Some invarianceproperties of the wide-bandambiguity function
R. A. Altes ESL Incorporated,Sunnyvale, California 94086 (Received 10 December 1971)
Certainchanges in pulseshape do not affectthe resolutioncapability of a sonarsystem. These changes can be expressedas mathematical transformations that leave invariant important properties of thewide-band am- biguityfunction. Such transformations provide a largenumber of signaland filter functions, all of whichsatisfy a givenrange-Doppler resolution requirement. The sonardesigner can then choose the signaland filter that bestsatisfy other system constraints (e.g., ease of signalgeneration and filter synthesis,clutter suppression capability, and peak power).
SubjectClassification: 15.2, 15.3.
INTRODUCTION by examiningthe relativemagnitude of When echoesare detectedby meansof a correlation oo 2 process,the capabilitiesof a sonarsystem can be de- scribedin termsof the wide-bandcross-ambiguity func- Ix•,(•,•; 3,•)12= (3g)•f• tion. Although this function dependsupon the sonar signal-filterpair, there are transformationsof signal = s• (2) and filter that will not affect certain propertiesof the f• u(t)u*Es(t+r)3dt2, ambiguityfunction. Such transformationssuggest the existenceof alter- where s=3/•, r=•(•--•). Again, s>0. If s=l and native waveformsthat will satisfya particularrange- r=0, i.e., if •=3 and ?= •, I x•l • assumesits maximum Doppler resolutionrequirement. Having found alter- possiblevalue for a given signalenergy. The function natives, the designercan then choosethe signal and Ix•(r,s) l 2 is called the wide-bandauto-ambiguity filter that best satisfyother systemconstraints2 function; I x•,(r,s) I • is the wide-bandcross-ambiguity function. Invariance transformations are also important to the studyof animalecholocation. Dolphins, for example, The auto-ambiguityfunction is an important special are capable of using a bewilderingvariety of echo- case of the cross-ambiguityfunction and corresponds location waveforms. Suppose,however, that all these to a systemthat optimizesoutput signal-to-noiseratio. waveformsare related by signal transformationsthat It is possible,however, that interferencecan take forms leave invariant a certain system capability. There other than white noise, and in such cases a cross- would then be strong evidence that the invariant correlationoperation may be advisable? capabilityis of primary interestto the animal. A signal-filterpair is Dopplertolerant •-• if max (3) I. DEFINITIONS
When a signalu(t) is reflectedfrom a planar target for a large range of s values. which has a constantvelocity/•c, where c is the speed A signal-filterpair is Dopplerresolvent • if of sound, the energy normalizedecho has the form •u[-3(tq-•)3, where3= (lq-/•)/(1--/•) and • is the time delayor rangevariable? Since IBI <1, s>0. A corre- lation processorforms the function evenif s is only slightlydifferent from unity. Two signal-filter pairs (u•,vO, (u•,v•) are equally f () [3()3• • Doppler resolventif where v(t) is called the filter function. The asterisk symbolizescomplex conjugation. for any given value of s. In the presenceof additive white noise,maximum In the above definitions, output signal-to-noiseratio is realized when v(t) =•tu[-3(t+•)3. The target parameters(X,i) are gener- max l 2 ally unknown a priori. The receiver must therefore form a seriesof hypotheses(•,?), which can be tested is the maximumoutput powerof a filter with a specific
1154 Volume 53 Humber 4 1973 INVARIANCE PROPERTIES
