Signal Processing 24 (1991) 299 317

299Elsevier

Parallel modified spatial smoothing algorithm forcoherent interference cancellation

Sun Park and Chong Kwan UnCommunications Research Laboratory, Department of Electrical Engineering . Korea Advanced Institute ofScience andTechnology . P .O. Box 150, Chongyangni . Seoul, Korea

Received 30 July 1990Revised 28 March 1991

Abstract. In array processing, the spatial smoothing technique and its variations are known to be effective in combattingcoherent interferences . However, they are disadvantageous in that they either reduce the effective array aperture or require theformation of covariance matrices which causes numerical difficulties when finite-precision computations are involved and givenarray data are ill-conditioned . In this paper, we present a data-domain spatial preprocessing algorithm, by which the effectivearray aperture is expanded without forming covariance matrices . Also, we propose a parallel spatial smoothing technique inwhich spatial subarray data are rearranged before processing . The incorporation of the data-domain spatial preprocessingalgorithm and the parallel spatial smoothing technique is simple, and constitutes a parallel modified spatial smoothing techniquewhich is a parallel implementation method of the modified spatial smoothing technique . The proposed parallel modified spatialsmoothing method is highly fast, numerically stable, and capable of nulling out coherent interferences . Since the proposedmethod can readily he combined with least-squares solving systems using orthogonal transformations, one can take fulladvantages of systolic /wavefront arrays to get high throughput .

Zusammenfasvung . Bci der Array-Verarbeitung ist bekannt, dab die Technik der ortlichen Glattung and ihrc Variationeneffektiv bci der Unterdruckung koharenter Interferenzen sind . Sic haben jedoch den Nachteil in dem Sinn, call sie entwederdie effektivc Arrayapertur verkleinern oder die Bestimmung von Kovarianzmatrizen erfordern, was wiederum auf numerischeProbleme fuhrt, wenn die Rechnungen mit endlicher Wortlange durchgefuhrt werden oder die Arraydaten schlecht konditioniertrind . In dieser Arbeit stellen wir einen Vorverarbeitungsalgorithmus vor, der im ortlichen Datenhereich arheitet, mit dem dieeffektive Arrayapertur vergrolert werden kann ohne Kovarianzmatrizen aufstellen zu mussen . Wir stellen such eine parallelsortliche Glattungstechnik vor, bet der Daten ortlicher Teilarrays vor der Verarbeitung umgeordnet werden . Die Verbindungdes Algorithmus zur Vorverarbeitung mil der parallelen Glattungstechnik ist einfach and stellt eine modifizierte paralleleortliche Glattungsmethode dar, die eine parallele Implementierung der modifizierten ortlichen Glattungsmethode ist . Dievorgeschlagene Methode ist auBerst schnell, numerisch stabil and in der Lage, koharente Interferenzen zu eliminieren . Da siceinfach mit Least-Squares Lbsungssystemen, die orthogonalc Transformationen verwenden, kombiniert werden konnen,bekommt man alle Vorteile systolischer oder Wellenfrontarrays, um einen hohen Durchsatz zu erhalten .

Resume. La technique de lissage spatial et ses variations sent connues en traitement de reseaux pour titre efficaces pourcombattre les interferences coherentes . Toutefois, elles sent desavantageuses en ce qu'elles reduisent I'ouverture mile do researton requierent la formation de matrices de covariance qui sent a la source de problemes numeriques lorsque des calculs enprecision finie sent en jeu et que les donnees du reseau scant and conditionnees . Nous presentons darts cet article un algorithmede pre-traitement spatial clans le domaine des donnees grace auquel l'ouverture mile du reseau est accrue sans former dematrices de covariance . De plus, nous proposons one technique de lissage spatial parallele dans laquelle les donnees spatialesde soul-reseaux sont r6-arrangees avant traitement . L'incorporation de I'algorithme de pre-traitement spatial clans Ie domainedes donnees et de Ia technique de lissage spatial parallele est simple et constitue une technique de lissage spatial parallelemodifie qui constitue une methode d'implantation parallele de la technique de lissage spatial modifie . La methode de lissagespatial parallele modifie est tres rapide, stable numeriquement, et capable d'annuler des interferences coherentes . Du fait quela methode proposee peut titre combine facilement avec des systemes de resolution aux moindres carries utilisant des trans-formations orthogonales . i t est possible de tirer pleinement parti d'architectures systoliques pour obtenir un traitement rapide .

Keywords. Adaptive array, coherent interference, spatial smoothing, parallel processing, QR decomposition, systolic array .

0165-1684/91/$0350 :', 1991 - Elsevier Science Publishers B .V . All rights reserved

300

S. Park, C. K. Un / Smoothing for interference cancellation

1. Introduction

Early optimum array processors were designedunder the assumption that signal sources are plane-wave and uncorrelated with each other [5, 13, 24,29, 30] . Because of the sophisticated signal andjamming environment in recent years, some of theassumptions turn out to be no longer valid . Aboveall, there may exist interfering signals that arehighly correlated with the desired signal, whosepresence would completely destroy the nulling cap-ability of adaptive arrays. In practice, due to thepresence of multipath propagation or smart jam-mers, even coherent (i .e., perfectly correlated)interferences can exist .

Several methods have been proposed for the can-cellation of the coherent interferences with differentdegrees of success [6, 25] . As a practical approachto remove the coherence between sources, Evanset al. suggested a subaperture sampling technique,which is also known as the spatial smoothing (SS)technique [2] . Recently, Shan and Kailath studiedit further [ 17] . Although this method is effective incombatting coherent interferences, it reduces theeffective array aperture and requires a considerableamount of computations. Furthermore, because ofthe possible ill-conditioning of the sample covari-ance matrix, the method often requires double-pre-cision computations, which will significantlydecrease the beamformer throughput . The samplecovariance matrix frequently becomes ill-condi-tioned when strong jammers coexist with the weakdesired signal or noise [9] . More recently, So et al .proposed a parallel spatial processing (PSP)scheme, by which high throughput is expectedwhereas the signal cancellation phenomenon canbe avoided [22] . In order to achieve highthroughput with the method, however, there is nochoice other than only simple gradient-descent-type adaptive algorithms, such as the least mean-square (LMS) algorithm . To successfully incorpor-ate high performance least-squares (LS)-typeadaptive algorithms in the parallel spatial process-ing scheme, all the subbeamformers have to sharethe same covariance matrix, which seems to be adifficult task .

