Blind Equaliza ion in Dynamic PMD Channels
Using Va ia ional Au oencode s wi h LSTM
Jos´
e N´
u˜
nez-Kasane a, Bo is Ka ano , Alex Al a ado and Gab iele Liga
1Depa men o Elec ical Enginee ing, Eindho en Uni e si y o Technology, Eindho en, The Ne he lands
*j.i.nunez.kasane a@ ue.nl
Abs ac : We p opose a new a ia ional au oencode -based blind equalize o pola iza-
ion demul iplexing, and assess i in a dynamic pola iza ion channel. We demons a e a 0.4
dB SNR gain and doubled ole ance o s a e-o -pola iza ion d i compa ed o CMA-RDE.
© 2024 The Au ho (s)
1. In oduc ion
Fibe bi e ingence leads o simul aneous coupling and il e ing e ec s on he ansmi ed pola iza ion channels, a
channel impai men e e ed o as pola iza ion mode dispe sion (PMD). One o he main unc ions o a cohe en
op ical ecei e is he pola iza ion channel demul iplexing in he p esence o PMD. PMD can be desc ibed as
he combina ion o wo phenomena: a) a local o a ion o he s a e o pola iza ion (SOP) o he op ical ield; b)
a di e en ial g oup delay (DGD) be ween he local p incipal s a es o pola iza ion. Bo h hese phenomena a e in
gene al ime- a ying, wi h he SOP o a ions ep esen ing he main componen o a dynamic PMD channel. SOP
luc ua ions o e di e en imescales ha e been epo ed in he li e a u e, spanning om days [1] o µs [2].
Cohe en ecei e s equi e channel equaliza ion o ack he SOP changes and compensa e o he link DGD luc-
ua ions. Blind equaliza ion schemes a e widely used o his pu pose, wi h he cons an modulus algo i hm (CMA)
being he mos popula equaliza ion me hod [3]. Fo mul i-modulus cons ella ions, he adius-di ec ed equalize
(RDE) [4] combined wi h CMA (CMA-RDE) achie es be e con e gence. Recen ly, new machine lea ning ech-
niques we e in oduced o pola iza ion demul iplexing, such as he a ia ional au oencode -based linea equalize
(VAE-LE) [5]. The VAE-LE uses he a ia ional in e ence p inciple [6] o ain he 2 ×2 ini e impulse esponse
(FIR) mul i-inpu mul i-ou pu il e . The VAE-LE has been shown o ou pe o m CMA o high-o de modula ion
o ma s [5], bu i s pe o mance has no been cha ac e ized in ime- a ying pola iza ion channels [1,2,7].
In his pape , we p opose an ex ension o he VAE-LE blind equalize in [5], inco po a ing a long-sho - e m
memo y (LSTM) ne wo k. The goal o his LSTM ne wo k is o mi iga e apid pola iza ion changes in dynamic
op ical ibe channels. Ou esul s demons a e ha he p oposed a chi ec u e signi ican ly imp o es he SOP
acking, leading o an imp o ed ou age p obabili y and inc eased ole ance o ime- a ying pola iza ion channels
compa ed o CMA-RDE.
