Signal Processing techniques to optimize the detection of radio pulsar signals
Abstract
[ANGLÈS] Radio Pulsars are neutron stars that emits high polarized electromagnetic pulses with a very accurate and stable periodicity. Adding the fact that those pulses have a wideband nature, so they can be received almost everywhere, make Radio Pulsar signals a perfect candidate for navigation systems. The challenge, however, is that the Radio Pulsar signal is degraded and submerged in Additive White Gaussian Noise when it is propagated through the ISM, so that make them difficult to detect. The aim of the project is the design of the optimum receiver in order to estimate the time of arrival of the Radio Pulsar Signal and make possible the real-time navigation. In this contribution I introduce the detection theory applied to radio pulsars and show the different signal processing techniques to improve the detection with the minimum computational time possible. Although the previous work has centered in the Signal to Noise Ratio of the pulsar signals, our focus will be the improvement of the Energy to Noise Ratio. We will see as Epoch Folding and the increment of the Bandwidth of the receiver are the best solutions. Moreover, Integration in Time appears to be one of the most promising techniques to simplify the computation of the whole process. Also I demonstrate how the detection performance is not affected by any digital or analogue filter, so Low-Pass Filtering will not have any effect in the receiver but to eliminate spurious signals. Some experiments with simulated wideband signals are shown in order to prove the optimality of the signal processing techniques. Finally, summarizing all the results obtained in the Thesis, I propose two optimum receivers for pulsar-based navigation system applications. It is known that the interstellar medium has a frequency dependent transfer characteristic, so the higher frequencies of a signal arrive earlier than the lower frequencies. This will cause the pulse profile of the radio pulsar signal to appear dispersed in time. Up until now, the researchers have been performing de-dispersion techniques in order to obtain the original power pulsar profile spending more than the 80% of the actual processing time. The prove to avoid de-dispersion without decreasing the detection performance are presented. Furthermore, some simulations with data from the pulsar PSR B0329+54 recorded by the Westerbork Observatory are shown. PSR B0329+54 is one of the strongest pulsar signal visible in the northern hemisphere with one of the lowest Dispersion Measure, so it will be easier to perform the Signal Processing techniques and see the profile of the pulse. We will see how the results are not the ones expected, because the Radio pulsar signal happens to be very Narrowband. Finally, I am going to give a possible explanation of what is happening and in what part of the acquisition the wideband nature of the rotating star pulses are lost.
Full text
Signal and In o ma ion P ocessing
(EEMCS, TU Del )
Signal P ocessing Techniques o op imize he
De ec ion o Radio Pulsa Signals
Daniel He nando Po e o
Supe iso s
Specialisa ion
Type o epo
Da e
D . I . Richa d Heusdens
D . Nikolay D. Gaubi ch
Signal P ocessing
Mas e O Science Thesis
Augus 28, 2014
Signal P ocessing echniques o op imize he
de ec ion o Radio Pulsa Signals
Mas e O Science Thesis
Fo he deg ee o Mas e o Science in Telecommunica ions Enginee ing
a Poly echnic Uni e si y o Ca alonia unde he E asmus exchange
p og amme a Del Uni e si y o Technology
Facul y o Elec ical Enginee ing, Ma hema ics and Compu e Science (EEMCS)
Del Uni e si y o Technology
Escola Tecnica Supe io d’Enginye ia de Telecomunicacions de Ba celona (ETSETB)
Poly echnic Uni e si y o Ca alonia
Daniel He nando Po e o
Augus 22, 2014
Co e Image: Fou an ennas o he A acama La ge Millime e /submillime e A ay
(ALMA) gaze up a he s a - illed nigh sky, in an icipa ion o he wo k ha lies ahead.
The Moon ligh s he scene on he igh , while he band o he Milky Way s e ches ac oss
he uppe le .
Poly echnic Uni e si y o Ca alonia
Escola Tecnica Supe io d’Enginye ia de Telecomunicacions de Ba celona
Supe iso :
G ego i Vazquez
Del Uni e si y o Technology
In elligen Sys emsDepa men
Mul imedia and Signal P ocessing (MSP) G oup
Signal and In o ma ion P ocessing (SIP) Lab
Supe iso s:
D . I . Richa d Heusdens
D . Nikolay D. Gaubi ch
Copy igh 2014 TU Del , UPNA
All igh s ese ed.
Abs ac
Key Wo ds: ENR: Ene gy o Noise Ra io, GENR: Gene alised Ene gy o Noise Ra io,
TOA= Time o A i al, Addi i e Whi e Gaussian Noise, An enna, De-Dispe sion.
Radio Pulsa s a e neu on s a s ha emi s high pola ized elec omagne ic pulses wi h
a e y accu a e and s able pe iodici y. Adding he ac ha hose pulses ha e a wideband
na u e, so hey can be ecei ed almos e e ywhe e, make Radio Pulsa signals a pe ec
candida e o na iga ion sys ems. The challenge, howe e , is ha he Radio Pulsa signal is
deg aded and subme ged in Addi i e Whi e Gaussian Noise when i is p opaga ed h ough
he ISM, so ha make hem di icul o de ec .
The aim o he p ojec is he design o he op imum ecep o in o de o es ima e he
ime o a i al o he Radio Pulsa Signal and make possible he eal- ime na iga ion. In his
con ibu ion I in oduce he de ec ion heo y applied o adio pulsa s and show he di e en
signal p ocessing echniques o imp o e he de ec ion wi h he minimum compu a ional ime
possible. Al hough he p e ious wo k has cen ed in he Signal o Noise Ra io o he pulsa
signals, ou ocus will be he imp o emen o he Ene gy o Noise Ra io. We will see as
Epoch Folding and he inc emen o he Bandwid h o he ecei e a e he bes solu ions.
Mo eo e , In eg a ion in Time appea s o be one o he mos p omising echniques o simpli y
he compu a ion o he whole p ocess. Also I demons a e how he de ec ion pe o mance is
no a ec ed by any digi al o analogue il e , so Low-Pass Fil e ing will no ha e any e ec
in he ecep o bu o elimina e spu ious signals. Some expe imen s wi h simula ed wide-
band signals a e shown in o de o p o e he op imali y o he signal p ocessing echniques.
Finally, summa izing all he esul s ob ained in he Thesis, I p opose wo op imum ecep o s
o pulsa -based na iga ion sys em applica ions.
I is known ha he in e s ella medium has a equency dependen ans e cha ac e -
is ic, so he highe equencies o a signal a i e ea lie han he lowe equencies. This
will cause he pulse p o ile o he adio pulsa signal o appea dispe sed in ime. Up un-
il now, he esea che s ha e been pe o ming de-dispe sion echniques in o de o ob ain
he o iginal powe pulsa p o ile spending mo e han he 80% o he ac ual p ocessing ime.
The p o e o a oid de-dispe sion wi hou dec easing he de ec ion pe o mance a e p esen ed.
Fu he mo e, some simula ions wi h da a om he pulsa PSR B0329+54 eco ded by
he Wes e bo k Obse a o y a e shown. PSR B0329+54 is one o he s onges pulsa signal
isible in he no he n hemisphe e wi h one o he lowes Dispe sion Measu e, so i will be
easie o pe o m he Signal P ocessing echniques and see he p o ile o he pulse. We will
see how he esul s a e no he ones expec ed, because he Radio pulsa signal happens o be
e y Na owband. Finally, I am going o gi e a possible explana ion o wha is happening
and in wha pa o he acquisi ion he wideband na u e o he o a ing s a pulses a e los .
Acknowledgemen s
W i ing he Acknowledgemen o my Mas e Thesis means ha I am inishing my s udies.
I s ill do no ealize he impo ance o ha ac , ha in a b ie pe iod o ime I will ha e
o ace he i s days o he es o my li e. I should eel sca ed, o doub ul wi h he hings
ha awai me in he u u e, bu u h be old I am eage o s a and c ea e my own pa h.
Fi s o all I would like o hank D . Richa d Heusdens o gi ing me he oppo uni y
o wo king in ha p ojec wi h him. I ha e enjoyed all he momen s o my esea ch, e en
he ha d ones. I ha e lea ned how o wo k alone and a he same ime, being pa o a
g oup. I also wan o hank him o le me being pa o he weekly discussions abou he
p ojec , whe e a he beginning I was a li le bi los bu wi h he help o him and D .
Nikolay Gaubi ch I became one o hem. I also wan o exp ess my g a i ude o D . Nikolay
Gaubi ch o he good ad ices gi en du ing all my Thesis.
Se e al people ha e con ibu ed o make his semes e an un o ge able one. I would like
o s a wi h he people om he Lib a y G oup. Al hough he i s mon h we did no know
each o he , a he end we became a amily. O cou se I am alking abou Roge , Paula,
Benede a, C is ina, Mikel, Pa i and Ricca do. I hope we see each o he soon. The second
g oup o people a e he Ma cusho /Roland c ew. I will be di icul o o ge all he dinne s,
bee s, con e sa ions we ha e sha ed. A he end I do no wan o o ge he es o he
In e na ional G oup ha has con ibu ed o he mos amazing pe iod o my li e. In special,
many hanks o Roge , my lib a y ma e and bes iend in ha E asmus ha has helped me
o being cons an in ha long-dis ance ace ha is he Thesis.
I will no end his Acknowledgemen s wi hou saying hanks o my g oup o iends o
Ba celona. Al hough we ha e no seen each o he o 6 mon h, ou iendship is so s ong ha
he dis ance has no had any e ec on us. I am e e ing o Xa i, Cesc, Julia, Ca les, Gonzalo,
Pablo, Da id, Ped o and Anna. Also I wan o gi e special hanks o Da id, Ped o and Xa i
o gi ing me ad ices whene e I needed and o Anna o always being he e no ma e wha .
Finally, I need o hank my pa en s, my sis e , and he es o my amily, he mos im-
po an people o my li e. I is only hanks o hem ha I ha e managed o become he
pe son I am oday, o which I will be e e nally g a e ul.
Del , The Ne he lands
Del Uni e si y o echnology Daniel He nando Po e o
Augus 28, 2013
Table O Con en s
1 In oduc ion 1
1.1 In oduc ion o he Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mo i a ions..................................... 1
1.3 Thesisgoals..................................... 1
1.4 Thesiscon ibu ions ................................ 2
1.5 Ou line ....................................... 2
2 Radio Pulsa Signals 4
2.1 Pulsa desc ip ion and emission p ope ies . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Pulsa s ................................... 4
2.1.2 Radio Pulsa signal cha ac e is ics . . . . . . . . . . . . . . . . . . . . 6
2.2 P opaga ione ec s................................. 8
2.3 Pulsa -Based Na iga ion Sys em . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Na iga ion sys em challenge . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Pulsa sConclusions ................................ 10
3 De ec ion Theo y applied o Radio Pulsa Signals 11
3.1 Basic De ec ion o de e minis ic signals . . . . . . . . . . . . . . . . . . . . . 11
3.2 Gene alised Likelihood Ra io Tes applied o Radio Pulsa Signals . . . . . . 14
3.3 Conclusions o he De ec ion Theo y . . . . . . . . . . . . . . . . . . . . . . . 20
4 Radio Pulsa Signal Model 22
4.1 Fil e ed Analogue Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 A/Dcon e e ................................... 24
4.3 Gene alised Ene gy To Noise Ra io . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Theo e ical Signal P ocessing Techniques 28
5.1 EpochFolding ................................... 28
5.1.1 In eg a ioninTime ............................ 30
5.2 Low-PassFil e ing ................................. 32
5.3 Downsampling ................................... 36
5.4 O e sampling.................................... 41
5.5 Unde sampling ................................... 48
5.6 Signal P ocessing Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 Radio Pulsa Signal PSR B0329+54 obse a ions 53
6.1 PSRB0329+54 ea u es .............................. 53
6.2 Radio Pulsa Signal PSR B0329+54 da a acquisi ion om WSRT . . . . . . . 54
6.2.1 P ocess o isualize he signal . . . . . . . . . . . . . . . . . . . . . . . 55
6.2.2 Rep esen a ion o he adio Pulsa Signal . . . . . . . . . . . . . . . . 57
7 Signal P ocessing expe imen s 60
7.1 Simula edda a ................................... 60
7.1.1 EpochFolding ............................... 62
7.1.2 Low-PassFil e ing............................. 64
7.1.3 Whi eningP ocess............................. 66
7.1.4 Downsampling ............................... 68
7.1.5 Inc ease he Bandwid h o he signal . . . . . . . . . . . . . . . . . . . 69
7.2 Radio Pulsa signal B0329+54 om WSRT . . . . . . . . . . . . . . . . . . . 71
7.2.1 Epoch Folding + High Pass Fil e . . . . . . . . . . . . . . . . . . . . 71
7.2.2 Downsampling + High-Pass Fil e . . . . . . . . . . . . . . . . . . . . 75
7.2.3 In eg a ioninTime ............................ 76
7.3 Expe imen al Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8 P oposed Recep o 80
9 Summa y and Fu u e wo k 83
9.1 Summa y ...................................... 83
9.2 Fu u eWo k .................................... 85
Bibliog aphy 86
Chap e 2
Radio Pulsa Signals
2.1 Pulsa desc ip ion and emission p ope ies
2.1.1 Pulsa s
Pulsa s a e highly magne ized, pe iodically o a ing neu on s a s ha emi a beam o elec-
omagne ic adia ion. They we e i s disco e ed by Jocelyn Bell on No embe 28 h, 1967.
A i s and o a sho pe iod o ime scien is s hough he elec omagne ic emission was
coming om an ex a- e es ial ci iliza ion, bu his heo y was soon ejec ed. Up o now
o e 1500 pulsa s ha e been de ec ed in ou Galaxy [3], bu i is expec ed ha housands
mo e will be disco e ed du ing he nex ew yea s.
In spi e o mo e han ou decades o in ensi e esea ch he e a e s ill many open ques-
ions in pulsa as onomy, and hus i would be a ai s a emen ha hese neu on s a s a e
unde s ood only poo ly. On he one hand, s udies so a ha e allowed us o cha ac e ize he
p ope ies o he emi ed signals a e hei a el h ough he In e s ella Medium (ISM),
bu on he o he hand he comple e desc ip ion o he in e nal s uc u e o a pulsa emains
as a complex issue. A he bes o ou knowledge, he answe o ques ions such as how many
pulsa s a e he e in he Galaxy, wha is hei bi h a e, how a e isola ed millisecond pulsa s
p oduced, how many pulsa plane a y sys ems exis o many o he s a e ei he unknown o
simply he e is a lack o gene al scien i ic ag eemen abou hem.
