Urkowitz: Maximum Likelihood Angle Estimates

CONCLUSIONS

The concepts of Markov chains have been applied tothe narrow-beam communication system acquisitionproblem. Application of Markov techniques allows theexpected value (8) and standard deviation (9) of theacquisition time to be determined.The expected acquisition time for a communication

link consisting of terminals having limited knowledge ofrelative angular locations has been determined. It hasbeen shown that the expected acquisition time is de-pendent upon the probability of detecting a signal innoise (pa) and the probability of looking in the correctdirection (PA). It has been shown that the expectedacquisition time is essentially independent of the systemfalse alarm probability provided that the false alarmprobability is not large."3 Curves of the expected num-ber of samples (ri) as a function of probability of detec-tion and pointing probability are shown in Fig. 5, inwhich the example given in the text is plotted. It isnoted (for the example considered) that a very sharpminimum exists in the expected number of samples re-quired to establish the communication link. It shouldalso be noted (Fig. 5) that the pointing probability must

13 This is uisually the case since small changes made in power andthreshold levels yield large changes, or reductions, in false alarmprobability (see Fig. 3).

be quite large if short acquisition times are to beachieved. This implies searching with as broad a beamas possible (making the search beamwidth approach theangular uncertainty). The effect upon pa of searchingwith a broad beam cannot be neglected since pa is afunction of search beamwidth [see (19)].

It has been shown that the expected acquisition timeof the random search and the programmed (raster scan)search are approximately equal (1<TIR/TrP< 2). Thusindications are that the particular form of scanningprocedure chosen will probably depend strongly uponthe standard deviation of the acquisition time andhardware considerations, in addition to the expectedacquisition time. Mote optimized forms of search which,for example, make use of the probability density func-tion of the uncertainty volume and/or use sequentialdetection techniques, etc., deserve further considerationfor reducing acquisition time. The concepts of Markovchains as outlined in this paper may be used to greateradvantage when investigating these more complexmodels of the acquisition procedure.

ACKNOWLEDGMENT

The author wishes to express his appreciation to Dr.H. Mills and Dr. M. Handelsman of Advanced MilitarySystems, RCA, for their many helpful suggestions.

The Accuracy of Maximum Likelihood Angle Estimatesin Radar and Sonar

HARRY URKOWITZ, SENIOR MEMBER, IEEE

Summary-By extending the results of Kelly, Reed, and Root,'formulas are derived for the variances of maximum likelihood esti-mates of azimuth, and azimuth and elevation, jointly, by dense,discrete and discrete-continuous apertures for the strong signalcase. The accuracy of angle measurements depends upon

1) total signal energy captured by the aperture,2) the mean-square aperture size,3) carrier frequency,4) the mean-square signal bandwidth.

Mean-square quantities are the second moments about the centroids.The actual signal form and aperture form do not matter, except asthey affect the mean-square quantities.

When joint estimates of azimuth and elevation are made, theerrors are generally coupled. Minimum variances are obtained whenthe errors are uncoupled. This condition is obtained, in the narrow-band case, when the two-dimensional illumination function is factor-able into the product of one-dimensional functions.

Manuscript received October 16, 1963.The author is with the Philco Scientific Laboratory, Blue Bell, Pa.' E. J. Kelly, I. S. Reed and W. L. Root, "The detection of radar

echoes in noise, II," J. Soc. Ind. and Appl. Math., vol. 8, pp. 481-510; September, 1960.

The formulas for the dense and discrete apertures are identical inform, the various factors being discrete or continuous analogs of oneanother in which integrations are replaced by summations. Theformulas for the discrete-continuous array differs in form by thepresence of terms which reflect the anisotropy of the beam patterns.

INTRODUCTION

N GAUSSIAN NOISE, the signal ambiguity func-tion plays an important part in obtaining a maxi-mum likelihood estimate of signal parameters. In

fact, in white noise, the estimate is obtained by con-structing the ambiguity function as a function of theparameters to be estimated and the values which maxi-mize the function are taken as the desired estimate.Woodward2 has shown how this is done for delay andDoppler and has derived formulas for the variances of

2 P. M. Woodward, "Probability and Information Theory withApplications to Radar," Pergamon Press, New York, N. Y., ch. 6;1953.

