IEEE TRANSACTIONS ON MILITARY ELECTRONICS

quencies corresponding to about 0.15 the wind velocity.A smaller component having a slightly higher velocityis attributed to wind-blown spray. The spectral widthsof the sea and cloud returns each correspond to velocityranges of approximately two knots.

CONCLUSIONS

Measurements have shown that TRADEX radarclutter is derived from two sources.

1) Sea Clutter: Clutter is primarily due to normal seareturn at ranges less than 13 nmi and elevationangles less than 3°.

2) Cloud Clutter: Clutter due to cloud returns canextend in range beyond the 54-nmi ambiguousrange interval for elevation angles less than 50 andcan exist for elevation angles as large as 25° atshort ranges. The large amount of cloud clutter,particularly at UHF, is attributed to coherentscattering from cloud droplets.

The magnitude of both sea and cloud clutter is nearlythe same at UHF and L band. The velocity spectra ofboth sea- and cloud-clutter returns have widths ofabout two knots. The cloud-clutter spectrum is cen-tered at a velocity corresponding to the radial compo-

nent of the surface wind. The sea-return spectrumvelocity is approximately 0.15 that of the cloud return.

Cloud clutter magnitude and location vary somewhatwith the density and type of cloud cover. No othersignificant day-to-day variation of radar clutter wasobserved, probably due to the uniformity of wind andsea conditions at the radar site.

ACKNOWLEDGMENT

The author wishes to thank L. A. Blasberg, RadioCorporation of America, for many helpful discussionsand suggestions during this study. The assistance ofD. Batman and J. E. M\lorriello in programming compu-tations used for this study is greatfully acknowledged.

REFERENCES[11 Curry, G. R., Introduction to the TRADEX radar system, TR

357, MIT Lincoln Lab., Lexington, Mass., Jul 1964.[21 Curry, G. R., Propagation measurements made, 3 JanuLary 1963.

Internal Memo., MIT Lincoln Laboratory, Lexington, Mass.,Jan 1963.

[3] Kerr, D. E., Propagation of Short Radio TVaves, New York: Mc-Graw-Hill, 1951.

[4] Skolnik, M. I., Introduction to Radar Systems, New York: Mc-Graw-Hill, 1962.

[5] Battan, L. J., Radar Meteorology, Chicago, Ill.: University ofChicago Press, 1959.

[6] U. S. Air Force, Handbook of Geophysics, New York: Macmillan,1960.

On Detection and Estimation of Wave Fieldsfor Surveillance

HARRY URKOWITZ, SENIOR MEMBER, IEEE

Abstract-This paper considers the detection and estimation of a

signal field in the presence of a noise field. The wave field, which is a

continuous space-time function, is converted into a discrete set oftime functions by an array of transducer elements which convert thephysical field quantities into other quantities appropriate for process-ing. The resulting set of time functions makes up a vector randomprocess. A generalization of the one-dimensional Karhunen-Loeveexpansion applied to the vector random process yields a series rep-resentation with uncorrelated coefficients. The effects of complexelement weighting and of internal noise are considered in describingthe noise and signal vector processes. If the noise field is Gaussian,the conditional probability density functions of the vector processes,under the hypotheses of noise alone and of signal pulse noise, are

straightforwardly written, leading directly to the likelihood ratio fora completely known signal. The operation to obtain a test statisticbased upon the likelihood ratio is interpreted as a set of filteringoperations, time-varying in the general case where the noise field isnot wide-sense stationary. When the noise field is wide-sense

Manuscript received July 3, 1964.The author is with General Atronics Corp., Wyndmoor, Pa.

stationary, the field may be described by a spectral density matrixwhose elements are the cross-spectral densities of the total noise atthe transducers taken in pairs. The operation to obtain the teststatistic is now interpreted as a set of filtering operations describedby a filtering matrix. This filtering matrix is given by the product ofthe inverse transposed noise spectral density matrix and the matrixof signal transforms. This result is a generalization of a similar resultfor one-dimensional waveforms.

Various special cases are considered, including that in which ex-ternal noise is negligible compared to internal noise and in which thesignal components differ only in time delay. In this case, the result isa simple weighed element phased array.

It is shown that an appropriately defined signal-to-noise ratio(SNR), even when the noise field is not wide-sense stationary, ismaximized by the optimum processing considered above.

When the signal process is of known form but contains unknownparameters, the formulation enables one to obtain the test statisticwith maximum likelihood estimates of the signal parameters. As ex-pected, the processing is the same as the previously described proc-essing, iterated for the various possible values of the unknownparameters, leading to estimator-detector receiver structures.

44 January

Urkowitz: Wave Fields Surveillance

INTRODUCTION

r vHERE IS a well developed theory of the detectionand estimation of signals in noise where the signalsand noise are expressed as functions of the single

variable time [1]-[5]. The theory has been most ex-tensively developed for signals of known form inGaussian noise. In many situations, however, natureprovides us with wave fields, which are space-time func-tions. Examples are electromagnetic waves (as in radarand communication) and sound waves (as in sonar). Ofcourse, one does not deal directly with the wave fieldsbut with the result of converting the physical fieldquantities into other quantities more appropriate forprocessing. It is convenient for the purposes of thispaper to imagine that the continuous space-time func-tion has been converted by an array of transducers intoa discrete (and finite) set of time functions, one timefunction for each element of the array. In other words,the array "samples," in space, the wave field to producethe discrete set of time functions. This set of timefunctions can be considered as a vector process and theinterest in this paper is in Gaussian vector processes,i.e., each component function of the vector process isa scalar Gaussian random process.

In applying decision theory to the detection prob-lem associated with one-dimensional signals, the Kar-hunen-Loeve expansion [1],1 [6] has been used toobtain discrete representations of continuous functions.This representation is particularly convenient becauseit gives a series with uncorrelated coefficients. TheKarhunen-Loeve expansion has been extended to vectorstochastic processes [7]-[9], and this representation isthe one which fits the needs of the present paper. Sucha vector process representation has been applied byvarious authors [10]-[14] to problems in communica-tion systems, chiefly the problem of diversity reception.Balakrishnan [8], using Hilbert space notation, has acompact treatment of a particular detection problemin which only linear operations are allowed in thedetection process. An alternative representation basedupon sampling models has been devised by Kailath[15]-[17] and has been very successful in handlingproblems where the signals have been randomly cor-rupted by passage through a randomly time-varyingtransmission medium. Kailath also points out that hisrepresentation can be extended to vector processes anddoes this specifically in [18]. Again, the emphasis ap-pears to be upon the diversity situation.