These neu on s a s ha e magne ic ields o he o de o 108 o 1015 G (Ea hs magne ic
ield magni ude a i s su ace anges om 0.25 o 0.65 G), and as a esul o Maxwells equa-
ions an elec ic ield is induced. Cha ged pa icles a e accele a ed o he magne ic poles o
he pulsa by his elec ic ield, and as hey a e a elling h ough a magne ic ield a beam
o elec omagne ic adia ion o high magni ude is emi ed alongside he magne ic axis.
4
Figu e 1: Ro a ing pulsa model and i s emission. C edi B. Sax on/NRAO/AUI.
A ep esen a ion o his phenomenon can be shown in Figu e 1. I is clea ly seen ha he
magne ic axis and he o a ional axis a e no necessa ily he same. This misalignmen causes
he in ensi y o he elec omagne ic adia ion o a y in a pe iodic ashion when ecei ed
om a ixed line o sigh . Indeed, he beam is only seen om Ea h as i sweeps pas ou
line o sigh once o e e y o a ion o he neu on s a , which leads o he pulsed na u e
o i s appea ance. In addi ion, bo h he angle be ween spin axis and magne ic axis and he
equency o he o a ion is unique o each pulsa . So, his in o ma ion becomes an exclusi e
signa u e. [17]
Two pa s o he spec um o pulsa emissions a e good candida es o na iga ion, mainly
he adio spec um and he high-ene gy spec um, such as X ay and γ- ay. The choice o
wha kind o spec um use can be made p ima ily using h ee c i e ia: quali y o he e-
cei ed signal, equipmen and pulsa a ailabili y. While high-ene gy pho ons, by de ini ion,
gi e a be e SNR, emi ance in he adio spec um gene ally equi e much less om he
ecei e . Fu he mo e, pulsa s emi ing s ong and usable signals in he adio spec um a e
signi ican ly mo e plen i ul. Thus, while he SNR su e s in adio spec um, he na iga ion
sys em can be ealized wi h less o a bu den due o ecei e equipmen . [19] Fu he mo e,
a na iga ion sys em based on adio pulsa s could ind use in ehicle na iga ion on Ea h
as well as o e acking on o he plane s whe e high-ene gy pulsa signals a e blocked by
he a mosphe e. Hence, he esea ch o his hesis is a ge ed owa ds de ising a na iga ion
sys em based on adio pulsa s. Such a sys em has use ul possibili ies ha a e no o e ed by
he high-ene gy pulsa based sys em.
5
2.1.2 Radio Pulsa signal cha ac e is ics
In his sec ion i is going o discuss he main cha ac e is ics ha make pulsa s unique com-
pa ed o o he s ella en i ies. The i s and mo e impo an one is he pe iodici y. Signals
coming om pulsa s a e highly pe iodic. Each pa icula pulsa has i s own pa icula pe-
iodici y which is di e en om he o he ones. The mos apidly o a ing neu on s a
cu en ly known is PSR B1937+21 wi h a pe iod o only 1.56 ms. In con as , he longes
pe iod obse ed o any adio pulsa so a is 8.5 s o PSR J2144-3933.
The pulse pe iod o all pulsa s slowly decays, supposedly un il hey come o a ull s op.
This decay is e y slow, and qui e s able, and has al eady been de e mined o mos pulsa s.
I anges om 10−13 s/s up o 10−19 s/s, which makes i a e y slow decay. This gi es he
pulsa s hei cha ac e is ic equency s abili y, bu ha decay can be used o na iga ional
pu poses as well, due o ela i is ic e ec s, as he pulse decay will appea as e o slowe
compa ed o he expec ed decay a ea h. Ne e heless, he ema kable ac abou he pulsa
pe iodici y is ha i is ex emely p ecise. In some cases (millisecond pulsa s), he egula i y
o he pulsa ion is as p ecise as an a omic clock. This s abili y allows millisecond pulsa s o
be used in es ablishing epheme is ime o building pulsa clocks. Due o his ac , pulsa s a e
ideal o ime-o -a i al (TOA) based na iga ion sys ems, as i will be explained in u he
sec ions. [18]
The second main cha ac e is ic o he adio pulsa signals is hei spec um. Pulsa
emissions a e known o occupy a e y wide band o he elec omagne ic spec um. Howe e ,
based on he loca ion o he equency ange o he emission i is common o classi y hem
in o X- ay pulsa s (3 ∗1016 o 3 ∗1019 Hz) o adio pulsa s (3 kHz o 3 THz), al hough some
pulsa s ha e been ound o emi in isible ligh , gamma ays o e en all o he equency
bands a o emen ioned, making a possible o al emission spec um om 3 kHz o 3 ∗1020 Hz.
One in e es ing esul o his issue is ha no ma e which equency a ecei e is uned a
i will s ill be able o ecei e he signal. Tha Wideband na u e o he pulsa signals will be
he i s and mos impo an assump ion used in his Thesis o design he op imal ecep o .
Ano he in e es ing ac is ha i you cu some pa o he spec a (Pe: Fil e ing he signal),
he in eg a ed powe p o ile o he pulsa will no change. Howe e , he ampli ude o ha
p o ile will be smalle . Tha will be an impo an ac o ake in o accoun in he Chap e
5. No e ha o a pa icula pulsa he pulse shape a ies as a unc ion o he obse ing
equency, as s a ed in Figu e 2. [17]
6
Figu e 2. Mul i- equency pulse p o ile o wo pulsa s: (a) B1133+16 (b) J2145-0750. C edi Lo ime
and K ame , EPN da abase [5].
Pulsa s emi he s onges signals a he lowes equencies. A inc easing equencies,
he signal le els will exhibi a decay, which di e s om pulsa o pulsa . Howe e , a lowe
equencies he backg ound noise empe a u e on ea h is qui e high, and e en in space, in-
e e ence caused by he plane s and he sun a e ela i ely s ong. Selec ing a high obse ing
equency is he e o e bene icial om his poin o iew.
Pulsa s a e one o he mos pola ized adio sou ces. They usually ha e linea pola iza-
ions, bu in some cases hey can be ecei ed wi h ci cula o ellip ical pola iza ions. The
S okes Pa ame e s desc ibe he pola iza ion o he signal, bu o he pulsa s applica ion we
a e going o use he I pa ame e .
I=E2
0=|Ex|2+|Ey|2
I is now clea ha he I S oke pa ame e is ela ed o he o al in ensi y o powe o he
elec omagne ic adia ion, i.e. is he ac ual pulse p o ile. As onomical obse a ions almos
always eco d he whole ou S okes pa ame e s so comple e in o ma ion abou he s a e o
pola iza ion o he signal is achie ed. The e a e a ela ion be ween he I S oke Pa ame e
and he o al powe o he elec omagne ic adia ion. Tha can be seen in he nex equa ion
whe e he o al adia ed powe by he s a is showed
7
PT=Z ZS
|Eθ|2+|Eφ|2
ηdS(W) (2.1)
Whe e θand φ e e o he sphe ical coo dina es, ηis he cha ac e is ic impedance o he
medium and S is a sphe ical su ace emula ing he he adio elescope an enna. To conclude,
in o de o ob ain he shape o he pulse p o ile o a pa icula pulsa he adio elescope will
acqui e bo h Vx( ) and Vy( ) ol age signals. Then, hey will be summed oge he ollowing
he exp ession |Vx|2+|Vy|2. [17]
2.2 P opaga ion e ec s
A pulsa signal a els e y la ge dis ances on i s way o eaching ou plane . Pulsa s a e
loca ed a se e al hund ed o in o he cases, se e al housand ligh yea s away om Ea h.
The signals pass h ough he in e galac ic space, which is scien i ically known as he In-
e s ella Medium (ISM) and a e a ec ed by di e en e ec s: Dispe sion, Scin illa ion, and
Sca e ing. These e ec s a e discussed and analysed in he ollowing ex . Fu he mo e, a
b ie e iew o he ac ual de-dispe sion echnique is p esen ed in o de o know i s p ope ies
and analyse i s in luence in he de ec ion.
Scin illa ion is a p ocess whe e inhomogenei ies o he e ac i e index o he medium
(caused by s ong a ia ions o elec on densi ies) p oduce phase modula ions on he p opa-
ga ing pulsa signal. Tha leads o a luc ua ion o he in ensi y on a a ie y o bandwid hs
and ime-scales. This e ec is modelled as a hin sc een o i egula i ies midway be ween
he Ea h and he pulsa [18]. I has been demons a ed ha his e ec is highly equency
dependen . In e e ence can occu only i he phases o he wa es do no di e by mo e han
abou 1 adian. Then, as he phases a e equency dependen , he e is a limi a ion in band-
wid h o he in e e ing wa es. This means ha wa es ou side he scin illa ion bandwid h
∆ ∞ 4will no con ibu e [3]. The mos powe ul me hod o deal wi h he Scin illa ion is
he a e age o di e en ecei ed pe iods. As s a e be o e, al hough he di e en pulses can
a i e wi h a e y di e en in ensi y, he in eg a ed powe p o ile happens o be e y s able.
The e o e, a e some olds we can assume ha he scin illa ion e ec p o oked by he ISM
is gone.
Sca e ing is basically a adia ion e ec ela ed o mul ipa h en i onmen s. In he hin-
sc een model in oduced be o e his e ec can be ela ed di ec ly o he a iable pa h leng hs.
F om he ecei ed poin o iew, he pulse shape will be b oadened since no only he di ec -
pa h componen eaches i , bu also se e al delayed e sions o i ha a el h ough di e en
pa hs. This will cause he appea ance o he cha ac e is ic exponen ial ails, wi h he con-
secu i e educ ion in he SNR. No e ha his e ec is also equency dependen , wi h a much
lesse impac when obse ing high equencies. [17]. Fo a equencies highe han 600 MHz
he sca e ing disappea . The e o e, in o de o a oid his e ec we will a oid low- equencies
in he applica ion o de ec he adio pulsa signal.
The in e s ella medium has a equency dependen ans e cha ac e is ic: highe e-
quency signals a i e ea lie han lowe equency signals, e en hough he ime o ansmis-
sion was he same. This will cause he pulse p o ile in a b oadband ecei e sys em o appea
smea ed ou in ime, and will change he pulsa signal shape. The phenomenon is called
Dispe sion. The Dispe sion depends on one e m, and his is he Dispe sion Measu e (DM)
8
[17]. The dispe sion measu e is cons an only o a ce ain measu emen ime and posi ion.
In o he wo ds, as he in e s ella medium is no homogeneous, he dispe sion measu e will
change depending on whe e and when he obse e has aken he measu emen s. The e o e,
e e y Pulsa has a di e en Dispe sion Measu e.
Dispe sion can be emo ed by he p ocess o de-dispe sion. The e a e wo known me h-
ods o de-dispe se he ecei ed signal, he i s me hod de-dispe ses he signal in he ime
domain and is called incohe en de-dispe sion, he second me hod employs equency do-
main ope a ions and is called cohe en de-dispe sion. The compu a ional equi emen s o
de-dispe sing uni a e e y high [13]. Ac ually, all he acquisi ions om Radio Pulsa Signals
a e de-dispe sed in o de o be p ocessed wi h a be e SNR. Tha echniques equi e mo e
han he 80% o he compu a ional ime needed o ecei e, p ocess and de ec he signal.
La e on and unlike a lo o esea ches hink, we will see as he de-dispe sion p ocess can be
a oided. Tha is due he ac ha dispe sion doesn’ change he ene gy o he Radio Pulsa
Signal bu only he shape. In ac , he e ec o he In e s ella Medium (ISM) is desc ibed
as a phase only il e , as by he Fou ie T ans o m delay in ime domain is equi alen o
phase shi in he equency domain.
2.3 Pulsa -Based Na iga ion Sys em
Pulsa based na iga ion is no a no el a ea o esea ch, and migh once o e he possibili y
o man o sa ely a elling dis ances much beyond Ea h. Up un il now he ocus has been on
X- ay based pulsa na iga ion, whe eas ecen s udies ocus on he possibili y o using adio
pulsa s. The adio equency ange had been neglec ed in he pas because he pulses we e
assumed o be oo weak o de ec wi h an ennas o a easonable size. Nowadays, howe e ,
due he eally good pe o mance o he Ma ched Fil e as a de ec o [1] [16] and he as e
e olu ion o he ins umen a ions on pulsa ecei e s, he goal o using he Radio Pulsa o
eal ime na iga ion appea as a p omising end. Fu he mo e, no only can be used o
localize a spacec a bu o localize a ge s on Ea h. The e o e, pulsa -based na iga ion
sys em can be a subs i u e o he ac ual na iga ion sys ems as GPS o Galileo.
[7] p o ides an o e iew o he wo k ha has been done on pulsa na iga ion and shows
his new di ec ion in pulsa -based na iga ion esea ch. Since pulsa signals o e such a high
s able pe iodici y, he idea is o use hem as beacons o Time o A i al (TOA) based na i-
ga ion. The e wo kinds o na iga ion algo i hms ha use he ex emely accu a e pe iodici y
o Pulsa s. The i s is he Dopple Shi ed Na iga ion, which uses he Dopple e ec in he
es ima ed TOA’s in o de o localize he a ge . The o he echnique is called Pe iod Decay
me hod and uses he pe iod decay o he pulsa s. These wo echniques a e explained in
de ail in [18].
Al hough some esea ch has been made abou building he ac ual pulsa na iga ion sys-
em s ill no p ac ical implemen a ion has been done. In ollowing chap e s we will p opose
a ecep o ha includes he op imal de ec o o he pulsa case and he signal p ocessing
echniques ha imp o e he de ec ion pe o mance.
2.3.1 Na iga ion sys em challenge
Al hough seemingly simple in p inciple, he e a e se e al hu dles ha a e needed o be o e -
come in ealizing such a na iga ion sys em. The main challenges when using adio pulsa s
9
o na iga ion a e he ollowing:
1. The ex emely weak pulsa signal s eng h ha is being used o na iga e. The e-
cei ed signal is comple ely subme ged in Addi i e Whi e Gaussian Noise, so he Signal-Noise
Ra ios a e e y small on Ea h. This is because he pulsa signals a e emi ed many ligh
yea s away om Ea h, leading o addi ion o noise and dis o ion due o he p opaga ion
channel as will be discussed la e .