Recently, a modified spatial smoothing (MSS)technique has been suggested by Evans et al . inorder to achieve a larger effective array aperture[2] . Williams et al . [26] and Pillai and Kwon[15] provided a proof for it . Although the methodprovides a larger effective array aperture com-pared with the SS technique, it also requiresforming covariance matrices and a considerableamount of computations . Since high speed com-putations require the use of a short computationwordlength, it is desirable to avoid the potentialloss of accuracy in the computations of covari-ance matrices .

In this paper, we present a parallel modifiedspatial smoothing (PMSS) technique which isa parallel implementation method of the MSStechnique. In the proposed method, the arraydata are processed in two steps . First, a data-domain spatial preprocessing (DDSP) algorithmis applied, which is proposed to expand the effec-tive array aperture without forming covariancematrices. Then, a parallel spatial smoothing (PSS)technique which is a parallel implementationmethod of the spatial smoothing technique isfollowed to decorrelate the coherence of signalsources. The proposed method is numericallystable and fast because it does not form covari-ance matrices and processes the array data inparallel. Furthermore, it can easily be realizedusing the QR decomposition least-squares minim-ization, and thus be implemented with systolicarrays .

Following this Introduction, in Section 2, weprovide a comparison of the nulling performancesof the SS and MSS techniques . The MSS tech-nique is shown to be more effective than the SStechnique in decorrelating coherent signals anddetecting near-field targets . In Section 3, we pro-pose the DDSP algorithm by which the effectivearray aperture is increased . In Section 4, wedescribe the PSS and PMSS techniques which areparallel realization methods of the SS and MSStechniques, respectively. Finally, in Section 5, westate our conclusions.

2. The modified spatial smoothing technique

Consider a uniformly spaced linear array of Momnidirectional sensor elements which receives Knarrowband, stationary random signals. Theobserved signal vector can be represented as

K

x(t)= Y' s,(t)a(O,)+n(t),

(1)

where s,(t) and 0, represent the signal waveformand the direction-of-arrival (DOA) of the ith sig-nal, respectively, n(t) is the noise vector, and a(0,)is the steering or direction vector for the ith signal,which is defined as

a(0,) °- [I . elf" n1 e t °° =, . . . , e'("f-I'°"r']T, (2)

where T denotes transpose and a is the centerfrequency of signals . The inter-sensor path delay r,is given by r, °=(d/v) sin 0,, where d representssensor spacing and v is wave velocity . Noise at eachsensor element is assumed to be uncorrelated withthe signals, and is modeled as a spatially whiteGaussian process with unknown variance a;,, suchthat

E[n(t)nH(t)] = U 2'1.

(3)

where E[ ] is the expectation operator, H denotescomplex conjugate transpose, and I is the identitymatrix . Using vector and matrix notations, (1) canbe rewritten as

x(t) = As(t) + n(t) .

(4)

where

A °=[a(Oi), a(O2), . . . , a(OK)]

(5)

and

s(t) _A [SI (t), S2(t), . . . , SK(t)1r

S. Park, C.K Un / Smoothing for interference cancellation

301

(6)

The spatial covariance matrix of this observed sig-nal vector is defined as

R,,°=E[x(t)xH(t)]=ASAH+aI,

(7)

where S°=E[s(t)sH(t)] is the spatial correlationmatrix of signal sources .

2.1. Spatial smoothing technique

If the signal sources are perfectly correlated witheach other, the spatial correlation matrix Sbecomes singular [17] . The principal idea of thespatial smoothing technique is to restore the rankof the matrix (R,.,-a21) to the number of signalsources impinging upon the array by an appropri-ate preprocessing of the array data .

Let an array of equi-spaced L (=-N+M-1) sen-sors be divided into N overlapping subarrays ofsize M as shown in Fig . 1 . We may write the nthsubarray signal as

xn(t) -[xJt), x„+I(t), . . . , x„+"f-I(t)]'

=AD' - 's(l) +n(t),

(8)

where

D °=diag[e"`°", e'~", . . , e 'o~=x] .

(9)

The spatially smoothed covariance matrix is thengiven by

It can be shown that the spatial signal correlationmatrix S has its full rank K if and only if N>,K[16-18] . If Shas rank K, the signal space does notcollapse in the presence of coherent signals, andthe noise eigenvectors of the spatially smoothedcovariance matrix Rss are orthogonal to the col-umns of direction matrix A . Thus, the spatialsmoothing technique gives nulls upon all the inter-ference directions as long as the total number ofsensor elements L satisfies the following condition :

L=N+ M-1 2K.

(12)

2.2. Modified spatial smoothing technique

With the spatially smoothed covariance matrixRss given by (10) in hand, we may introduce a

Vol 24. No 3. September 1991

N

Rss =-J

E[x„(t)xn (t)]N„- 1

= ASA H +ant, (10)

where

1

'`N

niS° X D'"-[)SD" (11)

302

S. Park, C. K. Un /Smoothing for interference cancellation

x1

I x21st SUBARRAY

Tx

2nd SUBARRAYI

It can be shown that the matrix S has its full rankK if and only if N3 iK [15, 26] . Therefore, thetotal number of sensor elements required in theMSS technique reduces to

L=N+M-1>,~K.

(15)

2.3. Decorrelation performance

Let us now compare the decorrelation perfor-mances of the SS and MSS techniques . For simpli-city, we assume that two equal-strength signalsources are impinging upon an (N+2) elementarray composed of N subarrays of size three . Theinter-sensor path delays which are determined bythe DOAs of the signals are assumed to be r, andT2, respectively . Also, assume that the signalstrength is a. , and that two signals are correlatedwith each other by an amount of correlationcoefficient r, with phase difference (fi . The spatialSignal Processing

I

Nth SUBARRAY

Fig . I . Subaperture sampling from an overlapping sensor array .

modified spatially smoothed covariance matrix

signal correlation matrix S given by (11) for theRMSS as

SS technique is represented by

RMSS°z[Rss+JRsSJ]=ASA1+a,,I,

(13)

r

a2

asrejwhere the asterisk denotes complex conjugate, and

S-Lasr e -

J

as ]

(16)

J is the exchange matrix whose cross-diagonal ele-ments are ones and all others are zeros . Here, the where a is a complex constant whose value is givenmodified spatially smoothed signal correlation by the sum of N uniformly spaced terms on thematrix S is given by

unit circle, as shown in Fig. 2, such that

S° 1

[D In-oSDt1-nt

a° 1 N e'

(17)2Nn ,

Nn-I

+D(2-n-A")S*Dtn+a,-21]

(14) where y/°-c%kr,-T2) . Consequently, with the SStechnique, the inter-signal correlation is reduced byan amount of lal .

xN+1

TXN+M-1

IMAGINARY AXIS

DIRECTION OFDISTRIBUTION

REAL AXIS

Fig . 2. Locations of zeros on the unit circle (the spatialsmoothing technique was used) .