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the estimates. For more general maximum likelihoodestimates, Kelly, Reed, and Root1 introduced a moregeneral version of the ambiguity function which isparticularly suited to the purpose of this paper. Al-though the quantity in question was not called an am-biguity function in Kelly, et al.,1 in the case of whiteGaussian noise it is essentially proportional to theWoodward ambiguity function2 if the unknown param-eters are delay and Doppler. For those readers whowish to refer immediately to Kelly, et al.,1 the functionunder discussion is Q(a, ao0), introduced in Section LII.An angular ambiguity function was introduced by

Urkowitz, Hauer and Koval3 and its role in angularestimation was explained. With a slight generalization,the Kelly, et al., ambiguity function' can be shown to beequivalent for the type of noise treated here and willserve the purpose of determining accuracy. This gen-eralization is accomplished by treating the signal as aspace and time function rather than as a time functionalone.

This paper considers the following three types oflinear (or planar) apertures:

1) The dense aperture.2) The discrete aperature.3) The discrete-continuous aperture.

A dense aperture is really an idealization. It consists of acontinuous aperture in which the wave captured byeach elementary length (or area) is separately availableto be processed. This kind of aperture and the space-time wave appropriate to it have been described byUrkowitz, Hauer and Koval.' Of course, no physicalaperture is dense, but the formulas appropriate to it arestraightforwardly derivable and are very similar inappearance to those for the discrete aperture. Further-more, simple physical interpretations are possible. Thediscrete aperture is more realistic; it consists of isotropicelements which have no mutual coupling. The discrete-continuous aperture4 consists of a number of linear radi-ators lying in a line; each radiator is called an elementand the energy incident on each element is collected toform a single output.The accuracies of maximum likelihood estimates are

described by the variances of these estimates; theformulas for the variances are based upon appropriate(and slight) generalizations of formulas given in Kelly,et al.' The first task, then, is laying the groundwork insignal description, using what is needed from Kelly,et al. All of the analysis is based upon the assumption ofa strong signal.The variance of any particular estimate depends, in

general, upon how accurately other parameters areknown. At first, only one angular parameter is consid-

3 H. Urkowitz, C. A. Hauer, and J. F. Koval, "Generalizedresolution in radar systems," PROC. IRE, vol. 50, pp. 2093-2105;October, 1962.

4H. Urkowitz, "The angular ambiguity function of a discrete-continuous array," PROC. IEEE, vol. 51, p. 1775; December, 1963.

ered as unknown so that the discussion is somewhatsimplified. The effect of the other angular parameter isinvestigated afterwards.

THE SIGNAL DESCRIPTION AND THE GENERALIZEDAMBIGUITY FUNCTION

Consider the complex signal s(t) given by

s(t) -a(t) exp (jwot), (1)

where a(t) is the complex envelope or modulation func-tion and wo is 27r times the carrier frequency fo. To acertain extent, the choice of coo is somewhat arbitrary,but it is convenient to define fo as Woodward does.2Letting S(f) be the Fourier transform of s(t), fo isdefined by

f I S(f) I2dfvi 00

fo =-.r 0

S(f) I ldf-00

SinceS(f) = A (f + fo),

and

tS(f) df = A(f) df,_00 -00

(2)

(3)

(4)

it is readily found that

(5)f A(f) (2df = 0.oo

Now, consider a wave whose time variation is describedby (1), incident upon a linear aperture lying in the xdimension. It is assumed for the present that all param-eters are known except the azimuth angle 0, or equi-valently, u=sin 0/c. The signal captured by a denseaperture, written as a space-time function, is

s(t, x; u) - I(x)a(t - ux) exp [joio(t - ux) + jlj, (6)

where A is a generally unknown phase angle. For thedense aperture, s(t, x; u) is treated as a continuous func-tion of x.

For the discrete-continuous aperture, the spacedimension is designated by an index i representing aparticular element. Thus in Urkowitz,4

s4(t; u) = a(t - uxj)Gj(wou) exp [jwo(t - uxi) + jy/], (7)

whered2-xi

Gi(wou) = IIl-xi

ri(x) exp [-jwouxidx. (8)

In writing (7) it has been assumed that the elements aresmall enough for a(t-ux) to be the modulation appear-ing at time t at all points of the ith element. For thespecial case of the discrete aperture, one has

Gi(cou) =Gi(O) =Gi. (9)

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Urkowitz: Maximum Likelihood Angle Estimates

Before the generalized ambiguity function is defined,a word of caution is in order. The notation of Kelly,et al.,1 is followed here, the letter Q being used. InUrkowitz, et al.,4 the letter Q is used for a slightly dif-ferent quantity. Here, the important formulas of Kelly,et al.,' are put into a form more suitable to the presentpurpose.

