An Introduction to Matched Filters

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1960 IRE TRANSACTIONS ON INFORMATION THEORY 311

An Introduction to Matched Filters*GEORGE L. TURINt

Summary-In a tutorial exposition, the following topics arediscussed: definition of a matched filter; where matched filtersarise; properties of matched filters; matched- filter synthesis andsignal specification; some forms of matched filters.

I. FOREWORDN this introductory treatment of matched filters, anattempt has been made to provide an engineeringinsight into such topics as: where these filters arise,what their properties are, how they ma y be synthesized,etc. Rigor and detail are purposely avoided, o n the theorythat they tend, o n first contact with a subject, to obscurefundamental concepts rather than cla!ify them. Thus,for example, although it is not a ssumed that the reader

is conversant with statistical estimation and hypothesis-testing theories, the pertinent results of these are invokedwithout mathematical proof; instead, they are justifiedby an appeal to intuition, starting with simple casesand working up to greater and greater complexity. Sucha presentation is admittedly not sufficient for a com-pletely thoro ugh understanding: it is merely a prelude.It is hoped that the interested reader will fill in the gapshimself by consulting the cited refer ences at his leisure.Of course, one must always start somewhere, and hereit is with the assumption that the reader is alreadyfamiliar with the elements of probability theory andlinear filter theory-that is, with such things as prob-ability density functions, spectra, impulse response func-tions, transfer functions, and so forth. If he is not, refer -ence to Chapter s 1. and 2 of [57] and Chapter 9 of [7]mill probably suffice.The bibliography, although lengthy, is not meant tobe complete, nor could it be. Aside from the inevitableinability of the author to be familiar with the entireunclassified literature on the subject, there is an extensivebody of classified literature, much of it precedent to theunclassified literature, which of course could not be cited.Of the latter, it must be said regretfully, large portionsshould not have been classified in the first place, orshould long since have been declassified.

No bibliography can satisfactorily reflect the influenceof personal conversations with colleagues on an authorsthoughts about a subject. The present author wouldespecially like to acknowled ge the many he has hadover the years with Drs. W. B. Davenport, Jr., R. M.Fano, I. E. Green, Jr., and R. Price; this, without attrib-uting to them in any way the inadequacies of whatfollows.

II. DEPINITION OF A MATCHED FILTERIf s(t) is any physical waveform, then a filter which ismatched to s(t) is, by definition, one with impulse response

h(r) = Ics(A - T), (1)where lc and A are arbitrary constants. In order to envisagethe form of h(t), consider Fig. 1, in part (a) of which isshown a wave train, s(t), lasting from t, to t2. By reversingthe direction of time in part (a), i.e., letting T = --t,one obtains the reversed train, s( -T), of part (b). Ifthis latter waveform is now delayed by A seconds, andits amplitude multiplied by 16, he resulting waveform-part (c) of Fig. l-is the matched-filter impulse responseof (l).

The transfer function of a matched filter, which isthe Fourier transform of the impulse r esponse, has theformH(j27rf)= CmL(T)e-2fTJ-m

= lc ms s(A - ~)e-~~ dT-03= jce-i2fA s - s(T)eiZfi &,,--m (2)s(t)----$2-t

h(r)=y(A-r)

A-12 .I-A-t,(cl

Fig. l-(a) A wave train; (b) the reversed train; (c) a matched-filter impulse response.1 For so me types of synthesis of h(T)-for example, as the impulseresponse of a passive, linear, electrical networlr-A is constrainedby rcalixability consideration s to the region A 2 tz. If tz = m,approximations are sometimes-necessary. The problems of realizationwill be considered more fully later.* Manuscript, received by the PGIT, January 23, 1960.t Hughes Research Laboratories, Malibu, Calif.


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312 IRE TRANSACTIONS ON INFORMATION THEORY Junwhere the substitution 7 = A - T has been made ingoing from the third to the fourth member of (2). Now,the spectrum of s(t), i.e., its Fourier transform, is:

Z(j27rf) = l: s(t)e--i2rrft cit. (3)Comparison of (2) and (3) reveals, then, that

H(j24) = kX( -j2af)e-iZA = hS*(j2rf)e-i2A. (4)That is, except for a possible amplitude and delay factorof the form Jce-zcfA, the transfer function of a matchedfilter is the complex conjuga te of the spectrum of thesignal to which it is matched. For this reason, a matchedfilter is often called a (conjuga te filter.Let us postpone further study of the characteristicsof matched filters until we have gained enough familiaritywith the contexts in which they appear to know whatproperties are important enough to investigate.III. WHEIZE MATCHED FILTERS ARISE

make the instantaneous power in y.(A) as large as possiblcompared to the average power in n(t) at time A.Assuming that n(t) is stationary, the average power in(t) at any instant is the integrated power under thenoise power density spe ctrum at the filter output. IG(j2,f) is the transfer function of the filter, the outputnoise power density is (N,/2) 1 G(prrf) 1; the outputnoise power is therefore(5

Further, if x(j2rf) is the input signal spectrum, thenX(j2nf)G(j2rf) is the output signal spectrum, and y.(A)is the inverse Fourier transform of this, evaluated at = A; that is,

y,(A) = /- ,S(j27rf)G(j27r~)~~~~f. (6-mThe ratio of the square of (6) to (5) is the power ratiowe wish to maximize:

A. M em-Square CriteriaPerhaps the first context in which the matched filtermade its appearance [31], [55] is that depicted in Fig. 2.

[S m 22 X(j2d)G(j27rf)ei cZfpz -m m 1. (7Nil s -m I GW7d I2 dft

Recognizing that the integral in the numera tor is real [it iy,(A)], and identifying G(j2af) with f(x) and X(j2nf)ei2fAwith g(x) in the Schwarz inequal ity,.r,..,,.,.(1)Fig. a--Pertain ing to the maximization of signal-to- noise ratio.

Suppo se that one has received :I waveform, x(t), whichconsists either solely of a white noise, n(t), of powerdensity N,/2 watts/cp$ or of n(t) plus a signal, s(t)-say a radar retune-of known form. One wishes to deter-mine which of these contingencies is true by operatingon x(t) with a linear filter in such a way tha t if s(t) ispresent, the filter output at some time t = A will beconsiderably greater than if s(t) is absent. Now, sincethe filter has been assumed to be linear, its output, y(t),will be composed of a noise component y,(t), due to n(t)only, and, if s(t) is present, a signal component y,(t),due to s(t) only. A simple way of quantifying the re-quirement that w(A) be considerably greater when s(t)is present than when s(t) is absent is to ask that the filter

one obtains from (7)

Since 1 S(j27rf) /2 is the energy density spectrum of s(l)the integral in (9) is the total energy, E, in s(t). Thenp 5 g. (10II

It is clear on inspection that the equality in (8), andhence in (9) and (lo), holds when f(x) = kg*(x), i.e.,whenG(j2,f) = IcX*(j2af)e-i2 nA. (11

Thus, when the filter is matched to s(t), a maximum valueof p is obtained . It is further easily shown that the equalityin (8) holds only when f(z) = Jcg(x), so the matchedfilter of (11) represents the only type of linear filter whichmaximizes p.Notice that we have assumed nothing about the statis-tics of the noise except that it is stationary and white,with power density N,/2. If it is not white, but ha

2 This is a density spe ctrum: that is, if s(t) is., e.g., a voltage some arbitrary power density spectrum / N(&-/) 1) waveform, X(j27rf) is a voltage density, and its integral from fl derivation similar to that given above [9], [15] leads toto f:! (plus that from -12 to -fl) is the part of the voltage in s(t) the solutionoriginating in the band of frequencies from fl to Iz.3 This is the dou ble-ended density, covering posilive andnegative frequencies. The single-ended physical power density(positive frequencies only) is thus N 0.


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1960 Turin: An Introduction to Matched Filters 313One can convince himself of this intuitively in the fol-lowing manner. If the input, z(t), of Fig. 2 is passedthrough a filter with transfer function l/N(j2af), thenoise component at its output will be white; however,the signal component will b e distorted, now having thespectrum x(j27rf)/N(j27rf). On the basis of our previousdiscussion of signals in white noise, it seems reasonable,then, to follow the noise-whitening filter with a filtermatched to the distorted signal spectrum, i.e., with thefilter IcX*(j2nf)e- /N*(j2af). The cascade of the noise-whitening filter and this matched filter is indeed thesolution (12) .4So far we have considered only a detection problem:is the signal pre sent or not? Suppo se, however, we knowthat the signal is present, but has an unknown delay,t,, which we wish to measure (e.g., radar ran ging). Thenthe first of the output waveforms in Fig. 2 obtains, butthe peak in it is delayed by the unknown delay. In ord erto measure this delay accurately, we should not onlylike the output waveform, as before, to be large at t =A + t, but also to be very small elsewhere.More generally, we may frame the problem in themanner depicted in Fig. 3. A sounding signal of knownform, s(t), is transmitted into an unknown filter, theimpulse response of which is U(T). [In Fig. 2, this filteris merely a pair of wires with impulse response 6(t), theDirac delta function.] At the output of the unknownfilter, stationary noise is added to the signal, the sumbeing de noted by x(t). We desire to operate on x(t) witha linear filter, whose output, y(t), is to* be as faithful aspossible an estimate of the unknown impulse re sponse,perhaps with &me delay A.