On the other hand, the spatial signal correlationmatrix S given by (14) for the MSS technique isrepresented by

a

rsr eimIS vgr e -imf*

where /3 is a complex constant whose value is givenby the sum of two groups of N uniformly spacedterms on the unit circle, such that

'3 1

(ex°-'lw2N„

+e -12.*+mt e -ih-of) .

Observe that one group of terms of /3 lies aroundthe unit circle along the counter-clock-wisedirection, whereas the other group lies along theclock-wise direction . This is shown in Fig . 3. Theintersignal correlation is now reduced by anamount of / . By comparing (17) with (19), wecan conclude that the MSS technique provides abetter decorrelation capability than the originalspatial smoothing technique because we have I/3lal (see Appendix A) .

IMAGINARY AXIS

Fig . 3 . Locations of the two groups of zeros on the unit circle(the MSS technique was used) .

S. Park, C. K. Un ,% Smoothing for interference cancellation

303

DIRECTION OF

DISTRIBUTION-GROUPI

2.4. Nulling performance againstnear-field interferences

Until recently, signal sources have been assumedto be sufficiently far from the array, thus makingthe plane wave assumption valid . In practical appli-cations of adaptive beamforming, however, thearray may be too close to signal sources for thefar-field assumption to be valid. Readers arereferred to [3, 8, 27, 28] for detailed informationson near-field source problems. Here, we comparethe detection performances of the SS and MSStechniques when the target lies in the near-fieldzone .

When the array lies in the Frauhofer zone [19],the propagation of wave from a far-field source canbe modeled as shown in Fig . 4(a) . Now considerthe propagation of wave from a near-field sourcewhen the array lies in the Fresnel zone [ 19] asshown in Fig . 4(b), where the wave propagationhas been modeled as being spherical . Let y denotethe distance from the field point source to somereference point of the array aperture . Here, we takethe location of the first sensor element as a refer-ence point, and denote the DOA of the signalsource by 0. Then, the incremental distance d l fromthe wave surface to the second sensor element canbe found from the relation [19]

(y+A 1)'=y'Cos'0+(d+ysinf) z .

(20)

It immediately follows that

r

/d\

d\2 rz

4,=Y 1+2 sinol ) l+ (l ) IJ

-y.

(21)

Then, using the Maclaurin

/

expansion, we canapproximate (21) as

r

/ `A 1 y

[l + sin of d l+

1cos' o(d

z

) ] yY) 2!

Y)

DIRECTION OF

=dsin 0+Z cos' 0 .

(22)DISTRIBUTION

2y

-GROUP2 Let us consider a three-element array of only onesubarray for simplicity. We assume that a desiredsignal source arrives from the broadside direction

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304

S. Park, C.K. Un / Smoothing for interference cancellation

PLANEWAVE

(b)

ARRAY NORMAL

ARRAY NORMAL

NEAR-FIELD SOURCE

Fig . 4 . (a) Wave propagation model for a far-field source . (b) Wave propagation model for a near-field source .

without the loss of generality, and take the second

response (MVDR) beamformer when the spatialsensor element as a reference point . We then have

smoothing technique is used and is given byan observed data of the form

Rss g E[x(t)x"(t)1x (t)= [e i0e t ei0e] TSp(t) + n(t),

(23)

where so(t) is the waveform of the desired signalsource, and n(t) is the mutually independent noisevector whose variance is a. . The phase delay r Acaused by the near-field effect is given by

d - __0v o -

(24)v 2yv

Since there is only one subarray, the covariancematrix for this input vector would be equivalent tothat of the minimum variance distortionlessSignal Processing

INTER-SENSOR PATH DELAY

SENSOR-1

SENSOR-2 ARRAY AXIS

S

WAVE SURFACE

INTER-SENSOR PATH DELAY

A

1

;

1= as ~* 1 ~* + anir,

(25)I

1

where as is signal power, and ;-el"°. If the signalsource lies at infinite distance from the array, thecovariance matrix would be

1

1

11 1 1 + anL

(26)1

l

1

Since the optimum weight vector of the adaptivebeamformer is dependent upon the covariancematrix, we may compute the error in the covariancematrix caused by the near-field effect by taking theFrobenius norm of the difference matrix R,x,-Rss[11] . The squared Frobenius norm of this differ-ence matrix is

lRRx - Rss I,F=2a°(I1- ;1 2 +11-~*12)

=8a4(1-cos OA) .

(27)

This expression means the larger signal power and/or the smaller distance from the target to the arraywould result in a poorer performance due to theincreased near-field effects .

When the MSS technique is used, we have acovariance matrix of the form

RMSS = 2' [Rss+JRssJ] .

(28)

Since Rss given by (25) is a centro-symmetricmatrix, pre- and post-multiplications of theexchange matrix J on both sides of Rss give nochange in the structure of the matrix Rss , that is,JRS5J=Rss . Therefore, (28) reduces to

RM55 - as

1

cos OA

l

cos On

1

cos 0,I

cos 0,

1

+ a;I.

(29)

Then, the squared Frobenius norm of this differ-ence matrix is immediately given by

IR-_ - RMSS .I1 - 4a°(1 -cosy )2 .

(30)

Defining a real constant p °=1 -cos 0e, we cancompare the squared Frobenius norms of (27) and(30) as follows :

IIR, - Rss1IF - IIR,. - RMSSIIF

=4C4p(2 - ~) .

(31)

S. Park, C. K. Un / Smoothing for interference cancellation

305

Because q lies in the range of [0, 2], (31) alwayshas a positive value, so that we finally have

11 R= - RMSS iF, IIR,, - Rss IF

(32)

The equality is satisfied if and only if 0n=0, whichis the case of an infinite-distance source . Con-sequently, we may conclude that the MSS tech-nique can enhance better the detection capabilityfor near-field targets than the original spatialsmoothing technique [14] .

To illustrate the performance advantage of theMSS technique over the original spatial smoothingtechnique, we performed simulation in which inter-ference sources are in the near-field zone . In thesimulation we assumed that two coherent interfer-ences arrive from directions -50° and -30°, andtwo noncoherent ones from 10° and 60°, and thatwave propagation is modeled as being spherical[19] . We considered a communications systemwhose carrier frequency is 100 MHz (i .e ., .&/2=1 .5 m) with a fifteen-element linear uniform arrayof half wavelength spacing. The distance frominterference sources to the array aperture wasassumed to be 200 m (i .e ., 66.677) which satisfiesthe near-field condition y<2(2 2 /A, where S2denotes the effective aperture length [3, 19] . Asshown in Fig . 5, the spatial smoothing techniquecannot give a deep null on the near-field interfer-ences that are close to the broadside direction (e.g.,10°) . On the contrary, the MSS technique stillyields exact deep nulls upon the interferences thatare close to the broadside direction as well as uponthose far off the broadside direction .

