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Beamforming: A VersatileApproach to Spatial FilteringBarry D. Van

A beamformer is a processor used in conjunction with anarray of sensors to prov ide a versatile form of spatial filtering.The sensor array c ollects spatial samples of prop agating wavefields, whic h are processed by the beamform er. The objectiveis to estimate the signal arriving from a desired direction inthe presence of noise and interfering signals. A beamformerperforms spatial filtering to separate signals that have over-lapping frequency content but originate from different spatiallocations. This paper provides an ove rview of beamform ingfrom a signal processing perspective, with an emphasis on re-cent research. Da ta independent, statistically optimum, adap-tive, and partially adaptive beam form ing are discussed.

1. I N T R O D U C T I O NThe term beamforming derives from the fact that early

spatial filters were designed to form pencil beams (seepolar plot in Fig. 1 I) in order to receive a signal radiatingfrom a specific location and attenuate signals fro m otherlocations. Forming beams seems to indicate radiation ofenergy; however, beamforming i s applicable to eitherradiation or reception of energy. In this paper we discussformation of beams for reception.

Systems designed to receive spatially propagating sig-nals often encounter the presence of interfe rence signals.If the desired signal and interferers occupy the same tem-poral frequency band, then temporal filtering cannot beused to separate signal from interference. However, thedesired and interfering signals usually originate from di f-ferent spatial locations. This spatial separation can be ex-ploited to separate signal fr om interference using a spatialfilter at the receiver. Implementing a temporal filter requiresprocessing of data collected over a temporal aperture.Similarly, im plementing a spatial filter requires processingof data collected over a spatial aperture.

Several applications that employ spatial fil tering of dataare listed in Table 1 1. Fig. 1 I illustrates a microwave com-munications antenna that employs a continuous spatialaperture to accomplish spatial filtering with a single an-tenna. Fig. 1.2 depicts a low frequency-towed sonar arrayin which the spatial aperture i s obtained through a dis-crete spatial sampling by an array of sensors. When thespatial sampling i s discrete, the processor that performsthe spatial filtering i s termed a beamformer. Typically a4 IEEE ASSP MAGAZINE APRIL 1988

Veen an d Kevin M. Buckley

igure 1 I A continuous spatial aperture provides onenechanism for spatial filtering. Illus trated is a parabolicnicrowave antenna system. The antenna dish provides;he spatial aperture over which energy is collected. This3nergy is reflec ted t o the antenna feed. The dish and feediperate as a spatial integrator . The energy from a far fieldsource located directly in front of the antenna arrives a t;he feed temporarily aligned [i.e., all source-to-feed pathengths are equal1 and is coherently summed. In general,?nergy from other sources will arrive a t the feed viaiariable length paths, and add incoherently. A polar plotif typical antenna beampattern (i.e., power gain v s . , in;his case, azimuth angle1 is shown for a selected fre-iuency and for the elevation angle at which the antenna isminted.

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TABLE 1.1ARRAYS N0 BEAMFORMERS PROVIDE AN EFFECTIVE AND VERSATILE MEANS OF SPATIALFILTERING. THIS TABLE LISTS A NUMBER OF APPLICATIONS OF SPATIAL FILTERING, GIVES

EXAMPLES OF ARRAYS AN0 BEAMFORMERS, AND PROVIDES A FEW KEY REFERENCES.

Application Description ReferencesRADAR phased-array RADAR; air traffic Brookner [19851; Haykin 119851;

SONAR source local ization and classification Knigh t et al. [19811; Owsley

Communica tions directional transmission and Mayhan 119761; Com pton119781; Adams e t al. 119801

contro l; synthetic aperture RADAR Munson et al. (19831

[I9851

reception; sector broadcast insatellite communications

Imaging ultrasonic; opt ica l; tomographic Macovski [19831; Pratt I19781;

Geophysical earth crust mapping; oil exploration Justice [I9851ExplorationAstrophysical high resolution imaging of the Readhead [19821; Yen [I985Exploration universeBiomedical fetal heart moni toring ; tissue

Kak [I9851

Wi dr ow e t al. 119751; Gee et al .[19841; Peterson et al. [I9871yperthermia; hearing aids

beamformer linearly combines the spatially sampled timeseries from each sensor to obtain a scalar output timeseries in the same manner that an FIR filter linearlycombines temporally sampled data. Two principal advan-tages of spatial sampling wi th an array of sensors are dis-cussed below.