2. The equi emen o ecei e, p ocess and de ec he adio pulsa signal in a ew seconds
in o de o pe o m a eal ime na iga ion. Up o now he p ocessing ime o localize he
signal i is highe han 10 minu es. Fu he mo e, i is needed eally big an ennas (+10 m
diame e dish-an ennas) o ecei e he adio Pulsa Signal wi h enough SNR. Ne e heless,
due he imp o emen o he echnology (compu e s and de ices wi h highe compu a ional
cos s), he an ennas (possibili y o each highe Bandwid h) and he le o he ecep o
de ices, someday i will be possible o achie e he goal o a pulsa -based na iga ion sys em.
2.4 Pulsa s Conclusions
In his chap e I ha e in oduced a b ie explana ion abou wha is a pulsa and a classi ica-
ion o hem depending on he signal hey emi . We also ha e seen he main cha ac e is ics
o he Radio Pulsa signals. Tha signals appea o be ex emely pe iodic pulses ha a i e
o he Ea h wi h a e y weak in ensi y and subme ged in Addi i e Gaussian Whi e Noise.
Howe e , due hei accu a e pe iodici y hey ha e been chosen as a pe ec candida es o a
eal- ime na iga ion sys em. Mo eo e , he Wideband ea u e o ha pulses has been shown.
Due ha Wideband na u e, we a e able o ecei e hem in a e y big equency ange, om
kHz o THz. Tha will be an impo an assump ion in o de o design he op imum signal
p ocessing echniques. Also i has been s a ed ha Pulsa signals a e highly pola ized, hence
an acquisi ion o wo o hogonal pola iza ions is enough o ob ain he powe pulse p o ile.
A e ha , he p opaga ion e ec s has been explained. The ISM causes se e al equency
dependen unwan ed e ec s on he wideband pulsa signal, such as dispe sion, sca e ing
and scin illa ion. Howe e , some p ac ical solu ions ha e been gi en in o de o a oid hose
e ec s. Fo example, choosing a adequa e obse a ion equency. Mo eo e , i has been in-
oduced wo de-dispe sion me hods. Al hough i looks like hey will be an impo an block
o ou ecep o , we will see as his s a emen is no ue.
Finally, i has been explained he goal o implemen ing a adio eal- ime pulsa -based
na iga ion sys em. The wo main challenge o his aim has been in oduced. Tha challenge
a e he low-in ensi y o he ecei ed pulsa signal and he equi emen o p ocess and de ec
he signals in ew seconds. A e ha , wo na iga ion algo i hms has been in oduced. Tha
me hods a e he Dopple Shi and he Pe iod Decay, and hey bo h use he ex emely pe-
iodici y o he adio pulsa signals as a key o localize a a ge .
In he nex sec ion, a summa y o he de ec ion heo y w i en by [1] and [22] will be
explained in o de o design a de ec o o es ima e co ec ly he ime o a i al o he pulsa .
10
Chap e 3
De ec ion Theo y applied o Radio
Pulsa Signals
De ec ion heo y deals wi h echniques o de e mine how good da a ob ained om a ce ain
model co esponds o a gi en da a se . An example o ha can be he ada s, whe e he
p esence o a a ge has o be de ec ed. Ano he example could be o de ec whe he a 0 o 1
has been sen in a communica ion sys em. In his Thesis we will only deal wi h he de ec ion
o a pulsa signal o e one pe iod in p esence o noise. Fu he mo e, we will assume ha
he pulsa signal is de e minis ic, so he de ec ion will be easie o pe o m. O he wise he
p ocess will be a de ec ion wi h andom p ocesses.
In his sec ion I will summa ize he de ec ion heo y applied o adio pulsa signals done
by Richa d Heusdens in [1] and [22]. As s a ed, Radio Pulsa emi a high pola ised pulses
ex emely pe iodic. Bu , as hese s a s a e loca ed millions o km om he Ea h hey a -
i e wi h a e y low in ensi y. Mo eo e , when we ecei e hose signals only noise can be
obse ed due he ac ha hey a i e subme ged in Addi i e Whi e Gaussian Noise wi h a
e y low Signal o Noise Ra io. In o de o achie e ou goal o using he Radio Pulsa signal
o na iga ion applica ions we should assu e ha we es ima e in a co ec way he ime o
a i al o he pulses.
Fi s o all, in o de o ind he op imum de ec o o ou applica ions a basic de ec ion
heo y o de e minis ic signals is in oduced. The ac ha we know exac ly how he adio
pulsa signal is will be e y impo an o choose a de ec o . Then, he de ec ion heo y o
adio pulsa s signal applica ion will be explained. As we can guess, besides de ec he pulsa ,
we will need o es ima e he ampli ude o he pulsa p o ile in o de o implemen a empla e
and he Time O A i al o localize ou a ge .
3.1 Basic De ec ion o de e minis ic signals
The de ec ion o de e minis ic signals is he simples case because he p io we know abou
he signal play in ou a ou . The main idea behind he de ec ion p ocess is he s a is ical
hypo hesis es ing. Gi en a da a se and di e en hypo hesis ou aim will be de e mine
which model i s he da a bes . Due he ac ha we wan o de ec one signal ( he one
o he pulsa we wan o use o localiza ion), we will only conside in his Thesis wo Hy-
po hesis. The i s hypo hesis H0is he case when only andom noise is ecei ed. In he
second Hypo hesis H1 he de e minis ic signal is ecei ed in p esence o he same andom
p ocess. We will assume ha he andom p ocess is an addi i e gaussian noise wi h 0 mean
11
and co a iance ma ix Φz. So, we can model ou case as:
H0:y(n) = z(n)n= 0,1,2, ..., N −1
H1:y(n) = s(n) + z(n)n= 0,1,2, ..., N −1
Being y(n) he disc e e ecei ed signal, s(n) he disc e e de e minis ic signal and z(n)
he noise p ocess modelled as N∼(0,Φz). The ocus will be pu in he p obabili y densi y
unc ions o he bo h hypo hesis. Tha will be use ul in o de o choose one o he o he
hypo hesis depending i he ecei ed belongs o he pd o he i s o second model. In he
Hypo hesis H0only noise is ecei ed. Then we can s a e ha pd ois:
pd 0(y) = 1
(2π)N
2|Φz|1
2exp(−1
2(y−µ)TΦ−1
z(y−µ)) = 1
(2π)N
2|Φz|1
2exp(−1
2yTΦ−1
zy)
Being µ he mean o y when we only ecei e noise. As he H1will be he hypo hesis
when we ecei e he de e minis ic signal subme ged in gaussian noise, he p obabili y densi y
unc ion o hese model will be also gaussian wi h a non-ze o mean. The e o e,
pd 1(y) = 1
(2π)N
2|Φz|1
2exp(−1
2(y−µ)TΦ−1
z(y−µ)) = 1
(2π)N
2|Φz|1
2exp(−1
2(y−s)TΦ−1
z(y−s))
In he nex igu e 3 we can obse e he p obabili y densi y unc ions o one bo h hy-
po hesis. They a e almos iden ical due he ac ha hey ha e he same andom Gaussian
p ocess. The only di e ence is ha he one o he hypo hesis 1 is shi ed s (being s he mean
o he Hypo hesis H1).
Figu e 3. Dis ibu ions o he Hypo hesis H0and H1. C edi D . Richa d Heusdens
The de ec o will consis on choosing one o he dis ibu ions depending on he da a se
we ecei e, so a h eshold will be needed. Looking a he igu e 4 we can see how he e a e
wo possible mis akes we can make when we assess he de ec ion wi h he h eshold. The
i s one is called miss (II) and is p oduced when you choose o he hypo hesis H0and, in
ac , he de e minis ic signal is ecei ed. The o he possible mis ake is called alse ala m
12
(I) and consis on deciding ha you ha e ecei ed he de e minis ic signal (Hypo hesis H1)
al hough i is no ue. These e o s a e una oidable o some ex en bu may be aded o
agains each o he by adjus ing he de ec ion h eshold. I is no possible o educe bo h
e o s a he same ime once he p obabili y densi y unc ions a e se . As a consequence, a
ypical app oach o design an op imal de ec o is o ix one e o p obabili y and minimize
he o he . In he nex igu e we can obse e he Hypo hesis es ing e o s and hei a e-o .
Figu e 4. Hypo hesis es ing e o s and hei ade-o adjus ing he de ec ion h eshold. C edi D . Richa d
Heusdens
To assess ha p obabili ies we will assume a simples model whe e we only ecei e one
sample wi h ampli ude s and ha he noise p ocess z is s ill a gaussian p ocess and whi e
wi h a iance σ2
z1. So, z1∼N(0, σ2
z1). So he p obabili y densi y unc ions will be:
pd 0(y1) = 1
√2πσ2
z1
exp(−y2
2σ2
z1
)
pd 1(y1) = 1
√2πσ2
z1
exp(−(y−s)2
2σ2
z1
)
Now i is possible o compu e he p obabili y o alse ala m as he p obabili y ha y1is
bigge han he h eshold and in ac we a e no ecei ing he desi ed signal. Tha can be
s a ed as:
PF A =P(y1> γ |H0) = Z∞
γ
1
p2πσ2
z1
exp(−y2
2σ2
z1
)dy
=Q(γ
σz1
)
(3.1)
Being Q(y) he Q- unc ion o complemen a y cumula i e dis ibu ion unc ion ela ed o
he complemen a y (Gauss) e o unc ion by:
Q(y) = 1
2e c(y
√2)
13
o he di e en shape ans o ma ions. The bes esul s a e ob ained o he smoo hed pul-
sa p o ile. All-pass il e ing, howe e , has no e ec on he pe o mance, as expec ed. As he
andomized signal is he one wi h a mo e independen a iables (a eally high and na ow
peak in he co ela ion unc ion), D . Heusdens has p o ed how he ac ha he TOA is
maximal o s a is ically independen andom a iables does no imply ha his esul s in
he bes es ima ion. Mo eo e , as expec ed, he TOA es ima ion pe o mance inc eases wi h
he ENR.
Figu e 6. MLE o he TOA τo. C edi Richa d Heusdens
3.3 Conclusions o he De ec ion Theo y
F om ha sec ion we ha e ob ained a lo o impo an conclusions in o de o make easible
ou aim o eal ime na iga ion using Radio Pulsa Signals. Fi s , we ha e s a ed ha he
bes app oach o ou case will be he pe o mance o Neyman-Pea son de ec o . Tha is
because he Neyman-Pea son gi es he bes de ec ion pe o mance once he p obabili y o
alse ala m is ixed. Howe e , as we need o es ima e he Time O A i al o he signal, i is
no possible o use ha de ec o . So, i has been ound a subop imal de ec o , he GLRT.
Assuming ha he ecei ed signal is he adio pulsa pulse subme ged in whi e noise, he
GLRT will co ela e N imes (being N he leng h o on pe iod) ha ecei ed signal wi h he
shi ed eplica o empla e o he adio pulsa Dτs. A e doing ha , we will es ima e ou
20
TOA as he shi ed alue τ ha has gi en he highes co ela ion. Besides, we will decide
ha he signal has been de ec ed i ha co ela ion alue is abo e a h eshold γ. I he
noise is colo ed, we ha e seen as he De ec ion Pe o mance will depend on sΦ−1
zs. So, he
implemen a ion o he GLRT when he noise is colo ed will be a Whi ening P ocess ollowed
by N Co ela o s.
Fu he mo e, i has been ound he exp essions o he de ec ion pe o mance and he
es ima ion o he ime o a i al, s a ing ha we ha e o inc ease he ENR (i he noise
p ocess is whi e) o he GENR (i he noise p ocess is colo ed) as much as possible. In he
p e ious Thesis and Repo s he aim has been o ind he echniques ha imp o e he SNR
o he signal. Ne e heless, he SNR will no ha e any e ec in he de ec ion pe o mance.
Also i has been shown ha o a ENR-GENR o 16 dB, he de ec ion o he pulsa and he
es ima ion o he ime o a i al will be almos pe ec . So, in he nex chap e s ou aim
will be o ind which signal p ocessing echniques imp o es he GENR o he p ocess assum-
ing ha he Radio pulsa signals ha e a Wideband na u e. The de ec ion heo y explained
added o he signal p ocessing echniques will lead o a heo e ical op imum ecep o o he
de ec ion o adio pulsa signals. Tha de ec o is p oposed in he chap e 8.
The las conclusion and mo e impo an o ha sec ion is he ac ha he GLRT de-
ec o and MLE es ima ion o he TOA p ocesses does no depend on any all-pass il e ing
ope a ion. As he de-dispe sion p ocess can be modelled as a all-pass il e ing, ha means
ha pe o ming he de-dispe sion me hod will no ha e any e ec in he de ec ion pe o -
mance nei he he es ima ion o he TOA. Tha is a eally impo an esul since up o 80%
o he compu a ional cos dedica ed o p ocess he adio pulsa signal is spen in ha p ocess.
21
Chap e 4
Radio Pulsa Signal Model
Figu e 7. Block Diag am o he ecei e and he Analogue o Digi al Con e e
This sec ion ocus in he o iginal p ope ies o he Radio Pulsa signal and he Noise
p ocess. F om now on I will assume ha he ecei ed Radio Pulsa Signal is a de e minis ic
wideband signal wi h unknown ampli ude and Time O A i al subme ged in Addi i e Whi e
Gaussian Noise. The e o e, we can model i as:
y ( ) = as ( −τ) + z ( )
Being s( ) he Radio Pulsa signal, z he noise p ocess, a he unknown ampli ude and τ
he unknown Time O a i al. In he De ec ion Theo y sec ion we ha e seen how o es ima e
he Ampli ude and he ime o a i al o he signal. In he nex subsec ions we a e going
o assume ha he ampli ude and he ime o a i al a e known in o de o compu e in an
easy way he ENR and GENR o he signal. Hence, he signal be o e passing h ough he
analogue Low-Pass Fil e will be:
y ( ) = s ( ) + z ( )
22
4.1 Fil e ed Analogue Signal
Figu e 8. Powe Spec um o he AWGN and i s Au oco ela ion unc ion
F om now on, his Thesis will e e o he analogue il e ed signal as
ya( ) = sa( ) + za( )
Being ya he signal ecei ed and il e ed wi h an analogue Low-Pass Fil e , sa he de e -
minis ic wideband signal and za he Addi i e Whi e Gaussian Noise. As we assume ha he
Radio Pulsa signals a e comple ely known, we model hem as de e minis ic wi h an ene gy
a=R∞
−∞ |sa( )|2d =1
2π sR∞
−∞ |Sa(w)|2dw
The second e m o he equa ion is only ue i we a e wo king wi h angula equency
in ad/s. On he o he hand, he noise is a s ochas ic Whi e Gaussian signal N∼(0, σ2
za)
wi h 0 mean and a iance σ2
za. As s a e be o e, yais composed by he pulsa and noise
signal, Band-limi ed o he Bandwid h o he analogue low-pass il e . In his i s pa o
he Thesis we will assume ha he an enna is ideal wi h a Bandwid h equal as he cu -o
equency o he Analogue Low-Pass Fil e . La e on, we will obse e how unde ha as-
sump ion he analogue Low-Pass Fil e is useless om a de ec ion pe o mance poin o iew.