In the present paper, the emphasis is on surveillancesituations, such as radar and sonar, in which the signalis of known form with possibly unknown parameters.This is appropriate to the active radar or sonar case.The parameters of delay and Doppler have beentreated extensively; the introduction of the signal as awave field examined by a space extended array makes it

see p. 96.

possible to treat angular parameters as well. Indeed,when maximum likelihood estimation is considered, thecase of unknown angle and/or angle rate will be treated,although briefly. Previous treatments of detection ofwave fields include Stocklin [19], [20] who obtainsprobability densities by means of a space-time samplingplan and Bryn [21] who applies Rice's Fourier seriesrepresentation to the vector processes to obtain a filter-ing interpretation of likelihood ratio detection valid forwide-sense stationary processes of long duration. Bryn'swork is applied to the detection of random signals, i.e.,signals which are sample functions of random processes,appropriate to passive detection, rather than to thedetection of signals of known form with unknownparameters. In addition to these works, special mentionshould be made of some work of Vanderkulk [22 ] whichis not yet publicly available. He obtains optimumtime-invariant filtering operations using a SNR criterionapplied to long samples of wide-sense stationaryprocesses.The detection method discussed in this paper is that

of a hypothesis test whereby the system must, on thebasis of the observed data, choose between the hy-pothesis Ho that noise alone is present and the hypothe-sis H1 that signal plus noise is present.

THE SIGNAL REPRESENTATION AND THE TEST STA-TISTIC FOR A SIGNAL KNOWN COMPLETELY

Let a wave field have placed in it sensing elementsor transducers which convert the physical wave func-tion into quantities which can be processed. The wavefield is assumed to consist of the sum of a deterministiccomponent (called the signal component) of knownform and a random zero-mean component (called thefield noise component). The continuous (in space)physical field is converted by the transducers into afinite set of time functions (see Fig. 1). It is furtherassumed that each element introduces a complexweighting and that each has associated with it a noisesource statistically independent of the incident (that is,external) noise. The total noise component zj(t) at theith element is made up of two parts

zi(t) = Iizi/)(t) + z,12)(t) (1)where Ii is the complex weighting, zi(') (t) is the incidentnoise, and zi(2) (t) is the internal noise. The complexweighting considered here is a simple weighting whichintroduces a multiplication by |I1j and a phase shiftindependent of frequency into the complex envelope ofthe waveform passing through the transducer. Moregeneral weightings are considered in Appendix I, andthe simple complex weighting is explained. Similarly, ifsi(t) designates the incident signal component at theith element, the signal component to be processed isIisi(t). The time function yi(t) available at the ith ele-ment for processing is given by

yi(t) = Ais(t) + zi(t) (2)

1 965 45

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

/ // r -*

sL (t)s+ z(1(t) + yi(t) -Ils1(t)w+ zl(t)

z(2)(t)

</s2(t) + zI +y(t) -I2s2(t) + z2(t)

* *X ~ ~~~S,,t S

Sm zt YM(t) Is (t)+z (t)I| ~~M + ____t ______

EQUIVALENT INPUT 21(t)TIME FUNCTIONS Zm TIME FUNCTION TO BE

L_____ J PROCESSED

TRANSDUCER ELEMENTS WITHCOMPLEX WEIGHTINGS Ij ANDASSOCIATED NOISE GENERATORSz(2)(t). I M

Fig. 1. Conversion of a wave field into a discrete set of time functi

The signal component is written with the elemweight shown explictly so the array parameters canseparated from the signal parameters. This has b(done previously by the author [23]-[24] in deteriring the effects of array parameters in the measuremof angular parameters of a signal.Each signal component is of known form and, at fi

will be treated as exactly known. Later, the signaallowed to have a finite number of unknown paraiters. Each noise component is a zero-mean Gaussrandom process, not necessarily stationary. The crcovariance function Rij(t, t') of zi(t) and aj(t) is

Rij(t, t') = E[zi(t)zj*(t')]= I I/j*Rti()(t, t') + Rij(2)(t, t')

where the superscripts refer to the incident and internoises, respectively, and the asterisk indicatescomplex conjugate. The overall cross-covariance fution is the sum of the individual cross covariances sithe incident and internal noises are assumed to be statically independent. Both si(t) and z(t) are complex -

may be taken either as the analytic signal represertions (pre-envelopes) of these functions or as their ccplex envelopes. (See Kelley, et al. [5], Helstromand Zadeh [25] for a discussion of analytic representions and pre-envelopes.) Very often the internal n(sources are stationary and statistically independenione another. In this case,

Rij (21 (t, t'J) = Rij(2) (t- I') ai

where b5j is the Kronecker delta.It is convenient to use matrix representations for

sets of functions. The set of signals and the set of noare represented, respectively, by the following coluvectors:

Iisl(t)

12S2Qt)js(t) = s,t

_ IMSM(t)- z1(t)

() Z

ZM(t)

The set of covariance functions willthe covariance function matrix

(4)

(5)

be represented by

Rii(t) t') R12(t, t') . ..R2M(t, t')

R,) R21(t, t') R12(t, t') .R207

LRM(t, ti) * * Rmm(t, tI)

The vector Karhunen-Loeve expansion mentionedabove provides a series expansion with uncorrelatedcoefficients for the vector process. The main facts are

reviewed here with the notation to be used throughoutthis paper. Each noise component zi(t) can be expandedover the interval (- T, T) in a series as follows:

00

zi(t) = E Znffiin(t)n=l

(7)

where the fin)(t) are the eigenfunctions of the followingmatrix integral equation:

(8)rT

R(t) t')4(t')dt' = u24((t)-T

(3) In expanded form, (8) is

'nal M Tthe E J Rij(t, t')Oj(t')dt' = 20j(t)I"_ i=j=l -T

in which the eigenvalues are labelled a"2. The followingproperties hold:

M rT

zn = Zfzi(t)4in*(t)dt (10)i=l -T

m T

cj,in(t)qkim*(t)dt = S5nm ( 1)X==1 _T

E[nZmj = on26ttm (12)