error on the velocity hypothesis.When passivefilters positivevelocity mismatchtherefore equals the sensi- implementthe correlationprocess, the maximumfilter tivity of I x•(r,s) I • to a negativevelocity mismatch of responseis usedboth to detecta target and to estimate the samemagnitude. If positiveand negativevelocity its range and velocity. Delay hypothesesare auto- mismatchesare equally probable,the interchangeof maticallyimplemented by the passageof time. Targets signaland filter has no effect upon the expected doppler are said to be presentwhen filter output exceedsa resolutioncapability of the system. particular threshold.Velocity is estimatedby com- All the propertieslisted in thispaper can be proven by paringthe maximum responses for a numberof different direct substitutioninto one of the expressionsfor filters, x•(r,s), i.e., max I X•,(r,s)=sif_•v(t)u*[-s(t+r)]dt and choosingthe largest. If the real and imaginary parts of the complex signalu(t) are a Hilbert transformpair, the signalis = V(co)U* e-s'•dco. (7) 2•rs« said to be Analytic.4.8 The Fourier transform U(co) of an Analytic signalis one sided,i.e., U(co)is zero for co<0. For an Analytic signal,Parseval's theorem gives To prove the fourth property,for example,we change variablesin Eq. 7, giving
[X•,,(r,s)12 = U(co)U* e-J"'dco. (6) x =-- v(cos)e-'"co 2•r
II. INVARIANCE PROl•ERTIES Vs) (8) We are interestedin signal transformationsthat do not affectcertain properties of the wide-bandambiguity sothat, for a givenx)alue of s, function. Cross-ambiguityfunctions will be used (9) wheneverpossible, since auto-ambiguityresults im- max 12--max l 2. mediatelyfollow as a specialcase. I. If f(t)=u(-t) and g(t)=v(-t), then IX0/(r,s)l2 V. If f(t)=u(t+T) and g(t)=v(t+T), then = I Xv•(--r, s)]L Timereversal of bothsignal and filter functions leaves invariant the Doppler resolution max Ix0z(r,s)l,=max [X•(r,s)[ 2 propertiesof the wide-band ambiguity function. If positiveand negativerange errors are equallyprobable, the expected range resolution capability is also If F½)= U(w) exp(jcoT) and G(co)=V(co) exp(jcoT) unaffected. are substitutedinto the last expressionin Eq. 7, it is II. If F(w)=S(co) exp(jk logco)and G(w)= V(co) easilyshown that xoz(r,s) = x•,,(r-- T+ T/s, s), so that Xexp(jk logco),where F(co), U(co), G(M, V(co)are the time translation of signal and filter does not affect Fourier transformsof the Analytic signalsf(t), u(t), Doppler resolutionproperties. g(l), v(t), respectively,then IxoAr,s)12=lx(r,s)l VI. Supposethat u(t) and v(t) are either even or The ambiguityfunction is unaffectedwhen signal and odd; eitheru(t)=u(-t), v(t)=v(-t) or u(t)= -u(-t), filter spectraare multipliedby exp(jk logco). v(t)=-v(-t). If f(t)=2«u(t)[«-l-«Sgnt• and g(t) III. If f(t)=k•u(kt) and g(t)=kiv(kt), then = 2•v(t)]-«+« Sgnt-],where I X.(r,s)12= I X.(kr,s) I Sgnt=+1, t_>O, (10) and =--1, l<0, max [X,z(r,s)[2 =max I x•(r,s) 12. then max IXo•(r,s) [2_> max Ix•dr,s) l 2. An energyinvariant scaling of the signaland filter functionsdoes not affectDoppler resolution capability. IV. max, lx,,(r,s) la=max,[X,,v(r,1/s)lL If echoes Part of an even or odd Doppler tolerant wave- inducedby the signalu(t) are crosscorrelated with a form can be discarded, and the waveform will still filter function v(t), then interchangingsignal and filter retain its Doppler tolerance.It also follows that if functionshas the effect shownabove. If s corresponds a given causal waveform f(t) is Doppler resolvent, to a velocitymismatch tic, 1Is correspondsto a velocity then u(t)=2-•[f(t)+f(--t)• is at least as Doppler mismatchof --•c. The sensitivityof I Xvu(r,s)l2 to a resolventas f(t)?
The Journalof the AcousticalSociety of America 1155 R. A. ALTES
The above statement follows from the fact that, if PropertiesVII-X are especiallyuseful when v(t) and u(t) are evenor odd, max ]X•(r,s)[ := Ix,•(O,s) I: (14a)
x•.t(-r, s)=2 v(t)u*[-s(t--r)-ldt and max [X•/(r,s) l: = [X•/(0,s)I •. (14b)
= --2 v(--t)u*[---s(t+r)•dt A sufficientcondition for Eq. 14a is that U(co)and V(co)are real (or imaginery) (15a) --2 v(t)u*[-s(t+r)-]dt, (11) and so that U(w) and V(w) are positive(or negative) semidefinite. (15b) •[-x•(,,s) +xoA-*, s)• If Conditions15a and 15b are true, then I V(co)U*(o•/s){ = V (w)U* (co/s).Since r exp (- jwr) I = 1= exp(-
=x•(•,s). (12) {x"•(r's)I• 2• 1 fo• Iv(•)U* (•) Therefore, =lx,•(0,s) l •. (16) max Ix•(r,s) l By Property II, Eq. 16 is also true if signalsthat satisfyConditions 15 are multipliedby exp(jk log•). -• max I X•(*,s)+Xo•(--*, s)l If Condition15b is dropped,then Eq. 14a will hold true in the neighborhoodof s= 1, but not necessarily _<«max Ixo•(r,s)l+« max I xo•(--*, s)[ over the entire (r,s) plane.If Condition15a is true, then
