DTIC ADA568864: Complementary Waveforms for Sidelobe Suppression and Radar Polarimetry

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ChapterC-XV-1
ComplementaryWaveformsforSidelobeSuppressionandRadarPolarimetry
YuejieChi,AliPezeshki,andA.RobertCalderbank
1. Introduction
Advances in active sensing are enabled by the ability to control new degrees of
freedom and each new generation of radar platforms requires fundamental advances
in radar signal processing [1],[2]. The advent of phased array radars and space-time
adaptive processing have given radar designers the ability to make radars adaptable
on receive [2]. Modern radars are increasingly being equipped with arbitrary wave-
formgeneratorswhichenabletransmissionofdifferentwaveformsacrossspace,time,
frequency, andpolarizationonapulse-by-pulsebasis. Thedesignofwaveformsthat
effectivelyutilizethedegreesoffreedomavailabletocurrentandfuturegenerationof
radar systems is of fundamental importance. The complexity of the design problem
motivates assembly of full waveforms from a number of components with smaller
time-bandwidth products and complementary properties. By choosing to separate
waveformsinspace,time,frequency,polarization,oracombinationofthesedegrees
offreedom,wecanmodularizethedesignproblem. Modularityallowsustofirstan-
alyzetheeffectofeachcontrolvariableinisolationandthenintegrateacrossdegrees
offreedom(e.g.,timeandpolarization)toenableadaptivecontroloftheradar’soper-
ation.
Sixty years ago, efforts by Marcel Golay to improve the sensitivity of far infrared
spectrometry led to the discovery of complementary sequences [3]–[5], which have
thepropertythatthesumoftheirautocorrelationfunctionsvanishesatalldelaysother
than zero. Almost a decade after their invention, Welti rediscovered complementary
sequences(therearehisD-codes)[6]andproposedtousethemforpulsedradar.How-
ever, since then they have found very limited application in radar as it soon became
evidentthattheperfectautocorrelationpropertyofcomplementarysequencescannot
be easily utilized in practice. The reason, to quote Ducoff and Tietjen [7], is “in a
practicalapplication, thetwosequencesmustbeseparatedintime, frequency, orpo-
larization, which results in decorrelation of radar returns so that complete sidelobe
TheresearchcontributionspresentedhereweresupportedinpartbytheNationalScienceFoundation
underGrant0701226,bytheOfficeofNavalResearchunderGrantN00173-06-1-G006,andbytheAir
ForceOfficeofScientificResearchunderGrantFA9550-05-1-0443.
Y.ChiiswiththeDepartmentofElectricalEngineering,PrincetonUniversity,Princeton,NJ08544,
USA.Email:[email protected].
A.PezeshkiiswiththeDepartmentofElectricalandComputerEngineering,ColoradoStateUniversity,
FortCollins,CO80523,USA.Email:[email protected].
A.R.CalderbankiswiththePrograminAppliedandComputationalMathematics,PrincetonUniver-
sity,Princeton,NJ08544,USA.Email:[email protected].
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Complementary Waveforms for Sidelobe Suppression and Radar
5b. GRANT NUMBER
Polarimetry
5c. PROGRAM ELEMENT NUMBER
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cancellationmaynotoccur. Hencetheyhavenotbeenwidelyusedinpulsecompres-
sionradars.” Variousgeneralizationsofcomplementarysequencesincludingmultiple
complementary codes by Tseng and Liu [8] and multiphase (or polyphase) comple-
mentarysequencesbySivaswami[9]andFrank[10]sufferfromthesameproblem.
