 10-3 for different control gains of the polarization
control/ISI compensation algorithm and different values of the FIR filter half width ........ 78
Figure 4.32: BER vs. OSNR for the simplified and original ISI compensation algorithm
with g = 2-10 and different FIR filter half widths ................................................................. 79
10. List of figures & tables
127
Figure 4.33: Input referred DGD profiles for the emulation of PMD ................................. 81
Figure 4.34: Receiver sensitivity for the original ISI compensation algorithm with different
FIR filter half widths for different DGD profiles ................................................................ 82
Figure 4.35: Receiver sensitivity for the simplified ISI compensation algorithm with
different FIR filter half widths for different DGD profiles ................................................. 83
Figure 4.36: Developing of |det{Mi}| for the simplified ISI compensation algorithm with
different FIR filter half widths for different DGD profiles ................................................. 84
Figure 4.37: Receiver sensitivity (top row) and developing of |det{Mi}| (bottom row) for
different DGD profiles and different FIR filter half widths for the simplified algorithm
with optimized start-up sequence. ....................................................................................... 85
Figure 5.1: 800 Mb/s single-polarization synchronous QPSK transmission setup with real-
time digital coherent receiver .............................................................................................. 88
Figure 5.2: Single-polarization QPSK transmitter with Bookham optical QPSK modulator.
............................................................................................................................................. 88
Figure 5.3: LiNbO3 optical 90° hybrid and associated control unit .................................... 89
Figure 5.4: Differential photodiode pairs and amplifiers .................................................... 89
Figure 5.5: Commercial ADC board (left) and FPGA board (right) for digital signal
processing ............................................................................................................................ 90
Figure 5.6: BER vs. preamplifier input power for synchronous QPSK transmission with
self-homodyne detection using an ECL. ............................................................................. 91
Figure 5.7: BER vs. preamplifier input power for synchronous QPSK transmission with
self-homodyne detection using a DFB laser. ....................................................................... 92
Figure 5.8: Laser beating at the output of the coherent receiver frontend ........................... 92
Figure 5.9: BER vs. preamplifier input power for synchronous QPSK transmission with
intradyne detection using DFB lasers. ................................................................................. 93
Figure 5.10: Measured BER vs. optical preamplifier input power at 1.6 Gb/s data rate..... 94
Figure 5.11: Transformation of the 3x3 coupler outputs to I&Q signals ............................ 95
Figure 5.12: Synchronous QPSK transmission setup with clock recovery and symmetrical
3x3 coupler .......................................................................................................................... 96
Figure 5.13: Receiver sensitivity of a synchronous QPSK receiver with either a symmetric
3x3 coupler or a 90° hybrid. ................................................................................................ 96
Figure 5.14: Simulated and measured BER floors for different linewidth-times-symbol-
duration products ................................................................................................................. 97
Figure 5.15: 2.8 Gb/s polarization-multiplexed QPSK transmission setup with a real-time
FPGA-based synchronous coherent digital I&Q receiver ................................................... 98
Figure 5.16: Block diagram for the pattern generator and QPSK precoder implemented in
an FPGA .............................................................................................................................. 99
Figure 5.17: MK325 Xilinx Virtex-II ProX RocketIO characterization board used for
pattern generation and precoding ...................................................................................... 100
Figure 5.18: Polarization-multiplexed QPSK modulator. ................................................. 101
Figure 5.19: Polarization scrambler and the corresponding fast polarization changes
displayed on the Poincaré sphere....................................................................................... 102
Figure 5.20: Polarization diversity coherent optical receiver frontend. ............................ 103
Figure 5.21: Commercially available analog-to-digital converters (left) and an FPGA board
for digital signal processing (center). ................................................................................ 103
10. List of figures & tables
128
Figure 5.22: Influence of the carrier recovery filter width on the receiver sensitivity and
BER floor. .......................................................................................................................... 105
Figure 5.23: Measured BER vs. optical preamplifier input power at the coherent receiver