=max I x•(r,s)!. (13) u(t)=•u*(-t) andv(t)=•v*(-t), (17) a conditionwhich implies that ]x•(r,s) • is symmetric VII. If F(oJ)=oj-1U(1/oJ) and G(co)=o;-1V(1/co),then aboutthe s axis•ø;[x•(r,s)} •= }X•(--r, s)[L Since Ixo•(O,s)12= I x•(o,x?s)[•= I• 1156 Volume 53 Number4 1973 INVARIANCE PROPERTIES real signalsthat are either evenor odd, Eqs. 17 are particular resolutionproperty. The volume of the satisfiedfor both u(t) and U(t), so that Eqs. 14 hold narrow-bandauto-ambiguity function dependsonly for the transformationsin PropertyX, for s in the uponsignal energy•; all signalswith the sameenergy neighborhoodof unity. have the same volume. Narrow-band volume invariance An illustrationof PropertyX, asapplied to real,6dd is conceptuallyuseful because,if the ambiguityis signals,is shownin Figs. 1 and 2. Figure 1(a) showsan "pusheddown" in one area, it is boundto "pop up" optimum signal for velocity discrimination.7 Figure somewhere else. l(b) showsthe ambiguityfunction of the signal in Volume invariance, therefore,can sometimesbe a Fig. l(a). Figure l(c) showsthe r=0 profileof the helpful signaldesign concept. Let us considersome ambiguityfunction. Figure 2(a) showsthe Fourier signal-filtertransformations that do not affect wide- transformof thewaveform in Fig. 1(a). The correspond-band cross-ambiguityvolume. ing ambiguityfunction is shownin Fig. 2(b), and the XIIa. If a wide-band filter function has Fourier r=0 profileof this functionis shownin Fig. 2(c). Note transformV(co) and if the transmittedsignal U(co) that both ambiguityfunctions are symmetricalabout =co-iV(l/w), then the volume under the cross-am- the line r=O, that their maximum value occurs at biguityfunction is 2•rEa, where r=0 for the range of s values shown,and that the Doppleraxis behavioris identicalfor both signals. In order to apply propertiesVII-X to an arbitrary signalf(t), onecan forcethe signalto satisfyCondition 17 by formingthe functionu(t)=2-•Ff(t)-I-f*(-t)]. XI. A large set of transformationsdo not affect the This volumeis exactly the sameas the volumeunder behaviorof I x•(r,s)] 2 for the particularvalues of s the narrow-bandauto-ambiguity function. describedby s=so•, where so can be chosenby the If a signal'sFourier transformsatisfies the relation designer.If •9o=Vo/c< The Journal of the AcousticalSociety of America i157 R. A. ALTES O.•88 1.000 ]..012 (½) b'io. 1(a). A Legendrepolynomial of order25, truncatedat its last zerocrossings. (b) The wide-bandauto-ambiguity function of the signalin (a). (c) The Doppleraxis behaviorof the ambiguityfunction in (b). Letting x=sz, we have and W,•= 2•rEz, as in Property XIIa. 2•r If G(,.,)=,.,-W(1/o,) ana iv(o)=o,-•UO/o,), {V(•0) {"• = {GO/o)I d•= (23) z L2.Jo [V(sz)[ • I•(•)l-d•I•(•)[=•. (21) • Iu )l IF(x/•>{ d•= {F(•)['•,(24) so that W•Wto, • in Property XIIc, •d, if V(•) I• u(•)=•-w(1/•), • U(w), W• Wtt • in Pro•rty XIIb. The abovediscussion h• generallyneglect• r•ge (22) r•iution invari•ce. For a correctvelocity by.thesis • Jo r•ge resolutionis determin• by lx•(r,1)[ •, a cross- 11• Volume53 Number4 1973 INVARIANCE PROPERTIES TIME (a) ]xuu (ø' •) { 9. 0. 990 0. 998 1. 006 1. 014 (c) FIG.2(a). The fourier transform ofthe signal inFig. 1(a), written asa function oftime. (b) The w/de-band auto-ambigu/ty function ofthe signal in (a).(c) The Doppler axis beh•v/or of the ambigu/ty function in (b). correlationfunction. Two signal-filter pairs (u,v)(J,g) changed, the cross-correlation function remains can then be consideredequally range resolvent if invariant? Invarianceproperties of ]x•(7,1)]: are important Many invarianceproperties of the cross-correlationfor Doppler resolution studies by virtueof J. Speiser's functionare already well known. If [/(o•)and U(o•) have transformations?