In this chapter, we present a couple of examples to demonstrate how degrees of
freedom available for radar transmission can be exploited, in conjunction with the
complementarypropertyofGolaysequences,todramaticallyimprovetargetdetection
performance. The chapter is organized as follows. In Section 2, we describe Golay
complementarywaveformsandhighlightthemajorbarrierinemployingthemforradar
pulse compression. In Section 3, we will consider coordinating the transmission of
Golaycomplementarywaveformsintime(overpulserepetitionintervals)toconstruct
Dopplerresilientpulsetrains, forwhichtheambiguityfunctioniseffectivelyfreeof
rangesidelobesinsideadesiredDopplerinterval. Thisconstructiondemonstratesthe
valueofwaveformagilityintimeforachievingradarambiguityresponsesnotpossible
with conventional pulse trains. The materials presented in this section are based on
work reported in [11]–[13]. In Section 4, we will present a new radar primitive that
enablesinstantaneousradarpolarimetryatessentiallynoincreaseinsignalprocessing
complexity. In this primitive the transmission of Golay complementary waveforms
iscoordinatedacrossorthogonalpolarizationchannelsandovertimeinanAlamouti
fashiontoconstructaunitarypolarization-timewaveformmatrix.Theunitaryproperty
enablessimplereceivesignalprocessingwhichmakesthefullpolarizationscattering
matrixofatargetavailablefordetectiononapulse-by-pulsebasis(instantaneously).
Thematerialspresentedinthissectionarebasedonworkreportedin[14],[15],[11].
2. GolayComplementaryWaveforms
2.1. ComplementarySequences
Definition 1: Two length L unimodular1 sequences of complex numbers x((cid:96)) and
y((cid:96)) are Golay complementary if for k = −(L−1),...,(L−1) the sum of their
autocorrelationfunctionssatisfies
C (k)+C (k)=2Lδ(k) (1)
x y
where C (k) is the autocorrelation of x((cid:96)) at lag k and δ(k) is the Kronecker delta
x
function. Thispropertyisoftenexpressedas
|X(z)|2+|Y(z)|2 =2L (2)
whereX(z)isthez-transformofx((cid:96))givenbythepolynomial
L(cid:88)−1
X(z)= x((cid:96))z−(cid:96). (3)
(cid:96)=0
1TheoriginalcomplementarysequencesinventedbyGolayforspectrometrywerebinaryandpolyphase
complementarysequenceswereinventedlater. However,inthischapterwewillnotdistinguishbetween
themandrefertoallcomplementarypairsasGolaycomplementary.
Henceforthwemaydropthediscretetimeindex(cid:96)fromx((cid:96))andy((cid:96))andsimplyuse
xandy. Eachmemberofthepair(x,y)iscalledaGolaysequence.
Atfirst,itmayappearthatcomplementarysequencesareidealforradarpulsecom-
pression. However, as we will show in Section 2.2. the complementary property of
thesesequencesisextremelysensitivetoDopplereffect. Forthisreason,complemen-
tary sequences have not found wide applicability in radar, despite being discovered
almostsixtyyearsago.SomeoftheearlyarticlesthatexploretheuseofGolaycomple-
mentarysequences(ortheirgeneralizationstopolyphaseandmultiplecomplementary
sequences)forradarare[6],[8]–[10].
2.2. GolayComplementaryWaveformsforRadar
Abasebandwaveformw (t)phasecodedbyaGolaysequencexisgivenby
x
L(cid:88)−1
w (t)= x((cid:96))Ω(t−(cid:96)T ) (4)
x c
(cid:96)=0
whereΩ(t)isaunitenergypulseshapeofdurationT andT isthechiplength. The
c c
ambiguityfunctionχ (τ,ν)ofw (t)isgivenby
wx x
(cid:90)∞
χ (τ,ν)= w (t)w (t−τ)e−jνtdt
wx x x
−∞
L(cid:88)−1
= A (k,νT )χ (τ +kT ,ν) (5)
x c Ω c
k=−(L−1)
where w(t) is the complex conjugate of w(t), χ (τ,ν) is the ambiguity function of
Ω
thepulseshapeΩ(t),andA (k,νT )isgivenby
x c
L(cid:88)−1
A (k,νT )= x((cid:96))x((cid:96)−k)e−jν(cid:96)Tc. (6)
x c
(cid:96)=−(L−1)
If complementary waveforms w (t) and w (t) are transmitted separately in time,
x y
withaT sectimeintervalbetweenthetwotransmissions,thentheambiguityfunction
oftheradarwaveformw(t)=w (t)+w (t−T)isgivenby2
x y
χ (τ,ν)=χ (τ,ν)+ejνTχ (τ,ν). (7)
w wx wy
Since the chip length T is typically very small, the relative Doppler shift over chip
c
intervalsisnegligiblecomparedtotherelativeDopplershiftoverT,andtheambiguity
2Theambiguityfunctionofw(t)hastworangealiases(crossterms)whichareoffsetfromthezero-delay
axisby±T. Inthischapter,weignoretherangealiasingeffectsandrefertoχw(τ,ν)astheambiguity
functionofw(t). Rangealiasingeffectscanbeaccountedforusingstandardtechniquesdevisedforthis
purpose(e.g.see[16])andhencewillnotbefurtherdiscussed.