The zn are statistically independent random variables,since they are Gaussian and uncorrelated. The signalcomponent I1si(t) can also be expanded in terms of theeigenfunctions in(t), as follows:

m

Iisi(t) = E Sn(4in(t) (13)n.1

(9)

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Urkowitz: Wave Fields Surveillance

The likelihood ratio X [y(t)] is the ratio Pl/po, so thatM rT

Sn = E IIiS(t)kin*(t)dti=1 -T

Now consider the detection situation where one hasto choose between mutually exclusive hypotheseslabelled Ho and H1, respectively. Under Ho, noise onlyis present so that

y(t) = z(t) (15)

Each component yi(t) can be expanded as above in a

series with uncorrelated and, therefore, statisticallyindependent coefficients Yn

Yn = Zn (16)

with Zn described as above, that is,00

yi(t) = E yn4'in(t) (17)n=i

where

(18)

Also, the conditional exception and variance are, re-

spectively,

E[yn Ho] = 0

var[yn Ho] = 0n2 (1 9)

Under H1, both signal and noise are present, so that

(14) X[y(t)] = exp -E Sn 12/n22 n=i

K _

+ Re E ynSn*/unO2n=l

(25)

A convenient test statistic may be obtained by takingthe logarithm of X and keeping only the part involvingthe observed waveforms. Thus, the test statistic L isgiven by

L[y(t)] -L = Re E ynSn*/n2n

(26)

It is shown in Appendix II that, for large K, L may beexpressed as

M rT

L = Re E jyi(t)gi*(t)dti=I -T

where gi(t) satisfies the following equation:M T

E Rik(t, t')gk(t')dtf = Iisi(t)k=1 -T

(27)

(28)

Since (27) shows that L is obtained by a linear opera-

tion on y(t), a filtering operation is suggested. Let hi(t)be the complex impulse response of a linear time-invariant filter whose input is yi(t) and let

hi(t) = gi*(T -t) O < t < 2T

(20)y(t) = z(t) + s(t)Yn = Zn + Sn

This time, the coefficients yn are uncorrelated Gaussianrandom variables with means and variances given by

E[yn H1] = Sn

var[yn H1] = O-n2

The discrete set of coefficients yn is a representationfor the vector process y(t). The joint probability densityfunction of the yn may be considered as the probabilitydensity function of y(t). It is assumed that y(t) is suffi-

ciently well approximated by the first K coefficients.Then, using subscripts 0 and 1 to represent, respectively,Ho and H1, the following conditional probability densityfunctions may be written:

po[Y(t)] = PO[y1, , yK]

exp Y nI12ATn2] (23)[ 2 n=1 -1(3

pl[y(t)] = pl[y, -*, YK]

exp E Iyn - Sn 12/n2] (24)

2n=1

= 0 otherwise (29)

(21) Then, if ri(t) is the response of hi(t), ri(t) is given by

a00

ri(t) = yi(v)hi(t - v)dv

t

=yi(v)gi*(T - t + v)dvg-2T(22)

andT

ri(T) = )yi(v)g*(v)dv-T

It follows, then, thatM

L = Re E ri(T)i=l

(30)

(31)

(32)

This apparently simple way to get the test statisticL is really not simple at all; the solution of (28) is re-

quired. Solutions of (28) and their interpretations are

discussed later. Another point to repeat here is that theassumption is made that each si(t) is known exactly,including the phase if it is a band-pass waveform. Thesituation where the phase and other parameters are

unknown is considered later.

where

1965 47

m T

Yn = E yi(t)lPin*(t)dti-1 T

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

METHODS OF OBTAINING THE TEST STATISTICAn explicit expression for gi(t) can be obtained by

defining an inversion operator. Define an inversionoperator w(v, t) by the following equation:

I Tw(v, t)R(t, t')dt = I6(v - t')

-T(33)

where I is the identity matrix. When this is applied to(28), the result is

) Tg(t) = ww(t, u)s(u)du

-T(34)

The ith component of g(t) isM JT

gi(t) = E WWik(t, U)IkSk(u)duk=l -T

(35)

Then, (27) becomesM Mr T rT

L = Re E Y' j yi(t)Wki(U, t)Ik*sk*(u) dudt (36)k=1 i=1 -T -T

since Wik(t, U) =Wki*(U, t). It should be noted that (36)remains the same if u and t are interchanged. Then, it isnot difficult to show that

L = 2 Re EE Ik*sk* (u)k=1 i=1 -T

[Ju yi(t)Wki(U, t)dt] du

Fig. 2. Obtaining the test statistic in the general case.

5l(t)~ ~ ~ hi(tU>S>

F-~~~~~~~~-s2( Y2(t

Fig. 3. Alternative method of obtaining the test statistic.

(37)y1 (t)

The test statistic may be obtained by the use of a lineartime-variant filter [26]. Such a filter is described by itsresponse h(u, t) at a time u to a unit impulse applied attime t. The response x2(u) of such a filter to an inputx1(u) is given by

L

y2(t)

x2(u) = xi(t)h(u, t)dt_oo

(38) Fig. 4. Passive method of obtaining the test statistic.

Now, define a linear time-variant filter with impulseresponse hki(u, t) as follows:

hki(u, 1) = Wki(U, t) -T < t < u

= 0 otherwise (39)

Then, (37) becomes

L = 2 Re E E Ik*Sk* (u)k=1 i=1 -T

*.[ Yi(t)hki(U t)dt] du (40)

A block diagram of this operation for M=2 is shownin Fig. 2.

It should be emphasized that the filters of this and thefollowing figures are real; they are just described interms of a complex impulse response. The result of allthe operations is real, as is L, so that it is not necessaryto indicate on the figure that a real part is taken.