In spite of its increased effective array aperture,the MSS technique has shortcomings in that itforms covariance matrices which may require long-wordlength computations in order to prevent thepotential loss of informations contained in thearray data . Particularly, when the sample covari-ance matrix is ill-conditioned, the use of doubleprecision is inevitable, which will significantlydecrease the beamformer throughput . To increasethe effective array aperture without forming covari-ance matrices, we propose a DDSP algorithm in

V .1 24, No . 3. September 1991

306

S. Park, C. K. Un / Smoothing for interference cancellation

-50-90

-60

--30

0

30ANGLE (DEGREE)

Fig . 5 . Beam patterns of the spatial smoothing and MSS techniques when interferences are in the near field .

the next section . It provides numerical stabilitytogether with large effective array aperture .

3. Data-domain spatial preprocessing(DDSP) algorithm

3.1. Description of algorithm

Let an array of L (°=N+M- l) sensors bedivided into N overlapping subarrays of size M asshown in Fig. 1 . Also, let x"(t) denote the sensoroutput of the nth subarray for n = 1, . . . , N. Then,assuming K signal sources, we may write the nthsubarray output x"(t) as (8) . It can be shown thatthe complex conjugate output across the nth back-ward subarray is then given by

i"(t)=-Jx„*(t)

-[xn+M-I(t), xn+M -2(t), . . . . xv (t)]T=AD"_"-")s*(t)+Jn*(t) .

(33)

We now introduce new vectors {r"(t)}, for n=1,2, . . ,N,as

r"(t) ° z[x"(t)+i>,(t)] .

(34)Signal Processing

DESIRED SIGNALINTERFERENCES

INTPRFERENCES

60 90

Then, it can be shown that the spatially smoothedcovariance matrix of r"(t) is given by

o N Y E[r"(t)r~(t)]

1 N4N"~, {E[x„(t)x„(t)]+JE[x'*(t)x,T,(t)]J

+E[x"(t)xn (t)]J

+JE[xn(t)x4(t)]} .

(35)

If signal sources and noise are assumed to be bandlimited and have zero mean Gaussian characterist-ics without the loss of generality, it can be shownthat the last two terms of (35) vanish to zero (seeAppendix A of [9]) . Then, we may rewrite (35) as

Ni? = 1 = [R7.)+JRxr")J],

(36)4N" ,

where R'x)-E[x"(t)x°(t)] is given by

R;,'=AD` ')SD"'-")A"+o I. (37)

It can easily be shown that JA * =A D` -" , so that(36) is equivalent to

k= ;[ASA"+cr I],

(38)where the spatially smoothed signal correlation

matrix S is given by (14). This centro-symmetriccovariance matrix is the same as that for the MSStechnique [15, 26] except for the scaling constantwhose value gives no effect upon the beamformer

function . Therefore, roll) constitutes a new inputwhose spatially smoothed covariance matrix hasthe property equivalent to that of the MSStechnique .

It leads us to conclude that if we use rn (t) as aninput signal for subbeamformers whose outputs aresummed to yield a spatially averaged system out-put, we need only 2'K sensor elements to resolve Kcoherent signal sources as in the MSS technique .For the original spatial smoothing technique thenumber would be 2K [ 16-18] .

3 .2. Modified parallel spatial processing scheme

To overcome the signal cancellation problem, Suet al . recently proposed a parallel spatial processingscheme in which the system output is obtained byaveraging the outputs of subbeamformers [22] .Due to its parallel structure, the adaptive algorithmrequires only one adaptation of order M, thenumber of sensor elements in each subbeamformer .The original spatial smoothing technique requiresN times of order M adaptation, where N is thenumber of subbeamformers [2, 17] . Because theparallel spatial processing scheme performs spatialaveraging of subbeamformer outputs without for-ming an explicit covariance matrix, the applicationof our DDSP algorithm to their beamformer wouldbe simple, and constitutes an alternative realizationof the MSS technique . Let us examine the perfor-mance of the modified parallel spatial processingscheme that uses the proposed DDSP algorithm .

If we assume a coherent interference and abroadside look-direction signal, the ith sensor out-put is given by

x,( t)= (Ts e'w"+ae'°'+'q-xt-I mar ',

(39)

whereas and a, denote the amplitudes of the signaland the jammer, respectively . As in the spatialsmoothing technique, let an array of L

S. Park, C. K. Un / Smoothing for interference cancellation

307

(=-N+M-1) sensors be divided into N overlap-ping subarrays of size M as shown in Fig . 1 . Also,let x„(t) denote the output of the nth subarray forn= I, . . . , N, as (8) . Similarly, we can write thereversely ordered complex conjugate output of thenth subarray i„(t) as (33) . As described in (34),let us introduce new vectors {r„(t)}, and use thesevectors as new input signals as shown in Fig . 6 .Then, for given time instant t, the nth Frost sub-beamformer output y„(t) is given by

yn(t)=w''(t)r„(t-n+1) .

(40)

At the (t + n - I )th time instant, one may then have

y„(t+n-1)=wn(t+n-I)r„(t) .

(41)

The sum of delayed outputs constitutes a systemoutput at time instant t, and is given by

1 Ny(t) =- Y y,(t-N+n) .

(42)N .,-1

Accordingly, the overall system output y(t +N-1)

I "V

y(t+N-1)= -NY y„(t+n-1) .

(43)n=1

Substituting (39) and (40) into (43) yields

y(t+N-1)

1=Re {as e' }+or,e' t0 r

:, a(t+n)2N„-,

x[eI)tn+e'We-ia-Ilr<ann], (44)

where Re{ } is the real part of a complex value,and

uall +n)=~ w*(t+n-1) e'''-"°a"

(45)i-I

and

yi=2o„t+20+(M-I)coo r, .

(46)

As the adaptive beamformer reaches its steadystate performance surface, we may make anapproximation as

w*(t+n-1)

:w,-*(t+N-1), n=1, . . .,N-1 .