Fig. 3-Pertaining to mean-square estimation of an unknown im-pulse response.The un known filter may b e some linear or quasi-lineartransmission medium, such as the ionosphere, the charsc-teristics of which we wish to measure. Again, it may

represent a complex of radar targets, in which case u(7)may consist of a sequence of delta functions with unknowndelays (ranges) and strengths.A reasonable mean-squar e criterion for faithfulness ofreproduction is that the average of the squared differencebetween y(t) and u(t - A), integrated over the pertinentrange of t, be as small as possible; the average must betaken over both the ensemble of possible impulse re-4 The weak l ink in this heuristic argumen t is, of course, thatit is not obvious that an optimization perfor med on the output ofthe noise-whitening filter is equivalent to one perfor med on itsinput, the observed wavefor m; it can be shown, however, thatthis is so.

sponses of the unknown filter, and the en semble of possiblenoises. Using such a criterion [51], one arrives at anoptimum estimating filter which is in general relativelycomplicated. However, when the signal-to-noise ratio issmall, and it is assumed that nothing whatever is knownabout U(T) except possibly its maximum duration, theoptimum filter turns out to be matched to s(t) if thenoise is white, and has the form of (12) for nonwhitenoise. Further, in the important case when the soundingsignal, s(t), is also optimized to minimize the error iny(t), the optimum estimating filter is matched to s(t)for all signal-to-noise ratios and for all deg rees of a prioriknowledge ab out u(r), provided only that the noise iswhite.5B. Probabil istic Criteria

In the preceding discussion, we have confined ourselvesto mean-squar e criteria-maximization of a signal-to-noisepower ratio or minimization of a mean-squar e difference.The use of such simple criteria has the advantage ofnot requiring us to know more than a second-orderstatistic of the noise-the power-density spectrum. But,although mean-square criteria often have strong intuitivejustifications, we should prefer to use criteria directlyrelated to performance ratings of the systems in whichwe are interested, such as radar and communication sys-tems. Such performance ratings are usually probabilisticin nature: one speaks of the probabilities of detection,of false alarm, of error, etc., and it is these which we wishto optimize. This brings us into the realm of classicalstatistical hypothesis-testing and estimation theories.Let us first examine perhaps the simplest hypothesis-testing problem, the one posed at the start of the sectionon mean-squa re criteria: the observed signal, z(t), is either -ue solely to noise, or to both an exactly known signaland noise.6 Such a situation could occur, for example,in an on-off communication system, or in a radar d etectionsystem. Adopting the standard parlance of hypothesis-testing theory, we denote the former hypothesis, no iseonly, by HO, and the alternative hypothesis by H,. Wewish to devise a test for deciding in favor of H, or H,.There are two types of errors with which we are con-cerned: a Type I error, of deciding in favor of H, whenH, is true, and a Type II error, of deciding in favor ofH, when H, is true. The probabilities of making sucherrors are denoted by (Y and p, respectively. For a choiceof criterion, we may per haps decide to minimize theaverage of a and /3 (i.e., the over-all probability of error);this would require a knowledge of the a priori prob-abilities of HO and H,, which is generally available in acommunication system. On the other hand, perhaps onetype of error is more costly than the other, and we maythen wish to minimize an average cost [29]. When, as isoften the case in radar detection, neither the a priori

5 Another approach to this problem of impulse-response estima-tion appears in [25].6 Here, however? we shall not initially restrict ourselves a sbefore solely to additive combinatio ns of signal and noise.


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314 IRE TRANSACTIONS ONprobabilities nor the costs are known, or even definable,one often alternatively accepts the criterion of minimizing0 (in radar: maximization of the probability of detection)for a given, predetermined value of OL false-alarm prob-ability)-the Neyman-Pearson criterion [7].What is important for our present considerations isthat all these criteria lead to the same generic form oftest. If one lets pO(z) be the probability (density) thatif H, is true, the observed waveform, z(t), could havearisen; and pi(z) be the probability (density) that if H,is true, x(t) could have arisen; then the test has theform [7], [29], [33]:

accept H, ifaccept H, if

(13)

Here x is a constant dependen t on a priori probabilitiesand costs, if these are known, or on the predeterminedvalue of OLn the Neyman-Pearson test; most importantly,it is not dependent on the observation x(t). The test (13)asks us to examine the possible causes of what we haveobserved, and to determine whether or not the observa-tion is X times mor e likely to have occurred if H, i s truethan if H, is true; if it is, we accept H, as true, and ifnot, we accept H,. If X = 1, for exampl e, we choose thecause which is the more likely to have given r ise to s(t).A value of h not equal to unity reflects a bias on the partof the observer in favor of choosing one hypothesis orthe other.Let us assume now that the noise, n(t), is additive,gaussian and white with spectral density N,/2,3 andfurther that the signal, if present, has the known fo rms(t - to), t, I t < to + T, where the delay, to, and thesignal duration, T, are assumed known. Then, on ob-serving x(t) in some observation interval, I, which includesthe interval t, 5 t 5 t, + T, the two hypotheses con-cerning its origin are:

H o : x(t) = n(t), t in IH, : x(t) = s(t - to) + n(t), t in

Now, it can be shown [57] that the probability densityof a sample, n(t), of white, gaussian noise lasting froma to b may be expressed as7p(n) = Icesp[-$[q2(1)cG], (15)

where N,/2 is the double-ended spectral density of thenoise, and 1~ s a consta,nt not dependent on n(t). Hencethe likelihood that, if Ho is true, the observation x(t)could arise is simply the probability (density) that thenoise waveform can assume the form of x(t), i.e.,p&) = k exp [ -$; l x2(t) dt] ,

7 The space on which this probability density exists must becarefully defined, but the details of this do not concern us here.

(16)

INFORMATION THBORY Jthe region of integration being, as indicated, the obsetion interval, I. Similarly, the likelihood t,hat, if Htrue, x(t) could arise is the probability density that noise can assume the form n(t) = z(t) - s(t - to),

+ $- / s(t - &)x(t) dt - $ ,0 I 01where we have denoted $I s(t - t,) dt, the energythe signal, by E.On substituting (16) and (17) in (13) and taking logarithm of both sides of the inequality, the hypothetesting criterion becomes

accept H, if y(to) > X 7accept H, if y(tJ < Xwhere we have set

y(hJ = j 4t - to)dt) dt Iand

X = $ log x + 2E.Changing variable in (19) by setting T = to - t,obtains

y(k) = so s(- ~)x(to - T) dr.-TBut one immediately recognizes this last as the outat time to, of a filter with impulse response g(T) = s(-in response to an input waveform x(t). The filter specified is clearly matched to s(t), and the optimsystem called for by (18) therefore takes on the fof Fig. 4.

6a t----inX(i) FILTER y (11 SAMPLER__) (IMPULSE V (SAMPLE B THRESHOLD

RESPONSE: AT MO) ILEVEL; A) - McSoSC-TI )

Fig. 4-A simple radar dctectiou system.

8 Notice that s( -7), hence g(7), is nonzero only in the inte-7 5 7 _< 0,; hence the limits of integration. In orde r to reg(7), a delay of A > 2 must be inserted in g(T)-see (I) and foo1. This introduce s an equal delay at the filter or ltput, which then be sampled at time t 0 + A, rather than at t O.Notice also that I could be obtained by literally follothe edicts of (19). That is, one could mlllt,iply the incomin g wform, z(t), by a stored replica of the signal waveform, s(t), d&by to; the product, integrated over the observation intervalg(Eo), and is to be compared with the threshold, X. Slrch a deteis called a correlation detector. WC shall in this paper, howerefer solely to the matched-filte r versions of our solutions, the understandin g that th ese may be obtained by correlatechniques if desired.


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1960 Turin: An Introduction to Matched Filters 315The similarity between the solution represented inFig. 4 and the one we obtained in connection with Fig. 2is apparent: the matched filter in Fig. 4 in fact maximizesthe signal-to-noise ratio of y(t) at t = t,, the samplinginstant. That a mean-square criterion and a probabilisticcriterion should lead to the same result in the case ofadditive, gaussian noise is no coincidence; there is an

intimate connection between the two types of criteria inthis case.So far, we have considered that the signal componentof z(t), if it is present, is known exactly; in particular,we have assumed that the delay to is known. Supposenow that the envelope delay of the signal is known, butnot the carrier phase, 0 [50]. Assuming that a probabilitydensity distribution is given for 0, (13) reduces to

s22P~(x~P(~) d eaccept H, if PO(X) > A . (22)

accept H, otherwiseHere pl(x/O) is the conditional probability, given 8, thatif H, i s true x(t) will arise; p(0) is the probability densityof 8. If 0 is completely random, so p(0) is flat, then car ryingthrough the comput,ations for the case of white, gaussiannoise leads to the intuitively expected result that anenvelope detector should be inserted in Fig. 4 betweenthe matched filter and the sampler [38], [57]. Note, how-ever, that the matched filtering of x(t) is still the core ofthe test.If the envelope delay is also unknown, and no prob-ability distribution is known for it a priori other thanthat it must lie in a given interval, Q, a good test is [7]:

s22max PM~)P(~) deaccept H, if LA > X PO(Z) (23)

accept H, otherwisewhere it is implicit that the integral in the numeratordepends on a hypothesized value of the envelope delay.For a flat distribution of 0, (23) yields a circuit in whichthe envelope detector just inserted in Fig. 4 remains, bu tis now followed not by a sampler, but by a wide gate,open during the interval Q. For all values of l,, wit,hinthis interval, this gate passes the envelope of I to thethreshold; if the threshold is exceeded at any instant,then the maximu m value of the gate output will alsosurely exceed it, and, by (23), H, must then be accepted.