A slightly different receiver structure may be ob-tained by writing (36) in the following form:

M M rT

L = 2 Re E E Jyi(t)k=1 i=1 -T

Ik*Sk(T)Wik(t, )du dt (41)

The operation of (41) is indicated in block diagram formin Fig. 3. The time-variant impulse responses are theconjugates of the corresponding ones in (39).A completely passive structure may be obtained in

the following way. Write (40) in the form:

M T M

L = 2 Re E JIk*Sk*(U) E: xki(u)duk=l -T i=l

(42)

where Xki(U) is the response of hki(u, t) to yi(t). Now,

y,(t)

Y2 (t)

s*2(t)0 - MULTIPLIER

h ki (0)=Wki (t.u)

48 January

Urkowitz: Wave Fields Surveillance

let Xki(U) be passed through a linear, time-invariantfilter with impulse response given by:

hk (t) = Sk*(T -t) O < I < 2T

= 0 otherwise (43)Then, straightforward substitution yields the result thatL may be obtained by adding the outputs of such fixedfilters evaluated at time T. The structure is shown inFig. 4.An important special case occurs when the signal

components have the same form but differ in time posi-tion. In this case,

hi(t)-ht-=

where ti is the time delay associated with the ith ele-ment. The resulting structure is shown in Fig. 5 forM=2. The set of delays serves to "steer" the arraytoward an apparent signal source.

SPECIAL CASE OF STATIONARITY

Let the various noise functions from the transducerelements be jointly wide-sense stationary so that

Rij(t, t') = Rij(t - t')- Rij(r), all i, j (44)

Also let T be large so that the limits on the integrationin (28) may be taken as (- oc, xc) to get an approxi-mate solution. Equation (28) may be solved by takingthe Fourier transform of both sides. Then

L___ SAMPLEDh2( ARRAY AT TIME Ty2(t- STEERING

h13tu h(t) =s*(T-t), O'<t'<2T=0, OTHERWISE

Fig. 5. Receiver structure for signal componentswhich differ only in delay.

Fig, 6. Receiver structure for wide-sense stationiary noiseand samples of long duration.

1DELAY~

Iht) L

SAMPLEDAT TIME T

M

E Nik(f)Gk(f) = IJSi(f) (45)k=l

where Nik(f) is the cross-spectral power density of zi(t)and Zk(t), Gkf) is the transform of gi(t), and Si(f) is thetransform of si(t). From (29) it may be found that:

Gk(f) = Hk*(f) exp (-jfT) (46)

where Hk(f) is the transform of hk(t). When (46) is sub-stituted into (45), the result is:

h (t) = s*(T-t) ,o < t < 2T= O,OTHERWISE

Fig. 7. Weighted element phased array.

and N(f) is the noise spectral density matrix given by

N11(f)

N(f)=N21=

NMl(f)

N12(f) ... NlM(f) -

N22(f) * * * N2M(f)

*. NMM(f) -

(50)

N

E Nik*(f)Hk(f) = Ii*Si*(f) exp (-jfT) (47)k=l

The inversion of (47) may be accomplished by writingit in matrix form

N*(f)H(f) = S*(f) exp (-jwT) (48)

where

[Hi(f)

H(f) =H2(f)

L HM(f) I

IlS(f)

[I21S2(f)

_ I(f()=

Let r(f) be the inverse of N(f) and let its elements bePik(f). Then, since Nik(f) ==Nki*(f), so that N*(f)=NT(f), the subscript T indicating the transpose, (48)leads to

H(f) = rT(f)S*(f) exp (-jcoT) (51)or

M

Hi(f) = E rki(f)Ik*Sk*(f) exp (-jwT)k=l

(52)

The receiver structure corresponding to (52) is shown(49) in Fig. 6 for M=2. The result shown in (52) and the

structure of Fig. 6 have been obtained by Vanderkulk[22] using a SNR criterion.

1965 49

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

An interesting specialization is obtained when theincident noise field is negligible compared to the internalnoise and the internal noise sources are white with thesame spectral density. Of course, if the incident noise issuch that what appears at each element is statisticallyindependent of all the others, the effect is the same. Letthe spectral density be No. Then, approximately,

Rij(t, u) = Nobi0b(t - u)

wij(t, u) = Bij8(t - u)/No

(53)(54)

The associated structure for M=2 is shown in Fig. 7for the case of signal components which differ only intime delays. The result is simply a weighted elementphased array.

FALSE ALARM AND DETECTION PROBABILITIES

The false alarm and detection probabilities may beobtained if the distribution of the test statistic L isknown. Since L is obtained by the linear operation of(27) upon y(t), it is a Gaussian random variable. Thus,it is necessary to find the mean and variance of L undereach of the hypotheses. Under Ho, yi(t) =zi(t) and themean value is zero. Also under Ho (and under Hi), thevariance of L is given by

- M T -2

var L = E [ReE J zi(t)gi*(t)dti==l -T

=--E E fzi(t)gi*(t)dtM rT 2

+ E J Zi*(t)gi(t)dtiit=l -T

Loeve [6]2 gives the following result for a comscalar process z(t) (our notation):

E[z(t)z(t')] = 0

By extending the proof of this result to a complextor process z(t), one may arrive at the following res

E[z(t)z*(t')TI = R(t, t')

E[Z(t)Z(t')T] = 0

where the subscript T indicates the transpose. (Rthat z(t) is a column vector.) In particular,

E[z1(t)zj(t')] = 0, all i, jTherefore,

1 M rTvar L = - E zi(t)gi*(t)dt

2 i=l T

2

1 M M rT rT- E E RJR R(t, tI)gj*(t)gj(t')

2 i=l j=l T -T

2 See p 467.

(58)- E~- Jf Iisi(t)gi*(t)dt=b2 j-, -T

where (28) has been used. Incidentally, this shows thatthe integral in (58) is real. Under H1, from (27),

M rT

E[L Hi] = Re E E[yj(t)jgj*(t)dti=1 -T

M rT

= Re EJ- si(t)gi*(t)dii=1 -T

M fT= E Iisi(t)gi*(t)dt = 2bl

i=l _T(59)

since it was just established that this integral is real. Ifthe threshold value of L is Lo, the probability of falsealarm Pf (noise alone exceeds threshold) is

1 Lo_ \Pf= -erfc

2 \b\/2(60)

and the probability of detection Pd (noise plus signalexceeds threshold) is

1 /Lo-2b2\Pd = -erfc ( )

2 b-\2/(61)

In (60) and (61), the complementary error functionerfc is defined by

rxerfc(x)-=(2/-\/;) exp(-t2)dt

(55)

iplex

vec-ults:

(62)