Spatial discrimination capability depends on the size ofthe spatial aperture; as the aperture increases, discrimi-nation improves. The absolute aperture size i s not impor-tant, rather its size in wavelengths i s the critical parameter.A single physical antenna (continuous spatial aperture)capable of providing the requisite discrimination i s oftenpractical for high frequency signals since the wavelengthi s short. However, when low frequency signals are of in-terest, an array of sensors can often synthesize a muchlarger spatial aperture than that practical with a singlephysical antenna.

A second very significant advantage of us ing an array ofsensors, relevant at any wavelength, i s the spatial filter ingversatility off ered by discrete sampling. In many applica-tion areas it i s necessary to change the spatial filteringfunction in real time to maintain effective suppression ofinte rfer ing signals. This change i s easily implemented i n adiscretely sampled system by changing the way in whichthe beamformer linearly combines the sensor data.Changing the spatial filtering function of a continuous ap-erture antenna i s impractical.

The purpose of this paper i s to describe beamformingfrom a signal processing perspective, provide an overviewof beamformer design, and briefly discuss perform-ance and implementation issues with an emphasis on re-cent research. The paper begins wit h a section devoted to

defining basic terminology, notation, and concepts. Suc-ceeding sections cover data-independent, statistically op-timum, adaptive, and partially adaptive beamforming. Wethen provide a brief discussion of implementation issuesand conclude wi th a summary.

Throughout t he paper we use familiar methods andtechniques from FIR filtering to provide insight into vari-ous aspects of spatial filter ing wi th a beamformer. How-ever, i n some ways beamforming differs significantly fromFIR filtering. For example, i n beamforming a source ofenergy has several parameters that can be of interest:range, azimuth and elevation angles, polarization, andtemporal frequency content. Different signals are oftenmutually correlated as a result of mult ipath propagation.The spatial sampling i s often nonuniform and multi-dimensional. Uncertainty must often be included in char-acterization of individual sensor response and location,motivating development of robust beamforming tech-niques. These differences indicate that beamforming rep-resents a more general prob lem than FIR filte ring and as aresult, more general design procedures and processingstructures are common.

Rather than mak ing a futile attempt at attributing devel-opments due to many different researchers in beam-forming, we refer the reader to the following references:books-J. W. R. Griffiths et al., ed. [19731, Hudson [19811,Monzingo and Miller [19801, Haykin, ed. [19851, Compton[19881; special issues / E Transactions on Antennas andPropagation [19761, [19861, jo ur na l of Ocean Engineering[19871; tutorial- abriel [1976]; and bibliography- Marr[19861. Papers devoted to beamforming are often found inthe I E E E Transactions on: Antennas and Propagation,

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have any necessary receiver electronics and an ND con-verter if beamforming is performed digitally.

The second beamformer i n Fig. 2.1 samples the propa-gating wave field in bot h space and time and is often usedwhen signals of significant frequency extent (broadband)are of interest. The outp ut i n this case can be expressed as

(2.2)

where K - 1 s the number of delays in each of the J sensorchannels. If the signal at each sensor i s viewed as an input ,then a beamformer represents a multi-input single out-put system.

It i s convenient to develop notation which permits us totreat both beamformers in Fig. 2.1 simultaneously. Notethat (2.1) and (2.2) can be wri tte n as

y(k) = w H x ( k ) . (2.3)by appropriately defining a weight vector w and data vec-tor x(k). We use lower and upper case boldface to denotevector and matrix quantities respectively, and let super-script H represent Hermitian (complex conjugate) trans-pose. Vectors are assumed to be column vectors. Assumethat w and x(k) are N dimensional; this implies that N = KJwhen referring to (2.2) and N = J when referring to (2.1).Except for Section V o n adaptive algorithms, we wil l d ropthe time index and assume that it s presence i s understoodthroughout the remainder of the paper. Thus (2.3) i s writ-ten as y = wHx. Many of th e techniques described in thispaper are applicable to con tinuous time as well as discretetime beamforming.

The frequency response of an FIR filter w ith tap weightsw 0 5 p 5 J and a tap delay of T seconds is given by

1 K- 1y(k) = c c WtpXl(k - P)I= 1 p= o

I

p = l,.(@) = W p + e - ~ ~ T ( ~ - l ) (2.4a)

Alternativelyr(w) = wHd(w) (2.4b)

where w H = [w? w;. . .WT] and d ( w ) = [ I eJwTe12wT. .e ~ i l - l ) w T H1 . r(o) represents the response of the filter to acomplex sinusoid of frequency w and d(w) i s a vector de-scribing the phase of the complex sinusoid at each tap i nthe F IR filter relative to the tap associated with wl.