As he noise is he p incipal p oblem why we can’ co ec ly de ec he signal, we a e
going o ocus on i s p ope ies. As s a ed, he ecei ed noise is a con inuous ime Addi i e
Whi e Gaussian p ocess wi h 0 mean, a iance σ2
zaand spec al densi y No. The e o e, he
powe spec um o he noise is:
Sza(w) = No o |w| ≤ 2πB
0 o he wise (4.1)
whe e wis he angula equency in ad/s. I we compu e he in e se o he Fou ie
T ans o m o he Powe Spec um we will ob ain he au oco ela ion unc ion o he noise:
23
Rza( ) = 1
2πZ∞
−∞
Sza(w)ejw dw =1
2πZ2πB
−2πB
Noejw dw
=No
π sin(2πB ) = 2BNosinc(2πB )
(4.2)
The e o e, he a iance o he noise p ocess is σ2
za=Rza(0) = 2BNo, wi h B he Band-
wid h o he signal and No he Powe Spec al densi y o he noise. Looking a he Powe
Spec um and he Au oco ela ion unc ion o he noise p ocesses a e passing he signal
h ough some p ocessing echniques we will be able o know he a iance o he noise and
calcula e he a ia ions o he ENR.
4.2 A/D con e e
Jus be o e s a ing o p ocess he signal, i is passed h ough an he A/D Con e e wi h
sampling equency sin o de o wo k in he digi al domain. The eason o sampling he
signal is o ha e mo e acili ies o p ocess i wi h a lowe cos and a lexible digi al ha dwa e.
A he end o he A/D con e e we will ha e:
y(n) = ya(nTs) = sa(nTs) + za(nTs) = s(n) + z(n)
whe e Tsis he in e se o he sampling equency scalled sampling pe iod. Hence, he
disc e e signals sand za e ob ained by sampling saand zawi h he sampling Nyquis a e
s= 2B. We use his sampling equency in o de o a oid aliasing and keep he whi e
p ope y o he noise p ocess. La e on we will see how impo an is o ha e AWGN om a
p ocessing ime poin o iew.
As we ha e seen in he las sec ion is used he un-no malised equency was he a iable
o he ep esen a ions o he signals in he equency domain, being w= 2π . To change
be ween ime and equency domain i is used he Disc e e ime Fou ie T ans o m:
y(n) = 1
2π sR2π s
0Y(w)ejwnTsdw
Y(w) = P∞
n=−∞ y(n)e−jwnTs
Fi s o all, o assess he ENR o he p ocess ywe will ocus on he de e minis ic signal s.
As i is known, when a signal goes in o an A/D con e e , i loses in o ma ion, and i s isible
spec a will be limi ed o he sampling equency, 2π s. Besides, in he equency domain
appea s images o he signal e e y 2πk s(wi h ka in ege numbe , c ea ing aliasing i he
Nyquis Sampling F equency, o a bigge a e is no used). As we s a ed be o e, he Nyquis
sampling equency is used in his sec ion, so no aliasing will be ob ained. Following ha ,
he spec a o he disc e e signal compa ed o he spec a o he p e ious analogue signal is:
24
S(w) = P∞
n=−∞ s(n)e−jwnTs= sP∞
n=−∞ Sa(w+ 2πk s)
So we can ind an exp ession o he ene gy o he de e minis ic sampled signal sin com-
pa ison wi h he ene gy o he analogue il e ed signal saas:
s=X
n|s(n)|2
∗
=1
2π sZ2π s
0|S(w)|2dw
= s
2πZ2π s
0|∞
X
k=−∞
Sa(w+ 2πk s)|2dw
∗∗
= s
2πZπ s
−π s|Sa(w)|2dw
= s
2πZ∞
−∞ |Sa(w)|2dw
= ssa
(4.3)
* By Pa se al Theo em ha s a es ha he ene gy o he signal in ime is p ese ed in he equency domain
as well.
** Taking in o accoun ha Sa(w) do no ha e any con ibu ion in w /∈[−π s, π s]. No aliasing.
In his o mula has been assumed ha he bandwid h o he signal sadoes no exceed
hal o he o al isible spec a 2π s. As we can obse e, he ene gy o he sampled signal
inc eases wi h he sampling equency in compa ison o he analogue one. This esul shows
ha inc easing he s he ene gy o he p ocess is imp o ed, e en when he bandwid h o
he signal is no inc eased.
On he o he hand, o check he beha iou o he disc e e ime noise z, we a e going o
look o i s Au oco ela ion unc ion:
Rz(k) = z(n)∗z(n+k) = z(nTs)∗z((n+k)Ts)
=Rza(kTs) = 2BNosinc(2πBkTs)
= 2BNosinc(2Bπk
2B)
= 2BNosinc(kπ) = 2BNoδ(k)
(4.4)
Being δ(k) he K onecke del a. Tha happens because sampling he noise wi h he
Nyquis sampling equency p ese e he Whi eness o he Noise. We can also obse e
ha he a iance o he disc e e Noise will emain he same as be o e passing he sig-
nal h ough he A/D con e e no ma e he sampling equency used. The e o e, σ2
z=
Rz(0) = sNoδ(0) = 2BN0=σ2
za. Finally, compu ing he Powe spec um o he Noise as
he disc e e- ime Fou ie T ans o m o he Au oco ela ion Func ion, we ha e:
25
Sz(w) = sNo o |w| ≤ π s
0 o he wise (4.5)
As we can obse e, he shape o he Powe spec um o he noise emains equal, al hough
he alue o he disc e e powe spec al densi y is inc eased by he ac o s. Once ob ained
he alues o he ene gy o he sampled signal sand he a iance o he noise z, ENR is
compu ed:
ENR =s
σ2
z
= ssa
σ2
za
= ssa
2BNo
= ssa
sNo
=sa
No
(4.6)
In he nex subsec ion he e m GENR is in oduced. I is a Ra io o assess he ENR
when he noise is no whi e. As we ha e seen in he De ec ion Theo y, his Ra io assess
he de ec ion pe o mance o he GLRT De ec o . As we will see la e , one way o imp o e
he GENR is inc easing he obse a ion ime o he signal o be de ec ed, so sawill be
inc eased. The p oblem is ha we will ocus on de ec ing he Pulsa signal o e one pe iod,
so we will no be able o inc ease he ene gy o he signal in eg a ing o e a longe ime span.
4.3 Gene alised Ene gy To Noise Ra io
To assess he beha iou and he imp o emen o he signals a e pe o ming some signal
p ocessing echniques, he scien i ic use di e en a ios. The mos used is he SNR o Signal
o Noise a io, being he ela ion be ween he powe o he desi ed signal and he a iance
o Noise.
SNR =Ps
σ2
z
(4.7)
Al hough up un il now he esea che s ha e been using his adio o assess he Radio pul-
sa signals beha iou , in his epo we a e going o use o he wo a ios, he Ene gy o Noise
a io and he Gene alised Ene gy o Noise a io. As i has been explained in he De ec ion
Theo y chap e , he de ec ion pe o mance only depend on he Gene alised Ene gy o Noise
a io, and, in some cases, on he Ene gy o Noise Ra io. The e m Gene alised Ene gy o
Noise Ra io has no been used be o e, bu i desc ibe he ela ion be ween he signal and he
co a iance ma ix o he noise. Then, we a e going o e e o Gene alised Ene gy o Noise
a io o he signal y=s+zas:
GENR =sTΦ−1
zs(4.8)
26
wi h
Φz=E((z−z)(z−z)T)
he co a iance o he noise p ocess and s he desi ed signal. We can see ha i he
noise is AWGN wi h 0 mean and co a iance ma ix Φz=σ2
zI, so Φ−1
z=1
σ2
zI, he GENR
becomes he ela ion be ween he ene gy o he signal and he a iance o he noise. The e o e
GENR =sTΦ−1
zs=sTs
σ2
z
=s
σ2
z
=ENR (4.9)
The GENR akes in o accoun he colo o he noise o assess he de ec ion pe o mance.
F om now on, Gene alised Ene gy o Noise Ra io will be used o measu e up whe he he
di e en signal p ocessing algo i hms inc ease o no he de ec ion pe o mance.
27
Chap e 5
Theo e ical Signal P ocessing
Techniques
Figu e 9. Block Diag am o he Signal p ocessing echniques applied o he disc e e- ime signal
The signal om he Radio pulsa is ecei ed subme ged in AWGN and i is no possi-
ble o see o de ec i wi hou some p ocessing. In his chap e we in oduce some basic
signal p ocessing echniques o check i he ENR and GENR o he pulsa signal inc ease.
The algo i hms explained will be Epoch Folding, an a e age o he ecei ed signal a he
exac pe iod o he pulsa ; Low-Pass Fil e ing, o elimina e he high equencies o he sig-
nal; Downsampling, o change he sampling a e o he signals and he e o e, he leng h o
he da a; dec ease and inc ease he Bandwid h o he An enna (and he e o e, he cu -o
equency o he Analogue il e ); and O e sampling/Unde sampling, passing he analogue
signal o he A/D con e e wi h a sampling equency highe /smalle han he Nyquis one
s.
Downsampling can also be seen as a p ocess ha changes he bandwid h o he an enna
and he sampling equency o he A/D con e e , an in e es ing ea u e ha will help us
o inc ease he ENR o he o a ing s a pulse wi hou inc easing he obse ing ime. Tha
esul could be good o pe o m a eal- ime de ec ion o he pulsa o na iga ion applica ions.
5.1 Epoch Folding
Epoch Folding is a Signal P ocessing echnique used o dec ease he a iance o he unco -
ela ed noise while keeping he ene gy/powe o he desi ed signal. Epoch olding consis on
choosing a ange o pe iods, and a e age he da a a hose pe iods. The algo i hm assumes
ha we know he pe iodici y, T, o he signal. The i s s ep consis on b eaking he ecei ed
signal in in e als o ime T. Then, sum all hese clipped signals oge he and di ide he
esul an signal by he numbe o oldings, K. No ma e i he signal is na owband o
28
wideband, because we a e no clipping any equency spec um o he signal p o ided he
igh pe iod o pe o m he olding is used. So, he shape, ampli ude, ene gy and powe o
he signal will emain he same no ma e how many olds you do.
In he case o Radio Pulsa signals, al hough hey a i e wi h a e y p ecise pe iodici y
o he Ea h, hei ampli ude can a y signi ican ly o e ime due o he scin illa ion e ec s
p o oked by he ISM. Ne e heless, he a e aged pulsa p o ile emains e y s able, allowing
us o pe o m he Epoch Folding wi hou loosing any in o ma ion o he signal no in ime
nei he in equency domain. As i has been explained, he noise will be addi i e, whi e and
gaussian wi h 0 mean, a iance σ2
zand unco ela ed. Is he las ea u e he impo an o
he success o his echnique, because a e aging AWGN unco ela ed noise leads o a linea
dec ease o he noise a iance wi h he numbe o olds. Fu he mo e, he noise is s ill whi e
a e passing h ough he a e aging, so he GENR o he ecei ed da a will inc ease. In he
Figu e 10 we can obse e he p ocess o epoch olding.
Figu e 10. Epoch Folding algo i hm pe o med o a pe iodic signal subme ged in unco ela ed noise.
As we a e going o pe o m his algo i hm o a disc e e signal, le me conside as he
disc e e signal wi h KT ssamples, being K he numbe o olds and T s he numbe o
samples in one pe iod. The nex s ep is b eaking he da a in a sequence o disc e e signals
yko leng h L, being L= sT. As we ha e s a ed, yk=sk+zkwi h zk∼N(0, σ2
z) an
Addi i e Whi e Gaussian Noise and sk he desi ed signal wi h leng h L. So, pe o ming he
Epoch Folding we ha e:
x(n) = 1
K
K−1
X
k=0
yk(n)
=1
K
K−1
X
k=0
sk(n) + 1
K
K−1
X
k=0
zk(n)
≈s(n) + z(n)
(5.1)
29
F om he las exp ession we can obse e as he p e-whi ening ma ix will be: U=1
σzL−1.
Then, aking in o accoun ha he il e ed signal is p=Ls, he p ocess Up can be w i en
as
Up =1
σzL−1Ls =1
σzs.
Tha means ha no ma e wha il e you ha e in he digi al signal p ocessing chain, i
will no ha e any in luence in he GENR o he il e ed signal. As we ha e seen, ha happens
because he p e-whi ening p ocess Uis cancelling he e ec o he il e once mul iplied by
he il e ed signal. Finally, he ene gy o ha p ocess and as we s a ed be o e, he GENR
will be:
GENR =Up =X
n|1
σz
s(n)|2=1
σ2
zX
n|s(n)|2
=1
σ2
z
s=s
σ2
z
=sa
No
(5.11)
To sum up, we ha e p o ed ha il e ing he disc e e p ocess ydoesn’ change i s GENR.
The e o e, he de ec ion pe o mance is no al e ed o his ope a ion. So, assuming ha
he spec um o he signal is almos la , he ela ion be ween he loss o signal ene gy and
noise a iance will be he same.
I we look a he de-dispe sion sec ion o he de ec ion chap e , we can see as i has been
explained ha due o he dispe si e na u e o he in e s ella plasma, lowe - equency adio
wa es a el h ough he medium slowe han highe - equency adio wa es, which mani es s
i sel as phase dis o ion. Then, as i has been p o ed ha he il e ing p ocess does no a ec
he GENR and ha he de-dispe sion p ocess can be assumed as a il e , he de-dispe sion
will no ha e any e ec in he GENR o he adio pulsa signal. Tha ac ag ees wi h he
heo y w i en by Richa d Heusdens in [1] and p o ing ha he de-dispe sion does no a ec
he de ec ion pe o mance o he GLRT o any kind o noises.
Ano he su p ising hing o ha sec ion is he abili y o he whi ening ma ix U o e-
co e he high equencies cu o pby he Low-pass il e . Indeed, he ma ix U=1
σzL−1
no only s e ch he signal, bu eco e in a good way he shape o he high equencies o s.
So, we can s a e ha Up =s
σz. In he subsec ion 7.1.3 we will show he pe o mance o he
whi ening ma ix and how i eco e s he signal spec a in he high equencies.