2. Channel model and p oposed equalize
2.1. Channel model
In his wo k, we u ilize he dynamic PMD channel model p oposed in [7], which is ma hema ically desc ibed by
S( ,k) =
N
∏
n=1
Hn(k)Bn( ),(1) Bn( ) =
M
∏
m=1
Rm,nDm,n( ),(2)
S( ,k)is a 2 ×2 equency ( ) and disc e e- ime (k)-dependen ma ix, ep esen ing he block-wise a ying
equency esponse o he pola iza ion channel. S( ,k)consis s o Nsec ions, each comp ising a so-called hinge,
modeled by a ime-dependen 2 ×2 complex uni a y ma ix Hn(k), and a s a ic andom bi e ingence sec ion
desc ibed by he equency-dependen uni a y 2 ×2 complex ma ix Bn( ). The hinges model localized ime-
a ying pe u ba ions o he SOP along he op ical link, whe eas he bi e ingence sec ions desc ibe a s a ic
equency-selec i e pola iza ion o a ion. Each bi e ingence sec ion in (2) consis s o a conca ena ion o Msec-
ions each comp ising a complex 2 ×2 o a ion ma ix Rm,n ha sca e s he SOP iso opically on he Poinca ´
e
sphe e, and a delay elemen Dm,n( ) = diag(exp(jπ τm,n),exp(−jπ τm,n)), whe e τm,na e d awn independen ly
om a Gaussian dis ibu ion acco ding o [7, Sec. II]. The empo al d i caused by each hinge is desc ibed by
Hn(k) = J(˙
αn(k))·Hn((k−1)), whe e J(˙
αn(k)) = exp(−j˙
αn(k)·−→
σ), is a andom Jones ma ix wi h −→
σbeing he
Pauli enso [8, eq. (3)]. The pa ame e ˙
αnis a ec o o S okes inno a ion angles assumed o be an i.i.d. ze o-mean
Gaussian dis ibu ion wi h a iance σ2
=2π∆PT/N, whe e ∆Pis he pola iza ion linewid h and Tis he selec ed
SOP inno a ion pe iod [8].
The channel model used in his wo k only conside s uni a y pola iza ion e ec s, neglec ing o he impai men s
such as ch oma ic dispe sion and ibe nonlinea i y. Finally, whi e Gaussian noise is added a he channel ou pu .
2.2. VLSTM Equalize
The s uc u e o VAE-LE was ecen ly p oposed in [5] and is illus a ed in Fig. 1(a). The VAE-LE u ilizes he
e idence lowe bound (ELBO) o app oxima e he maximum likelihood (ML) channel es ima ion and equalize he
ecei ed symbols Rx,y
i, whe e xand y ep esen he dual pola iza ion, whe e each pola iza ion has he espec i e
in-phase (I) and quad a u e (Q) componen s wi hin ame i(numbe o symbols/symbol a e). The equalize inco -
po a es a complex- alued 2 ×2 bu e ly s uc u e wi h FIR il e s, whose ou pu a e he demul iplexed symbols
ˆ
Rx,y
i. The VAE-LE uses one il e sys em o equaliza ion whe e he LE il e weigh s a e ep esen ed by he enso
hi
es , which a e upda ed by he ELBO unc ion and also used o he channel es ima ion as p esen ed in [5, eq. (6)].
A maximum a pos e io i (MAP) so demappe is used o ansla e ˆ
Rx,y
iin o he p obabili ies o he co esponding
symbols ˆqx,y
i. The so demappe is ollowed by a MAP symbol de ec o p oducing he es ima ed symbol ˆ
Tx,y
i ha
goes o he SER he complex ec o Tx,y
io ansmi ed symbols.
SER
Es ima ion
VAE-LE
Linea
Equalize
Rx
i
Ry
i
So
Demappe
ˆ
Rx
i
ˆ
Ry
i
ELBO
hi
es
ˆ
hi
es
To
LSTM
F om
LSTM
hi
es
MAP
Symbol
De ec o
SER
ˆ
Tx
i
ˆ
Ty
i
ˆqx
i
ˆqy
i
Tx
iTy
i
(a)
VAE-LE
SERi−1
Es ima ion
LSTM VAE-LE
SERi
Es ima ion
P oposed
block
Rx
i−1
Ry
i−1
ˆqx
i−1
ˆqy
i−1
hi−1
es ˆ
hi−1
es
Rx
i
Ry
i
ˆqx
i
ˆqy
i
Rx
i
Ry
i
(b)
Fig. 1: S uc u e o he VAE-LE (a), and VLSTM (b).