Similarly, beamformer response i s defined as the ampli-tude and phase presented to a complex plane wave as afunction of location and frequency. Location i s in generala three dimensional quantity, but often we are only con-cerned with one or two dimensional direction of arrival(DOA). Throughout the remainder of the paper we do notconsider range. Fig. 2.2 illustrates the manner in which anarray of sensors samples a spatially propagating signal.Assume that the signal i s a complex plane wave withDO A 8 and frequency w . For convenience let the phasebe zero at the first sensor. This implies xl(k) = elwkandxdk) = eJw[k-AI'o)l,5 I 5 J.Al(8) represents the time delaydue to propagation from the first to the Ithsensor. Substi-tution into (2.2) results in the beamformer output

ISensor#IReference

I Y- - ly(k) = 2 '2' t p e - l W I A l ( o ) + P l - eIwkr(8,w) (2.5)I=1 p= o

where A , ( @ = 0. r(0, w ) i s the beamformer response andcan be expressed in vector form as

r(8, w ) = wHd(8, 0). (2.6)The elements of d ( 8 , w ) correspond to the complexexponentials e-lw[Al(o)+P'In general it can be expressed as

d(8, w ) = [IJ W T 2 ( o ) e l W T 3 ( 8 ) . . . elwTN(@I]H (2.7)where the 7,(8), 2 5 i 5 N, are the time delays due topropagation and any tap delays from the zero phase refer-ence to the p oint at which the i th eight i s applied. Werefer to d(0 ,w) as the array response vector. It i s alsoknown as the steering vector or direction vector. Non-ideal sensor characteristics can be incorporated intod ( 8 , w ) by multiplying each phase shift by a functionado, w ) , which describes the associated sensor response asa function of frequency and direction.

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The beampattern i s defined as the magnitude squaredof r(0,w). Note that each weight in w affects both thetemporal and spatial response of the beamformer. His-torically, use of FIR filters has been viewed as providingfrequency dependent weights in each channel. This inter-pretation i s accurate but somewhat incomplete since thecoefficients in each filter also influence the spatial filter ingcharacteristics of the beamformer. As a multi-input singleoutput system, the spatial and temporal filtering that oc-curs i s a result of mutual interaction between spatial andtemporal sampling.

The correspondence between FIR filtering and beam-forming i s closest when the beamformer operates at a singletemporal frequency w,) and the array geometry i s linearand equi-spaced as illustra ted in Fig. 2.3. Letting the sen-sor spacing be d, propagation velocity be c, and 0 repre-sent DOA relative to broadside (perpendicular to the array)we have T , (@ )= (i - 1) d/c) sin 0. In this case we identifythe relationship between temporal frequency w in d(w)(FIR filter) and direction 0 in d(O,wO) (beamformer) asw = wo(d/c) sin 0. Thus, temporal frequency in an FI R fil-

mm I

Figure 2.3 The analogy between an equi-spaced omni-directional narrowband line array and a single-channelFIRFilter is illustrated in this figure.

ter corresponds to the sine of direction in a narrowband,linear equi-spaced beamformer. Complete interchange ofbeamforming and FI R filtering methods i s possible for thisspecial case provided the mapping between frequencyand direction i s accounted for.

The vector notation introduced in (2.6) suggests a vectorspace interpretation o f beamforming. This point of view isuseful both i n beamformer design and analysis. We use ithere in consideration of spatial sampling and array geome-try. The weight vector w and the array response vectorsd(0, w ) are vectors i n an N dimensional vector space. Theangles between w and d(0, w) determine the responser(0,w) . For example, if for some (0 , w ) the angle betweenwand d(0, w) s 90" (i.e., if w is orthogonal to d(0, w ) ) , henthe response is zero. If the angle is close to o", hen theresponse magnitude will be relatively large. The ability t odiscriminate between sources at different locations and/orfrequencies, say (01,l) and ( 0 2 , 2 ) , s determined by theangle between their array response vectors, d(0, w , ) and