Considered as a function of the single real variable t,let s(t; a) be the complex signal where the componentsof the vector a are the quantities to be estimated, the"true value" vector being ao. Then the generalizedambiguity function Q(a, ao) is given by

T

Q(a, a0) = g*(t; a)s(t; a0)dt,-T

(10)

andf L fT

R(t, t'; x, x')g(t', x'; a)dt'dx'-L '- T

= s(t, x; a),-T<t< T

-Lx < L,(14)

where R(t, t'; x, x') is the space-time covariance functionof the noise field. If the noise is both space and time sta-tionary (at least in the wide sense),

R(t, t'; x, x') = R(t -t'; x -x').

For the discrete-continuous case,T

Q(a) a0) = E gf*(t; a)si(t; a0)dt,i T

(15)

(16)

where the asterisk indicates the complex conjugate, andg(t; a) is the solution of the integral equation

T

J R(t, t')g(t'; a)dt' = s(t; a),-T

-T<t<T, (11)

in which R(t, t') is the autocovariance function of thenoise. It is presumed, of course, that a sample of wave2T long is available. In Kelly, et al.,' there is a nor-malization so that Q(a, a) = 1. Because such normaliza-tion is not made here, Q(a, a) can be interpreted as asignal-to-noise ratio. In addition, the amplitude of thesignal is absorbed into s(t; a) and not brought outseparately as in Kelly, et al.1 Now, the variance of amaximum likelihood estimate of the kth parameter inGaussian noise is given by the negative of the k - k ele-ment in the inverse of a matrix C (called the errormatrix) whose elements Ci, are given by

1 a2 12

=c°

= Q(aa, a0)Q(0 0 aa,aa

where g,(t; a) is the solution ofT

E R(t, t'; xi, xj)gj(t'; a)dt' = si(t; a),i -T

-T < t < T. (17)

The only noise situation treated here is that in whicheach element of the aperture is a source of noise statisti-cally similar to, but statistically independent of, everyother element. In this case, one may write

R(t, t'; x, x') = R(t, t')3(x - x') (18)

and

R(t, t'; xi, xj) = R(t, t')Sij (19)for the dense and discrete-continuous apertures, respec-tively. Eqs. (18 and (19) amount to saying that thenoise is space-white.Then (14) and (17) become

rTfTR(t, t')g(t', x; a)dt' = s(t, x; a), - T < t < T (20)aT

and

a2Q 2 dQ aQ*=2 Re + Re

aai4aa a=co Q(a0, ao) dati aj a=aC

This is the unnormalized version of (49) in Kelly,(Remember that the amplitude is absorbed into 6

in the present notation. Also, the assumption is m;strong signal as compared to noise.)When the signal is a function of more than one

able (space and time, for example), (10) and (11lbe still considered to hold by considering t as a N

and the integrations to be over all the componentables. The autocovariance function is now a functvector variables. Considering the dense aperturemay write

QaL aTQ(a7 ao° = *(t, x; ax)s(t, x; a°)dt dx

-L -T

( T

. (12) R(t, tl)gi(tl; at)dt' = si(t; a) -T<t<T. (21)

In order to get computational formulas, it will be nec-essary to assume some form for the noise autocovariancefunction. The simplest situation is that in which thenoise is wide-sense stationary, white and broadband.Then, to a good approximation,

R(t, t') = 2No0(t- t'). (22)

Furthermore, at first the estimation of u alone will bemade, assuming knowledge of all the other parameters.Then, using (6) and (7), the basic formulas to be em-ployed are: for the dense aperture

1 T rLQ(u, uo) = (x) 2a(t - uox)a*(t - ux)2Nex x-T -L

* exp [jSwo(u-uo)x]dx dt, (23)

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IEEE TRANSACTIONS ON MILITARY ELECTRONICS

and for the discrete-continuous case,

Q(u, uo) = 2 : Gw(coouo)Gi*(wou)a(t-uOx)a*(t-ux)

*exp [jcoo(u-uo) xi] dt. (24)

Letting u represent the desired estimate, one has

1= - C

var X

where

C=2Re 2Q(U, uo) + 2 aQ(u, uo) 2

au2 u=UO Q(uo, uo) auU=Un

From (23) and (24), it is easy to write

Q(uo,uo) = 2N0F I I(x) 12| a(t - uox) 12dx dt

or1 to

Q(uo, uo) = E JI Gj(coouo) 12| a(t -uOx) 12dt. (28)

(The limits on the integrals are understood.) If T islarge enough, the double integral in (27) and thesummed integration in (28) are expressions for the totalenergy ET captured by the aperture. Thus,