9 Here, for lack of knowing the true envelope delay, we haveessentially used test (22), in which we have assumed that the en-velope delay has that value which ma ximizes the integral in thenumerator. This procedure is part and parcel of the estimationproblem, which we have already considered briefly in the discussionof mean-sq uare criteria, and which we shall later bring into thepresent discussion on probabilistic criteria.

As before, a matched filtering operation is the basic partof the test.We have been discussing, mostly in the radar context,the simple binary detection problem, signal plus noiseor noise only? Clearly, this is a special case of the generalbinary detection problem, more germane to communica-tions, signal 1 plus noise or signal 2 plus noise?, wherewe have taken one of the signals to be identically zero.Let us now address ourselves to the even more generaldigital communicati ons problem, whi ch of .M possiblesignals was transmitted?Let us first consider the case in which the forms ofthe signals, as they appear at the receiver, are knownexactly. In this situation, a good hypothesis test, of which(13) is a special case, is of the form:

accept t,he hypothesis, H, , for which1. (24)~~,p,(x) (m = 1, . . . , M) is greatest

Here p,(x) is the probability that if the mth signal wassent, z(t) will be received. Th e pns are constants, in-dependen t of z(t), which are determined solely by thecriterion of the test. If the criterion is, for example,minimization of the over-all probability of error, EL, isthe a priori probability of transmittal of the mth signal[56], [57]. The pms may possibly also be related to thecosts of making various types of errors. If neithe r a prioriprobabilities nor costs are given, the p ms are generallyall equated to unity, and the test is called a maximum-likelihood test [7].If the noise is additive, gaussian, and white with spectraldensity N,/2, then following the reasoning which led to(17), we have

x2(t) dt + + y,(O) - $ , (25)0 01where, letting s,(t) be the mth signal,

s,(- T)x(t - 7) dr. (26)We have arbitrarily assumed that the signals are receivedwith no delay (this amounts to choosing a time origin),and that they all have the same duration, T. E, is theenergy in the mth signal.Taking logarithms for convenience, test (24) reduces to:

accept the hypothesis, H, , for whichy,(O) + 2 log P, - 2E, > is greatest

(27)

I0 Puote that if a correlation detector (see footnote 8) is to beused here, it must compute the envelope of y(lo) separately for eachvalue of to in Q. That is, (19) and its quadrature component mustbe obtained,, squared and summ ed for each to by a separate multipli-cation and integration process; to is here a parameter (parametrictime). In general, the advantages of using a matche d filter to obtain&to), for all to in a, as a function of real time a re obviously great.In a few circumstances, however-such as range-gated pulsedradar-the correlation technique may be more easily implemented,to a good approximation.


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316 IRE TRANSACTIONS ON~~(0) is, as we have seen, the output at t = 0 of a filtermatched to am, in response to the input z(t). Therefore,the optimum receiver has the configuration of Fig. 5,where the biases B, are the quantities (N,/2) log pL, - 2E,.The output of the decision circuit is the index m whichcorresponds to the greatest of the M inputs.

Fig. 5-A simple M-ary communications receiver.

-bOECISION

If the forms of the signals I, as they appear at thereceiver, are not known exactly, but are dependent on,say, L random parameters, &, . . . , oL, whose jointdensity distribution CJ(&, . . . , 0,) is known, then (24)takes the form:accept the hypothesis, H, , for which 1

i , (28)I.pm(x/& , 1 . , 19,) d0, . . . d0, is greatest]where pm@/&, . a , e,) is the conditional probability,given el, . . s , eL, hat if N, is true, the observed receivedsignal, z(t), will arise.Suppose there is only on e unknown parameter, thecarrier phase, 8, which is assumed to have a uniformdistribution. Then, as in the on-off binary case, it turnsout that the only modification required in the configura-tion of the optimum receiver is that envelope detectorsof a particular type must b e inserted between thematched filters and samplers in Fig. 5.Test (28) may also be applied to the case of multipathcommunications. For example, one may consider the caseof a discrete, slowly varying, multipath channel in whichthe paths are independent, and each is charactcrixed bya random strength, carrier phase-shift, and modulationdelay; if P is the number of paths, there are then 32random parameters on which the signals depend. It can

11Unlike the on-off binary case, in which an y monotone-in-creasing transfer characteristic can be accommo dated in the enve-lope detector by an adjustment of the threshold, in this case, exceptwhen all the biases are equal, the transfer characteristic must b e ofthe log 10 type [7], [38], [50]. In the r egion of small signal-to-noiseratios, however, such a characteristic is well approximated by asquare-law detector.

INFORMATION THEORY Junbe shown [50] that, for a broad class of probability distributions of path characteristics, an optimum-receiver arangement similar to l?ig. 5 still obtains. Instead of simple sampler following the matched filter, howeverthere is in general a more complicated nonlinear devicethe form of which depends on the form of the distribution,404, - . . , e,), inse rted in (28) . If, for instance, it assumed that the modulation delays of the paths aknown, but that the path strengths and phase-shiftshare joint distributions of a fairly general form [50the device combines nonlinear functions of several sampletaken at instants corresponding to the expected patdelays, of both the output of the matched filter and its envelope. Again, if all the paths are assumed a priorto have identical strength distributions, their carriephase-shifts assumed uniformly distributed over (0, 27and their modulation delays assumed uniformly distrbuted over the interval (t,, tb),12 he form of the nonlineadevice which should replace the mth sampler in Fig. is given, to a good approximation, by Fig. 6 [12], [50The exact form of the nonlinear device, F,,, is determineby the assumed path-strength distribution.

Fig. 6-A nonlinear device for use in Fig. 5 when the path delayare unlmown.

The case in which the path modulation delays aknown, and the carrier phase-shifts and strengths varrandomly with time (i.e., are not restricted, as beforto CLslow variations with respect to the signal duration)has also been considered for Rayleigh-fading paths [36Again, at least to an approximation, the results can interpreted in terms of matched filters. The operationperformed on the matched filter outputs now vary wittime, however.In order to use test (28), it is necessary to have parameter distribution, a( &, . . . , 0,). It may be, howevethat a complete distribution of all the parameters is navailable. Then it is often both theoretically and practcally desirable to return to a test based on the tenthat the parameters for which no distribution is giveare exactly known; in place of the true values of theparameters called for by the test, however, estimates aused. We have already come across such a technique connection with (23); in that case, an a priori distributionof one parameter, phase, was used, but another parametemodulation delay, was estimated to be that value whicmaximized the integral in the numerator.g

12These distributions of phase shift and modulation delay mnot, of course, be totally due to random ness in the channel, but mpartially reflect a distribution of error in phase-bas e and time-bassynchronization of transmitter and receiver; the original assumpti oof knowledge of the CXXJC~orms of the signals as they appear the receiver implied perfect synchronization ,


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1960 Turin: An Introduction to &latched Filters 317Let us, as a final topic in our discussion of probabilisticcriteria, then, consider this problem of estimation ofparameters. Clearly, estimation may b e an end in itself,as in the case of radar ranging briefly considered in thesection of mean-square criteria; on the other hand, itmay be to the end discussed in the preceding paragraph.The step from hypothesis testing to estimation mayin a sense be considered identical to that from discrete

to continuous variables. In the hypothesis test (24), forexample, on e inquires, TO which of M discrete causes(signals) may the observation z(t) be ascribed? If oneis alternatively faced with a continuum of causes, suchas a set of signals, identical except for a delay which mayhave any value within an interval, the equivalent testwould clearly be of the form:accept the hypothesis, H, , for which