SIGNAL-TO-NOISE RATIO

A convenient measure of SNR is the ratio E [L HI ] 2/var L. This is evidently 4b2. It is shown now that thisis the maximum SNR obtainable by a linear operationupon the input. Let the linear operation upon y(t) in theinterval (- T, T) be designated as follows:

M aT

U = Re E yi(t)qi*(t)dti=l -T

(63)

It is now required to find the form of qi(t) so that asuitably defined SNR is maximized. The signal outputwill be defined as the square of the peak value of Uwhen signal only is present. Equivalently, this is the

ecall square of the peak value of E[U| H1], that is, the ex-pected value of U under H1. Now, the real part of acomplex number has a maximum value when the argu-ment is zero. Thus,

M rT 2

(Max E[U H1])2= Iisi(I)qi*(t)dt (64)i=1 -T

The noise contribution will be defined as the variance of

didt' the output when signal and noise are present.v M zT (6

var U =E Re E J z(t) qi* (t) dt (65)i=l -T

50 January

9Jrkowitz: Wave Fields Surveillance

In the light of the discussion of the previous section,1 M M T T

var u = E2 Rij(ty t')qi*(t)qj(t')dtdt' (66)2 i=1 j=1 -T -T

The SNR, then, is given by

(max E[U| H1])2(SINl)

var UM rT 2

2 E Iisi(t)qi*(t)dti=l -T~1 -T (67)

E J fi(t)qi*(t)dti=1 -T

whereM rT

ft(t)-E J Rj(t, t')qj(t')dt' (68)j=l -T

Now, Schwarz's inequality [27] shows that, for A (t)and B(t) meeting certain very mild restrictions, thefollowing is true:

JE Ai(t)Bi(t)dt 2-< Bi(t)dt 2 (69)

Zf Ai(t)j2dt

Schwarz's inequality applies because the operations in(67) are recognizable as inner products and the de-nominator is positive. The maximum value of the left-hand fraction of (69) is obtained when equality holds;this, in turn, occurs when numerator and denominatorintegrands are proportional, that is,

Ai(t)Bi()- Ai(t) 2 (70)where the proportionality constant has been omitted.When this is applied to (67), the result is that themaximum SNR is obtained when

E f Rij(t, t')qj(t')dt' - Iisi(t) (71)

Clearly, this means that

qj(t) = gj) (72)and the test statistic L gives maximum SNR.The above result should cause no astonishment. It is

a generalization of a familiar result for one-dimensional,wide-sense stationary processes. If the vector noiseprocess in the discussion above is wide-sense stationary,the SNR can be expressed in spectral terms, althoughthere is some question as to just when the peak signalvalue occurs. Vanderkulk [22] used this spectral ap-proach to express the SNR for wide-sense stationarynoise and stated the result given by (52), although hedid not give a derivation.

SIGNALS WITH UNKNOWN PHASE ANDOTHER PARAMETERS

ESTIMATOR-DETECTOR RECEIVERSUp to this point in this paper, it has been assumed

that the signal is known exactly, i.e., there are no un-known parameters such as Doppler shift, angle of ar-rival, carrier phase (if the signal is a band-pass signal),etc. When there are unknown parameters, the simplelikelihood ratio criterion must be modified. There are atleast two viewpoints applied in making the modifica-tion. One viewpoint is to consider an unknown parame-ter as a random variable having a probability densityfunction. The likelihood ratio X(y) is now considered asa function of this random parameter and is averagedover its possible values to get a more general likelihoodratio. This is often done when the carrier phase is un-known.3 When the carrier phase is assumed to be uni-formly distributed in the interval (0, 27r), the resultingreceiver structure remains the same except that theenvelope of the result after processing is taken and thenthe output is sampled. The new test statistic L' is theenvelope of the summation in (26), i.e.,

L' = E ynSn*/0'n2 (73)

The other point of view is the principle of maximumlikelihood [1], [5].4 This viewpoint treats the condi-tional probability density pi[y(t)] under hypothesis H1as a function of the unknown parameters. Then, whenthe observation y(t) is inserted into the likelihood ratio,different values of the likelihood ratio will be obtainedfor different assumed values of the unknown parame-ters. It is reasonable to pick the largest numerical valueof likelihood ratio (or of test statistic) as representingthe most likely set of signal parameters which could beresponsible for the particular observed set of timefunctions. This numerical value is then compared to thethreshold for the detection decision. The maximumvalue of the test statistic also gives a set of signalparameters which may be considered as those mostlikely to have produced y(t). These values are themaximum likelihood estimates of the parameters.Parenthetically, it should be remarked that maximumlikelihood estimates are also obtained from the likeli-hood function, which is the name given to Pi [y(t) ]when it is considered a function of the unknown parame-ters.

Except for carrier phase, this paper treats unknownparameters as quantities to be estimated to provide thetest statistic. Let Ol be a vector representing the set ofunknown parameters. Then, the conditional probabilitydensity under hypothesis H1 can be written explicitly interms of a -p' [y(t), a]; so may the likelihood ratio and

8 See Helstrom [2], ch V, for a thorough discussion.' In Kelly [51, see especially pt II.

1965 51

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

* L(al)

-1 L(2)t

SAMPLEDAT TIME T

EACH VECTOR a HAS TWO COMPONENTSEACH COMPONENT HAS TWO POSSIBLE VALUES

Fig. 8. Estimator-Detector structure.

the test statistic L of (26), now written L(a). L(a) isdetermined for the various values of a, and Lmax(a) ischosen as the test statistic. The set a such that

L(a) = Lmax(a)

is the maximum likelihood estimate of az.A glance at (26) or (27) shows that L is an operation

performed upon y(t). The various values of L(a) areobtained by assuming different values of a for theoperation to be performed upon y(t). Thus, the sn areconsidered as functions of Oa

L(oa) = Re E ynSn*(a)/on2 (74)n

Equivalently,M pT

L(ct) = Re E f yi(t)gi*(t, at)dt (75)i=l -T

where gi*(t, a) satisfies the following equation:M rT

E Rfj(T , u)gj(u, a)du = Iiss(t, a) (76)j=1 -T

Note that gi(t) and si(t) have been written explicitly asfunctions of a.