In he ollowing chap e I am going o explain he p ocess o changing he bandwid h
and a e o he signal. The ope a ions a e called downsampling, ha means dec ease he
bandwid h and he sampling equency o he signal.
5.3 Downsampling
Downsampling is signal p ocessing ope a ion ha change he a e o he signal while keeping
he ela ion Bandwid h-Sampling equency s able. Tha means ha , o example, down-
sampling a signal wi h a Bandwid h Band sampling equency s= 2Bby a ac o o
36
Mwill be he same as Low-pass il e ing he analogue signal wi h a bandwid h B/M and
sampling i wi h a equency sM= s/M = 2B/M. So i he signal be o e downsampling
has been sampled wi h he Nyquis sampling equency, he signal a e he downsampling
will also ha e a Nyquis sampling equency. Hence, he p ope y o whi eness o he noise
a e changing he a e will no be al e ed. As I will explain in he nex sec ion, his is e y
impo an o pe o m he de ec ion wi hou a lo o compu a ional cos . In he nex igu e
we can obse e a ep esen a ion o he Downsampling p ocess.
Figu e 14. Downsampling p ocess in he ime domain applied o a sinusoid. I will be he same as dec easing
he bandwid h o he ecei e and he sampling equency o he con e e .
I we pe o m he downsampling algo i hm o he ecei ed signal y=s+z, we will ob-
ain yd= +W. Hence, ywill be he upsampled e sion o yd. The Downsampling p ocess
by a ac o o M can be di ided in wo s eps:
1) A low-pass il e wi h a cu -o equency c=B/M o elimina e he highes equen-
cies in o de o a oid any aliasing in he nex s ep.
2)Decima ing yd(n) = y(nM) he signal by a ac o o M
As we ha e done he low-pass il e ing wi h a ac o o M be o e, now we only ha e o
explain he e ec s o decima ion. So, we will s a applying he decima ion block o he
il e ed signal y (n) = p(n) + (n). The il e ed signal has a bandwid h o B/M, so i will
no be aliasing a e applying he decima ion p ocess. Fi s , I am going o ocus in he noise
p ocess W(n) = (Mn) = z(Mn). Looking a he au oco ela ion unc ion o he downsam-
pled noise:
RW(k) = W(n)∗W(n+k) = (nM)∗ ((n+k)M)
=R (kM) = sNo
Msinc(πkM
M)
= sNo
Msinc(πk) = sNo
Mδ(k)
(5.12)
37
The a iance o he p ocess Wwill be σ2
w=RW(0) = sNo/M =σ2
=σ2
z/M, he same
as he p ocess and M imes smalle han in he p ocess z. So i is clea ha he decima -
ing algo i hm doesn’ change he a iance o he noise. Now, looking a he powe spec um
o he noise we can see how he noise became whi e again a e he decima ion, al hough
he powe spec al densi y has dec eased in a ac o o M:SW(w) = PkRW(w)e−jwkTs=
No s
MPkδ(k)ejwkTs=No s
Me0=No s
M. Then, he powe spec um will be la in all he band-
wid h:
SW(w) = sNo
M o |w| ≤ π s
M=π sM
0 o he wise (5.13)
Being sM= s/M he new sampling equency and Bd=B/M he new Bandwid h o
he Band-Limi ed Signal. As we can obse e, he noise a e he downsampling is s ill whi e,
so he de ec ion pe o mance will depend on he ENR.
In o de o assess he ene gy o he pulsa signal a e he downsampling, we ha e o
ake in o accoun ha downsampling by a ac o o M a signal wi h a bandwid h Band a
sampling equency scan be seen as he same p ocess as dec easing he bandwid h o he
ecei e and he sampling equency o he A/D con e e by M. I is some hing i ial as
he Decima ion p ocess can be seen as an A/D con e e . In he nex igu e we can see a
ep esen a ion o a ecep o wi h a p ocessed signal 2wi h M imes less Bandwid h han s.
The e o e, i is easy o see how ha disc e e signal 2will be absolu ely he same p ocess as
, he downsampled signal.
Figu e 15. Scheme o he p ocess equi alen o Downsampling he signals sand z. I consis on dec easing
he bandwid h o he signal and he sampling equency by a ac o o M.
We can see ha in he Figu es 15 and 9 ha he ene gy o he downsampled pulsa signal
will be he same as he ene gy o he new p ocess 2(being 2 he p ocess wi h a analogue
low pass il e cu -o equency cM= 2πB/M and sampling equency sM) ( = 2). In
o de o p o e ha , he powe spec um o he RW2(k) is compu ed:
38
RW2(k) = W2(n)∗W2(n+k) = W2(nTsd)∗W2((n+k)Tsd)
=RW2a(kTsd) = 2BNo
Msinc(2πBkTsd
M)
=2BNo
Msinc(2BπkM
2BM )
=2BNo
Msinc(kπ) = 2BNo
Mδ(k)
(5.14)
Being Tsd he in e se o he sampling equency sd = 2B/M. So, he a iance o W2will
be σ2
W2=RW2(0) = sNoδ(0)/M = 2BN0/M =σ2
W. Finally, compu ing he Powe spec um
o he Noise as he disc e e- ime Fou ie T ans o m o he Au oco ela ion Func ion, we ha e:
SW2(w) = sNo
M o |w| ≤ π s
M
0 o he wise (5.15)
As we can see, W2is he same signal as he downsampled noise p ocess W. Tha p o e
he hing ha Downsampling by M is he equi alen p ocess as educing he Bandwid h o
he ecei e and he sampling equency by a ac o o M. So, knowing ha he downsampled
pulsa signal will be he same p ocess as 2, we can compu e he ene gy o 2.
Fi s , we s a o see i wha is he ene gy o a2in compa ison wi h he ene gy o sa:
a2=Z∞
−∞ | a2( )|2d =Z∞
−∞ |S a2(w)|2dw
=Z2πB
M
−2πB
M|S a2(w)|2dw ∗∗∗
=1
MZ2πB
−2πB |Ssa(w)|2dw
=sa
M
(5.16)
*** Assuming ha he spec um o he analogue pulsa signal is almos la in a bandwid h o B, so clipping
i by a ac o o Mmeans ha ing he o al spec um o he il e ed analogue signal ( a) di ided by M.
Then, i we compu e he ene gy o 2we ha e:
39
2=X
n| 2(n)|2
=1
2π sMZ2π sM
0|S 2(w)|2dw
= sM
2πZ2π s2
0|∞
X
k=−∞
S 2a(w+ 2πk sM)|2dw
= sM
2πZπ sM
−π sM|S a2(w)|2dw
= sM
2πZ∞
−∞ |S a2(w)|2dw
= sM a2= s a2
M
= ssa
M2=s
M2=
(5.17)
As I said be o e, = 2, so =p
M=s
M2. Due o he Wideband Na u e o he pulsa
signal, he ene gy will dec ease in an o de o M2when you downsample he signal. The e-
o e, i will dec ease in an o de o Mwhen you low-pass il e (as we ha e seen in he las
sec ion) and in an o de o Mwhen you decima e. Howe e , he a iance o he noise has
only dec eased in an o de o Mdu ing he downsampling. Tha is due o he ac ha he
a iance is powe and i akes in o accoun he ene gy pe sample. I we compu e he ENR
o he downsampled p ocess yd= +Wwe will ob ain:
ENR =
σ2
W
=
p
M
σ2
=
s
M2
σ2
z
M
= ssa
M sNo
=sa
MNo
(5.18)
Then, a e downsampling he pulsa signal he ENR becomes smalle . To assess he
de ec ion pe o mance we a e going o look a he GENR.
GENR = TΦ−1
W = T
σ2
W
=
σ2
W
=sa
MNo
=ENR (5.19)
As we we e expec ing, he alue o GENR is he same as he ENR because he noise p o-
cess Wis whi e. So, we can s a e ha he downsampling and/o dec easing he obse ing
bandwid h o he ecei ed signal makes he GENR and ENR dec ease. Hence, de e io a e
he de ec ion pe o mance. I we ealize ha inc easing he obse ing bandwid h (inc ease
he cu -o equency o he analogue low pass il e and he sampling equency) is he in-
e se ope a ion o he downsampling, we can s a e ha inc ease he obse ing Bandwid h
imp o es he GENR and he de ec ion pe o mance. So, om now on we ha e wo di e en
ways o imp o e he de ec ion pe o mance:
1) Epoch Folding
40
2) Inc ease he obse ing bandwid h o he ecei e and he sampling equency o he
A/D con e e
Some expe imen s abou he imp o emen o he GENR due he inc emen o he Band-
wid h o he signal will be shown, as well as he pe o mance o he o he explained algo-
i hms. As we can’ pe o m a Epoch Folding wi h a eally big numbe o olds because we
need o make easible he eal- ime na iga ion, he main solu ion o imp o e he de ec ion
pe o mance will be inc ease he bandwid h o he an enna. Now ha solu ion doesn’ allow
us o inc ease he GENR oo much because o he echnological limi s. Ne e heless, e e y
yea he bandwid h o he ecei e is inc easing wi h echnological imp o emen s, so he
de ec ion pe o mance will be inc easing along ime un il i will be easible o do a eal- ime
na iga ion.
To inish his signal p ocessing backg ound I am going o in oduce he o e sampling and
unde sampling echniques. Tha consis in inc easing o dec easing he sampling equency
o he A/D con e e wi hou changing he bandwid h o he an enna. So, as we can expec ,
he noise will be colo ed al hough i will keep cons an he gaussian and 0 mean ea u es. We
will see how unde sampling doesn’ a y he de ec ion pe o mance unde some assump ions,
and how depending on he way you pe o m o e sampling, you can imp o e he de ec ion
pe o mance.
5.4 O e sampling
Figu e 16. Scheme o he p ocess o O e sampling. I consis on keeping he bandwid h o he ecei e while
inc easing sampling equency by a ac o o M o e he Nyquis one.
O e sampling is a signal p ocessing echnique ha consis in inc easing he sampling
equency while keeping he Bandwid h o he signal. Tha means sampling he il e ed ana-
logue signal sao, wi h Bandwid h B, wi h a sampling equency highe han 2B. As we can
see in he igu e 16, o e sampling will inc ease he sampling equency abo e he Nyquis
one. One o he easons o pe o m o e sampling is ha he ene gy o sowill inc ease in
compa ison wi h swhe eas he a iance o he o e sampled noise zowill no change. In ha
41
sec ion we will assume ha he Bandwid h o he An enna is BM. La e on, we will see
how impo an is ha in a de ec ion pe o mance poin o iew.
Fi s o all, o assess he ene gy o he signal and he a iance o he noise, we shall no ice
ha he ene gy o he o e sampled analogue signal yaowill be he same as he analogue sig-
nal o he i s chap e s ya. Tha is because we a e band-limi ing i wi h he same bandwid h
B. Hence, sao=sa. The a iance o he noise zaowill also be he same as he a iance o
za. So, ocusing in he o e sampled p ocess yo=so+zoa e he sampling, we will ha e a
signal wi h a isible spec um o so= sM= 2BM band-limi ed wi h a bandwid h B.
Be o e calcula ing he ene gy o he o e sampled pulsa signal and he a iance o he
o e sampled noise, le ’s compu e how he au oco ela ion unc ion and powe spec um o
z and analogue il e ed signal zaoa e:
Szao(w) = Sza(w) = No o |w| ≤ 2πB
0 o he wise (5.20)
Rzao( ) = Rza( ) = 1
2πZ∞
−∞
Sza(w)ejw dw =1
2πZ2πB
−2πB
Noejw dw
=No
π sin(2πB )=2BNosinc(2πB )
(5.21)
So, he a iance o he noise p ocess zaois σ2
zao=Rzao(0) = 2BNo=σ2
z. In o de o
compu e he au oco ela ion unc ion and he Powe Spec um o he analogue unp ocessed
noise z , we ha e o ake in o accoun ha is he same p ocess as zao, bu wi h M imes
mo e Bandwid h.
Sz (w) = No o |w| ≤ 2MπB
0 o he wise (5.22)
Rz ( ) = 1
2πZ∞
−∞
Sz (w)ejw dw =1
2πZ2πMB
−2πMB
Noejw dw
=No
π sin(2πMB )=2MBNosinc(2πMB )
(5.23)
As we can see, he a iance o ha noise p ocess is σ2
z =Rz (0) = 2MBNo=Mσ2
z.
Hence, M imes highe han he a iance o he Low-Pass Fil e ed analogue p ocess. Tha is
due he e ec o he Low-Pass Fil e ha is cu ing he Bandwid h o he ecei ed analogue
signal by a ac o o M . In he igu e 17 we can obse e he shape o he Powe Spec um
and he au oco ela ion unc ion o he noise z .
42
Figu e 17. Rep esen a ion o he Powe Spec um and he au oco ela ion unc ion o he analogue noise p o-
cess z .
Now, i we sample he signal yaowi h a sampling equency so= sM= 1/Tso, he
esul an p ocess will be yo(n) = yao(nTso) = sao(nTso)+zao(nTso) = so(n)+zo(n). In o de
o assess he ene gy o he o e sampled pulsa signal, i s spec a is shown:
So(w) = ∞
X
n=−∞
so(n)e−jwnTso= so
∞
X
k=−∞
Sao(w+ 2πk so) (5.24)
So we can ind an exp ession o he ene gy o he de e minis ic signal soin compa ison
wi h he ene gy o he analogue il e ed signal saoand s :
so=X
n|so(n)|2
=1
2π soZ2π so
0|So(w)|2dw
= so
2πZ2π so
0|∞
X
k=−∞
Sao(w+ 2πk s)|2dw
= so
2πZπ so
−π so|Sao(w)|2dw
∗
= so
2πZπ s
−π s|Sao(w)|2dw
= so
2πZ∞
−∞ |Sao(w)|2dw
= sosao=M ssa=Ms
(5.25)
* Assuming ha he spec um o he disc e e- ime o e sampled signal is has a bandwid h o π sbecause o
he analogue low-pass il e .
** Because he ene gy o he analogue p ocess s is M imes highe han he ene gy o saoand sadue he
e ec o he Low-Pass Fil e and assuming ha he ecei ing pulsa signal has a Wideband na u e. Also
so=M s
43
Figu e 18. Powe spec um o he O e sampling Noise.