1 f (29)F(m)p,(x) (m, < m 5 m,) is greatestwhere the notation is the same as that used in (24),except that m is now a continuous parameter. Only o neparameter has been explicitly shown in (29), but clearlythe technique may be extended to many parameters.To consider but one application of (29), let us againinvestigate the radar-ranging problem, in which the obser-vation is known to be given byx(t) s(t to) n(t) to f (30)t in Iwhere s(t) is of known form, nonzero in the interval0 5 t 5 T. Again let us imagine n(t) to be stationary,gaussian and white with spectral density N,/2. Then alittle thought will convince one that the probability thatx(t) will be observed, if the delay h as some specific valuet, in the interval (a, b), is of the form of the right-handside of (17). Placing this in (29), identifying m with t,,and letting p(m) be independen t of m (maximum-likeli-hood estimation) [7], it be comes clear that we seek thevalue of t, for which the integrals (19) and (21) are maxi-mum. Rememberin g our previous interpretation of (al),it follows that a system which estimates delay in thiscase is one which passes the observed signal, x(t), intoa filter matched to the signal s(t), and estimates, as thedelay of s(t), the delay (with respect to a reference time)of the maximum of the filters output. Thus, in optimumradar systems, at least in the ca se of nddit.ive, gaussiannoise, a matched filter plays a central part in both thedetection and ranging operations. This is a dual r81e wehave previously had occasion to note heuristically.gThe application of the technique of estimation of un-known signal parameters for use in a hypothesis test hasbeen considered for ideal multipath communication sys-tems [al], [36], [37], [41], [48], [SO]. In fact, even if suchestimation techniques are not postulated explicitly tostart with, but some a priori distribution of the unknownparameters is assumed, it is on occasion found that opera-tions which may b e interpreted as implicit a posteriori

estimation procedures appear in the hypothesis-testingreceiver [21], [36]. Sgain, matched filters, or operationsvery close to matched filtering operations, arise promi-nently in the solutions.It is perhaps not necessary to emphasize that the pointof the foregoing discussion of where matched filters ariseis not to paint a detailed picture of optimum detectionand estimation devices; this has not been done, nor couldit, in the limited space available. Rather, the point isto indicate that matched filters do appear as the core ofa large number of such devices, which differ largely inthe type of nonlinear operations applied to the matched-filter output. Having made this point, it behooves us tostudy the properties of matched filters and to discussmethods of thei; synthesis, to which topics the remainderof this paper will be devoted.It seems only proper, however, to end this section witha caveat. The devices based on probabilistic criteria inwhich we have encountered matched filters were derivedon the assumption that at least the additive part of thechannel disturbance is stationary, white and gaussian.The stationarity and whiteness requirements are not tooessential; the elimination of the former generally leadsto time-varying filters, and the elimination of the latter,to the use of the noise-whitening pre-filters we havepreviously discussed [ll], [30]. The requirement of gaus-siamless is not as easy to dispense with. We have se enthat, within the bounds of linear filters, a matched filtermaximizes the signal-to-noise ratio for any white, additivenoise, and this gives us some confidence in their efficacyin nongaussian situations. But due care should nonethelessbe exercised in implying from this their optimality fromthe point of view of a probabilistic criterion.

IV. PROPERTIES OF ~~STCHED FILTERSIn orde r to gain an intuitive grasp of how a mat,chedfilter operates, let us consider the simple system of Fig. 7.

s (4 y(t)S*(j2rf) L

SIGNAL-GENERATING MATCHEDFILTER FILTERn(t)

WHITENOISE

Fig. Y-Illustrating the properties of matched filters.

A signal, s(t), say of duration T, may be imagined to begenerated b y exciting a filter, whose impulse r esponse isS(T), with a unit impulse at time t = 0. To this signal isadded a white noise waveform, n(t), the power densityof which is N,/2. The su m signal, x(t), is then passedinto a filter, matched to s(t), whose output is denotedby y(t).This output signal may be resolved into two components,

!-l(t) = ?h(t) + Y7d0, (31)


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318 IRE TRANSACTIONS ONthe first of which is due to s(t) alone, the second to n(t)alone. It is these two components which we wish to study.For simplicity of illustration, let us take all spectra to becentered around zero frequency; no generality is lost inour results by doing this.The response to an input s(t) of a linear filter withimpulse response h(7) is

.I m h(7)s(t - T) dT.-mIf h(7) = s(-T), as in Fig. 7, then

-,(Q = s s( - T)s(t - T) CIT. (32)-mThis is clearly symmetric in f, since

y.(-t) = j-;s(-T))S(-t - T) dT(33)

=sm s(t - T)S(-7) dT = y,(t),-cc

where the second equality is obtained through the substi-tution 7 = t + 7. For the lo w-pass case we are considering,y,(t) may look somethin g like the pulse in Fig. 8. Theheight of this pulse at the origin is, from (32),s(T) dT = E, (34)

where E is the signal energy. By a pplying the Schwarzinequality, (8), to ( 32), it be comes apparent] that 1 y.(t) 1cannot exceed y(O) = E for any t.The spectrum of y.(t), that is, its Fourier transform,is 1 X(j2rf) 1. This is easily seen from the fact that v.(t)may be looked on, in Fig. 7, as the impulse response ofthe cascade of the signal-generating filter and the matchedfilter. But the over-all transfer function of the cascade,which is therefore the spectrum of y.(t), is just ) s(j27rf) 1.The spectrum corresponding to the y.(t) in Fig. 8 willlook something like Fig. 9.In Figs. 8 and 9 we have denoted the widths ofy*(t) and j S(j27rf) 1 by 01and /3, respectively. It has beenshown [13] that for suitable definitions of these widths,the inequality,

CUP a constant of the order of unity, (35)holds. The exact value of the constant depend s on thedefinition of width and need not concern us here.The important thing is that the width of the signalcomponent at the matched-filter output in Fig. 7 cannotbe less than the order of the reciprocal of the signalbandwidth.In view of the above results, one might question theoptimal character of a matched filter. In the precedingsection we saw, for example, that in the radar case withgaussian noise we were to compare y(t) with a thresholdto determine whether a signal is present or absent (seeFig. 4). Now, if a signal component is present in s(t),we seemingly should require, for the purpose of assuring

INFORMATION THEORY

tY,(f)

June

E

fti? a

---we-- 0Fig. B-The signal-com ponent output of (32).

t1st j2d)PPf 0

Fig. O-The spectrum corresponding to y,(t) in Fig. 8.

that the threshold is exceeded, that the signal componentof y(t) be made as large as possible at so me instant. Again,if s(t) has been delayed b y some unknown amount t,,so that the signal component of y(t) is also delayed,we saw that we could estimate t, by finding the locationof the maximum of y(t); in order to find this maximumaccurately, one would i magine we would require that thesignal component of y(t) be a high, narrow pulse centeredon t,. In short, it appears that for efficient radar detectionand ranging we really requir e an output signal com-ponent which looks very much like an impulse, ratherthan like the finite-height, nonzero-width, matched-filter output pulse of Fig. 8. Furthermore, we couldactually obtain this impulse by using an inverse filter,with transfer function l/S(jZrf), instead of the matchedfilter in Fig. 7. (To see this, note from Fig. 7 that theoutput signal compouent wou ld then be the impulseresponse of the cascade of the signal-generating filterand the inverse filter, and since the product of the trans-fer functions of these two filters is identicall y unity, thisimpulse response would itself be an impulse.) We arcthus led to ask, Why not use an inverse filter ratherthan a matched filter?The answer is fairly obvious once one considers theeffect of the additive noise, which WChave so far explicitlyneglected in this argument. Any physical signal musthave a spectrum, S(j27rf), which approache s zero forlarge values of f, as in Fig. 9. The gain of the inversefilter will therefore become indefinitely large as J ---f 03Since the input noise spectrum is assumed to extendover all frequencies, the power in the output noise com-ponent of the inverse filter will be infinite. Indeed, theoutput noise component will override the output signal-component impulse, as may easily be seen by a comparis:mof their spectra: the former spectrum increases withoutlimit as f --f ~0, while the latter is a constant for all f


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1960 Turin: An Introduction to Matched Filters 319Thus, one must settle for a signal-component outputpulse which is somewhat less sharp than the impulsedelivered by an inverse filter. One might look at thisas a compromise, a modification of the inverse filterwhich, while keeping the output signal component asclose to an impulse as possible, efficiently suppressest,he noise outside the signal band. In order to under-stand the nature of this modification, let us write the

signal spectrum in the formx(j2?rf) = / S(j2af) [ emi () , (36)

where #(f) is the phase spectrum of s(t). Then the inversefilter has the form

Now, since the input noise at any frequency has randomphase anyway, we clearly can achieve nothing in theway of noise suppression by modifying the phase charac-teristic of the inverse filter-we would only distort theoutput signal component. On the other hand, it seemsreasonable to adopt as the amplitude characteristic ofthe filter, not the inverse characteri stic l/l S(j2~f) 1of (37), but a characteristic which is small at frequencieswhere the signal is small compared to the noise, and largeat the frequencies where the signal is large comparedto the noise. In particular, a reasonable choice seems tobe 1 s(j27rf) ].I3 Then the comp romise filter is of the formH(j27rf) = 1 X(j27rf) 1ei = S*(jZ?rf), (33)

a matched filter. [The last equality in (38) follows from(36) and the fact that, for a physical signal, j X(j27rf) Iis even in j.]The compromise solution of (38), of course, as we havealready seen, maximizes the height of the signal-com-ponent output pulse with respect to the rms noise output.This maximum output signal-to-noise ratio turns out tobe perhaps the most important parameter in the calcu-lation of the performance of systems using matched filters[20], [as], [33]-[35], [39], [52], [53], [57]; it is given bythe right-hand side of (10) :

2EPO= N (39)0Note that p0 depend s on the signal only through its energy,E; such features of the signal as peak power, time duration,waveshape, and bandwidth do not directly enter theexpression. In fact, insofar as one is considering onlythe ability of a radar detection system or an on-off com-munication system to combat white gaussian noise, itfollows from this observation that all signals which ha vethe same energy are equally effective.14 .

13This is indeed the solution Brennan obtains in deriving weightsfor optimal linear diversity combinati on [2]; here we have essentiallya case of coherent frequency diversity.r* As we shall see, for more complicated communication systemsin which more than one signal waveform may be transmitted,paramete rs go verning the similarity of the signals enter theperform ance calcula tions [20], [52].