ETQ(uo, Uo) = 2N (29)

and

a(t - uox)a*"(t - uox)dt = - f I A(f) I2df. (34)

(33) follows from (5) which comes from (2), the defini-tion of carrier frequency. Thus,

(25) au U=Uo -w2No j x I I(x) 12 A (f) 2df dx

and

() 2Q(26) Ou2 u="

(35)

= - X 2| I(x) 12(O2 + Wo2) A(f) 2dfdX. (36)

(27) It is convenient to define the mean-square radian band-width Br2 by

fo2 A(f) 2df

Br2 = I (37)A (f) 2df

the horizontal centroid Xc by

XxI 1(x) 2dx

xc =-f I(x) I2dx

(38)

and this may be considered as a signal-to-noise ratio.

VARIANCE OF THE ANGULAR ESTIMATE

A. Dense A perture

Straightforward analysis yields the following:

OQ 1 1I(x) a(t - uox)du u=,O 2No

*jwoxa*(t - uox) - xa'(t - uox) } dx dt (30)

and

a2Q 2 f J(x) 2a(t - uox){x2a*/(t - uox)an2 "=0 2No

- 2jwox2a*'(t - uox) - Wo2x2a*(t - uox)}dx dt. (31)

In (30) and (31), the primes indicate differentiation withrespect to the whole argument. Now, if T is presumedlarge, one may write

f a(t-uox) 12dt = f A(f) 12df (32)

fa(t-uox)a*'(t-uox)dt = jr c A(f) 12df = 0, (33)

and the mean-square aperture length Lr2 by

fx2 1(x) 2dx

Lr 2 = -XC2.

I 1I(x) 2dx(39)

Using (25), (26), (29), (38), and (39), the result is

1 ETWO2 2X2W2var=2- { (1 + Br2,Io2)Lr2 + Br2Xc2 wo2}. (40)var X No

Eq. (40) shows quite clearly the effects of carrier fre-quency, bandwidth, size (appropriately defined), etc.One may note that the bandwidth results in a second-order effect, because it is the ratio of bandwidth to car-rier frequency which counts, and this ratio is usuallysmall compared with unity. For a given physical length,Lr is made large by concentrating the illumination func-tion toward the ends, like a simple interferometer. Thepresence of the term containing the centroid Xc impliesthat something is to be gained by choosing the origin tomake Xc as large as possible. In any event, the effect isusually small because the ratio of bandwidth to carrierfrequency is usually small.

42 January

6Urkowitz: Maximum Likelihood Angle Estimates

B. Discrete ApertureUsing (9) and (24) and (32) to (34), straightforward

calculation yields

= ZaIE IG f2xiA(f) I2df (41)ouU= 2No

and

02Q

OU2 U-UO

-2NXi Gi (w2 A (f) (42)2No i

For the discrete aperture, the horizontal centroid Xcand the mean-square length Lr2 are given by the discreteanalogs of (38) and (39),

where I(x, y) is the two-dimensional illumination func-tion; for the discrete array,

sik(t; U, p)

= a(t - uxi - Pyk)Gik exp [jco(t - uxi - pyk) +±j7], (47)

where Gik is the complex pattern associated with theelement whose horizontal index is i and whose verticalindex is k. Furthermore, when the noise is white,

Q(u, u0; P, Po)

2N= I(x, y) 12a(t - uox - poy)a*(t - ux - py)

*exp -jwo[(u - uo)x + (p - po)y] I dx dy dt (48)

and

Q(u, uo; P, Po)

T,EE2No i k-Gik | 2a(t-uOxi- Poyk) a* (t-UXi-yPk)

Xi2 Gi 2

Lr2 -X 2. (44)E Gi 12

After routine substitution and the use of (32) to (34)with (29), there results

ETwO2

A_= (1 + Br2/W02)Lr2 + Br2XC2/C)O2}. (45)

var ?2 No

In form, (45) is identical with (40). The quantities Lrand X, are merely the discrete or continuous analogs ofone another. This similarity holds true throughout thispaper; it is therefore unnecessary, in the rest of thispaper, L uerove LWo riLs oI results. One type of aperturemay be analyzed and the same formulas used for theother type with integrals replaced by appropriate sums.

C. Discrete-Continuous A rray

The formulas for the discrete-continuous array are

fairly complicated and difficult of physical interpreta-tion; they are therefore relegated to the Appendix so

that the flow of the arguments will not be interrupted.