Figure1. TheperfectautocorrelationpropertyofaGolaypairofphasecodedwave-
forms
functionχ (τ,ν)canbeapproximatedby
w
L(cid:88)−1
χ (τ,ν)= [C (k)+ejνTC (k)]χ (τ +kT ,ν) (8)
w x y Ω c
k=−(L−1)
where in this approximation we have replaced A (k,νT ) and A (k,νT ) with the
x c y c
autocorrelationfunctionsC (k)andC (k),respectively.
x y
Along the zero-Doppler axis (ν = 0), the ambiguity function χ (τ,ν) reduces to
w
theautocorrelationsum
L(cid:88)−1
χ (τ,0)= [C (k)+C (k)]χ (τ +kT ,0)=2Lχ (τ,0) (9)
w x y Ω c Ω
k=−(L−1)
wherethesecondequalityfollowsfrom(1). Thismeansthatalongthezero-Doppler
axis the ambiguity function χ (τ,ν) is “free” of range sidelobes.3 This perfect au-
w
tocorrelation property (ambiguity response along zero-Doppler axis) is illustrated in
Fig. 1foraGolaypairofphasecodedwaveforms.
Offthezero-Doppleraxishowever,theambiguityfunctionχ (τ,ν)haslargeside-
w
lobesindelay(range)asFig. 2shows. ThecolorbarvaluesareindB.Therangeside-
3TheshapeoftheautocorrelationfunctiondependsontheautocorrelationfunctionχΩ(τ,0)forthe
pulseshapeΩ(t). TheGolaycomplementarypropertyeliminatesrangesidelobescausedbyreplicasof
χΩ(τ,0)atnonzerointegerdelays.
x 10−6
4 50
4.5 40
5 30
5.5 20
6 10
Delay(sec)6.5 0
7 −10
7.5 −20
8 −30
8.5 −40
9
−3 −2 −1 0 1 2 3
Doppler(rad)
Figure2.TheambiguityfunctionofaGolaypairofphasecodedwaveformsseparated
intime
lobespersistevenwhenapulsetrainisconstructedinwhichw(t)=w (t)+w (t−T)
x y
is transmitted several times, with a pulse repetition interval (PRI) of T seconds be-
tweenconsecutivetransmissions. Inotherwords,theambiguityfunctionofaconven-
tionalpulsetrainofGolaycomplementarywaveforms,wherethetransmitteralternates
between w (t) and w (t) during several PRIs, also has large range sidelobes along
x y
nonzeroDopplers. ThepresenceofDoppler-inducedrangesidelobesintheambiguity
functionmeansthatastrongreflectorcanmaskanearby(inrange)weaktargetthatis
movingatadifferentspeed. Figure3showsthedelay-Dopplermapattheoutputofa
radarreceiver(matchedfilter),whenaconventional(alternating)pulsetrainofGolay
complementarywaveformswastransmittedover256PRIs. Thehorizontalaxisshows
Doppler and the vertical axis depicts delay. Color bar values are in dB. The radar
scenecontainsthreestationaryreflectorsatdifferentrangesandtwoslow-movingtar-
gets, which are 30dB weaker than the stationary reflectors. We notice that the range
sidelobes from the strong stationary reflectors make it difficult to resolve the weak
slow-movingtargets. ThesensitivityofGolaycomplementarywaveformstoDoppler
effecthasbeenthemainbarrierinadoptingthesewaveformsforradarpulsecompres-
sion.AswewillshowinSection3,thisissuecanberesolvedbyproperlycoordinating
thetransmissionofGolaycomplementarywaveformsovertime.