(47)Vol . 24, Nn. 3, September 1991

308

1/ (x1+JX ;)

1stSUBBEAMFORMER

y1 (k)

WEIGHTCOPY

IX2

1/y (X2+JXi)

Consequently, we have an approximate value ofthe system output as

y(t+N-1)

:Re{as e'°"Of +a1 e1 ( "°" 0 ta(t+N) I2N

x Y, ~eitn-nw,T,+eiWe-i(~-Dcwr~)

(48)

Observe that the last term in (48) is the summationof two groups of N uniformly spaced terms on theunit circle, as shown in Fig. 3, which asymptoticallyapproaches zero for a large number of subarraysN, and thus we have

lim y(t+N-1) Re{ase'°'°r},

(49)'V-c

which is a maximum likelihood estimate of thedesired signal .

To illustrate the enlarged effective array aper-ture, we performed simulation for an environmentwhere two coherent and two noncoherent interfer-ences exist. We assumed that coherent interferencesSignal Processing

S. Park, C. K. Un / Smoothing far interference cancellation

2nd

SUBREAMFORMER

WEIGHTCOPY

SYSTEM OUTPUT

Fig . 6. Application of the DDSP algorithm to the parallel spatial processing scheme .

%2 (XN+JN)

z

XN

1

NthI SUBBEAMFORMER

YN(k-N+1)

arrived from directions -50° and -30 = , and nonco-herent ones from 10' and 60` . In Fig . 7 beam pat-terns of the parallel spatial processing scheme withand without the DDSP algorithm is shown . Onecan see that with an 8-element array, the originalparallel spatial processing scheme cannot give nullsupon coherent interferences, but the modified par-allel spatial processing scheme with the DDSPalgorithm puts nulls upon coherent interferences aswell as noncoherent ones .

4. Systolic implementation of beamformer

4.1. Least-squares filtering based onQR decomposition

In adaptive beamforming, array sensor outputsare combined by the recursively updated adaptiveweight w(t) to yield the system output signal e(t),which is represented as

e(t)=xT(t)w"(t)= y x,(t)w`(t),i= t

(50)

where M is the number of sensor elements in an

0 .

array. In an MVDR beamformer, it is desired tominimize the output signal power while maintain-ing a fixed gain for the desired look direction [ 1, 4] .Without the loss of generality, we may assumethat one receives the desired signal that arrivesfrom the broadside direction with unity gain as~" wi(t)=1 . With these assumptions in hand, wehave

M-Ie(t)= /r [xi(t)-xM(t)]w,*(t)+x4(t) .

(51),-I

By denoting y(tj) °=-x,M(ti), (51) for a given timeinstant g may be represented as

Denoting w*(t„) by w(n) and gathering (52) up totime t„ yields the following equation :

S. Park, C. K. Un / Smoothing for interference cancellation

DESIRED SIGNAI,INTERFERENCES

INTERFERENCEF

-40-

I

-50~ti 1 -90

-60

-30

0

30

60

90ANGLE (DEGREE)

Fig . 7 . Beam patterns of the parallel spatial processing scheme with and without DDSP algorithm (L=8) .

and

309

y(n) =[y(tl),y(t2), . . .,Y(tl)]T .

(57)

Now, we wish to find a weight vector which liesin the range space of the given data matrix Z(n),and, at the same time, is as near as possible to agiven observation vector y(n) . When the 2-norm isused as a performance measure, the optimumweight vector can be represented as [10)

w(n)= {wECM-II IIZ(n)w(n)-y(n)12

=minimum;, (58)

where C" _' is the (M-1)-dimensional complexspace, and II 112 denotes the 2-norm or Euclideannorm. This solution is unique, and has minimallength if and only if Z(n) has full rank [201 . If thereexist coherent interferences, however, the rangespace of Z(n) collapses to a one-dimensional sub-space . In that case, neither columns nor rows ofZ(n) are linearly independent of each other .Accordingly, there may exist an infinite number oftrivial solutions wtr(n) which are projections ontothe nullspace of Z(n), and thus satisfy

Z(n)wtr(n) =0 .

(59)Vol . 24 . No . 3, September 1991

e(n) = Z(n)w(n) - y(n), (54)

where

e(n)=[e(t1), e(t2), . . . , e(t.)]T, (55)

Z(n)=[ZT(t, ), ZT(t2), . . . , zT(t..)] I (56)

e(t) =zT(tt)w*(t)-y(ti), (52)

where the ith element of z(tj) is given by

zt(ti)ax/(tf)-XM(tf) . (53)

This means that we cannot find an optimum solu-tion whose projection lies entirely on the row spaceof Z(n) [21] . Furthermore, the general solutionobtained in this environment yields completely zeroresidual undesirably .

Once Z(n) is guaranteed to have its full rank,the method of orthogonal transformations, suchas the QR decomposition least-squares (QRD-LS)minimization method, can be used to circumventthe numerical difficulties arising in solving (58)directly [7] .

4.2. Parallel spatial smoothing (PSS) technique

Let a linear uniform array of L ( °= N+M-1)sensors be divided into N overlapping subarrays ofsize M as shown in Fig . 1 . Also, let xn (t) denotethe output of the nth subarray for n=1, . . . , N,Then, we may write the sensor output of the nthsubarray as (8) . Now, we define new vectors{g„(t)} for n=1, . . . . N, such that

gn(t)=xmod(n+,_2, . +,(t), (60)

where mod(a, b) stands for modulo b operationson a . We illustrate these vectors for n=1, 2, . . . , NT, . . . , in Table 1, where T is an integernumber. The NT-sample covariance estimate of theSignal Processing

first vector g,(t) can easily be shown to be1 NT

R (1 j= I g1(t)g'(t)NT, ,1 NT

_-

xm0 (,-7,M+1(t)NT,-,

HX xmod(r-[ ,N), I (t) •

Introducing an integer number l that satisfies t=(i-1)N+n, we may rewrite (61) as

R =1 E L xn((t-1)N+n)

Xx4((I-1)N+n) .