The LSTM is a ype o ecu en neu al ne wo k de-
signed o ime-dependen asks, capable o lea ning long-
e m dependencies [9], we used his p ope y o es ima e
he incoming il e ap weigh ˆ
hi
es o di e en ames. We
assumed ha wi hin a ame, Rx,y
iis co ela ed wi h Tx,y
iby
(1). Based on his assump ion, he e is no co ela ion be-
ween di e en ames. So, we use he VAE-LE s uc u e
p e iously desc ibed wi h he addi ion o an LSTM block
(yellow block), a e he ime ame i−1 o he VAE-LE,
as illus a ed in Fig 1(b). The esul ing VAE-LE+LSTM
(VLSTM) scheme, add esses he PMD channels impai -
men s a ame iby using he p e iously ap weigh il e
hi−1
es as he hidden s a e o he LSTM block, and also us-
ing he ecei ed symbols o he nex ime ame Rx,y
ias
he LSTM inpu . This gene a ed a new ˆ
hi−1
es , which se es
as he new s a ing ap weigh il e o he new ame i
ep esen ed as he dashed a ow in Fig.1(a).
3. Resul s
In his sec ion, we compa e he pe o mance o 3 blind equaliza ion schemes: CMA-RDE, VAE-LE, and VL-
STM. The nume ical simula ion se up is based on (1) and (2). The model’s inpu is a single-channel pola iza ion-
mul iplexed signal employing a uni o m 64-QAM modula ion o ma pulse shaped using a oo - aised cosine wi h
0.1 oll-o and an o e sampling ac o o 2 samples pe symbol. The DGDs in he Dm,n( )ma ices we e com-
pu ed assuming a symbol a e o 40 GBd, and a PMD coe icien o 0.1 ps/√km. The op ical ibe link consis s o
10 spans o 100 km each, wi h hinge sec ions a he end o each span. Each span is hen spli in o 1,000 bi e in-
gence segmen s. All equalize s we e ained using ames o 15,000 symbols. The lea ning a es o he VAE-LE
and VLSTM models we e ini ialized a l =4×10−3, and l =10−3 o he CMA-RDE. All equalize s lea ning
a es we e educed by 50% e e y 20 ames, while he LSTM lea ning a e was educed by a ac o o 1,000.
In his wo k, we ocus on he pola iza ion acking pe o mance o equalize s, omi ing hei singula i y beha -
io , which has been co e ed o CMA-RDE and VAE-LE in p e ious s udies [10]. Fig. 2(a) shows he con e gence
beha io o he 3 blind equaliza ion schemes analyzed o a nea -s a ic and a dynamic scena io wi h ∆P1 ad/s and
103 ad/s, espec i ely. Fo he CMA-RDE scheme, a e 100 ames, he loss unc ion o he scheme swi ches om
CMA o RDE, eaching a s eady s a e wi hin 106 ames. We de ine s eady s a e he poin a which he windowed
SER o e he las 5 ames does no dec ease by mo e han 20%. Mo eo e , he VAE-LE equalize exhibi s a as e
con e gence (43 ames), bu also shows highe s eady-s a e SER alues compa ed o he CMA-RDE scheme,
whe e he impac o he SOP d i is no iceable a e eaching he s eady s a e wi h a highe SER oscilla ion in he
050 100 150 200
10−3
10−2
10−1
100
(a)
F ame index i
SER
200 220 240
10−3
CMA-RDE
VAE-LE [5]
VLSTM
∆P=1 ad/s
∆P=103 ad/s
100101102103104
10−3
10−2
(b)
∆P[ ad/s]
SER
104
10−2
×2×1.3
CMA-RDE
VAE-LE [5]
VLSTM
Fig. 2: Resul s o he blind equaliza ion schemes analyzed in his wo k a SNR=25 dB: a) SER s ame index o e 200
ames wi h a windowed SER o 5 ames, and b) SER s SOP d i speed (∆P).
21 21.522 22.523 23.5
10−3
10−2
10−1
100
(a)
0.4 dB
0.5 dB
Pou =2×10−3
SNR [dB]
Pou [%]
CMA-RDE
VAE-LE [5]
VLSTM
100101102103104
10−3
10−2
10−1
100
(b)
∆P[ ad/s]
Pou [%]
103
10−2
×2.4
×2
CMA-RDE
VAE-LE [5]
VLSTM
Fig. 3: Ou age p obabili y o he 3 di e en equalize s analyzed in his wo k, an SER h eshold o 10−2and, o : a) di e en
SNRs and a ixed ∆P=103 ad/s; b) di e en ∆P alues o SNRs ha gi es Pou =10−3wi h ∆P=0 ad/s.
dynamic scena io compa ed wi h he nea -s a ic one. The VLSTM scheme achie es he bes pe o mance among
he 3 analyzed equalize s in bo h channel condi ions, showing bo h as e con e gence (46 ames), and lowe
s eady-s a e SER.