The general effects of spatial sampling are similar totemporal sampling. Spatial aliasing corresponds to an am-biguity in source locations. The implication is that sourcesat different locations have the same array response vector,e.g., for narrowband sources d(Ol ,w0) = d ( 0 2 , w o ) . Thiscan occur if the sensors are spaced too far apart. If thesensors are too close together, spatial discrim ination suf-fers as a result of the smaller than necessary aperture;array response vectors are not well dispersed in the Ndimensional vector space. Another type of ambiguity oc-curs with broadband signals when a source at one locationand frequency cannot be distinguished from a source at adifferent location and frequency, i.e., d(&, w, ) = d(02, J .For example, this occurs in a linear equi-spaced arraywhenever w1 sin O 1 = w2 sin 0 2 . The addition of temporalsamples at one sample prevents this particular ambiguity.)

A primary focus of this paper is on designing responsevia weight selection; however, (2.6) indicates that responsei s also a function of array geometry (and sensor character-istics if the ideal omnidirectional sensor model is invalid).In contrast with single channel filtering where AID con-verters provide a uniform sampling in time, there i s nocompelli ng reason to space sensors regularly. Sensor loca-tions provide additional degrees of freedom in designinga desired response and can be selected so that over therange of ( 0 , ~ )f interest the array response vectors areunambiguous and well dispersed in the N dimensionalvector space. Utilization of these degrees of freedom canbecome very complicated due to the multidimensionalnature of spatial samp ling and the nonlinear relationshipbetween r(0, w ) and sensor locations. References discus-sing array geometry design for response synthesis includeUnz [19561, Harrington [19611, lshimaru [19621, Lo [19631,and Skolnik et al. [19641.

d(02, ~ 2 )COX 19731).

B. Second Order StatisticsEvaluation of beamformer performance usually involves

power or variance, so the second order statistics of the


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-80 -60 40 -20 0 20 40A m v a i Anelc~deereef

weight index

1

0 8 -07

1 0 6 -3 0 5 -% 04-3

03

0 2 -01 -0- ~ 2 4 rliIO

wngtl1 Indcxii4

-100 -80 -60 -40 -7.0 0 20 40 60 80 1w

-60--70-80-lw -80 -60 40 -20 0 20 40 60 80 Io0

response be unity for an arrival angle of 18". Energy ISassumed t o arrive a t the array from several interference

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response converges to a statistically optimum solution.Partially adaptive beamformers reduce the adaptive algo-rithm computational load at the expense of a loss (designedto be small) in statistical optimality.

Ill. DATA INDEPENDENT BEAMFORMINGThe weights in a data independent beamformer are de-

signed so the beamformer response approximates a de-sired response independent of the array data or data statis-tics. This design objective-approximating a desiredresponse- is the same as that for classical FIR fi lte r design(see, for example, Parks and Burrus [19871).We shall ex-ploit the analogies between beamforming and FIR filteringwhere possible in develop ing an understanding of the de-sign problem and in presenting design procedures. Wealso discuss design problems specific to beamforming.

The first part of this section discusses forming beams ina classical sense, i.e., approx imating a desired response ofunity at a point of direc tion and zero elsewhere. Methodsfor designing beamformers having more general forms ofdesired response are presented in the second part.A. Classical Beamforming

Consider the problem of separating a single complexfrequency component from other frequency componentsusing the J tap FIR filter illustrated in Fig. 2.3.If frequency

i s of interest, then the desired frequency response i sunity at w n and zero elsewhere. A common solution to thisproblem i s to choose w as the vector d(w,). This choice canbe shown to be optimal in terms of minimizing thesquared error between the actual response and desiredresponse. The actual response is characterized by a mainlobe (or beam) and many sidelobes. Since w = d(w,), eachelement of w has uni t magnitude. Tapering or windowingthe amplitudes of the elements of w permits trading ofmain lobe or beam w idth against sidelobe levels to formthe response into a desired shape. Let T be a J by J diago-nal matrix with the real-valued taper weights as diagonalelements. The tapered FIR filter weight vector i s given byTd(w) . A detailed comparison of a large number o f taper-ing functions is given in Harris [1978].