One may relate the output signal-to-noise ratio, p,,,to that at the input of the filter. Let the noise bandwidthof the matched filteri.e., the bandwidth of a rectangular-band filter, with the same maximum gain, which wouldhave the same output noise power a s the matched filter-be denoted by B,. Then, for simplicity, one may thinkof the amount of input noise power within the matchedfilter band as being given by Ni, = B,No. Furtherlet the average signal power at the filter input be Pi, =E/T, where T is the effective duration of the signal.Then, letting pi = Pi,/Ni,, (39) becomes

p,, = 2B,Tp; . (40)In this formulation of p,,, it is apparent that the matchedfilter effects a gain in signal-to-noise power ratio of 2B,T.This last result seems at first to contradict our previousobservation that, in the face of white noise, the signalbandwidth and time duration do not directly influence po.There is no contradiction, of course. For a given totalsignal energy, the larger T is, the smaller the inputaverage signal power Pi, is, and hence the smaller piis. Any increase in the ratio of p0 o p% aused by spreadingout a fixed-energy signal is thus exactly offset by adecrease in pi. Similarly, any increase in the ratio ofp,, to pi occasioned b y increasing the signal bandwidthis also offset by a decrease in pi; for, the larger the signalbandwidth, the larger BN, and hence the larger N,,that is, the greater the amount of input noise taken inthrough the increased matched-filter bandwidth.However, in relation to the foregoing argument let ussuppose that one is combating band-limited white noiseof fixed total power, rather than true white noise, whichhas infinite total power. That is, suppose the interferenceis such that its total power Ni, is always caused , malevo-lently, to be spread out evenly over the signal bandwidthB,, whatever this bandwidth is. Then, in (40), pj is nolonger dependent on B,, and it is clearly advantageousto make the signal and matched-filter bandwidths aslarge as possible; the larger the bandwidth, the morethinly the total interfering power must be spread, i.e.,the smaller the value of N, in (39) becomes. In thissituation, of all signals with the same energy, the onewith the largest bandwidth is the most useful.An interesting way of looking at the functions of thepair of filters, &(j2,f) and S*(ja?rf), in Fig. 7 is from thepoint of view of coding and decoding. The impulseat the input to the signal-generating filter has componentsat all frequencies, but their amplitudes and phases aresuch that they ad d constructively at t = 0, and canceleach other out elsewhere. The transfer function S(j2xf)codes the amplitudes and phases of these frequencycomponents so that their sum become s some arbitrarywaveform lasting, say, from t = 0 to t = T, such as isshown in Fig. 10. Now, what we should like to do at thereceiver is to decode the signal, i.e., restore all theamplitudes and phases to their original values. We haveseen that we cannot do this, since it would entail aninverse filter; we compromise by restoring the phases,


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320 IRE TRANSACTIONS ONso that all frequency components at the filter output havezero phase at the same time (t = 0 in Fig. 8) and addconstructively to give a large pulse. This pulse has anonzero width, of not less than the order of the reciprocalof the signal bandwidth, because we are not able to restorethe amplitudes of the components pr operly.

Fig. 10-A signal waveform.

In %od ing, then, we have spread the signal energyout over a duration 2; in Ldecoding, we are able tocollapse this energy into a pulse of the order of PT timesnarrower, where p is some appropriate measure of thesignal bandwidth. The squashing of the energy into ashorter pulse leads to an enhancement of signal-to-noisepower ratio by a factor of the order of BjvT, alreadynoted in comlection with (40).We thus see that the time-bandwidth product of tehsignal or its matched filter, which we shall henceforthdenote by 1M/, is a very important parameter in thedescription of the filter. It also turns out to be an indexof the filters complexity, as can be seen by the followingargument. It is well known [43], that roughly 21Tiv in-dependen t numbers a re sufficient (although not alwaysnecessary) to describe a signal which has an effectivetime duration 7 and an effective ban dwidth T/v. It followsthat the complete specification of a filter which is matchedto such a signal requires, at least in theory, no more than2TW numbers. Therefore, in synthesizing the filter theretheoretically need be no more than 22W ele ments orparamct(ers specified, whence the use of the 77% productas a measure of complexity. (See footnote 20, however.)Hopefully, this section has provided an intuitive in-sight into the nature of matched filters. It is now timeto investigate the problems encoun tered in their synthesis.

V. MATCHED-FILTIGH SYNTHZSISAND SIGNAL SPECIFICATION

In considering the synthesis of matched filters, onemust take account of the edicts of two sets of constraints:those of physical realizability, and those of what mightbe called practical realizability. The first limit what onecould do, at least in theory; the second are more realistic,for they recognize that what is theoretically possibleis not always attainable in practice-they define thelimits of what one can do.The constraints of physical realizability are relativelyeasily given, and may be found in any g ood book onnetwork synthesis [19]. Perhaps the most important for

INFORMA TION THEORYus, at least for electrical filters, is that expressed infootnote I, that the impulse re sponse must bc zero fornegative values of its argument.15 If the signal to whichthe filter is to be matched stopsi.e., falls to zeroand remains there thereafter-at! some finite time t, (seeFig. l), then we have seen that by introducing a finitebut perhap s large delay, A 2 t,, in the impulse response,we can render the impulse response physically realizable.We must of course then wait until t = A 2 t, for thepeak of the output signal pulse to occur; put anothe rway, we camlot expect an output containing the fullinformation about the signal at least until the signal hasbeen fully received. Suppose, however, that t, is infinite,or it is finite but we cannot afford to wait until the signalis fully received before we extract information about it.Then it may easily be shown, for example, that in orderto maximize the output signal-to-noise ratio at someinstant t < t,, we should use that part of the optimumimpulse response which is realizable, and delete that partwhich is not [58]. Thus, if the signal of Fig. l(a) is to bedetected at t = 0, the desired impulse response is pro-portional to that of Fig. 1 (b) ; the best we can do in render-ing this physically realizable is to delete that part of itoccurring prior to 7 = 0. The output signal-to-noiseratio at the in&ant of the output signal peak (in thiscase, t. = 0) is still of the form of (39), but now E mustbe interpreted not as the total signal energy, but onl yas that part of the signal energy having arrived by thetime of the output signal peak. Of course, we are nolonger dealing with a true matched filter.The constraints of practical realizability are notso easy to formulate: they are, rather, based on engi-neering experience and intuition. Let us henceforthassume that we are conccrncd with a true matched filter,i.e., that the impulse response (1) is physically realizable.Even then, we arc aware that the filter may not be practi-cally realizable, because too many elements may b e re-quired to build it, of because excessively flawless elementswould be needed, or beca,use the filter would be toodificult to align or keep aligned, etc. In this light, theproblem of realizing a matched filter changes from Hereis a desirable signal; match a filter to it to Here is aclass of filters which I can satisfactorily build; whichmembers of the class, if any, correspond to desirablesignals? In the former case we might here have neglectedthe question of how it was decided that the given signalis desirable; it would per haps have s&iced merely todiscuss how to realize .the filter. But from the latter pointof view the choice of a practical filter becomes inextricablyinterwoven with criteria governing the desirability of asignal, and one would therefore do well to design boththe filter and the signal together to do the best over-alljob. For this reason it is worthwhile to devote some timeto answering the question, What is a desirable signal?

16 Note that such a constraint is not neces sary for optical filters[5], and filters-such as may bc programmed on a computer-whichuse parametr ic rather t han real time.


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1960 Turin: An Introduction to Matched Filters 321The answer, of course, depends on the application. Letus first consider the case of radar detection and ranging.From the point of view of detection, we noted in thelast section that in the face of white gaussian noise allsignals with the same energy are equally effective, whileif the interfering noise is band-limi ted and is constrainedto have a given total power, then of all signals of the sameenergy those with the largest bandwidth are the mosteffective.From the point of view of ranging there arc at leastthree properties of the signal which we must consider:accuracy, resolution, and ambiguity. Let us first discussthese for the case of a low-pass signal. In this case, thematched-filter output has the form of the pulse in Fig. 8,but delayed by an unknown amount and immersed innoise. In order accurately to locate the position of thedelayed central peak of the output signal pulse, we shouldlike both to make p0 of (39) large and also to make or,the width of this peak, small. This latter requirementimplies making the bandwidth large [see (35)], whether

the noise is truly white, or is of the band-limited, fixed-power variety. Further, if several targets are present, sothat the signal component of the matched filter outputconsists of several pulses of the form of Fig. 8, delayedby various amounts (see Fig ll), then making CY erysmall will allow closely adjacent targets to be resolved.That is, if two targets have nearly the same delay, as,for example, targets 3 and 4 in Fig. 11, making 01smallenough will lead to the appearance of two distinct, peaksin the matched-filter output, rather than a broad hump.

Fig. 11-The signal-componentoutput of a matched filter for aresolvablemultitarget or multipath situation.