(62)

Here, without the loss of generality, we mayassume that signal sources are narrowband, andthat signal and noise fields are temporally station-ary. Then, for large T, we can approximate (62) as

1 N

1 (T-I)N

R ^-N E,T

k~a xn(k)xn(k)],

(63)

where k A(7- I )N is an integer number whose timesequence is given by k=0, N, 2N, . . . , ( T-1)N.We note that the last term in the bracket of (63) isthe T-sample covariance estimate of xn (t) amonggiven NT samples, which, for sufficiently large T,asymptotically approaches to the covariance esti-mate for the nth subarray in the spatial smoothing

310 S. Park, C.K Un / Smoothing for interference cancellation

Table I

The time sequence of {g,(t)}

Time, t

g,(1) g2(t) g,(t) g {t)I

x,(I) xz(I) x,(1) x,.,(1)2

xx(2) x3(2) x,,,(2) x,(2)

x,.(N) x,(N) ,(N) s (N)N+I

x,(N+1)N+2

x,(N+2)

2N

x,,,(2 .N)

x,(N+1)x,(N-2)

x 1 (2N)

x,(N+ 1)x, + ,(N+2)

x,-,(2.7%')

x,,.(N+ 1)x,(N+2)

xN ,(2N)

NT- T+I

x,(NT-T+I)

NT-I

x,v ,(NT-1)NT

xN(NT)

x i(NT-T+I)

x,,,(NT-1)x,(NT)

x„(NT-T+1)

,(NT- I)x, ,(NT)

XN(NT-T+1)

x,_ r(NT-1)x, ,(NT)

technique [17] . Therefore, Rt' can be regarded asthe spatially smoothed covariance matrix in whichT time samples are used among given NT timesamples, and thus will always be nonsingular evenwhen there exist coherent interferences [17] . Simi-larly, it can be shown for large T that

R t ozR12, t . . xR'`)z . . --R(N) .

(64)

Furthermore, summing {Rt"1 } for n= l, . . . , N, wehave

NR -A-- Y R1ro

N„- I

1 N

.

1 Nr

g"(t)gHU)

(65)N J, NT ,_ j

From Table l, one can see that

N

Y_ g"(t)gn(t)= L x"(t)x"(t) .

(66)"-1

"=l

Accordingly, (65) can be rewritten as

I N

1 NTR =-

x"(t)xH(t) I .

(67)N"=1 NT,-,

Note that the last term in the bracket of (67) isnow a complete (i .e ., NT-sample) covariance esti-mate for the nth subarray . Hence, as T becomeslarger, (67) approaches to the spatially smoothedcovariance matrix which is given by (10) . Con-sequently, we conclude that {g"(t)} constitute newinput vectors each of which yields a nonsingularcovariance matrix, and the spatially averaged co-variance matrix of {g„(t)} is exactly identical tothe spatially smoothed covariance matrix of Shanet al . [17] .

We note that since each of {g"(t) } yields a non-singular covariance matrix, {g„(t)} can be pro-cessed in parallel, and hence high throughput canbe achieved. The system output is obtained simplyby summing all the subbeamformer outputs . In

S. Park, C.K. Un /Smoothing for interference cancellation 311

Fig. 8, we show a schematic diagram for the PSStechnique . Any type of adaptive algorithms can beused in each subbeamformer .

Recently, McWhirter et al . suggested a recursiveQRD-LS algorithm in which the least-squares resi-dual is directly obtained at each stage of the recurs-ive process without the explicit computation of theweight vector [12, 23] . It provides an outstandingnumerical robustness against roundoff errors evenwhen the given data are ill-conditioned . Further-more, due to the parallel/pipelined systolic archi-tecture, the method provides high throughput .Since the proposed PSS technique does not formcovariance matrices, it can easily be realized withsystolic arrays described in [12, 23] .

4.3. Parallel modified spatial smoothing(PMSS) technique

Since the proposed PSS beamformer performsspatial smoothing without forming explicit covari-ance matrices, the incorporation of the DDSPalgorithm described in Section 3 would bestraightforward and constitutes a parallel reali-zation of the MSS technique .

Let an array of L (-°-N+M-I) sensor elementsbe divided into N overlapping subarrays of size Mas shown in Fig . l . Now, we compose vectors{go(t)} with subarray data as (60) . According tothe DDSP algorithm which expands the effectivearray aperture . we introduce new subbeamformerinput vectors {r"(t)} as

r"(t)=° =[g"(t)+g"(t)] .

(68)

These vectors differ from (34) in that we now useg„(t) instead of x"(Q in carrying out the DDSPalgorithm, by which spatial smoothing is per-formed . Then, by summing all the outputs of sub-beamformer whose inputs are }r„(t)}, we canobtain the nulling performance of the MSS tech-nique without forming covariance matrices . In Fig .9, we show the implementation of the proposedPMSS technique. The constraint preprocessor(CP) in Fig. 9 for imposing the unit gain constraint

Vol . 24, No 3. September 1991

Signal Prncesslng

RECURSIVE

TRIANGULARORD-LS

SYSTOLICMINIMIZATION

RRAY

SYSTEM OUTPUT

Fig . 8 . Schematic diagram of the PSS technique .

TRIANGULARSYSTOLIC

RRA

TRIANGULARSYSTOLIC

RRAY

TRIANGULARSYSTOLIC

RRAY

e n (t)

eN (t)

.!~

SYSTEM OUTPUT

Fig . 9 . Schematic diagram of the PMSS technique .

9 1 (t)

92(t)

9n(t)

9N(t)

9 1(2)

92(2)

9n(2)

9N(2)

91(1)

92(1)

9n(1)

9N(1)DOR

~~ALGORITHM

9 1 ( ) J9 1 (t)I I 92(t)

(t)

9 n(t)+J9;, (t)

9N(t) J9N (t)+

J92

I . .

+ICONSTRAINT

PREPROCESSINGr 1 (t) r2(t) rn(t) rN (t)I

I

C P

C P

. . f

C P

C PIZ 1(t) Y1(t) Z2(t) Y2(t) Zn(t) Yn(t) ZN(t) YN(t)

\l

I

\/ f1 st 2 nd n th

312 S. Park, C.K. Un / Smoothing for interference cancellation

91(t) 92(t) 9n (t) 9N (t)

g1 (2) 92(2) gn(2) 9N(2 )

91(1) 92(1) 9n (1) 9N(1)

NV U,1 st 2 nd n th N th

SUBBEAMFORMER SUBBEAMFORMER SUBBEAMFOR ER SUBBEAMFORMER

r,(' ) (t)

r.( 2) (t)

zn')(t)

zn2)(t)

z ^0 (t)

z("-,)(t) Yn(t)

Fig . 10 . Constraint preprocessor for imposing unit gain on thebroadside direction .

on the broadside look direction can be implemen-ted as shown in Fig . 10 .

Let the ith elements of data vectors r„(t), z„(t)and g„(t) in Fig. 8 be denoted by r ;,°(t), z;,`) (t) andg;,°(t), respectively. Then, according to (68) and(51), the ith elements of data vectors rn (t), zn(t)

and yn(t), for n=1, 2N, are represented asfollows

The proposed architecture provides numericalstability, high throughput and high nulling cap-ability against coherent interferences .