Hence, as he sampling equency sois bigge han he Nyquis one, he ene gy o he
o e sampled pulsa signal is also highe by a ac o o M. To check he beha iou o he
disc e e ime noise zo(n), we a e going o look o i s Au oco ela ion unc ion:
Rzo(k) = zo(n)∗zo(n+k) = zo(nTso)∗zo((n+k)Tso)
=Rzao(kTso)=2BNosinc(2πBkTso)
= 2BNosinc(2Bπk
2BM )
= 2BNosinc(kπ
M)
(5.26)
Being Tso= 1/2BM. Tha happens because sampling he noise wi h a sampling e-
quency highe han he Nyquis one elimina es he whi eness p ope y o he noise. Ne e -
heless, he a iance o he sampled noise will emain he same as be o e passing he signal
h ough he A/D con e e no ma e he sampling equency used. So, he o e sampled
noise zowill be he same as he a iance o he no mal p ocess z.
σ2
zo=Rzo(0) = sNo= 2BN0=σ2
z(5.27)
Finally, compu ing he Powe spec um o he o e sampled noise as he disc e e- ime
Fou ie T ans o m o he Au oco ela ion Func ion, we ha e:
Szo(w) = soNo=M sNo o |w| ≤ π s
0 o he wise (5.28)
44
Whe e we can obse e as he noise is no whi e anymo e. Mo eo e , we can see ha in
he Figu e 19 whe e i is shown he powe spec um o he colo ed o e sampled noise. Now
ha we ha e he ene gy o he pulsa signal and he a iance o he noise, we can compu e
he ENR:
ENR =so
σ2
zo
= sosao
sNo
=M ssao
sNo
=Msao
No
=Msa
No
(5.29)
Bu as I explained be o e ha does no mean ha he de ec ion pe o mance is imp o ed.
To assu e ha , we ha e o compu e he GENR. As he noise is no colo ed, we will do he
same p ocedu e as in he Low-Pass Fil e ing sec ion o calcula e i .
In he Low-Pass Fil e ing sec ion we ha e seen as he digi al il e s had no e ec in he
GENR o he p ocess. So, in o de o being able o compu e he co a iance ma ix o he
o e sampled noise, needed o assess he GENR, some ma hema ical s u will be shown.
Knowing ha sampling and Low-Pass il e a e lineal ope a ions and looking a he equa-
ions s a ed in ha sec ion, we can w i e he o e sampled signal as:
yo(n) = so(n) + zo(n) = sao(nTso) + zao(nTso) = Ls (nTso) + Lz (nTso)
Being L he il e ing ma ix aken om he il e h(nTso), and h(nTso) he sampled e -
sion o he analogue low-pass il e h( ) wi h cu -o equency o B Hz. Do no con use he
p ocesses s (nTso) o z (nTso) wi h s ( ) and z ( ), as he i s ones a e he sampled e sion
o he second ones. Tha means ha s (nTso) and z (nTso) a e disc e e signals. Hence, as
p o ed be o e, he ene gy o s (nTso) will be so imes he ene gy o ( ).
Ano he impo an ea u e o s (nTso) o z (nTso) is ha hey ha e a Bandwid h MB,
so he sampling equency sowill be, in ac , he Nyquis sampling equency o heses p o-
cesses. As he eade al eady knows om he p e ious chap e s, ha mean ha he noise
p ocess z (nTso) is whi e wi h he same a iance as he analogue p ocess z ( ). Tha is due
he ac ha sampling he noise does no a y he a iance.
So, he a iance o ha p ocess will be σ2
z (nTso)=σ2
z ( )= 2MBNo. Tha o mula ion
will be e y use ul when compu ing he GENR o he o e sampled signal. Then, aking in o
accoun he esul s we had in he sec ion 5.2, he co a iance ma ix o he o e sampled noise
and p e-whi ening ma ix U o he noise p ocess zowill be:
Φzo=E(zozT
o) = E(zao(nTso)zao(nTso)T) = E(Lz (nTso)z (nTso)TLT)
=LΦz (nTso)LT∗
=σ2
z (nTso)LLT(5.30)
* Knowing ha he p ocess z (nTso) is whi e gaussian noise wi h a iance σ2
z (nTso)= 2BMNo.
45
eal pulsa signal will no ha e a comple ely la spec um, so he aliasing will no be added
linea ly o he spec um and he pe o mance will go down. .
Finally, he In eg a ion in Time has been in oduced. Tha echnique will educe he
de ec ion pe o mance. Howe e , i also will highly dec ease he numbe o samples o ou
da a. The e o e, In eg a ion in Time may be an in e es ing echnique in o de o dec ease
he compu a ional complexi y o he whole ecep o .
52
Chap e 6
Radio Pulsa Signal PSR
B0329+54 obse a ions
6.1 PSR B0329+54 ea u es
In o de o p o e he heo e ical backg ound s a ed in he las chap e , he esul s o some
expe imen s wi h simula ed and eal da a a e shown. Bu , i s o all, I will explain he
ea u es o he Radio Pulsa signal PSR B0329+54 and he way i has been eco ded.
Table 2. Pa ame e s o he Radio Pulsa B0329+54
PSR B0329+54 is a neu on s a si ua ed app oxima ely 2,643 ligh -yea s away om he
Ea h in he cons ella ion o Camelopa dalis and i was c ea ed 6.74 millions o yea s ago.
In 1979, wo ex asola plane s we e announced o be o bi ing he pulsa (being classi ied
as pulsa plane s). La e obse a ions howe e uled ou his idea. These adio pulsa emi s
one o he s onges pola ized pulses ecei ed in he no h hemisphe e wi h a pe iodici y o
0.71451866398 s. Fu he mo e, he ac ha i has a eally low dispe sion and an almos
negligible spin down, makes i a good candida e o pe o m expe imen s. The Dispe sion
Measu e o ha pe iodic signal is 26.776 cm−3pc, a low enough alue ha allow us o a oid
pe o ming he de-dispe sion p ocess.
Due he in ensi y o ha Radio Pulsa we can make ou i s pulsa p o ile a e some
oldings. The pulsa has an a e age lux densi y a he obse a ion equency o= 1400
MHz o 203 mJy and an a e age lux densi y o 1650 mJy in a o= 400 MHz. As we can
obse e in he nex igu e, he s a has h ee nes ed cones o emission and a cen al co e
emission. Also we can see how he pulsa is isible and almos iden ical in all he obse ed
equencies, om 117 MHz o 1170 MHz.
53
Figu e 21. P o iles o he Radio Pulsa B0329+54 wi h di e en obse ed equencies.
This neu on s a can be classi ied as a no mal pulsa as i is no a millisecond pulsa .
Howe e , he pe iod o ha pulse is enough o being able o pe o m hund eds o olds
wi hou losing many ime. Al hough his pulsa is no s ongly a ec ed by dispe sion, i is
known ha i scin illa e a lo . The e o e, he ampli ude o he pulsa signal will a y o e
ime. Tha is no some hing we ha e o wo y abou due he ac ha he in eg a ed pulsa
p o ile (a e olding) is qui e s able. In he EPN da abase we will be able o ind in eg a ed
adio pulsa p o iles o he B0329+54 obse ed in a di e en equencies and eco ded wi h
a di e en Bandwid h.
6.2 Radio Pulsa Signal PSR B0329+54 da a acquisi ion om
WSRT
On Feb ua y 2nd 2012, he Wes e bo k Syns hesis Radio Telescopes obse a o y eco ded
da a om he adio pulsa PSR B0329+54 o a o al obse a ion ime o 140 s. The Wes e -
bo k Syn hesis Radio Telescope (WSRT) is an ape u e syn hesis in e e ome e nea camp
Wes e bo k, no h o he illage o Wes e bo k, Midden-D en he, in he no heas e n Ne he -
lands. I consis s o 14 dish-shaped an ennas. The ope a o in he con ol oom has a good
iew o he dishes in he a ay. By means o a a ie y o compu e s i is possible o he
ope a o o con ol he elescopes, ecei e s, and e e y hing in he obse ing sys em. In he
con ol oom a e ins umen s, which con e he signals o digi al in o ma ion o be ead and
p ocessed by a compu e . The so wa e ha has been specially de eloped o his pu pose
is so cle e ha i makes he 14 dishes look like one la ge dish.
The acquisi ion was a demo obse a ion planned in o de o ob ain es da a wi h a high
bandwid h. The signal was eco ded a he obse ing equency o 1330 MHz wi h a Bandi-
wd h o 20 MHz. A e ha , he signal was sampled wi h a sampling equency o 40 MHz
54
in o de o use he Nyquis sampling equency and p ese e he whi eness p ope y o he
ecei ed noise.
Taking in o accoun he pe iod o he pulsa , 196 comple e pe iods can be ex ac ed om
he da a. This acquisi ion was s o ed in o 14 iles in .dada o ma wi h a o al weigh o 10.4
GB. Each ile consis s o 4096 by es o heade and hen 800000000 by es o X and Y pola -
iza ion eal ol age signal samples in e lea ed. The o ma used o each sample is signed
in ege wi h li le endian by e o de ing: he digi al dynamic ange goes om -127 o 127,
bu no in o ma ion abou he ampli ude o he ol age signal can be ex ac ed om he e.
The heade con ains all he in o ma ion abou he acquisi ion besides he numbe o he ile
being open. Each ile has 10 seconds o da a and he acquisi ion is consecu i e, meaning ha
om he ending o one ile o he beginning o he nex no da a is los . The signal is o med
as samples o eal ol age signals in X and Y pola iza ion in e lea ed (XYXYXYXYXY).
Those signals has been collec ed by he WSRT wi h 14 Telescopes. As i has been s a ed
be o e, some beam o ming echniques has been applied in he con ol oom in o de o ha e
only one eco ded ou pu signal. The pu pose o ha is o ake p o i o he ea u es o he
Telescopes o WSRT and o inc ease he SNR and ENR o he signal wi h he beam o ming
echniques. Apa om ha , he sampled ol age signal is no p ocessed in any o he way.
Tha means ha i is an almos pu e, aw signal acquisi ion.
In he nex sec ions we will assess he GENR o he eco ded pulsa signal wi h he help
o he empla e aken om he EPN da abase. Tha empla e will allow us o compu e he
a iance o he noise sub ac ing he pulsa signal om he acquisi ion. Bu be o e ha ,
some adjus men s o he empla e ha e o be done in o de o ha e he same ampli ude and
ime o a i al han he ecei ed pulsa signal.
6.2.1 P ocess o isualize he signal
As I ha e s a ed in he las pa ag aph, we ha e 14 iles .dada wi h 140 s o eco ding o he
pulsa B0329+54. Taking in o accoun o he pe iod, we can ex ac 196 comple e pe iods
o ha da a. Howe e , as we a e going o pe o m Epoch Folding wi h all hose pe iods,
is impo an o check i he spin down o he pulsa signal will a y ou pe iod o e ime.
Be o e i has been s a ed ha he spin down o he adio pulsa is e y small (in he o de
o 10−15 ss−1), bu no o ha eason we ha e o o ge i . In he case we pe o m Epoch
Folding o e a long ime wi hou changing he ini ial pe iod, we will ha e a misaligned o
pulsa signals added oge he in he w ong posi ion. Tha will lead o a bad pe o mance o
Epoch Folding and a change o he pulsa p o ile. As we do no wan ha o happen due
he ac ha we need a clean pulsa p o ile o pe o m he Ma ched Fil e , we a e going o
check he ue pe iod o he pulsa signal.
[17] explains he p ocedu e o calcula e he co ec pe iod o he Radio Pulsa Signal
depending on how many olds you pe o m. Du ing he i s olds he P ue will be almos
he same as he o iginal pe iod. Bu , a e 50 olds he olded signal s a d i ing o he
igh , he p o ile s a s o b oaden and he pulsa signal p o ile s a s o blu . So, as we
can see in [17], he co ec pe iod o 196 oldings should be P ue = 0.7145579 s and he
numbe o samples pe pe iod will be 28582316. Tha change in he pe iod will be a p oblem
in o de o ead he da a. The ini ial da a has been eco ded in o de o ha e a pe iod o
0.71451866398 s. So, in e e y ile he e a e 14 exac pe iods o 28580746 samples. Now, i we
change he leng h o he pe iods, we will ha e o spli he eading o he 14 h pe iod o e e y
ile in wo s eps. Finally, ha will p o oke o ha e 195 whole pe iods in all he eco ded
da a, ha ing o le wi hou eading a li le po ion o he 196 h pe iod. Al hough i look
55
like a p oblem, a he end we will see as he inal olded pulsa p o ile will be e y accu a e.
Fi s , he wo pola ized ol age shas been in eg a ed in o de o ha e he pulsa powe
p o ile. To do ha , bo h ol ages signal will be combined in his way:
Pp o ile =|Vx|2+|Vx|2
Tha p o ile will be a signal wi h a ange o alues be ween [0,16128]. As i is e y
complex om a compu a ional poin o iew o wo k wi h a signal wi h 195 pe iods, we
will pe o m he nex echniques o one pe iod. So, in case we wan o pe o m he Epoch
Folding, ha will be he momen . Tha is o say, i we wan o pe o m Epoch Folding, we
will pe o m i in his pa o he p ocess o adequa e he signal. So, om now on, ou signal
will be modelled as:
y(n) = s(n) + z(n)
Being s he pulsa signal and z Addi i e Whi e Gaussian noise wi h unknown mean o e
one pe iod. Al hough is no some hing c i ic, i will be clea e o wo k wi h he exac mean
o he adio pulsa signal, so we ha e o ind he mean o he p ocess y. To do ha , we ha e
o ake in o accoun ha he ecei ed noise p ocess ha e 0 mean, so he mean o he o al
signal will be:
E(y) = E(s+z) = E(s) + E(z) = E(s)
Hence, we need o know he mean o he adio pulsa signal in o de o compu e he
mean o he whole p ocess. To do ha , we need o use he empla e o he pulsa signal
B0329+54 aken om he EPN da abase. Tha empla e is a signal wi h 1024 samples wi h
an ampli ude Aand a ime o a i al τ. In o de o know he exac ampli ude, and he e o e,
he mean o he ecei ed signal, i s we ha e o es ima e he alues o he ampli ude and
ime o a i al o he empla e. Fu he mo e, as ou signal yha e 28582316 samples, i s
we ha e o upsample he empla e. The nex s ep was es ima ing he ime o a i al o he
adio pulsa signal. As i has been s a ed in he de ec ion heo y, he bes way o do i is
pe o ming N ma ched il e s o he empla es and he adio pulsa signal and see in wha
posi ion he alue is maximum. A e doing ha , i was clea ha he signal was shi ed
335000 samples, so I shi ed he empla e ha numbe o samples o being able o compu e
he be e ENR possible. The las s ep o being able o ha e he eal mean o he signal
was es ima ing he ampli ude o he adio pulsa ecei ed signal. As s a ed in he de ec ion
heo y, he alue o he MLE o he ampli ude will be ˆa=<y,p>
<p,p>
Wi h ha p ocesses, we can s a e ha he esul ing empla e pis he bes es ima ion o
he ecei ed pulsa . This p ocess has o be epea ed i we use a olded signal o i we change
he numbe o oldings. The ime o a i al will no a y wi h he olding, hough, so we do
no ha e o compu e again he ma ched il e . So, i ins ead o wo king wi h he unp ocessed
signal we s a wo king wi h a K olded signal, we ha e o es ima e he ampli ude o he
empla e again.