COMBINED AZIMUTH AND ELEVATION ESTIMATES

Consider a plane aperture upon which a plane wave isincident. Let 0 and 0 be the azimuth (bearing) and ele-vation angles, respectively, and let u =sin 6/c andp=sin +/c. Analogous to (6) and (7), for the signal cap-tured by the array, the following formulas apply. Forthe dense array,

s(t, x, y; u, p)

=I(x, y)a(t -ux- py) exp [joo(t - ux - py) +j4P]j (46)

*exp -jcoo[(u-uO)xi+(p-po)yk] } di (49)

for the dense and discrete arrays, respectively.To find the variances of joint estimates, the elements

of C, given by (12), are needed. (The formulas for thediscrete array only are derived here.) Some symbology isneeded first. For the planar array, as distinct from thelinear array, the horizontal centroid Xc and the mean-

square length Lr2 are defined as follows:

E 5,xi Gik I'i k

Xc - IGik2

i k

E EXi21 Gk 2

E E | Gik 2i k

(50)

(51)

No confusion should arise in deciding whether to use

(50) and (51) rather than (43) and (44). The latter are

used when one has a linear array while the former are

used for a planar array. In a similar manner, the verticalcentroid Y, and mean-square height Hr2 are defined as

follows:

E ykI Gik I2i k

Yc E E Giki k

(52)

E7 E: Yk' Gi*k |I2k 2

Hr2 =

ZEE |GGik 12 -c .

i k

(53)

Z xi Gi 12

xc =

E G(432

1964 43

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IEEE TRANSACTIONS ON MILITARY ELECTRONICS

It is also convenient to define a coupling coefficient A.between the horizontal and the vertical directions asfollows:

E E Xiyk Gik|i k

A, SE IGikL 2i k

-Xc Yc(l +Br2/W02)-1. (54)

Then, after some manipulation, it is found that

Cii = _ T ° [(1 + Br2/WO2)Lr2 + Br2Xc2/WO2]No

C22 = ET [(1 + Br2/WO2)Hr 2 + Br2 Y2/WO2]No

E TWO02C12 = C21 = - [(1 + Br2/W02)AJ. (55)

No

The determinant Cl of the matrix C is

FT2WO4 2/2 222- 2

C [(1 + Br2/Wo2)2(Lr2Hr2 -Ac2)N02

+ (1 + Br2/W02)(Br2/W02)(Lr2Yc2 + Hr2Xc2)]. (56)

Then,

C22var Xu = -

Cil

varfp= - _. (57)clWhen A,==0, var 1= -1/C1l and var p= -1/C22, andthese are the minimum variances. This justifies theterm coupling coefficient for A,. Examination of (54)reveals that if Br<<wo (narrow-band case) and if Gik isseparable into the product of functions of each indexseparately, then A, = 0. Since the narrow-band conditionis usually satisfied, one may say that azimuth andelevation estimates are uncoupled if the two-dimen-sional illumination function is separable.

DISCUSSION

This paper shows how azimuth and elevation esti-mates affect each other and how these effects may beeliminated by symmetrical illumination functions. Themost important assumption made in all of this is thatthe other parameters (delay, Doppler, etc.) are known.In general, these are not known and errors in estimatingthem may result in increasing the variances of the angleestimates. Of course, the effects of these other param-eters enter through the error matrix C whose elementsare given by (12). Since this paper's interest has beenonly in angle estimates, other estimates have beendeliberately ignored; the position has been taken thatthe error matrix could easily be built up by anyone who

desires to do so. This has been done by Connolly5 for alinear dense array.

This paper also shows that the signal shape and theillumination function shape are important only as theyaffect certain mean-square quantities. The signal influ-ences accuracy through the ratio of the mean-squarebandwidth to the carrier frequency and this is usuallysmall so that the signal bandwidth usually gives asecond-order effect. The primary factors are signal-to-noise ratio, carrier frequency and mean-square aperturesize. For a given physical length and captured signalenergy, the mean-square aperture size is maximized byconcentrating the illumination at the ends of the aper-ture.When joint azimuth and elevation angle estimates are

made, the errors are, in general, coupled. The effect ofone error on the other is determined by the couplingcoefficient. When the coupling coefficient is zero, thevariances of the estimates are minimum and independ-ent of each other. For a narrow-band signal, the esti-mates will be uncoupled if the two-dimensional illumina-tion function is separable into the product of one-dimensional functions.Another interesting point which was mentioned ear-

lier is the identity in form of the formulas for the denseand discrete arrays; the symbols are merely the discreteor continuous analogs of one another. This does not holdtrue for the discrete-continuous array, whose varianceformulas are in the Appendix. These formulas are diffi-cult to interpret physically, but examination of (63)seems to indicate that element anistropy will increasethe variance. While the dense array is an idealizationand has no physical counterpart, the discrete array is agood approximation to actual arrays in many situations.