The requirements of accuracy and resolution thusdictate a large-bandwidth signal. A more stringentrequirement is that of lack of ambiguity. To explainthis, let us again consider Fig. 8, the output signal com-ponent in response to a single target. As shown, thereis but one peak, so even if the peak is shifted by an un-known delay there is hope that, despite noise, the amountof the delay can be determined unambiguously. However,suppose now that the waveform of Fig. 8 were to havemany peaks of equal height, as would happen if thesignal s(t) were periodic [see (32)]. Then even withoutnoise, it might not be totally clear which peak representedthe unknown target delay; that is, there would be ambig-

uity in the target range, a familiar enough phenomenonin, say, periodically pulsed radars.For the purposes of radar ranging then, what we requirein the low-pass case is a large bandwidth signal, s(t),for which, from (32),y*(t) = Ia s(ds(t + 7) d7 (41)-m

has a narrow central peak, and is as close to zero aspossible everywhere else. The right-hand side of (41) mayalso be written as the real part of2 soe / S(j27rf) j2 ej2 df, (42)

where X(j27rS) is, as usual, the spectrum of s(t). We see,therefore, that we are here concerned with a shaping ofthe energy density spectrum of the signal.In the case of a band-pass signal with random phase,all we have said still holds, but now not with respectto y,(t), but with respect to its envelope; yS(t) itself nowhas a fine oscillatory structure at the carrier frequency.This envelope is expressible as the magnitude of (42)[57]. In other words, we now want the envelope of thematched-filter output, in the absence of noise, to havea narrow central peak and be as close to zero as possibleelsewhere. As before, the minimum width attainable bythe central peak is of the order of the reciprocal of thesignal bandwidth.We have thus far in the present paper neglected thepossibility of doppler shift in our discussion of radarsystems. It is worthwhile to insert a few words on thistopic now; we shall limit ourselves, however, to con-sideration of narrow-band band-pass signals, which arethe only ones for which we can meaningfully speak ofa doppler shift of frequency. Suppose, in addition tobeing delayed, the target return signal may also be shiftedin frequency. It turns out then that whe n additive, white,gaussian noise is present, the ideal receiver should containa parallel bank of matched filters much like that in thecommunicati on receiver of Fig. 5. Each filter is matchedto a frequency-shi fted version of the transmitted signal,there being a filter for each possible doppler shift. (Ifthere is a continuum of possible doppler shifts, we shallsee hat the required conti nuum of filters is well approxi-mated by a finite set, in which the frequency shifts areevenly spaced by amounts of t.he order of l/T, the recip-rocal of the duration of the transmitted signal.) The(narrow-band) outputs of the bank of matched filtersare then all envelope detected and subsequently passedinto a device which decides, according to the edicts ofhypothesis-testing and estimation theories, whether ornot a target is present, and if present, at what delay(range) and doppler shift (velocity).Wow, as in the case of no doppler shift, the detectabilityof the signal is still governed by (39) and (40). But amodification is required in our previous discussion of thedemands of high accuracy, high resolution, and low ambig-


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322 IRE TRANSACTIONS ONuity. Generalizing this discussion to the case of nonzerodoppler shift, it is clear that we require the following:each target represented at the receiver input should exciteonly the filter in the matched-filter bank which corr espondsto the target doppler shift (velocity), and, further, shouldcause a sharp peak to appear in this filters output envelopeonly at a time corresponding to the delay of the target,and nowhere else. In order to state this req uirementmathematically, let us compute the output envelope attime t of a filter matched to a target return with dopplershift 4, in response to a target return with zero dopplershift a nd zero delay. (We lose no generality in assumingthese particular target par ameters.) We first note thatif the (double-sided) spectrum of the transmitted signalis &(j2af), then for the narrow-band band-pass signalwe are considering, the spectrum of the signal after under-going a doppler shift 4 is approximately X[j27r(f - +)]in the positive-frequency region.16 A filter matched to thisshifted signal therefore ha s the approximate transfer func-tion X*[j2,(f - 4)], again in the positive-frequency region.Then the spectrum of the response of this filter to a non-doppler-shifted, nondelayed signal is approximatelyx(j2rf) s*[j2,(f - +)] (f > 0); the response itself, attime t, is the real part of the complex Fourier transform

x(t, 4) = 2 la Li(j27~f)S*[j27r( f - +)]ei2* df. (43)The envelope of the respon se is just j x(t, +) j, and it isthis which we require to be large at t = 0 if 4 = 0, andsmall otherwise. That is! we require that / x(t, 4) / havethe general shape shown in Fig. 12: a sharp central peakat the origin, and small values elsewhere [23], [44], [57].Xote that (42) is a special case of (43) for I#I = 0; hence,the magnitude of (42) corresponds to the intersectionof the I x(4 4) I surface in Fig. 12 with the plane + = 0.It follows therefore from our discussion of (42) that thecentral peak in Fig. 1 2 cannot be narrower than the orderof l/W in the t direction. The use of an uncertainty relationof the form of (35) similarly reveals that the widthof the peak cannot be less than the order of l/T in the4 direction, where T is the effective signal duration [57].(From this is implied a previous statement, that wh en acontinuum of doppler shifts is possible, a good approxi-mation to the ideal receiver involves the use of matchedfilters spaced apart in frequency by I/?; for, a targetreturn which has doppler shift + will cause responses infilters matched to signals with doppler shifts in an intervalat least l/T cps wide centered on 4.) More generally,one may show that the cross-sectional Liarea of thecentral pe ak of the surface in Fig. 12-i.e., the size ofthe (t, 4) region over which the peak has appreciableheight-cannot be less than the order of l/TW [44].Thus, in order to attain a very sharp central peak it is

I6 In the negative-fr equency region, the shifted spectru m has theapproximate form S[j2~(f + I$)].

INFORMATION THEORY Ju

Fig. 12-A desirable jx(t, +)I function.

necessary-but not sufficient, as we shall see-to mathe TW product of the signal very large. Eliminatioof spurious peaks in j ~(t, 4) j away from the (t, 4) origis much more complicated, and has been studied elsewhf231, [451, [571.So much for radar signals. The requirements on comunication signals are somewhat different, in some walaxer and in others more stringent. In the simplest caon-off communication, in which we are concerned wonly one signal, the situation is obviously much lradar, except that in general one need not worry abodoppler shifts of the transmitted signal. In particular, gaussian noise, the detectability of the signal, and henthe probability of error, will in this case depend solon p0 of ( 39) and (40).It should be expressly noted, however, that the probleof ranging which we encountered in the radar case not entirely absent in the communication case. Fexample, in on-off communication, in order to samthe output of the matched filter at the time its sigcomponent passes through its maximum, we must knwhen this maximum occurs. In many transmission medhowever, the transmitted signal is randomly delaythus necessitating a ranging operation, in communicatioparlance called synchronization. We are not interestin this synchronization time per se, as in radar ranginand are therefore not necessarily interested in eliminatinlarge ambiguous peaks in y.(t) of (41). If such subsidiapeaks are of the same height as the central one, as woccur if the signal is periodic, we will be just as hapto sample one of them, rather than the central peOn the other hand, we should not like to synchronize and sample a peak appreciably smaller than the cenpeak, for then we would lose in signal-to-noise raThus, roughly, we require that a spurious peak in ybe either very large or nonexistent.


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1960 Turin: An Introduction to Matched Filters 323We may no longer even require the peaks in y.(t)to be narrow, for we are not interested in determiningthe exact location of the peak, but only in sampling thematched-filter output at or near this peak. Clearly, anerror in the sampling instant will be less disastrous ifthe peak is broad than if it is narrow. On the other hand,a narrow peak is often desirable in a multipath situation.There are oft,en several independent paths between the

transmitter and receiver, which represent independentsources of information about the transmitted signal. Itbehooves us to keep these sources separate, i.e., to beable to resolve one path from another: this is the sameas the radar resolvability requirement, illustrated inFig. 11. From the frequency-domain viewpoint, requiringthe matched-filter output pulse width to be small enought,o resolve the various paths is the sa me as requiring thesignal bandwidth to be large enough to avoid nonselec-tive fading of the whole frequency band of the signal.Such are the considerations which must be given tothe choice of a signal for an on-off communication system,or to each signal individually in multisignal systems.However, in systems of the latter type, such as in Fig. 5,one must also consider the relationships between signals.In the system of Fig. 5, for example, it is not enough tospecify that each signal individually have high energyand excite an output in its associated matched filterwhich consists, say, of a single narrow pulse at t = 0;if this were sufficient, we could choo se all the signals tobe identical, patently a ridiculous choice. We must alsorequire that the various signals be distinguishable. Moreprecisely, if the signal component of x(t) in Fig. 5 is,say, s,(t), then the signal component at the output ofthe ith filter should be as large as possible at t = 0, andat the same instant the signal components of the outputsof all other filters should be as much different fromthe ith filter signal output as possible.The phrase Las much different needs defining: onewishes to specify M signals so that the over-all prob-ability of error in reception is minimized. Unfortunately,this problem has not been solved in general for the phase-coherent receiver of Fig. 5. For the special case of binarytransmission (M L 2), however, it turns out that if thetwo signals are a priori equally probable, one should useequal-energy an tipodal signals, i.e., sl(t) = -.sZ(t) (201,[52]. For then, on reception of the ith signal with zerodelay, the output signal component of its associatedmatched filter at t = 0 is, from (32),