56
F om now on, we can isualize and wo k wi h he signal yand he empla e p. In he
nex sec ion I will show he isualiza ion o he adio pulsa signal, as well as compu e he
ampli ude, mean, ene gy o he pulsa signal, he a iance and spec a o he noise and he
ENR o he whole p ocess.
6.2.2 Rep esen a ion o he adio Pulsa Signal
In his sec ion I am going o show he plo s and he ea u es o he adio pulsa signal
B0329+54 desc ibed be o e. Fi s o all, we a e going o assume ha ou unp ocessed signal
is y=s+z, being s he pulsa signal and z he Addi i e Whi e Gaussian Noise. Also we ha e
he Templa e p, wi h he same ime o a i al, shape and ampli ude as he ecei ed pulsa
signal. The p ocess desc ibed in he las sec ions ha e been pe o med in his empla e in
o de o ha e a eplica o he ecei ed pulsa signal. So, we can s a e ha p=s. The e o e,
he GENR and ENR o he p ocess can be assessed. In ac , he alue o he GENR will
be he same as he ENR due he ac ha he noise is whi e. We will see ha in he nex
igu es, whe e he Powe Spec um o he noise p ocess will be shown. So, inally we can see
as he GENR:
GENR =ENR = 39.9dB
In he nex igu e we can obse e he plo s o he noisy signal and he Templa e in he
ime domain.
Figu e 22. Rep esen a ion o he unp ocessed noisy pulsa signal (le ) and he adio pulsa signal Tem-
pla e ( igh ).
F om ha unp ocessed noisy pulsa signal we can see as he adio pulsa signal is com-
ple ely subme ged in noise. Mo eo e , he e a e 4 in e e ence wi h a big ampli ude ha
has no in e es . As we will see in he esul s o he expe imen s in he Chap e 7, hose
in e e ence will disappea wi h he pe o mance o he Epoch Folding echnique. Then, i
we look a he Templa e we can see as he es ima ing ampli ude is a ound 30 while he
maximum ampli ude o he noise is abou 2000. Tha shows how small a e he unp ocessed
pulsa signal i we compa e i wi h he noise. I we keep looking o he empla e, we can
57
see he powe pulsa p o ile, and how i has he same ea u es ha ha e been desc ibed in
he las sec ions. Now, a e looking he signal in he ime domain, he powe spec um o
he noise p ocess zand he noisy signal yis compu ed in o de o see he equency domain
cha ac e is ics. Due o he ac ha wi h Ma lab is di icul o show he eal spec um o
he signal, I am going o compu e he powe spec um o he eal da a. To compu e i , he
pe iodog am app oxima ion is used:
1) Compu e he FFT o he ime-domain signal and ake he absolu e alue
2) Di ide i by he squa e oo o he numbe o samples
3) Apply an a e age il e
4) Do he squa e o he il e ed signal
Finally, we ha e 1
L|Sxx( )|2, being Sxx he a e age FFT o he signal s. This is a good
app oxima ion o he powe spec um, and he e o e, o he spec um o he signal. Fi s o
all, I will show he noise spec um in o de o see he whi eness p ope y and he alue o
he powe spec al densi y.
Figu e 23. Rep esen a ion o he Powe Spec um o he unp ocessed noise.
Looking a he Figu e 23 we can ake some conclusions. The i s one is ha , as we we e
expec ing, he noise powe spec um is comple ely la , so he unp ocessed noise p ocess is
whi e. Howe e , we can see a small peak in he cen e o he spec um due he li le di e -
ence be ween he Templa e and he Real Pulsa Signal. Tha is because he noise ha e been
compu ed as y−p=y−s=z, so any small di e ence be ween he eal pulsa signal and
he empla e will be e lec in he noise powe spec um. Anyway, his li le in e e ence is
negligible in he compu a ion o he a iance o he noise. Ano he conclusion we can obse e
is he ac ha he ampli ude o he powe spec um is equal o he a iance o he noise.
Tha p o es he ac ha he powe spec um o he whi e sampled noise is sNo, exac ly
he same alue as he a iance i he noise is whi e. Now, he powe spec um o he whole
58
signal is shown in o de o see i he signal is wideband as we a e expec ing. Compu ing i
wi h he same p ocess as he used o he noise powe spec um, we ha e:
Figu e 24. Rep esen a ion o he Powe Spec um o he Radio Pulsa noisy signal.
Looking ca e ully a he igu e 24, we can see as he bandwid h o he pulsa signal is 200
Hz and wi h an ampli ude highe han he noise spec um . So, he signal ecei ed om he
WSRT looks e y na owband. Tha esul is he opposi e ha he one we we e expec ing
due he ac ha he adio pulsa signals has a wideband na u e. Mo eo e , looking a hose
kind o Bandwid h ( he o de o MHz) we should see an almos comple ely la spec um.
F om his igu e we canno s ill conclude ha he adio pulsa signal eco ded by he WSRT
has a na owband na u e because we a e looking a he powe spec um o he signal and
no he eal spec um. Howe e , I will p o e in he Chap e 7 he ac ha he pulsa signal
om he WSRT has an almos na owband na u e.
59
Chap e 7
Signal P ocessing expe imen s
In his sec ion, expe imen s p o ing he pe o mance o he algo i hms explained in he Chap-
e 5 wi h Wideband Signals subme ged in Addi i e Whi e Gaussian Noise will be shown. To
do ha , I am going o di ide i in wo subsec ion. Fi s , I will pe o m simula ions wi h ake
da a c ea ed by Ma lab. In ha pa I am going o p o e ha he GENR o he Wideband
Signal inc eases when you pe o m Epoch Folding and when you imp o e he bandwid h o
he ecei e . As s a ed in De ec ion Theo y, inc easing he GENR will mean inc easing he
de ec ion pe o mance and he es ima ion o he Time O A i al o he Wideband Signal.
Fu he mo e, I will show ha Low-Pass il e ing doesn’ change he de ec ion pe o mance
and ha i dec eases wi h Downsampling.
A e ha , I will pe o m he same algo i hms o a Real Radio Pulsa Signal PSR
B0329+54 p o ided by he Wes e bo k Syn hesis Telescope. We will see as he esul s wi h
ha eal da a will no be he ones we a e expec ing. I am going o p o ide an explana ion
o hose esul s.
7.1 Simula ed da a
In ha subsec ion we a e going explained how I ha e c ea ed a Wideband Fake Radio Pulsa
Signal subme ged in whi e noise o apply he 5 algo i hms explained in ha epo . A -
e ha , we a e going o assess he ENR and GENR o he signal a e e e y p ocessing
s ep o p o e he heo y s a ed be o e. The da a has been implemen ed wi h Ma lab. Fu -
he mo e, he o he signal p ocessing echniques also has been pe o med wi h ha So wa e.
Fi s o all, i has been c ea ed a simula ed da a wi h ea u es simila o a Radio Pulsa
signal swi h leng h o 2000 samples and a bandwid h o 10 kHz. The simula ed da a ha
each he p ope ies o he s a pulses (Wideband, Time-Limi ed and accu a e pe iodici y)
is he Addi i e Whi e Gaussian Noise wi h a du a ion o ew samples. In his case, a AWGN
signal o leng h 100 samples o e a 2000 samples backg ound has been implemen ed wi h an
ene gy o s= 1.01 ∗106. The ac ha his ake pulsa signal is an addi i e whi e gaus-
sian noise makes he spec um Wideband. Mo eo e , he spec um is almos la , impo an
p ope y o show he linea imp o emen o he GENR o he signal a e applying he signal
p ocessing echniques. In he nex igu e we can obse e he ake o a ing s a sampled pulse
in he ime domain.
60
Figu e 25. Rep esen a ion o he simula ed adio pulsa signal in ime domain wi h a o al leng h o 2000
samples and a sampling equency o 20 kHz.
As I said, his signal has a wideband na u e. Due o he ac ha wi h Ma lab is di icul
o show he eal spec um o he signal, I am going o compu e he powe spec um o he
simula ed da a. So, we will ha e 1
L|Sxx( )|2, being Sxx he a e age FFT o he signal s. In
he nex igu e we can see he Powe Spec um o he simula ed da a. The ed line shows he
a e aged Powe Spec um while he blue one shows he same signal bu wi hou pe o ming
he a e age il e . The o al bandwid h o his signal is 10 kHz in o de o ha e he Nyquis
sampling equency o 20 kHz.
Figu e 26. Rep esen a ion o he Powe Spec um o he simula ed adio pulsa signal wi h a o al bandwid h
o 10 kHz.
61
Looking a he Figu e 32 we can s a e ha he whi ening ma ix is eco e ing he high
equencies o he signal s, elimina ed by he il e . Also we can obse e ha he ampli ude
has dec eased in σ2
z. Tha is due he whi ening p ocess ha e a no maliza ion by a ac o σ2
z,
he a iance o he noise be o e applying he il e .
7.1.4 Downsampling
In ha sec ion we a e going o assess he GENR when you downsample a signal. To do i
easie , I am going o pe o m he downsampling o he simula ed noisy signal o L= 2000
wi h he same ac o s Mas he ones in he Low-Pass Fil e ing expe imen . In he chap e
5.3 I ha e concluded ha downsampling dec eases he ENR and he GENR by a ac o o
M. Hence, he de ec ion pe o mance is dec eased.
So, i we ake in o accoun ha yd= +W, being yd he downsampled noisy signal,
he ake downsampled pulsa signal and W he downsampled noise p ocess, he ENR o he
downsampling signal will be ENRd= /σ2
W=s
M2/σ2
z
M=ENR/M. Mo eo e , as he noise
p ocess zis Addi i e Whi e Gaussian Noise, he downsampling will no change he whi e
ea u e on he p ocess W. The e o e, he alue o he GENR will be he same as he alue
o ENR.
Figu e 33. Rep esen a ion o he Gene alised Ene gy To Noise Ra io o a di e en alues o he downsam-
pling ac o M in he linea domain
As we we e expec ing, he de ec ion pe o mance dec ease when M inc eases. Is no de-
c easing in a linea way because he spec um o he simula ed pulsa signal is no comple ely
la . Mo eo e , as we a e looking a signals wi h a leng h up o 200 samples, he accu acy
o he a ios in no pe ec . Howe e , i you look ca e ully, he alue o he GENR is a ound
10 when M= 10 and a ound 120 when M= 1, so he dec emen is almos lineal.
68
7.1.5 Inc ease he Bandwid h o he signal
In he las sec ion i has been concluded ha dec easing he Bandwid h o he ecei e /an enna
dec eases he ENR, GENR and he de ec ion pe o mance. So, we can demons a e ha in-
c easing he Bandwid h o he an enna imp o es he de ec ion pe o mance knowing ha
he opposi e echnique, downsampling, dec ease he GENR. Anyway, in his sec ion I will
show he esul s o how inc easing he Bandwid h o he signal, and he e o e, he samples
pe pe iod, imp o e he a ios used in his Thesis.
Be o e showing he esul o he expe imen s done wi h simula ed da a, i s I will sum
up he heo y o how inc easing he Bandwid h o he ecei e /an enna imp o es he GENR.
I we ake a look a he nex igu e, we can see he schemes o bo h cases. The one wi h a
Bandwid h Band he one wi h a Bandwid h MB. The sampling equency will always be
he Nyquis one. Hence, wo imes he Bandwid h o he signal.
Figu e 34. Rep esen a ion o he ecei e scheme o he signal wi h a Bandwid h BHz (le ) and he one
wi h he signal wi h a Bandwid h MB Hz ( igh ).
As we can obse e, he analogue Low-Pass il e will no change any hing as i s cu -o
equency is he same as he Bandwid h o he an enna. Howe e , i is shown in o de o
elimina e he spu ious. Looking a he le igu e, we can see as is he same scheme as he
unp ocessed signal shown in he chap e 4. The e o e, we know ha he ENR and he GENR
o ha signal will be:
GENR =ENR =sa
No
Now, looking a he second Scheme, we can see as he ene gy o he signal saMwill be M
imes highe han he ene gy o sadue i has M imes mo e Bandwid h. In addi ion, as he
sampling equency is also M imes bigge in he second scheme, he ene gy also will inc ease
in a ac o M. So, inally we can s a e ha :
sM= sosaM=M ssaM=M2 ssa
The noise p ocess zM hough, only will inc ease i s a iance in a ac o o M i we com-
pa e o he noise p ocess z. Tha is due he ac ha he A/D con e e doesn’ inc ease
he a iance o he noise no ma e he sampling equency used. Then, i will be inc eased
by Mdue he bigge Bandwid h o zM. So, he a iance o he noise p ocess zMwill be:
69
σ2
zM=Mσ2
z=M sNo
Finally, and aking in o accoun ha he p ocess zMis whi e wi h au oco ela ion unc-
ion RzM(k)=2MBNoδ(k), he GENR will be:
GENR =ENR =sM
σz2
M
=M2 ssa
M sNo=Msa
No
So, i can be seen as he GENR is inc eased by a ac o o M. Now, o p o e ha I will
show he esul s o he expe imen s done. Fi s o all we will conside he signal wi hou he
Bandwid h inc eased. Tha will be a simula ed signal wi h Bandiwd h 1 kHz, a sampling
equency o 2 kHz and leng h 200 samples. Then, i has been c ea ed he signals wi h he
same p ope ies bu wi h a inc eased Bandwid h. The new signals will ha e a Bandwid h o
B=MkHz wi h a sampling equency o 2MkHz and leng h 200Msamples. To do ha
expe imen , i has been chosen di e en alues o M be ween 1 and 10, being he signal wi h
M= 10 he one I ha e shown in he sec ion 7.1.1. So, a e compu ing he GENR o he
di e en signals, we can see he e olu ion o i wi h he inc emen o M.
Figu e 35. Rep esen a ion o he Gene alised Ene gy To Noise Ra io o a di e en alues o M. Being M he
ac o o inc emen o Bandwid h
Looking a he igu e 35 we can see how inc easing he Bandwid h o he ecei e im-
p o es in a linea way he GENR. Hence, he de ec ion pe o mance will also be inc eased.