If the process is a band-pass one, and if there is anunknown carrier phase which is considered random, thetest statistic is

L'(a) = YSn*(C)n|

M rT

= E f yi(t)gi*(t, a)dt (77)i=1 -T

As before, (75) can be interpreted as a set of filteringoperations. If the vector has m parameters to estimate

and if the rth component has qr possible values, the totalnumber of operations to be performed for each elementin the array is

m

HJqrr=l

Figure 8 shows the structure of a two-element detectorestimator when there are two parameters to be esti-mated and each parameter has two possible values. Thisfigure requires simple modification if the signal is aband-pass function having unknown carrier phase inaddition to those indicated. If one were to treat thephase as a random variable as mentioned above, theproper receiver structure would be the same as that ofFig. 8 with the addition of an envelope detector in eachoutput line before sampling to give L'(ax).An example of this is shown in Fig. 9 which shows an

estimator-detector structure for a signal emanatingfrom a small target having unknown angle, angle rate,and range rate, together with a random carrier phase.The unknown angle is provided for by variable timedelays. Of course, each specific setting must be heldlong enough to ensure that all of the signal in theinterval (-T, T) is processed. A target having tan-gential motion with respect to a reference point in thearray will show a variable Doppler frequency shiftacross the array. The two inputs to each mixer are farenough apart in frequency to prevent overlap of un-wanted frequency components, and the image rejectionfilters reject the unwanted components. Range rate orradial motion of the target causes an overall Dopplershift, and the possible values of this shift are providedfor in the matched filters following image rejection.The envelope detectors provide L'.

52 January

Urkowitz: Wave Fields Surveillance

t ) E - V( 8, fL~~h(tDjfET 2,d2)

*(-) SAMPLEDf2(6) ATTIME TENV DET = ENVELOPE DETECTORS

®9 - MIXER

fi(gr) HETERODYNING WAVE AT ELEM)NTiTO PROVIDE FOR ANGLE RATE 8r

h(t,fdi) - S*(T-t,fdi) , O <t 2T-0 , OTHERWISE

Fig. 9. Estimator-Detector for unknown angle, angle rate, and range rate,and random carrier phase.

ACCURACY OF THE ESTIMATES

Formulas for the accuracy of the maximum likeli-hood estimates may be obtained by using results ob-tained previously by the author [24], which are exten-sions of results previously obtained by Kelly, et al., [5].These results are valid for the strong signal case.

In (75), yi(t) has a signal component and a noise com-ponent and the signal component, of course, has associ-ated with it the true value of az. We indicate the set oftrue values by aW. Then,

yi(t) - Is (1 a0) + zj'(t)

where zi'(t) is the modified noise including elementweighting and internal noise. When this is substitutedinto (54), there will be two components; the one con-taining the signal is called the generalized ambiguityfunction Q(a, ac) and is given by

M (TQ(CX ct°) = E JI,si(t, ax°)gi*(t, oa)dt (78)

i=l -T

assumption that the noise components are jointly wide-sense stationary and that T is large, so that, approxi-mately,

M OQ(,ot°-) = E J Isi(t) oto)gi*(t7 ot)dti=l -X

M * o

=E JIiSi(fj at°)Gi*(f, as)dfi=i Coo

(80)

(81)

where Gi(f, a) is the Fourier transform of gi(t, a).Taking into account the modifications to include ele-ment weighting and internal noise, inversion of (45)gives

M

Gj(f, a-) = E 1kPik(f)Sk(f, a)k=l

(82)

Therefore,

M M o

Q(ea, °) = E XE IiIk*rki(f)Si(f, aO)Sk*(f, a)df (83)k=l i=l -oo

The variance of a maximum likelihood estimate of thekth parameter in Gaussian noise is given by the negativeof k, k element in the inverse of a matrix C (called theerror matrix) whose elements C,q are given by

1 022Q(aO, a0) dOaida Q(a, a)Ca O (79)

In the general case, where the noise is not wide-sensestationary and T is not large, the solution of (78) to getan explicit form for Q will be very difficult. A somewhatmore explicit form may be obtained by making the

Before one can proceed further, the elements of the in-verse of the spectral density matrix must be known. Thesimplest case is that of uncorrelated noise sources ateach element, all source having flat power density spec-tra of the same intensity. This case has been treated bythe author [24].

It is interesting to compare (78) with (59) and (58).Clearly,

Q(a, ca0) = 2b2 (84)

which, except for the factor 2, is the SNR.

531965

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

DISCUSSION

This paper has treated the detection and parameterestimation of a deterministic wave field in the presenceof a noise field. Not unexpectedly, the theory turns outto be a fairly straightforward vector generalization ofthe detection of one-dimensional signals in noise. Also,not unexpectedly, the necessary processing turns out tobe more extensive. This, perhaps, is the most prominentfeature striking the reader. Each element in the trans-ducer array requires, in general, a number of processingoperations equal to the total number of elements, evenif the signal field has no unknown parameters. If thesignal field has unknown parameters in addition, theprocessing is still more extensive. However, by the exer-cise of ingenuity, it should be possible to reduce thecomplexity considerably.A more serious shortcoming of the theory is the

necessity for knowing completely the noise covariancefunction matrix. In general, the various cross-covari-ance functions may have to be estimated right alongwith the processing, with all the attendant difficulties,both theoretical and practical, of such estimates. Never-theless, as in one-dimensional theory, one at least has adescription of optimum processing.

APPENDIX I

ELEMENT WEIGHTING

Neglecting nonlinearities, the transducer elementsintroduce effects which may be described as time-invariant filtering operations. Let the ith weighting bedescribed by an impulse response vi(t), with Fouriertransform Vi(f). Let zi(l)(t) be the input to the trans-ducer. Then, the transducer output zi'(t) due to theincident noise will be

sN(t) I IS REPLACED BY s,(t) _ It) _

vi(t) IS EFFECTIVE IMPULSE RESPONSEOF TRANSDUCER ELEMENT

A_ F33fh-t IS REPLACED BY h

hj(t) = vi tT-t),O't < 2T,OTHERWISE

Om IS REPLACED BY 3 _

Fig. 10. Modification to provide for transducer filtering action.

before multiplication. In the other figures, an appropri-ate matched filter is placed in cascade with the signalmatched filter.A special case of interest occurs when the element

introduces a weighted time delay independent of fre-quency.