%0x = j-L S:(T) dr = E (i = 1, 2), (44)where E is the signal energy. The kth filter signal outputat t = 0 is clearly

sm_msk(-T)s,(-T) dr = -E (i # M, (45)

which is easily shown to be as much different from(44) as possible.If one considers the band-pass case in which the carrierphases are unknown, we have see n that the optimumsystem in Fig. 5 is modified by the insertion of envelopedetectors between the matched filters and samplers. Inthis case a reasonable conjecture, which has been estab-lished for the binary case [20], 1521, s that the M signalsbe envelope-orthogonal. That is, if the ith signal isreceived, the envelopes of the signal-component outputsof all but the ith filter should b e zero at the instantt = 0. Now, if the spectrum of the ith signal is denotedby S,(j2~f), then the envelope of the signal-componentoutput of the &h filter is:

2 m X*k(j2=f)Xi(j2=f)ei*t df . (46)This, evaluated at t = 0, must therefore be zero for allk # i. There are several ways of assuring that this beso, the most obvious being the use of signals with non-overlapping bands, so that fif(j27rf) X,(j2,f) = 0.Another way in volves the use of signals which are rectnng-ular bursts of sine waves, the sine-wave frequencies ofthe different signals being spaced apart by integralmultiples of l/T cps, where T is the duration of thebursts. A third method is considered elsewhere in thisissue [49].If random multipath propagation is involved and themodulation delays of the paths are known, then we havenoted that the samplers in Fig. 5 should in general sampleboth the matched-filter outputs and their envelopes atseveral instants, corresponding to the various pa th delays.We may in this case conjecture, again for the situationof unknown path phase-shifts (i.e., only envelope sam-pling), that for an optimum set of signals, (46) shouldvanish for k # i, but now at values of t correspondingto all path delay diferences, including zero. For, supposethe ith signal is sent. Then the spectrum

is received, where tl and 0, are, respectively, the modula-tion delay and carrier phase-shift of the lth path [50].

I7 Since he writing of this paper, Dr. A. V. Balakrishn an hasinformed the author that he has proved the following long-standingconjecture concerning the general case of M equiproba ble signals.If the dimensiona lity of the signal space (roug hly2TW) is at leastM - 1, then the signals, envisa ged as points in signal space, shouldbe placed at the vertices of an (M - 1)-dimensional regular simplex(i.e., a polyhedron, each vertex of which is equally distant fromevery other vertex); in this situation, (44) holds for all i, and theright-hand side of (45) becomes -E/(M - 1). The problem of asignal space of smaller dimensiona1it.y than M - 1 has not beensolved in general.I* If the signals sj(t) are of finite duration, then their spectrae.rtend over all f, and nonoverlappi ng bands are therefore notpossible. An approximation is achieved by spacing the band centersby amounts large compared to the bandwidths.


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324 IRE TRANSACTIONS ON INFORMATION THEORY JuThe envelope of the output signal component of the /cthmatched filter is then

12 &e-j lrn X~(j2~f)~i(j2~f)e2 ft-1 df . (48)Since the output envelopes of the matched filters are tobe sampled at t = t, (T = 1, . . . , L), it seems reasonab leto require that in the absence of noise these samplesshould all be zero for k # i. That is, (48) should be zerofor all k f i at all t = t,. For lack of knowledge of theOLs, a sufficient condition for this to occur is that (46)be zero at all t = t, - ft.An extension of this a rgument to the case of unknownmodulation delays considered in connection with Fig. 6leads similarly to a requirement that, for all Ic # i, (46)vanish over the whole interval -A < t 5 A, whereA = t,, - t,, t, and t, being the parameters r eferred toin Fig. 6. This may not be possible with physical signals,and some approximation must then be sought. Here isanother unsolved problem.

A final desirable property of both radar and com-munication signals which is worth mentioning is onearising from the use of a peak-power limited transmitter.For such a transmitter, operation at rated average poweroften points to the use of a constant-ampli tude signal,i.e., one in which only the phase is modulated. A signalof this sort is also demand ed by certain microwave devices.Having thus closed parentheses on a rather lengthydetour into t he problem of signal specification, let usrecall the question on which we opened them. We haddecided that the constraints of practical realizability hadlimited us to the consideration of filters which can, infact, be built. Looking at any particular class of suchfilters-for example, those with less than 1000 lumpedelement#s, with coils having Qs less than 200-we ask,Which members of this class are matched to desirablesignals? We have gotten some idea of what constitutesa desirable signal. Let us now briefly examine a fewproposed classes of filters and see to what extent thisquestion has been answered for these classes.

VI. SOME FORMS OF MATCHED FILTERSIt is not intended to give herein an exhaustive treatmentof all solutions obtained t,o the problem of matched-filterrealization; indeed, such a treatment would be neither

possible nor desirable. We shall, rather, concentrate onthree classes of solutions which seem to have attractedthe greatest attention. Even in consideration of these weshall be brief, for details are adequately given elsewhere,in many cases in this issue.A. Tapped-Delay-Line Filters

Let us first consider matched filters for the class ofsignals generatable as the impulse response of a filter ofthe form of Fig. 13. The spectrum of a signal of thisclass has the form

X(j2nf) = F(j2rf) 2 G,(j2Tf)eeizrfAi,i=o (which is, of course, the transfer function of the filter Fig. 13. It is immediately clear that a filter matched this signal may be constructed in the form of Fig. for the transfer function of the filter shown there is

H(j2af) = F*(jZrf) 2 G*(j2~f)e-i2rfA-Aii=o= S*(j2,f)e-2rfAn. (

The filters of Figs. 13 and 14 are thus candidates for tfilter pair appearing in Fig. 7.

Fig. 13-A tapped-dela y-line signal generator .

Fig. 14-A tapped-del ay-line matched filter.

If F(j2rf) and Gi(j2rf) are assigned phase functionwhich are uniformly zero, then F(j2rf) = F(j2rf) aGt(j2rf) = Gi(j2rf), and the two filters of Figs. 13 a14 become identical except for the end of the delay lwhich is taken as the input. [The same identity of Figs. and 14 is obviously also obtained, except for an unimportant discrepancy in delay, if F(j27rf) and G,(j2,f) halinear phase functions, the slopes of all the latter beequal.] The advantages of having a single filter whican perform the tasks both of signal generation asignal processing are obvious, especially in situations suas radar, wher e the transmitter and receiver are physicalat the same location.Having defined a generic form of matched filter, we still left with the problem of adjusting its characteristic[F(j2nf), G,(j2,f) and Ai, all i] to correspond to a


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1960 Turin: An Introduction to Matched Filters 325sirable signal. A possibility which immediate ly comes tomind is to set

G,(j27rf) = ai (51)

Ai = &For, the signal which corresponds to such a choice has,from (49), the spectrum

i- c1 lzS(j27rf) = j2w z= aze--12rfw2w), / f , 5 w ) (52)i 0, IfI > w>and the signal itself therefore has the form

sin 7r(2Wt - i)s(t) = 2 ai rr(2Wtq-.t= (53)It is well known [43] that any signal limited to the bandj f / L: W can be represented in a form similar to (53),but with the summation running over all values of i;the a;s are in fact just the values of s(t) at t = i/2lV.In (53) we therefore have a band-limited low-pass signalfor whichai , i = 0, ..a ,n (540, other integral values of i

Although s(t) is not uniformly zero outside of 0 < t


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32 G IRE TRAN~SACTIONX ON INFORMATION THEORYfor n = 10. An equivalent band-pass case is obtainedmerely by using a band-pass equivalent of the F(j27rf)in (55); the resulting signal is a sequence of sine-wavepulses of some frequency f,,, each pulse having either 0or 180 phase. The search for a desirable signal is nowsimply the sea rch for a desirable sequence, b,, O,, . . . , O,,of +1s and -1s.

Fig. 15-A signal correspon ding to (55).

Notice that s(t) is a constant-amplitude signal, a prop-erty we have already seen to be important in some situs-tions. For such signals, for the ba nd-pass case implicitin (43), we easily may show that

where T is the signal duration, equal to (n + 1) A inFig. 15. This is the cross section of the 1 x(t, 4) ( surfacein the 4 .direction; it h as the shape shown in Fig. 16.In view of our discussion of the properties of 1 x(t, +) /,this shap e seems reasonably acceptable: a central peakof the minimum possible width, with no seriousspurious peaks.The cross section of ( x(t, I$) ] in the t direction--i.e.,1 x(t, 0) I-can be made to ha ve a desirable shape byproper choice of the sequence b,, b,, . . . , b,,. Classes ofsequences suitable from this point of view have, in fact,been found. The members of one such class [l], [47]have the desirable property that the central pea k has awidth of the order of A [the reciprocal of the bandwidthof s(t)] and is (n + 1) times as high as any subsidiarypeak. The sequence of Fig. 15, of length n + 1 = 11,is in fact a member of this class and has the ] x(t, 0) Ishown in Fig. 17. Other members of the class with lengths1, 2, 3, 4, 5, 7, and 13 have been found. Unfortunately,no odd-length sequences of length greater than 13 exist;it is a strong conjecture that no longer even-length se-quences exist either.If we for the momen t redefine 1 x(t, 0) ) as the responseenvelope of the matched filter to the signal made periodicwith period (n + 1) A, we find t,hat at least for the odd-length sequences we have just discussed, j x(t, 0) 1 retainsa desirable (albeit necessarily periodic) shape; for example,the new, periodic j x(i, 0) j function for the sequencein Fig. 15 is shown in Fig. 18. This property is of interestin radar, where the signal is often repeated periodicallyor quasi-periodically.Another class of bin ary sequences, b,, b,, . . . , b,,which ha ve periodic I x(t, 0) ) functions of the desirable

.June

Fig. IG-Th e 1x(0, +)I function for the signal in Fig. 15.