3. DopplerResilientPulseTrains
Waveformagileradarsarecapableofchangingtheirtransmitwaveformonapulse-
by-pulsebasis,andhencearenotlimitedtotransmittingthesamewaveform(orwave-
x 10−5 N=256, Alternating sequence with output in dB
1.2
90
80
1.3
70
1.4
60
1.5 50
Delay(sec)1.6 40
30
1.7 20
10
1.8
0
1.9
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
Doppler(rad)
Figure3. Delay-Dopplermapattheoutputofaradarreceiver(matchedfilter),when
aconventional(alternating)pulsetrainofGolaycomplementarywaveformsistrans-
mitted.
formpair)overtime.Itisnaturaltoaskwhetherornotitispossibletoexploitthisnew
degreeoffreedomtoconstructaDopplerresilientpulsetrain(orsequence)ofGolay
complementarywaveforms, forwhichtherangesidelobesofthepulsetrainambigu-
ityfunctionvanishinsideadesiredDopplerinterval. Followingthedevelopmentsin
[11]–[13],weshowinthissectionthatthisisindeedpossibleandcanbeachievedby
coordinatingthetransmissionofaGolaypairofphasecodedwaveforms(w (t),w (t))
x y
intimeaccordingto1’sand−1’sinaproperlydesignedbiphasesequence.
Definition2: ConsiderabiphasesequenceP = {p }N−1,p ∈ {−1,1}oflength
n n=0 n
N,whereN iseven. Let1representw (t)andlet−1representw (t). TheP-pulse
x y
trainw (t)of(w (t),w (t))isdefinedas
P x y
1N(cid:88)−1
w (t)= [(1+p )w (t−nT)+(1−p )w (t−nT)]. (10)
P 2 n x n y
n=0
The nth entry in w (t) is w (t) if p = 1 and is w (t) if p = −1. Consecutive
P x n y n
entriesinthepulsetrainareseparatedintimebyaPRIT.
The ambiguity function of the P-pulse train w (t), after ignoring the pulse shape
P
ambiguityfunctionanddiscretizingindelay(atchipintervals),canbewrittenas[13]
1 N(cid:88)−1 1 N(cid:88)−1
χ (k,θ)= [C (k)+C (k)] ejnθ+ [C (k)−C (k)] p ejnθ (11)
wP 2 x y 2 x y n
n=0 n=0
whereθ =νT istherelativeDopplershiftoveraPRI.Thefirsttermontheright-hand-
side of (11) is free of range sidelobes due to the complementary property of Golay
sequencesxandy. ThesecondtermrepresentstherangesidelobesasC (k)−C (k)
x y
isnotanimpulse. Wenoticethatthemagnitudeoftherangesidelobesisproportional
tothemagnitudeofthespectrumS (θ)ofthesequenceP,whichisgivenby
P
N(cid:88)−1
S (θ)= p ejnθ. (12)
P n
n=0
By shaping the spectrum S (θ) we can control the range sidelobes. The question is
P
howtodesignthesequenceP tosuppresstherangesidelobesalongadesiredDoppler
interval. OnewaytoaccomplishthisistodesignP sothatitsspectrumS (θ)hasa
P
high-ordernullataDopplerfrequencyinsidethedesiredinterval.
3.1. ResiliencetoModestDopplerShifts
Let us first consider the design of a biphase sequence whose spectrum has a high-
ordernullatzeroDopplerfrequency. TheTaylorexpansionofS (θ)aroundθ =0is
P
givenby
(cid:88)∞ 1
S (θ)= f(t)(0)(θ)t (13)
P n! P
t=0
wherethecoefficientsf(t)(0)aregivenby
P
N(cid:88)−1
f(t)(0)=jt ntp , t=0,1,2,... (14)
P n
n=0
TogenerateanMth-ordernullatθ = 0,weneedtozero-forceallderivativesf(t)(0)
P
uptoorderM,thatis,weneedtodesignthebiphasesequenceP suchthat
N(cid:88)−1
ntp =0, t=0,1,2,...,M. (15)
n
n=0
This is the famous Prouhet (or Prouhet-Tarry-Escott) problem [17],[18], which we
nowdescribe.