4.4. Simulation results

To demonstrate the numerical properties andconvergence speeds of adaptive beamforming tech-niques, we compare the output residual powers ofthe three adaptive beamforming techniques, thespatial smoothing technique using the recursive

r,0) (t)

S. Park, C.K On / Smoothing for interference cancellation

313

rn(M

' ( t)M

( t )least-squares (RLS) algorithm (the SS-RLS beam-former), the parallel spatial processing schemeusing the Frost algorithm (the PSP beamformer),and the proposed PMSS technique using the QRD-LS minimization (the PMSS beamformer). Theoutput residual power at time instant t is definedas

Pd t) °=le(t)-d(t)12,

( 72)

where d(t) is the input desired signal at time t ande(t) is the corresponding beamformer output . Weassume that there arrive four coherent interferencesfrom -50 = , -30°, 10° and 60' . Also, we assumethat the input interference-to-noise ratio (INR) is40 dB, the mantissa wordlength is 24 bits and thenumber of sensor elements is 10, In Fig . 11, wecompare the output residual powers of the threeadaptive beamformers . The SS-RLS beamformerconverges in several time samples . On the contrary,the PSP beamformer has relatively slow conver-gence speed compared with the SS-RLS beam-former, as shown in Fig . 11. The proposed PMSSbeamformer has the same convergence speed as theSS-RLS beamformer as shown in Fig . 11 . We notethat the PMSS beamformer cannot give outputsduring the initialization interval which normallytakes N (i .e ., the number of subarrays) timesamples .

Next, we compare the output residual powersof the adaptive beamforming techniques when the

-50+ 1 V , r ,o

100

150

200ADAPTATION SAMPLES

Fig . 11 . Output residual powers of the three adaptive beam-formers (input INR=40 dB, wordlength-24 bits) .

So1 . 24, No . 3, September 1991

M-'+I)`rn)(t)

p2(g z) (t) +R

(t)),

i=1, . . . , M, (69)

z°(t) r:'(t)-rnst)(t),

1 =1, . .,,M-1 (70)

and

j' (M o r(M)(t) (71)

314

-500

S. Park, C. K. Un / Smoothing for interference cancellation

„iuiLs rcimn,m.,

Fig. 12 . Output residual powers of the three adaptive beam-formers (input INR = 40 dB, wordlength=16 bits) .

mantissa wordlength is reduced to 16 bits . Asshown in Fig. 12, the SS-RLS beamformer is extre-mely sensitive to the finite wordlength effect asexpected . On the other hand, the PSP and PMSSbeamformers are numerically robust as shown inFig. 12 .

Now, we compare the output residual powers ofthe adaptive beamformers when the input INR isincreased to 60 dB. As shown in Fig . 13, the PSPbeamformer converges more slowly than it doeswhen the input INR is 40 dB, and suffers numericalinstability due to the wide spread of eigenvalues .On the other hand, the proposed PMSS beam-former maintains fast convergence speed and goodnumerical stability as shown in Fig . 13 .

100

150

200ADAPTATION SAMPLES

Fig. 13 . Output residual powers of the PSP and PMSS beam-formers (input INR=60 dB, wordlength = 16 bits) .

Signal Processing

Finally, we illustrate the beam-pattern conver-gence speed of the PSP and PMSS beamformerswhen four noncoherent interferences arrive fromthe directions of 10°, 30°, 50° and 70° . The inputINR is assumed to be 40 dB, and the number ofsensor elements is 8 . The PSP beamformer con-verges slowly and yields high sidelobe levels asshown in Fig. 14. Notice that the method cannotgenerate null at 10° within 160 time samples . Onthe contrary, the proposed PMSS beamformer con-verges quickly and yields low sidelobe levels asshown in Fig . 15 .

5. Conclusions

In this work, we have studied methods toimprove the nulling capability of an adaptivebeamformer against coherent interferences, whilemaintaining numerical stability and highthroughput .

We have presented a DDSP algorithm, by whichthe effective array aperture is increased withoutforming covariance matrices . The proposedmethod generates new subbeamformer inputs byadding each subarray data vector to its reverselyordered complex conjugate signal vector. Then, bysumming the subbeamformer outputs, we canobtain the maximum likelihood estimate of thedesired signal, which is identical to the one that isobtained by the MSS technique . We can realize theMSS technique with only order M (the number ofantenna elements in a subarray) adaptation at eachsubbeamformer by combining the proposed DDSPalgorithm with the parallel spatial processingscheme. This method provides an enlarged arrayaperture as well as high throughput .

In addition, we have proposed a PSS techniqueby rearranging spatial subarray data vectors . Wehave shown that the proposed method can easily berealized using the QR decomposition least-squaresminimization and thus be implemented with sys-tolic arrays . Furthermore, since the proposed PSS

DESIRED SIGNAL : 00INTERFERENCES : 10° , 30° , 50 ° , 700

S. Park, C,K [in ; Smoothing for interference cancellation

315

GAIN (dB)

DESIRED SIGNAL : 00INTERFERENCES : 10° , 300 , 500 , 70 °

GAIN (dB)

0

-10

-20

-30

-40

g0 160

Fig . 14 . Convergence of the beam pattern with the PSP beamformer (input INR=60 dB . wordlength=24 bits) .

90 160

Fig . 15 . Convergence of the beam pattern with the PMSS beamformer (input INR=60 dB, wordlength=24 bits) .Vol, 24, No . 3 . September 1991

316

S. Park, C. K. Un /Smoothing for interference cancellation

method performs spatial smoothing without for-ming covariance matrices explicitly, the incorpor-ation of the proposed DDSP algorithm isstraightforward and constitutes a PMSS techniquewhich is a parallel implementation method of theMSS technique . The resulting beamformer yieldedhigh nulling capability, high throughput and goodnumerical stability, while nulling out coherentinterferences .

Acknowledgment

The authors wish to thank the reviewers of thispaper for their helpful suggestions .

Appendix A. Comparison of the decorrelationperformances of the spatial smoothing andmodified spatial smoothing techniques

In this appendix, we prove a theorem that thedecorrelation performance of the MSS technique isbetter than that ofthe spatial smoothing technique .

THEOREM. Let the effect of coherence of signalsin the spatial smoothing technique and the MSStechnique be reduced by an amount of a and p,

respectively, where (see (17) and (19))

1 Na=_Y

e;tn-t)W

(A.1)Nn-,

and

1 C.v~P=- ~, (ei(n-OW +e-i2(w+p) i<" ' ') .

2N n_,

Then we have Ipl 2 ,la1 2 .