Fig. 17-Th e jx(t, O)[ function for the signal in Fig. 15.

, Ix(t,O)l(periodic)

7.b tA-I;, ,710Fig. 18-The periodic jx(l, O)l f unction for the signal in Fig. 15

form shown in Fig. 18 are the so-called linear maximal-length shift-register sequences. These have lengths 2 - 1where p is an integer; the n umber of distinct sequencesfor any p is +,(av - 1)/p. where +(z), the Euler phi-function, is the number of integers less than x whichare prime to x [59]. The class of linear maximal-lengthshift-register sequences has been studied in great detail[lo], [16], [17], [42], [44], [46], [59], [61]. In general , theiraperiodic / x(t, 0) I functions leave something to bedesired [26].Other classes of binary sequences with two-level periodic] x(t, 0) I functions such as shown in Fig. 18 have b eeninvestigated [ 181, [60].So far we have discussed the desirability of certainbinary sequences for use in (55) only from the point oview of the form of the 1 x(t, 4) 1 function along thecoordinate axes. No gen eral solution has been given forthe form of the function off the axes, but certain conjectures can be made, which have been roughly corrobo-rated by trying special cases. It is felt [44], at least forlong linear shift-register sequences, that the central peak


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1960 Turin: An Introduction to Matched Filters 327of ) x(t, 4) / indeed has the minimum cross-sectionalarea of l,/TW, and has subsidiary peaks in the planeof height no greater than the order of l/d- ofthe height of the central peak [(n + 1) is the length ofthe sequence].No general conclusions ha ve been reached concerningthe joint desirability, from the viewpoint of (46), ofseveral of these binary codes.

The actual synthesis of matched filters with the charac-teristics of (55) is consider ed in detail elsewhe re in thisissue [24].As a final example of a choice of characteristics of thefilters of Figs. 13 and 14, let us considersin ?rfu -j a,aF(j27rf) = 7 e

G,(j27rf) = 1, all iA, = iA, A > a:

The impulse r esponse of the signal-generating filter in thiscase is a sequence of (n + 1) rectangular pulses of widtha, starting at times t = ;A. (Such a signal, o f course, ismore easily generated by other means.) The ambiguityfunction for the band-pass analo g of this signal, whichis often used in radar systems, is rather undesirable,having pronounced peaks periodically both in the t and 4directions [44], [57], but the signal has the virtue ofsimplicity. The filter wh ich is matched to the signal isoften called a pulse integrator, since its effect is to addup the received signal pulses; alternatively, it is calleda comb filter, since its transmission function, / S(j27rJ) I,has the form of a comb with roughly (A/a) teeth ofwidth l/(n + I) A, spaced by l/A cps [14], [54].B. Cascaded All-Puss Filters

Another way to get a signal-generating and matchedfilter pair is through the use of cascaded elementary all-pass networks [27], [48]. An elementary all-pass networkis defined here as one which has a pole-zero configurationlike that in Fig. 19. Such a network clearly has a transferfunction which has constant magnitude at all frequencies,f, on the imaginary axis.Now, supp ose that many (say, n) elementary all-passnetworks are cascaded, as in Fig. 20, to form an over-allnetwork with the uniform pole-zero pattern of Fig. 21.The over-all transfer function, G(j27rf), will have a con-stant magnitude; its phase will b e approximately linearover some bandwidth W, as shown by the upper curveof Fig. 22. If some of the elementary networks of Fig. 20are now grouped together into a network, A, with transferfunction GA(j2?rf), and the rest into a network, B, withtransfer function GB(j2?rf), both GA(j2~f) and G,(j2af)still have constant magn itudes, but neither necessarilywill have a linear phase function; however, the two phase

22We are again considering he low-pass case or convenience.The band-pass analog s obvious.

jzrf

x 0I 0x 0

Fig. 19-The pole-zero configuration of an elementary all-passnetwork.

I I 1&-J+--J...&-J+...J-IGA(j2rf) II

Gg(j2*f)G(jPTf)

Fig. 20-A cascade of elementary all-pass networks.

t jZrf

X 0X 0

Fig. 21-The pole-zero configuration for the over-all network ofFig. 20.

Fig. 22- -Phase functions corresponding to the networks in Fig. 20.

functions must add to approximately a linear phasewithin the band 0 5 f 5 W (see Fig. 22). Within thisband, the relationship between the A an d B transferfunctions ise i4nme-i$B(I) _ -iZrlla--e , (58)

where to is the slope of the over-all phase function, and#A and #B are the phase functions of the A an d B net-works, respectively.Suppose network A is driven by a pulse of bandwidthW, such as might be supplied by the pulse-forming net-work, F(jZrf), of (57) (a G l/W). Then the output


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328 IRE TRANSACTIONS ON INFORMATION THEORY Junsignal of network A will have the spectrum ,S(j2,f) =F(j2rf)e-? (I. From (58), therefore, over the bandwidthof njw1,

We hence have, at least approximately, a filter pairF( j2,f) G, (j2,f) and F* (j2,f) G, (j2~f) which are matched.The function of the filter F(j2~-f) is merely that ofband limitation; therefore F should be made as simpleas possible, preferably wit,h zero or linear phase. Theproblems of choosing GA(j2~/) so as to obtain a desirablesignal, and of actually synthesizing the filt,ers, are con-sidered elsewhere in this issue [-18].C. Chirp Filters

Let us now consider, as a final class of signals, signalsof the forms(t) = A(t) cos (I,& + 27rf,t + 2akt2), 0 5 t 5 T, (60)where A(t) is some slowly varying envelope. Such signals,whose frequency-more correctly, whose phase deriv-ative-varies linearly with time, have appropriately beencalled chirp signals.In general, the spectrum of a chirp signal has a verycomplicated form. For the case of a constant-amplitudepulse, A(6) = constant, an exact expression has beengiven [4]; this indicates, as one might expect, that inpractical cases t,he spectrum has approximately constant,amplitude and linearly increasing group time delay ofslope 1~over the band f. < f 5 lo + kT, and is approxi-mately zero elsewhere. The corresponding matched filterthen has approximately constant amplitude and linearlydecreasing group time delay of slope --X; over the sameband.Chirp signals and the synthesis of t,heir associatedmatched filters have been studied in great detail [3],[J], [6], [8], [22], [32], [44], [57]. The chief virtue of thisclass of signals is their ease of generation: a simple fre-quency-modulated oscillator will do. Similarly, chirpmatched filters are relatively easy to synthesize. 0 11 theother hand, from the point of view of the ambiguityfunction, j x(t, 4,) 1.e., of radars in which doppler shiftis important--a chirp signal is rat)her undesirable. Thearea in the (t, +) plane over which the ambiguity functionfor a chirp signal with a gaussian envelope is large isshown in Fig. 2 3 [&I, [57]. ,dlong the t and 4 axes thewidth of the \ x(t, 4) 1 surface is satisfa.ctorily small:l/W and l/T, respectively. But the surface, instead ofbeing concentrated nroulld the origin as desired, is spreadout along the line 4 = let; in fact,, the area of concentrationis very much greater than the minimum value, l/TW.This is equivalent to saying that it is very hard to deter-mine whether the chirp signal has been given a timedelay t or a frequency shift lit, a fact which is obviousfrom an inspect,ion of the signal waveform.

f-Wi -

Fig. 23-The ambiguou s area for a chirp signal.VII. Cox!Lusros

It must b e admitted that, in regard to matched filtersthe proverbial state of the art is somewhat less thansatisfactory. We have seen that there are two basic problems to be solved simultaneously: the specificat,ion of desirable signal or set of them, and the synthesis of t)heassociated matched filters. In re gard to the former, TXhave but the barest knowledge of the freedom with whichn-e can constrain the desirability functions (43) and,or (46) and still expect physical signals; we know evenless how to solve for the signals once both desirable a ndallowable constraints are set. It was partially for thesereasons that we attacked a different problem: of a seof physical signals corresponding to a class of filters whichwe can build, which are the most desirable? In e ven thiswe were stopped, except for some special cases.Sor can me assume that me are very sophisticated inthe area of actually constructing filters whi ch we sayon paper, we can build. It has become apparent that theTW product for a signal is a most important parameter;in general, the larger the product, the better the signal.But at the present time the synthesis of filters wit,h T 61products greater that several hundred may be deemedexceptional.Clearly, much mor e effort is needed in this field. Onehopes that the present issue of the TRANSACTIONS willstimulate just such an effort.

VIII. BIBLI~OK~PHY[l] Barker, R. H., Group synchronizing f binary digital systems -in Communication Theory, W. Jackson, Ed., Academ!Press,New York. N. Y.: 1953.PI[31[4[51LB1VIBl

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An Introduction to Matched Filters

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