Prouhet Problem: [17],[18] Let S = {0,1,...,N −1} be the set of all integers
between0andN−1.GivenanintegerM,isitpossibletopartitionSintotwodisjoint
subsetsS andS suchthat
0 1 (cid:88) (cid:88)
rm = qm (16)
r∈S0 q∈S1
forall0 ≤ m ≤ M? ProuhetprovedthatthisispossibleonlywhenN = 2M+1 and
thatthepartitionsareidentifiedbytheProuhet-Thue-Morsesequence.
Definition 3: [18],[19] The Prouhet-Thue-Morse (PTM) sequence P = (p )
k k≥0
over{−1,1}isdefinedbythefollowingrecursions:
1. p =1
0
2. p =p
2k k
3. p =−p
2k+1 k
forallk ≥0.
Theorem1(Prouhet)[17],[18]: LetP =(p ) bethePTMsequence. Define
k k≥0
S ={r ∈S={0,1,2,...,2M+1−1}|p =0}
0 r
(17)
S ={q ∈S={0,1,2,...,2M+1−1}|p =1}
1 q
Then,(16)holdsforallm,0≤m≤M.
Wehavethefollowingtheorem.
Theorem2: LetP = {p }2M−1 bethelength-2M PTMsequence. Then,thespec-
n n=0
trumS (θ)ofP hasan(M −1)th-ordernullatθ =0. Consequently,theambiguity
P
functionofaPTMpulsetrainofGolaycomplementarywaveforms(transmittedover
2M PRIs) has an (M −1)th-order null along the zero Doppler axis for all nonzero
delays.
Example1: ThePTMsequenceoflength8is
P =(p )7 = +1 −1 −1 +1 −1 +1 +1 −1. (18)
k k=0
ThecorrespondingpulsetrainofGolaycomplementarywaveformsisgivenby
w (t) = w (t)+w (t−T)+w (t−2T)+w (t−3T)
PTM x y y x
+w (t−4T)+w (t−5T)+w (t−6T)+w (t−7T). (19)
y x x y
Theambiguityfunctionofw (t)hasasecond-ordernullalongthezero-Doppler
PTM
axisforallnonzerodelays.
Figure 4(a) shows the ambiguity function of a length-(N = 28) PTM pulse train
of Golay complementary waveforms, which has a seventh-order null along the zero-
Doppleraxis.ThehorizonalaxisisDopplershiftinradandtheverticalaxisisdelayin
sec. Themagnitudeofthepulsetrainambiguityfunctioniscolorcodedandpresented
indBscale. Azoom-inaroundzero-DopplerisshowninFig. 4(b). Wenoticethatthe
rangesidelobesinsidetheDopplerinterval[−0.1,0.1]radhavebeenclearedout. They
areatleast80dBbelowthepeakoftheambiguityfunction. ThismeansthataPTM
pulse train can resolve a weak target that is situated in range near a strong reflector,
provided that the difference in speed between the weak and strong scatterers is rela-
tivelysmall. Figure5showstheabilityofthePTMpulsetraininbringingoutweak
slow-moving targets (e.g. pedestrians) in the presence of strong stationary reflectors
(e.g. buildings)forthefivetargetscenariodiscussedearlierinFig. 3. Thisexample
clearly demonstrates the value of waveform agility over time for radar imaging. We
notethattheGolaycomplementarysequencesusedforphasecodinginobtainingFigs.
x 10−5 N=256, PTM with output in dB

90
0.8
80
70
1
60
c) 1.2 50
e
s
ay( 40
el
D 1.4
30
20
1.6
10
1.8 0

−3 −2 −1 0 1 2 3
Doppler(rad)
(a)
x 10−5 Zoom around 0 for PTM Sequences

90
0.8
80
70
1
60
c) 1.2 50
e
s
ay( 40
el
D 1.4
30
20
1.6
10
1.8 0

−0.1 −0.05 0 0.05 0.1
Doppler(rad)
(b)
Figure4. Ambiguityfunctionofalength-(N =28)PTMpulsetrainofGolaycomple-
mentarywaveforms: (a)theentireDopplerband (b)Dopplerband[−0.1,0.1]rad