In spatial filtering one i s often interested in receiving asignal arriving from a known location point Bo . Assumingthe signal i s narrowband (frequency wJ , a common choi-ce for the beamformer weight vector i s the array reponsevector d(O,, U " ) . The resulting array and beamformer i stermed a phased array since the output of each sensor i sphase shifted prior to summation. Fig. 1.2 depicts themagnitude of the actual response when w = d(O,, wo ) .Asin the FIR filt er discussed above, beam wid th and sidelobelevels are the important characteristics of the response.Amplitude tapering can be used to control the shape ofthe response, i.e. , to fo rm the beam. Figs. 2.5a and b illus-trate the effect of amplitude tapering on the response.

The equivalence of the narrowband linear equi-spacedarray and FIR filter (see Fig. 2.3) implies that the sametechniques for choosing taper functions are applicable to

Y /. . . . . . . . . .~ , ~ ~ I . **.*.

. . . . . . . . . . . . . .. . . . . . . . . . . . . . .. .

X

Figure 3.1 Par t (a) depicts the array configuration anddefines azimuth angle Q, and elevation angle 0 for a 16 by16 rectangular planar array of sensors. The sensors arespaced at one-half wavelength along the x and y axes. Thenarrowband magnitude squared response is illus trated inIb l as a function of azimuth and elevation angles. Thearray is phase shift steered t o Q, = 30" and0 = 45". Theweights applied to each sensor channel are the product ofx axis and y axis Dolph-Chebyshev taper weights.

either problem (Schelkunoff [19431).Methods for choos-ing tapering weights also exist for more general array con-figurations. If the array i s narrowband and the sensors lieon a line, then methods evolving from continuous spatialaperture design can be employed. A desirable smoothamplitude distribution function from a continuous aper-ture is approximated by a step distribution in a discreteaperture (see lshimaru [I9621 for relationship betweencontinuous and discrete apertures).

If the array i s planar and factorable, then line array tech-niques can be used to synthesize the overall response asthe product of tw o linear array responses. A planar array inthe xy plane i s factorable if its response can be "factored"into the product of responses due to line arrays in the xand y directions. Fig. 3.1 depicts the beampattern for a16 x 16 narrowband planar array. The sensors are spacedby one-half wavelength in both the x and y directions andthe direction of interest is 30" bearing and 45 " elevation.The response i s synthesized as the product of Dolph-Chebyshev tapered line arrays in the x and y directions.

If the beamformer i s broadband and employs FI R filters,then tapering can be applied independently to both the

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the desired source.

sensor outputs and FI R filte rs as illus trated i n Fig. 3.2. Thetaper weights are chosen to shape the spatial responseand the FIR filter coeffic ients to present a desired temporalresponse. As noted i n Section I I , spatial and temporal re-sponse interact so the spatial and temporal responsescannot be synthesized completely independently. One ex-ample of where the structure of Fig. 3.2 i s used i s delaysum beamforming. Here the FI R filters approximate thepropagation delays (linear phase over the frequency bandof interest) and the taper weights are chosen to shape themain beam and side lobe struc ture of the spatial response.B. Gener al Data Independent Response Design

The methods discussed in this subsection apply to de-sign of beamformers that approximate an arbitrary desiredresponse. This i s of interest in several different applica-tions. For example, we may wish t o receive any signalarriving from a range of directions, in which case the de-sired response is unity over the entire range. As anotherexample, we may know that there i s a strong source ofinterference arriving from a certain range of directions, inwhich case the desired response i s zero in this range.These two examples are analogous to bandpass and band-stop FIR filtering. Although we are no longer formingbeams, it i s conventional to refer to this type of spatialfilter as a beamformer.

Consider choosing w so the actual response r(O,w) =wHd(O, w ) approximates a desired response rd(O,w ) .Ad hoc techniques similar to those employed in FI R filterdesign can be used for selecting w ; however, here weonly consider choosing w to minimize the weighted L,norm of the difference between desired and actual re-sponse. Weighted L, approximation i s utilized in severalestablished FI R filter design techniques. The most com-monly used norms are L, (minmax) and L2 (least squares).Specific techniques include (see Parks and Burrus [19871):

1) Window ing of an ideal filters unit pulse response(minimizes L2 norm over continuous w ) ;

2) Frequency response sampling and linear weightedleast squares (minimized L2 norm over discrete w ) ;


3) Minmax design w ith the Remez exchange algorithm4) Minmax complex and magnitude response design

FI R filter design corresponds to a polynomial approxi-mation problem since the frequency response (2.4b) i s thediscrete Fourier transform of the FIR filter weight se-quence. Several of the above methods exploit this poly-nomial structure.