(89)The impulse response hi'(t) of the filter matched tovi(t) is given by

hi'(t) = |II|I (T- t-T) O< t<2T= 0 otherwise (90)

that is, simple weighted time delays are inserted to"line up" all the signal components. For a simple delay,(87) leads to

Rij'(t, u) = Ii3j*Rij(')(t - Tj, u - Tj)In the case of a band-pass signal, the transducer ele-

ment may introduce a phase shift which is nearly inde-pendent of frequency. Then, vi(t) is given by

vi(t) = Ia(t) = Ii exp (iOi)a(t)

Straightforward analysis yields the result that the cross-

covariance function Rij'(t, u) of zi'(t) and zj'(t) isgiven by

wT eTRij'(t,u)= Rij(l)(t1,t2)Vi(t-tIIVj*(u t2)dttidt2 (86)

-T -T

where

(87)

The effect is to introduce a shift of As in the argument ofthe complex envelope of the signal (and noise) passingthrough the transducer and a multiplication by IiI.This case is the one treated in the text.

In this case,

-T < t < T (92)

and

zil'Q) Iizi(')(t)

Then,RTj(e s(ti t2) = E[z(1) (tla)zj* (1) (t2)

The signal comporient si'(t) available is

sT

si'(t) = si(t')vi(t -t')dt'-T

(88)

The modifications to Figs. 2-9 inclusive are indicatedin Fig. 10. The signal component into each multiplierin Figs. 2 and 3 is passed through an appropriate filter

R,ij'(t, u)-IIlj*E[zi(l)(t)zj*(')(u)]= Iilj*Rij(l) (t, u) (93)

It is from (92) and (93) that (2) and (3) are obtained.In any case, the cross-covariance function of the total

noise component of y(t) is given by

Rij(t, u) = Rij'(t, u) + Rij(2)(t, u)

(85) (91)

54 January

vi(t) = ii b(t ti)

T

Zil (t) = zi(i)(tl).vi(t tl)dt'T

sil(t) = Iisi(t)

6Urkowitz: Wave Fields Surveillance

APPENDIX I I

DERIVATION OF (27)

Using (7) and (12), it is found that

Rik(t, t') - Z a2_kn(t)0kn*(tI) (94)

Furthermore, gk(t') may be expanded in an orthonormalseries as follows:

gk(t) = Z gnckn(tf) (95)n

whereM rT

gn = gi(t)cpin*(t)dl (96)i=l -T

Using (13), (28) then becomes

M T

E Z Zno24in(t)gm 00kn*(tf)Pkm(t )dtk-1 n m T

= E Sni() (97)

Using (11), this becomes

E o2n2i(t)ggn = Z sn4Pin(t) (98)n n

so that

Sn = Un2gn (99)

(26) then becomes

L = Re E Yngn (100)n

Using (98) and (18), (100) may be written as

M M rT T

L-= ReZE E E yi(t)c0in*(t)dt 9k*(t')0kn(tf)dt (101)k-1 i=1 n T T

With some rearrangement, this becomes

rTL = Re F. yi(t)

i=1 -T

rT

E J gk*(tf)kkn(tf)dtj pin*(t)dt (102)n -T

When (96) and (98) are used, (27) follows.

GLOSSARY OF PRINCIPAL SYMBOLS

zi(t) =total complex noise component at ith ele-ment in the transducer array

z()(t) =incident noise at the ith elementzi(2) (t) = internal noise at the ith element

z(t) = total noise matrix (a column vector) whosecomponents are zi(t), i= 1,2, . , M

M=number of elements in the arraysi(t) =incident complex signal component at the

ith elementIs=complex weighting of the ith element

s(t) =signal matrix (a column vector) whose ele-ments are Iss(t)

Rij(t, t') =complex cross-covariance function of zi(t)and zj(t)

R(t, t') = covariance function matrix whose elementsare Rij(t, t')

yi(t) = Iss(t) +Zi(t)y(t) -z(t) +s(t) = matrix of observed complex

waveformszn=coefficients of Karhunen-Loeve expansion

of z(t)s= coeciffients of Karhunen-Loeve expansion

of s(t)yn=coefficients of Karhunen-Loeve expansion

of y(t)a,= variance of zn; Karhunen-Loeve eigenvalues

0jin(t) = Karhunen-Loeve eigenfunctions, i= 1,

po [y(t)] = conditional probability density function ofy(t) when the noise field alone is present

pi [y(t)] = conditional probability density function ofy(t) when both signal and noise fields arepresent

L, L [y(t) = test statistic of observed waveforms whensignal is known completely

Li = envelope of La =vector whose elements are unknown signal

parameters to be estimatedL(a) =test statistic when the signal contains un-

known parameters to be estimatedw(t, t') = inversion operator for R(t, t') [see (33)]hki(u, t) =impulse response of time-variant filter for

obtaining the inversion operatorSi(f) =Fourier transform of si(t)S(f) = transform matrix (column vector) whose

elements are IiS(f)N2j(f) = cross-spectral density of zi(t) and zj(t) when

they are wide-sense stationary, i, j = 1,

N(f) =spectral density matrix whose elements areNij(f)

r(f)= [Nq)-'H(f) =matrix of optimum filtering operations (a

column vector) when the noise is wide-sense stationary

b= (1/4) XSNR of the test statistic, signalknown completely

Q(a, a) =generalized ambiguity function involvingthe parameter vector ax and the set of "true"values ao.

1 965 55

IEEE TRANSACTIONS ON MILITARY ELECTRONICS

REFERENCES[1] Davenport, W. B., Jr., and W. L. Root, An Introduction to the

Theory of Random Signals and Noise, New York: McGraw-Hill,1958, ch. 14.

[2] Helstrom, C. W., Statistical Theory of Signal Detection, NewYork: Pergamon, 1960.

[3] Peterson, W. W., T. G. Birdsall, and W. C. Fox, The theory ofsignal detectability, IRE Trans. on Information Theory, vol IT-4,Sept 1954, pp 171-212.

[4] Middleton, D., and D. Van Meter, Detection and extraction ofsignals in noise from the point of view of statistical decisiontheory, J. Soc. Ind. Appl. Math. vol 3, Dec 1955, pp 192-253,vol 4, Jun 1956, pp 86-119 (in two parts).