In some situations, a radar or sonar system consists ofseparated and distinct arrays, or rather, subarrays.Each subarray has a single output which is the sum ofthe outputs of all elements of the subarray. The sub-arrays may be far apart, too. One is often interested inhow the outputs of the subarrays may be combined togive best angle estimates. The discrete-continuous arrayis particularly applicable to this case.

GLOSSARY OF PRINCIPAL SYMBOLS

s(t) =Transmitted signal.a(t) = Complex modulation function.A (f) = Fourier transform of a(t).

I(x), I(x, y) =Illumination functions.fo = Carrier frequency (cps).Wo = 2rfo.f= Frequency variable (cps).co= 27rf.0= Angle of a target with respect to

broadside in the plane containing the

I J. J. Connolly, "Fundamental accuracies in the simultaneousmeasurement of transverse and radial trajectories of radar targets,"Proc. 7th Nat'l Conv. Military Electronics, Washington, D. C.,pp. 64-68; September, 1963.

44 January

Urkowitz: Maximum Likelihood Angle Estimates

x axis and the line intersecting thephase center of the array and thetarget.

4 =Angle of a target with respect tobroadside in the plane containing they axis and the line intersecting thephase center of the array and thetarget.

u= (sin 01)/c.u +v = (sin 62)/C-

p-=(sin 01)/c.P+q-= (sin q52) /C.

c =Speed of propagation.X P, =Maximum likelihood estimates of u

and p, respectively.Uo, Po= True values of u and p, respectively.

s(t, x; u) =Vomplex signal as a function of spaceand time.

L= (1/2) Xphysical length of the linearaperture.

T= (1/2) X signal duration.Gi(wou) = Antenna pattern of i-th element of

discrete-continuous aperture.Q(a, ao) =Generalized ambiguity function in-

volving the parameter set a (a vector),one or more components of which areto be estimated, and ao, the set of"true" values.

R(t, t'; x, x') = Space and time noise covariance func-tion.

No = Spectral density of the white noiseconsidered in this paper.

E=Total signal energy absorbed by theaperture.

Br2= Mean-square radian bandwidth of thesignal.

Lrl = Mean-square aperture length.Hr2= Mean-square aperture height.Ci j = Elements of error matrix C.Xc=Horizontal centroid of the aperture.Y,=Vertical centroid of the aperture.Ac = Horizontal-vertical coupling co-

efficient.

APPENDIX

Variance of the Angular Estimate for aDiscrete-Continuous Array

Write (24) in the form (recalling that T is presumedlarge)

Q(u, uo) = 1 A|(f) |2G.(wouo)Gj*(wou)2No iJ*exp [j(wo -c)(u - uo)xi]df.

Then, straightforward analysis yields

aQ (.))fdf A(f) 21au U =Uo 2No

>E [Gj*(coouo)Gj*'(coouo) + jXi Gi(wouo) 12] (59)

and

,02Q Wo2 Eau2 =uo 2No iJ A(f) 12[Gi(wouo)Gi*t(Coouo)

+ 2jxiGi(coouo)G*I(ouo0) - (1 + Co2/W02)x,2

G,(wouo) 2]df. (60)

The centroid X,(uo) and the mean-square length Lr2(Uo)effective in the direction uo are defined by

E xi Gi(coouo) I2Xi (uo) =

E Gi(wouo) 12i

E Xi2 G (Coouo) 12Lr2(uo) = E G( )

i Gi(wouo) 2 - X(

(61)

(62)

When (25) and (26) are used, one gets

1 WO2ET{v= - (1 + Br2/co02)Lr2(Uo) + Br2Xc2(Uo)/Co2

va,ru NoV

1 )2 [Re E Gi(oouo)Gi*/l(wouo)E Gi(woo) 2

X

+ 2Xc Im E Gi(wouo)Gi*'(wouo)

-2 Im ZxiGi(ouo)Gi* (wouo)

E Gi(Coouo)Gi*'(Couo) 2

i GE Gi(WoU0) 12 J

(63)

The first two terms within the braces of (63) are likethose of (45). The remaining terms are the contributionsfrom the anisotropy of the pattern.

1964 45

(58)