The imp o emen is no comple ely linea due he ac I ha e done he simula ions wi h
signals o ew samples, so he accu acy o he p ocess is no pe ec . Anyway, i can be easily
seen as he a io is imp o ed.
70
7.2 Radio Pulsa signal B0329+54 om WSRT
In ha sec ion I will show he esul s o he expe imen s wi h he Real Radio Pulsa Signal
da a om he Wes e bo k Syn hesis Radio Telescope. The p oblem is ha his pulsa looks
beha e like a na owband signal, de ying i s wideband na u e. Howe e , only obse ing he
powe Spec um o he eal da a we canno conclude ha he pulsa signal da a is na ow-
band due he ac ha he Powe spec um we compu e is no he same as he eal spec um
o he signal. So, he expe imen s done in his subsec ion will be shown in o de o p o e
he ”na owband” na u e o his eco ded pulsa signal. The expe imen s done o he eal
da a has been:
1) Epoch Folding, in o de o see how his echnique inc ease he SNR and GENR no
ma e he spec um o he signal. Also o being able o see he blu ed pulsa p o ile and
how i dissapea a e pe o ming a High-Pass Fil e .
2) Downsample he signal o see how he SNR o he signal is imp o ed and he GENR
is no changed. In ac , a downsampling by a ac o o 27885 has been pe o med in o de
o see how he noise disappea and only he signal emains. Finally, a high pass il e will
p o e he almos na owband na u e o he pulsa signal aken om he WSRT.
3) In eg a ion in Time, in o de o see he Pulsa P o ile o e e y equency channel.
We will see as he shape o he pulsa can be seen o a high equencies al hough eally
a enua ed. So, he beha iou o he pulsa is close o a Na owband signal.
7.2.1 Epoch Folding + High Pass Fil e
Fi s o all, I pe o med he Epoch Folding wi h di e en alues o K in o de o assess he
GENR. The pe iods ha e been added e e y 28582316 samples. This numbe o samples is
used in o de o pe o m he Epoch Folding wi h he new pe iod compu ed in he sec ion
6.2.2. In he nex igu e we can obse e he e olu ion o he GENR o he signal depending
he numbe o olds. The assessmen o he a ios has been done e e y 14 olds, inishing
wi h K=195. The esul s we a e expec ing is an imp o emen o 10log(195) = 22.90dB in
bo h a ios.
71
Figu e 36. Rep esen a ion o he he GENR o he pulsa signals a e pe o ming Epoch Folding wi h K
olds.
The ENR has no been compu ed because as we ha e seen, he noise p ocess zis com-
ple ely whi e, so he GENR will be equal as he ENR. F om ha igu e we can conclude
ha Epoch Folding inc ease he a ios in a linea way. I we check a he GENR a e 195
olds we can obse e as:
GENRk=195 = 62.95dB =GENR + 23.05dB
So we can p o e as he inc emen is almos linea as we we e expec ing. In he nex
igu e is shown he olded pulsa signal a e 195 olds in he ime and equency domain.
We can see how he pulsa is isible al hough e y blu ed by he noise. Fu he mo e, in he
equency domain we can s ill see how he pulsa is na owband and i s powe spec um is
much highe han he noise one.
72
Figu e 37. Rep esen a ion o he Folded Radio pulsa signal a e 195 olds in ime domain (le ) and in
he equency domain (Powe Spec um)( igh )
We can obse e as he signal is s ill na owband wi h a Bandwid h o 200 Hz a e he
olding. F om now on, o make easy and in ui i e he expe imen s we a e going o wo k wi h
he Folded signal.
A e showing he e ec s o Epoch Folding , a high-pass il e wi h a cu -o equency
o 80 kHz has been pe o med in o de o show how he pulsa signal o he Figu e 37 dis-
appea . To s a wi h he expe imen , i s i has been c ea ed a High-Pass Fil e h(n) wi h
he So wa e Ma lab. The il e ha e has an ampli ude o 1 in he equencies be ween 80
kHz and 20 MHz and almos 0 om 0 Hz o 800 kHz. The nex igu e shows he ans e
unc ion o his il e .
Figu e 38. Rep esen a ion o he equency esponse o he High-Pass Fil e c ea ed wi h Ma lab.
73
The ollowing s ep was il e ing he olded signal y195 wi h ha il e . So, in o de o
see i he high pass- il e has had any e ec on he pulsa signal, he ep esen a ion o he
il e ed signal is shown.
Figu e 39. Rep esen a ion o he olded adio pulsa signal a e high-pass il e .
F om ha plo we can s a e ha he pulsa signal isible in he Figu e 39 is almos elim-
ina ed. We also can see ha he noise le el has dec eased. Tha can be explained i we ake
in o accoun ha he il e h(n) is elimina ing pa o he noise spec a, so he a iance o
he noise (powe /sample) is also educed. Ano he simula ion is pe o med o hose signals.
As we ha e s a ed, he empla e pis he same p ocess as he Folded Radio pulsa signal
s195. Also, we know ha i we co ela e ha Templa e wi h he whole noisy signal he esul
should be he au oco ela ion unc ion o he adio pulsa signal. Hence, a eally na ow
peak in he middle o he adio pulsa B0329+54 pe iod. Now, i we pe o m his co ela ion
be ween he High-Pass Fil e ed signal and he Templa e and we do no see any hing, ha
will mean ha he adio pulsa signal has been elimina ed wi h he il e ing. In he nex
igu e we can obse e ha ac .
74
Figu e 40. Rep esen a ion o he co ela ion be ween he olded signal and he Templa e (le ) and co -
ela ion be ween he olded signal a e high pass il e i and he Templa e( igh )
The co ela ion be ween he high-pass il e ed signal and he empla e is almos 0. Tha
means ha he signal does no ha e no hing in common wi h he empla e.
7.2.2 Downsampling + High-Pass Fil e
In his sec ion i is showed he pe o mance o Downsampling o he olded signal y195 by a
ac o o 27885. Tha is done in o de o ob ain a esul an signal wi h 1025 samples. Also
i has been done o compu e he ex eme case when he signal is low-pass il e ed wi h a
cu -o equency o 717 Hz. In he nex Figu e we can see he Downsampled pulsa signal
in he ime domain.
Figu e 41. Rep esen a ion o he olded adio pulsa signal a e downsampled i by 27885.
75
The Figu e shows he pulsa powe p o ile wi h he same le el as he empla e be o e
Downsampling. Tha means ha he Low-Pass il e is no cu ing any spec a o he pulsa
signal. Wi h ha esul we can ake wo conclusions. Fi s , he low-pass il e is no de-
c easing he ene gy o he pulsa signal. The second is ha he es ima ion o he ampli ude
o he empla e is eally accu a e.
Now, i we compu e he SNR o ha downsampled signal we can see as SNR = 24.60dB.
Tha means ha he SNR has inc eased in 36.2 dB, whe e i should be 0 dB i he signal
had a wideband na u e. Finally, he a high-pass il e has been applied in o de o see how
he Radio Pulsa Signal is almos elimina ed. Fi s , we ha e o ake in o accoun ha he
downsampled signal will ha e a Bandwid h o 717 Hz and a sampling equency o 1434 Hz.
So, he cu -o equency o he high pass il e is c= 250Hz. In he nex igu e we can see
he downsampled p ocess a e high pass il e i . As i can be obse ed, he signal has been
almos elimina ed by he il e , ano he p o e ha he da a om WSRT has been s o ed
elimina ing i s wideband ea u e.
Figu e 42. Rep esen a ion o he downsampled adio pulsa signal a e high-pass il e i .
7.2.3 In eg a ion in Time
The inal expe imen has been he In eg a ion in Time sepa a ing he signal in 33 equency
channels. Doing ha we will be able o see he In eg a ed Pulsa P o ile o he signal o a
di e en equencies. The channel 1 shows he lowes equencies and he channel 33 shows
he highes ones. In he nex igu e we can see he Powe Pulsa P o ile wi hou noise in he
i s channel. Tha is due he ac ha In eg a ion on ime highly inc eases he SNR o he
signal.
76
Figu e 43. Rep esen a ion o he Powe Pulsa P o ile in he i s F equency Channel (o 33).
The mos impo an hing o ha igu e in o de o check i he ecei ed signal is na -
owband is he ampli ude. I he da a beha e as a eal Pulsa , he o he equency channels
should show he same Pulsa Powe P o ile wi h almos he same ampli ude. This is due he
ac ha Radio Pulsa Signals a e eally Wideband, so he in ensi y o he pulsa should no
a y in a obse a ion Bandwid h o 20 MHz. In he nex igu e we can see he Pulsa P o ile
o he 4 h equency channel.
Figu e 44. Rep esen a ion o he Powe Pulsa P o ile in he ou h F equency Channel (o 33).
We can see as he Pulsa P o ile can s ill be obse ed bu wi h a lo o noise. Fu he mo e,
77
o all-pass il e ing. Then, as de-dispe sion can be modelled as an all-pass il e ing p ocess
because i only changes he phase o he ecei ed signal, i can be p o ed ha we can a oid
he de-dispe sion wi hou dec easing he de ec ion o TOA pe o mance.
A e ha , a e iew o he di e en signal p ocessing algo i hms has been shown in o de
o assess i he ENR/GENR o he signals a e imp o ed wi h hei applica ion. Some in e -
es ing conclusions can be ex ac ed. The i s one and mo e impo an is ha he de ec ion
pe o mance will no be a ec ed by he applica ion o any kind o analogue o digi al il e -
ing. The e o e and assuming he Radio Pulsa Signal as Wideband, no low-pass il e ing
nei he he de-dispe sion me hod will imp o e he de ec ion o TOA pe o mance. A second
conclusion is ha o imp o e he GENR/ENR i is necessa y o inc ease he Bandwid h
o he ecei e . Fu he mo e, i has been ound ha he limi a ion o Bandwid h in o de
o inc ease he ENR/GENR is he An enna since he il e ing does no a ec he de ec ion
pe o mance. Tha can be unde s ood due he ac ha i is no possible o econs uc any
signal ha he an enna has no ecei ed. Then, in o de o imp o e as much as possible
he GENR and he e o e, he de ec ion and TOA pe o mance, he aim will be o inc ease
as much as possible he Bandwid h o he An enna. O he signal p ocessing echniques as
Downsampling, O e sampling and Unde sampling has been ejec ed. Ne e heless, as known
om o he Thesis, he Epoch Folding imp o es he GENR and ENR linea ly wi h he num-
be o olds. Some expe imen s wi h simula ed signals ha e been pe o med in o de o p o e
he Signal P ocessing heo e ical echniques s a ed in he Thesis.
Ano he Signal P ocessing echnique has been in oduced in o de o assess i i s appli-
ca ion imp o es he de ec ion pe o mance. Tha me hod has been he In eg a ion in Time
o med by an a e age and a Downsampling p ocess. Tha algo i hm inc eases he SNR o
he signal bu we concluded ha i does no imp o e he de ec ion pe o mance. Howe e ,
i can be use ul due he ac ha i educes he leng h o he signal. Tha may dec eases he
compu a ional complexi y o he whole ecep o . In his Thesis is no p o ed he e iciency
o ha echnique, so i will no be included in he i s p oposed de ec o bu in he second.
La e on, a desc ip ion o he WSRT obse a o y is p esen ed, along wi h a PSR B0329+54
cha ac e iza ion. A e he pe o mance o some expe imen s wi h ha da a I ha e concluded
ha he signal has been eco ded in a way ha he Wideband Na u e o he Radio Pulsa
Signal has disappea ed. A e looking a he way hey usually eco d his signals, i may be
ha he loss o ha high equencies has been due he ac ha hey eco d he signals wi h
14 an ennas. So, hey compu e di e en Beam o ming echniques o con e he 14 ecei ed
signals in o 1. The e o e, we can’ use ha da a o p o e he heo e ical hings explained in
ha Thesis since he main assump ion has been he Wideband na u e o he Radio Pulsa
signals.
Finally wo ecep o s a e p oposed o de ec he adio pulsa signals. Tha analogue pa
o ha de ec o is o med by an an enna wi h he Bandwid h as la ge as possible, a Low-Pass
Fil e wi h a cu -o equency equal as he Bandwid h o he an enna and a A/D con e e
wi h an sampling equency wice he cu -o equency o he il e s (an enna). Then, in
he digi al domain wo di e en app oaches a e p esen ed depending on he e iciency o he
In eg a ion in Time echnique. The digi al blocks o he i s ecep o will be o med by
an Epoch Folding Block and a GLRT de ec o . The implemen a ion o ha de ec o will
be N di e en co ela o s in pa allel wi h di e en shi ed empla es. The digi al pa o
he second ecep o will be exac ly he same as he i s one bu including he In eg a o
be ween he A/D Con e e and he Epoch Folding in o de o educe as much as possible
he compu a ional complexi y o Epoch Folding and he de ec o . In he Figu es 59 and 60
84
we can obse e he p oposed de ec o s. As expec ed, he de-dispe sion block is no included
since i will no imp o e no he TOA nei he he de ec ion pe o mance.
9.2 Fu u e Wo k
Since he de ec ion and signal p ocessing heo y o de ec and make easible he eal ime
na iga ion wi h pulsa signal has been p esen ed in ha Thesis, some p ac ical expe imen s
a e needed in o de o p o e hem wi h eal Radio Pulsa Signals. The e o e, he u u e
wo k o do o he nex esea che s will be:
1) P o e wi h eal da a ha Fil e ing and de-dispe sion does no ha e any e ec in he
de ec ion pe o mance using he GLRT de ec o .
2) P o e wi h eal da a ha inc easing he Bandwid h o he ecei e leads o a be e
de ec ion and TOA pe o mance due he ac ha he GENR/ENR is inc eased.
3) Pe o m some expe imen s o assess he pe o mance o he In eg a ion in Time ech-
nique in o de o see i i dec eases he ime needed o ecei e, p ocess and de ec he adio
pulsa signal.
4) C ea ion o a dispe sed pulsa p o iles da abase: as onome s ha e always co ec ed dis-
pe sion in pulsa eco dings, so all he pulsa da abases show empla es o he de-dispe sed
e sions o hem. [5]
5) T y o eco d and p ocess (wi h he p oposed ecep o s) a adio pulsa signal wi h an
2-3 m diame e dish-shape an enna in o de o assess he ac ual p ocessing ime equi ed o
co ec ly de ec Radio Pulsa Signals.
85
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