• If U(•o)=o•-tF(logo•)and V(00) thesame spectral phase function, then [ X•,(7,1) l a is in- =•o-tG(logos), then variantto changes in this phase function •:and depends onlyupon the product I V(•o)[ I U(o•)[. The cross-corre- lationfunction is unaffectedby spectralshifts; u(t) [X•(0,s) [: = G(logw)F*(log•--logs)d0og• exp(j•od)and v(t) exp(jo•0t)have the same cross-correla- tionfunction as u(t) andv(t). If signaland filter are = I Za•(--logs,1) l • interchanged,[x• (7,1)I: = [x,•(- 7, 1)I:; theexpected rangeresolution capability of the systemis unaffected if positiveand negative range errors are equally likely. = g(t)f*(t)c i• '•'dt . (25) Time reversingsignal and filter functionshas the same effectas interchax•gingthem (seeProperty I). If the FromEq. 25,it canbe seen that timereversing f(O time-reversedwaveforms u(--t) anf v(--t) are inter- hasno effect u•n •z•O,s) l:. Thefunction f(0 is the The Journalof the AcousticalSociety of America ] 159 R. A. ALTES Fourier transformof F(co),i.e., results also prove useful for the identification of equivalentanimal echolocationsignals. tConstraintsof interestmight include carrier frequency, bandwidth, time duration, maximum power, and clutter (26) suppressioncapability. '[ aE.I. Kellyand R. P. Wishnet,"Matched-Filter Theory for High-Velocity, Accelerating Targets," IEEE Trans. Anytransformation of U(w) that time reverses f(t) will Microwave Theory Tech. 9, 56-69 0965). not affect ]xu•(0,s)[L U(co)--•,.,-•U(1/co)is sucha OR.A. Altes, "Suppressionof RadarClutter and Multi•ath transformation(Property VII). Effectsfor Wide-BandSignal," IEEE Trans.lnf. Theo• The right-handside of Eq. 25 is alsounaffected by 17, 344-346 (1971). 4A. W. Rihaczek,Principles of High-ResolutionRadar a translation,g(t), f(t)---* g(t+k), f(t+k). It follows (McGraw-Hill, New York, 1969). that any transformationof U(co)that makesf(t)---* sj. j. Kroszczynski,"Pulse Compression by Meansof f(t+k) will notaffect I x•(0,s) I:. Accordingto Eq. (26) Linear-Period Modulation," Proc. IEEE 57, 1260-1266 U(,.,)• U(co)exp(jk logco)is such a transformation (1969). eR. A. Aires and E. L. Titlebourn,"Bat Signalsas Optimally (PropertyI1). Doppler Tolerant Waveforms," J. Acoust. Soc. Am. The transformations listed above can be combined 48, 1014-1020 (1970). to provide further invariant transformations.For ?R.A. Aires,"Optimum Waveforms for SonarVelocity example,if ffft):kl•U(--k•t), then fl(t) and u(t) are Discrimination," Proc. IEEE 59, 1615-1617 (1971). SD.Gabor, "Theory of Communication,"J. lEE (London) equallyDoppler resolvent,by PropertiesI and III. 93, 429-457 (1946). UsingProperty V, kx«u(T--klt)also maintains Doppler *Thisobservation may explain the occasionalutilization of resolutionas measured by the auto-ambiguityfunction. "pulse pairs" by several speciesof echolocatingcetaceans. This transformationdemonstrates the equivalenceof tøSignalsthat satisfyEq. 17 are unaffectedby a timereversal two well-publicizedDoppler tolerant phase functions, (except for complex conjugation of both signal and filter impulseresponse, which doesnot affect the ambiguity [og(1--kt)and 1ogkt. •-• If function). Property I shows that the ambiguity functions of such waveforms are symmetrical about the line r=0. nL. Storch,"Synthesis of Constant-Time-DelayLadder Newtworks Using Bessel Polynomials," Proc. IRE occurs at r:0 for velocity errors of interest, then 42, 1666-1671 (1954). Up. M. Woodward,Probability and InformationTheory with Applicationsto Radar (Pergamon,Oxford, England, 1964), h (t) = t-tf• (l/t) exp( jk• log[t[ ) 2nd ed. =kt•t-tu(--kz/t) exp(jk• logI t I ) •aR.J. Purdyand G. R. Cooper,"A Note on the Volumeof GeneralizedAmbiguity Functions,"IEEE Trans. lnf. Theory hasthe same(or better)Doppler tolerance as u(t), by 14, 153-154 (1968). PropertiesVIII and IX. t•D. F. DeLong,Jr., and E. M. Hofstetter,"On the Designof Optimum Radar Waveforms for Clutter Rejection," IEEE IIL CONCLUSION Trans. lnf. Theory 13, 454-463 (1967). •sj. M. Speiser,"Ambiguity functions and continuousgroups," Thispaper has presented some signal transformations presentedat the IEEE International Symp. on Information Theory, Ellenville, N.Y., 1969. with which equivalentsonar waveforms can be gener- lSR.A. Altes, "Methodsof Wtde,-BandSignal Design for Radar ated. Suchtransformations provide a choiceof signals and Sonar Systems," Ph.D. Dissertation, University of and filters for any given systemspecification. The Rochester (1971). 1160 Volume53 Number4 1973