I -e 'NW

I-eiM

(A.2)

PROOF It can be shown that

la , 2-N2

Signal Pr«xs,ing

2 1 1 -cos Nyi

N2 1-cos W

(A.3)

Meanwhile, (3 can be rearranged to yield

Then the squared norm of p is given by

1p12 (I -cos Nyi+cos A -cos A cos Nyt)

2N2(1-cos yr)(A.6)

Accordingly, the difference of the two squarednorms 1a1 2 and ip1 2 is given by

Ia12-Ip12=(I-cos Nyt)(1 -cos A)2N2(1 -cos V)

0 .

(A.7)

The equality is satisfied when

Ny=2nit or (N+l)yt+2O=2mrt, (A.8)

where n and m are integer numbers . From (A.7)-(A.8), we finally have

IR2,<

Ia1 2 .

O

References

(A.9)

[1] S .P . Applebaum and D .J . Chapman, "Adaptive arrayswith main beam constraints", IEEE Traits . Antennas Pro-pagat ., Vol . AP-24, September 1976, pp . 650-661 .

[2] J.E . Evans, J .R. Johnson and D .F . Sun, "Application ofadvanced signal processing techniques to angle of arrivalestimation in ATC navigation and surveillance systems" .MIT Lincoln Lab ., Lexington, MA, Tech . Rep . 582, June1982 .

[3] A .J . Fenn, "Evaluation ofadaptive phased array antennafar-field nulling performance in the near-field region",IEEE Trans . Antennas Propagat ., Vol . AP-38, February1990, pp, 176 185 .

_ 1 I-e'NW+e-i~tw+m>1 a - 'Nw2N 1-e'W

1-e-' W

_ 1 1-cos NV+cos A -cos B

2N

1-cos yi-j sin yr

sin NV+sin A -sin Bl(A.4)

1-cos ip-j sin

lW

where A and B are real constants given by

A°=(N+1)w+2O and B °= yt+20 . (A.5)

[4] O .L . Frost. 111, "An algorithm for linearly constrainedadaptive array processing", Proc. IEEE, Vol . 60, August1972, pp. 926-935 .

[5] W.F . Gabriel, "Adaptive arrays - An introduction", Proc .IEEE, Vol . 64, February 1976, pp . 239-272 .

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[7] G.H. Golub and CF . Van Loan, Matrix Computations,Johns Hopkins Univ . Press. Baltimore, MD, 1985 .

[8] Y .J . Hong, C .C. Yeh and D.R . Ucci, "The effect of afinite-distance signal on a far-field steering Applebaumarmy Two-dimensional case', IEEE Trans. AntennasPropagat ., Vol. AP-36, April 1988 . pp. 468-475 .

[9] J .E . Hudson . Adaptive Array Principles, Peter Peregrinus,London, 1981 .

[10] CL. Lawson and R .J . Hanson. Solving Least-SquaresProblems, Prentice-Hall, Englewood Cliffs. NJ . 1974 .

[111 D .A . Linebarger and D .H. Johnson, "The effects of spatialaveraging on coherence and resolution", Proc . IEEE Inter-nat . Conj. Acoust . Speech Signal Process. . 1988, pp . 28652868 .

[121 J .G . McWhirter, "Recursive least-squares minimizationusing a systolic array", Proc . SPIE, Vol . 431, Real-TimeSignal Processing IV, 1983, pp . 105-112 .

[13] R.A. Monzingo and T .W. Miller, Introduction to AdaptiveArrays, Wiley, New York, 1980 .

[14] S . Park and C.K. Un, `Simplified realisation method ofmodified spatial smoothing technique - ', Electron. Lett,Vol. 24, No . 18, September 1988, pp . 1176-1177 .

[15] S .U. Pillai and B .H . Kwon, "Forward backward spatialsmoothing techniques for coherent signal identification",IEEE Trans. .Acoust. Speech Signal Process . . Vol. ASSP-37, January 1989, pp. 8-14 .

[16] T .1 . Shan . "Array processing for coherent sources", Ph.D .Dissertation . Stanford University, Stanford, CA, 1986 .

[17] T .1 . Shan and T . Kailath, 'Adaptive beamforming forcoherent signals and interference", IEEE Trans . Acoust .Speech Signal Process . . Vol . ASSP-33, June 1985 . pp. 527-536 .

[18] T .J . Shan, M. Wax and T . Kailath. "On spatial smoothingfor direction-of-arrival estimation of coherent signals" ..IEEE Trans . Acoust Speech Signal Process., Vol. ASSP-33 . August 1985 . pp . 806-811 .

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317

[19] B.D. Steinberg, Principles of Aperture and Array SystemDesign . Wiley, New York, 1976 .

[20] G .W . Stewart, Introduction to Matrix Computations .Academic Press . New York, 1973 .

[21] G . Strang, Linear Algebra and Its Applications, AcademicPress, New York . 1976 .

[22] Y .L . So, T .J. Shan and B. Widrow . "Parallel spatial pro-cessing : A cure for signal cancellation in adaptive arrays" .IEEE Trans. Antennas Propagat. . Vol. AP-34, March1986, pp . 347-354 .

[231 C .R . Ward, P .J . Hargrave and J .G. McWhirter, A novelalgorithm and architecture for adaptive digital beam£orm-ing", IEEE Trans. Antennas Propagat ., Vol. AP-34, March1986, pp . 338-346 .

[241 B . Widrow . P .E. Manley, L .J . Griffiths and B .B . Goode,"Adaptive antenna systems", Proc . IEEE, Vol . 55,December 1967, pp . 2143-2159.

[25] B . Widrow, K .M . Duvall, R .P. Gooch and W .C. Newman,Signal cancellation phenomena in adaptive antennas :

Causes and cures", IEEE Trans. Antennas Propagat ., Vol .AP-30. May 1982, pp . 469-478 .

[26] R .T. Williams, S .P. Prasad, A.K. Mahalanabis andL.H. Sibul, "An improved spatial smoothing technique forbearing estimation in a multipath environment IEEETrans . Acmu'i . Speech Signal Process ., Vol. ASSP-36,April 1988, pp . 425-43 L

[27] CC. Yeh, Y .J . Hong and D.R . Loci, "The performanceof a defocused adaptive array with a finite-distance signalsource", IEEE Trans . Antennas Propagat . . Vol. AP-36,April 1988, pp . 476-482 .

[281 C .C. Yeh, Y .J. Hong and D .R. Ucci, "The effect of afinite-distance signal on an Applehaum array" . IEEETrans. Antennas Propagat ., Vol. AP-33, September 1985,pp.1003-1008 .

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Vol. 24 . Na . 3 . SeNcmber 1991