for the two different control time constants of a) 12 μs and b) 3 μs and various polarization
change speeds. ................................................................................................................... 106
Figure 5.24: Receiver sensitivity penalty vs. scrambling speed for a bit error rate of 10-4107
Figure 5.25: Long-term BER measurements with polarization change speeds causing 1 dB
loss in receiver sensitivity. ................................................................................................ 107
Figure 5.26: BER (I&Q averaged) vs. optical power at the preamplifier input for different
speeds of polarization change. (a) for c = 3.0 μs (slow control) and (b) for c = 0.75 μs (fast
control). ............................................................................................................................. 108
Figure 5.27: Receiver sensitivity penalty vs. speed of polarization changes on the Poincaré
sphere. ................................................................................................................................ 109
Figure 5.28: BER vs. optical preamplifier input power for 0 dB and 6 dB of PDL. ......... 109
Figure 5.29: SiGe ADCs and CMOS chip package integrated in the synchronous QPSK
testbed. ............................................................................................................................... 110
Figure 5.30: Bit error rate vs. preamplifier input power for synchronous polarization-
multiplexed QPSK transmission at 10 Gb/s for different polarization states at the receiver
input. .................................................................................................................................. 111
Figure 5.31: Influence of the filter half width NCR in the carrier recovery circuit on the
BER performance of a 10 Gb/s polarization-multiplexed synchronous QPSK transmission
system ................................................................................................................................ 112
Figure 5.32: Comparison of 5 Gb/s single-polarization QPSK transmission against 10 Gb/s
polarization-multiplexed QPSK transmission ................................................................... 113
Figure 6.1: Milestones in synchronous QPSK transmission with real-time coherent digital
receivers ............................................................................................................................. 115
Table 2.1: Differential coding penalty for different square QAM constellations ............... 11
Table 3.1: Barycenter carrier recovery filter structures ....................................................... 39
Table 3.2: Required hardware components for different carrier recovery algorithms ........ 45
Table 4.1: Maximum tolerable linewidth for 10 Gbaud systems with different square QAM
constellations ....................................................................................................................... 71
Table 4.2: Required symbol rate and maximum tolerable linewidth to realize a 100GbE
(112 Gb/s) system with different square QAM constellations ............................................ 72
Table 4.3: Analog-to-digital converter requirements for a polarization-multiplexed QAM
transmission system for 100GbE ......................................................................................... 74
129
Acknowledgement
First of all I want to thank Prof. Dr.-Ing. Reinhold Noé for giving me the opportunity to do
research on this interesting topic of real-time coherent optical transmission. To work under
his supervision was both challenging and exciting. Many fruitful discussions and valuable
suggestions have substantially contributed to the success of this work. Moreover I would
like to thank him for his confidence to send me to several conferences as his representative.
The gained experiences and established contacts are invaluable.
I also cordially thank Prof. Dr.-Ing. Ulrich Rückert for the trustful cooperation within the
synQPSK project and for the assessment of this thesis.
Special thanks go to all my colleagues in the working groups “Optical Communication and
High Frequency Engineering” as well as “System and Circuit Technology”. Dr.-Ing.
Sebastian Hoffmann was a great officemate and was willing to discuss even the most
wacky ideas. Dr.-Ing. Suhas Bhandare was an invaluable source of help for any kind of
problem in the laboratory. Dr. Olaf Adamczyk was a great advisor and significantly
improved the quality of my publications. Dipl.-Wirt.-Ing. Ralf Peveling and Dipl.-Ing.
Christian Wördehoff were indispensable for the FPGA implementation of the algorithms
and additionally gave substantial moral support.
I would further like to thank all partners of the synQPSK project. All my experimental
work would have been impossible without the components provided by them. In this
context also the funding by the European Commission should be mentioned. Its continuing
support and patience in spite of obstacles and delays was not self-evident.
I am also grateful that the International Graduate School “Dynamic Intelligent Systems”
granted me a scholarship. Next to the skills and knowledge I gained in different lectures
and workshops they offered me the opportunity to built up a network of friends that spans
all around the world.
Heartfelt thanks go to my family. My whole life they gave me absolute love and support.
Finally I want to thank Luz-marina Guitard for her love and passion that help me to
become the person I want to be.