[5] Kelly, E. J., I. S. Reed, and W. L. Root, The detection of radarechos in noise, J. Soc. Ind. Appl. Math., vol 8, Jun 1960; pp 309-341, vol 8, Sep 1960, pp. 481-501.

[61 Loeve, M., Probability Theory, New York: Van Nostrand, 1955.[7] Kelly, E. J., and W. L. Root, Representations of vector-valued

random processes, Group Rept 55-21, MIT Lincoln Labora-tory, Lexington, Mar 1960.

[8] Balakrishnan, A. V., Estimation and detection theory for multiplestochastic processes, J. Math. Anal. and Appl., vol 1, Dec 1960,pp 386-410.

[9] Thomas, J. B., and L. A. Zadeh, Note on an integral occurring inthe prediction, detection, and analysis of multiple time series,IRE Trans. on Information Theory (Correspondence), vol IT-7,Apr 1961, pp 118-120.

[10] Thomas, J. B., and J. K. Wolf, On the statistical detection prob-lem for multiple signals, IRE Trans. on Information Theory, volIT-8, Jtul 1962, pp 274-280.

[11] Wong, E., Vector stochastic processes in problems of commu-nication theory, Ph.D. dissertation, Princeton University,Princeton, N. J., 1959. (Available for purchase from UniversityMicrofilms, Inc., Ann Arbor, Mich.; No. Mic 59-5243.)

[12] Wolf, J. K., On the detection and estimation problem for mul-tiple nonstationary random processes, Ph.D. dissertation,Princeton University, Princeton. N. J., 1960. (Available forpurchase from University Microfilms, Inc., Ann Arbor, Mich.;No. Mic 60-5074.)

[13] Wolf, J. K., On the detection and estimation problem for multi-

dimensional Gaussian random channels, Tech Rept 61-214,Rome Air Development Center, Griffiss AFB, N. Y., Nov 1961.

[14] Turin, G. L., On optimal diversity reception, IRE Trans. onInformation Theory, vol IT-7, Jul 1961, pp 154-166.

[15] Kailath, T., Correlation detection of signals perturbed by arandom channel, IRE Trans. on Information Theory, vol IT-6, Jun 1960, pp 361-366.

[16] Kailath, T., Communication via randomly varying channels,D.Sc. dissertation, Massachusetts Institute of Technology,Cambridge, 1961. (Photostat copies may be obtained fromMassachusetts Institute of Technology.)

[17] Kailath, T., Optimum receivers for randomly varying channels,Proc. Fourth London Symp. on Information Theory, 1961.

[18] Kailath, T., Optimum diversity combiners, Quart. Prog. Rept.58, MIT Research Laboratory of Electronics, Cambridge, 1960,pp 198-200.

[19] Stocklin, P. L., Limits of measurement of acoustic wave fields,Ph.D. dissertation, University of Connecticut, Storrs, 1962.Available for purchase from University Microfilms, Inc., AnnArbor, Mich; No. 62-4398.

[20] Stocklin, P. L., Space-time sampling and likelihood ratio proc-essing in acoustic pressure fields, J. Brit. IRE, vol 26, Jul 1963,pp 79-91; presented at the Symp. on Sonar Systems, Birming-ham, England, Jul 9-11, 1962.

[21] Bryn, F., Optimum signal processing of three-dimensional arraysoperating on Gaussian signals and noise, J. Acoust. Soc. Am.,vol 34, Mar 1962, pp 289-297.

[22] Vanderkulk, W., Optimum processing of active acoustic arrays,Institute for Defense Analyses Sonar Signal Summer Study,Jul 1963; vol 3, pt 1, issued Dec 1963. (This report is notpuiblicly available.)

[23] Urkowitz, H., C. A. Hauer, and J. F. Koval, Generalized resolu-tion in radar systems, Proc. IRE, vol 50, Oct 1962, pp 2093-2105.

[24] Urkowitz, H., The accuracy of maximum likelihood angle esti-mates in radar and sonar, IEEE Trans. on Military Electronics,vol MIL-8, Jan 1964, pp 39-45.

[25] Dugundji, J., Envelopes and pre-envelopes of real waveforms,IRE Trans. on Information Theory, vol 4, Mar 1958, pp 53-57.

[26] Zadeh, L. A., Frequency analysis of variable networks, Proc.IRE, vol 38, Mar 1950, pp 291-299.

[271 Lorch, E. R., Spectral Theory, New York: Oxford, 1962, p 58.

Matched-Filter Theory for High-Velocity,Accelerating Targets

E. J. KELLY AND R. P. WISHNER

Abstract-Two modifications of the conventional radar theory ofmatched filters and ambiguity functions are discussed. The firstmodification is to make the theory valid for high-velocity targets andwide-band signals, and the second is to include the effects of ac-celeration.' The ability of a radar to measure target acceleration haspreviously been discussed23 in terms of measurement accuracy forisolated targets. This paper is concerned with the form of data pro-cessing necessary for the measurement and with the effects of ac-celeration on the clutter problem.

Manuscript received July 23, 1964.The authors are with Lincoln Laboratory, Massachusetts In-

stitute of Technology, Lexington, Mass. (Operated with support fromthe U. S. Air Force.)

1 Throughout this report, velocity and acceleration mean theradial or apparent velocity and acceleration.

2 Kelly, E. J., The radar measurement of range, velocity and ac-celeration, IRE Trans. on Military Electronics, vol MIL-5, Apr 1961,51-57.

3 Manasse, R., Parameter estimation theory and some applica-tions of the theory to radar measurements, MITRE Tech Ser Rept3, Mitre Corp., Bedford, Mass., Apr 1961.

I. INTRODUCTIONITH RESPECT to the treatment of high-velocity targets, it is well known that a giventransmitted waveform returns, after reflection

from a "point" target approaching the radar at con-stant velocity, compressed in time by a certain factor.Thus, a sine wave appears to be shifted in frequency byan amount proportional to the transmitted frequency.When this sine wave (carrier) is modulated, the echoreturns with a higher carrier frequency and with aslightly compressed modulation. For example, a pulsetrain returns with a higher repetition frequency andshorter pulses than it had when it left the transmitter.These effects on the modulation are often small and areusually neglected, so that the conventional theory ofmatched filters and ambiguity functions is based on thisapproximation.

56 January