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Some Invariance Properties of the Wide-Band Ambiguity Function 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•<O,s) I •. The energyin- varient transformationsw-•U(1/w), w-•V(1/w) do not m•x [x•(•,•)l•= {x•(0,•)l • affect ambiguity function behavior along the r=0 profile if signal and filter are interchanged.In general, Eq. 14a will be true for s-• 1, i.e., on the main lobe of the transformationsF(•)=(• • 1)IU(w•) have the the ambiguity function, if Condition 15a is satisfied. effectthat I xf•(0,•)l•= I x•(O,s•)l •. This conclusion also follows from the two-dimensional viii. If f(t)=t-Xu(1/t) and g(t)=t-•v(1/t), then Taylor seriesexpansion of Ix•,(r,s) l • aboutthe point •xo/(0,s) [• = I x,• (0,l/s) I • = 1x• (0,s)I •.
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