Fig. 2(b), shows he a e age SER and s anda d de ia ion o di e en blind equaliza ion schemes as a unc ion
o ∆Pa a ixed SNR o 25 dB, whe e he a e age SER is ob ained wi h he las 50 ames, and he cha ac e iza ion
o he channel spanning om a nea -s a ic (∆P1 ad/s) o an ex emely dynamic SOP d i scena io [2]∆P10,000
ad/s. Fo he case ∆P1 ad/s, VAE-LE exhibi ed he wo s pe o mance wi h a highe a e age SER alue o 1.07×
10−3, while, RDE and VLSTM show compa able a e age SER alues o 8.2×10−4and 6.8×10−4, espec i ely.
This SER beha io changes o ∆P>103 ad/s, whe e all he equalize schemes each o a b eaking poin . In
his highly dynamic scena io, VLSTM shows a acking pe o mance imp o emen o a ac o o 2 and 1.3 in
∆Pcompa ed o he RDE and VAE-LE, espec i ely. These esul s show ha he LSTM block in Fig. 1(b) can
e ec i ely educe he impac o apid pola iza ion luc ua ions.
In Figs. 3(a)-(b), we cha ac e ize he ou age p obabili y Pou o he 3 equalize s, which is es ima ed as he
ac ion o ames abo e a gi en SER h eshold o e he o al numbe o ansmi ed ames. Only he ames
a e con e gence was eached we e aken in o accoun wi h a o al o 1,000 ames. Fo bo h scena ios, we chose
an SER h eshold o 10−2. Fig. 3(a) shows Pou as a unc ion o he SNR o ∆P=103 ad/s. We obse e ha
VLSTM dec eases signi ican ly Pou o SNR≥22 dB. A Pou =2×10−3(pu ple line), VLSTM achie es gains
o 0.4 dB and 0.5 dB SNR o e RDE and VAE-LE, espec i ely. Fig. 3(b) illus a es Pou o di e en alues o
∆P. To isola e he e ec o ∆Pon Pou , we ope a ed he di e en equalize s a di e en SNR alues such ha hey
exhibi he same SER=10−3in he s a ic channel case (∆P=0). These SNR alues a e 22.2 dB o he VLSTM,
and 22.6 dB o bo h VAE-LE and CMA-RDE. Fig. 3(b) shows ha VLSTM can ole a e ×2 o ×2.4 highe ∆P
han CMA-RDE and VAE-LE, espec i ely, a a Pou =2 ×10−2.
4. Conclusions
We p oposed a new blind equaliza ion scheme o pola iza ion demul iplexing and PMD compensa ion which
combines a VAE-LE wi h an LSTM block. This scheme shows imp o ed acking pe o mance in as SOP d i
en i onmen s wi h PMD compa ed o con en ional blind equaliza ion schemes such as CMA-RDE, and a plain
VAE-based equalize . Mino SER imp o emen s we e obse ed in slowly a ying SOP scena ios. Howe e , VL-
STM can signi ican ly educe he sys em ou age p obabili y in as -changing PMD channels. In such scena ios,
VLSTM can ole a e up o 2.4 imes highe SOP d i speeds compa ed o VAE-LE, and wice as as SOP o a-
ions compa ed o he CMA-RDE scheme. Fu he mo e, he VLSTM p o ides a 0.4 dB and 0.5 dB gain a ixed
ou age p obabili y o e CMA-RDE and VAE-LE schemes o he same ∆P, espec i ely.
Acknowledgemen s: This p ojec has ecei ed unding om he Eu opean Union’s Ho izon 2024 esea ch and inno a ion p og am unde he
Ma ie Skłodowska-Cu ie g an ag eemen No 101119983.
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