Excluding the cases for which beamformer design canbe reduced to equi-spaced line array geometries, beam-former design i s not a polynomial approximation problem.In general, the response in (2.6) is a weighted sum ofexponentials raised to non-integer powers. Thus, the L,methods (3 and 4) are not applicable since they are basedon the alteration theorem of polynomial approximation.The windowing method (1) i s based on the discrete timeFourier transform and i s also no t applicable. However, theL2 procedure using linear weighted least squares (2) i sapplicable.

To illustrate data independent beamformer design via Lzoptimization, consider minimizing the squared error be-tween the actual and desired response at P points ( O , , w , ) ,1 5 i 5 P. I f P > N, hen we obtain the overdeterminedleast squares problem

(minimizes L, norm over discrete w ) ;(minimizes L, norm over discrete w ) .

i

mi n IAHw - rdlL (3.1)wwhere

A = [d(Oq, w i ) d(O,, Wd*. .d (Op, WP) ] ;rd = [Td(O,, w , ) rd(O2, W2)..Td(OP, w d l H .

Provided AAH i s invertible (i.e., A is full rank), then thesolution t o (3.1) i s given as

W = A+rd (3.2)where A+ = (AAH)-A i s the pseudo inverse of A. Fig. 3.3depicts the response of a beamformer design using (3.2).

A note of caution is in order at this point. The whitenoisegain of a beamformer i s defined as the output powerdue to unit variance white noise at the sensors. Thus, thenorm squared of the weight vector, w H w , represents thewhite noise gain. If the white noise gain i s large, thenthe accuracy by which w approximates the desired re-sponse i s a moot point, since the beamformer output willhave a poo r SNR due to white noise contributions. If A i sill-conditioned, then w can have a very large no rm and stillapproximate the desired response. The matrix A i s ill-conditioned when the numerical dimension of the spacespanned by the d(O,w ) ,1 5 i 5 P, is less than N. or ex-ample, if only one source direction i s sampled, then thenumerical rank of A i s approximately given by the TBWPfor that direction. Low rank approximates of A and Ashould be used whenever the numerical rank i s lessthan N. This ensures that the norm of w wil l not be unnec-essarily large.

Specific directions and frequencies can be emphasizedin (3.1) by selection of the sample points ( O , , w , )and/or


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sensors space

unequally weighting of the error at each ( f ? , , w , ) . Parks andBurrus [I9871 discuss this in the contex t of F IR filtering.Kumar and Mu rthy [I9771 consider unequal weighting ofthe error to obtain minimax response design for beam-former weights in a linear narrowband array. I n general,guidelines for selection of the error weighting and ( O , , U , )are not available.

There are several alternatives to L optimization forgeneral data independent response design, includingmethods discussed in Butler and Unz [I9671 and Sanzgiriand Butler [19711.

IV. STATISTICALLY OPTIMUM BEAMFORMINGIn statistically optimum beamforming the weights are

chosen based on the statistics of the data received at thearray. Loosely speaking, the goal i s to optimize thebeamformer response so the output contains minimalcontributions due to noise and signals arriving from direc-tions other than the desired signal direction. We discussbelow several different criteria for choosing statisticallyoptimum beamformer weights. Table 4.1 summarizesthese different approaches. Where possible, equationsdescribing the criteria and weights are confined toTable 4.1. Throughout the section we assume that the datai s wide-sense stationary and that its second order statisticsare known. Determination of weights when the data statis-tics are unknown or time varying i s discussed in the fol-lowing section on adaptive algorithms.A. Multiple Sidelobe Canceller

The multiple sidelobe canceller (MSC) is perhaps theearliest statistically optimum beamformer. An MSC con-sists of a main channel and one or more auxiliarychannels as depicted in Fig. 4.la. The main channel canbe either a single high gain antenna or a data independent

TypeDefinitions

Optimum wa=R;rmaWeightsAdvantages Simple Directi on of

desired signalcan be unknown

Disadvantages Req Mus t generatereference signal

Max SNRx=s+n-array datas-signal componentn-noise componentR,= E{&}R,=E{nnH}output: y=wHx

RZ1R,w=h w

True maximizationof SN R

Must know R,and RSolve generalizedeigenproblem forweightsMonzingo andMiller [I9801

LCMVx-a rra y dataC- constraint matrixf- response vectorR,=E{xx~}output: y=wHx

rnin{wR,w}s.t.CHw=fw =R IC[C R;C]-f

Flexible and generalconstraints

Computation ofconstrainedweight vector

Frost 11972)

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main channel response

Vmain channel auxiliary branch response1auxiliary channels

a) b)

Figure 4.1 The multiple sidelobe canceller [MSCI con-sists of a main channel and several auxiliary channels asill ustrated in a1 The auxiliary channel weights are chosento cancel interference entering through the sidelobes ofthe main channel response. Part b l depicts the main chan-nel, auxiliary branch, and overall system response when aninterfere r arrives from direction 8,.

beamformer (see Section I l l ) . It has a highly directionalresponse, which i s pointed in the desired signal direction.Inter fering signals are assumed to enter through the mainchannel sidelobes. The auxiliary channels also receive theinterfering signals. The goal i s to choose the auxiliarychannel weights to cancel the main channel interferencecomponent. This implies that the responses to interferersof the main channel and linear combination of auxiliarychannels must be iden tical . The overall system then has aresponse of zero as illustrated in Fig. 4.l b. In general,requiring zero response t o all interfering signals i s eithernot possible or can result in significant white noise gain.Thus, the weights are usually chosen to trade off inter-ference suppression for white noise gain by minimizingthe expected value of the total ou tput power as indica tedin Table 4.1.

Choosing the weights to minimize output power cancause cancellation o f the des ired signal, since it also con-tributes to total outp ut power. I n fact, as the desired signalgets stronger it contributes to a larger fraction of t he totaloutput power and the percentage cancellation increases.Clearly this i s an undesirable effect. The MSC is very effec-tive in applications where the desired signal i s very weak(relative to the interference), since the opt imum weightswil l not pay any attention to it, or when the desired signali s known to be absent during certain time periods. Theweights can be adapted in the absence of the desiredsignal and frozen when it i s present.B. Use of Reference Signal

If the desired signal were known, then the weights cou ldbe chosen to m inimize the error between the beamformeroutput and the desired signal. Of course, knowledge ofthe desired signal eliminates the need for beamforming.However, for some applications enough may be knownabout the desired signal to generate a signal that closely14 IEEE ASSP MAGAZINE APRIL 1988

represents it. This signal i s called a reference signal. Asindicated in Table 4.1, the weights are chosen to minimizethe mean square error between the beamformer outputand the reference signal.

The weight vector depends on the cross covariance be-tween the unknown desired signal present in x and thereference signal. Acceptable performance s obtained pro-vided this approximates the covariance of the unknowndesired signal wi th i tself. For example, i f the desired signali s amplitude modulated, then acceptable performance i soften obtained by setting the reference signal equal tothe carrier. It i s also assumed that the reference signal i suncorrrelated with interfering signals in x. The fact thatthe direction of the desired signal does not need to beknown i s a distinguishing feature of the reference signalapproach.C. Maximization of Signal to Noise Ratio

Here the weights are chosen to directly maximize thesignal to noise ratio (SNR)as indicated in Table 4.1.A gen-eral solution for the weights requires knowledge of boththe desired signal, R,, and noise, R,, covariance matrices.The attainability of this knowledge depends on the appli-cation. For example, in an active radar system R , can beestimated during the time that no signal i s being trans-mitted and R, can be obtained from knowledge of thetransmitted pulse and direction of interest. If the signalcomponent i s narrowband, of frequency w , and direction8, then R, = c 2 d ( 0 , w ) d H ( 0 , w ) from the results in Sec-tion II . In this case the weights are obtained as

w = aRLd(8, 0) (4.1)where the (Y i s some non-zero complex constant. Substi-tution of (4.1) in to the SNR expression shows that the SNRi s independent of t he value chosen for a .D. Linearly Constrained Minimum Variance Beamforming

In many applications none of the above approaches i ssatisfactory. The desired signal may be of unknownstrength and may always be present, resulting in signalcancellation with the MSC and preventing estimation ofsignal and noise covariance matrices in the maximum SNRprocessor. Lack of knowledge about the desired signalmay prevent utilization of the reference signal approach.These limitations can be overcome through the applica-tion of linear constraints to the weight vector. Use of lin-ear constraints is a very general approach that permitsextensive control over the adapted response of thebeamformer. In this subsection we illustrate how linearconstraints can be employed to control beamformer re-sponse, discuss the optimum linearly constrained beam-forming problem, and present the generalized sidelobecanceller structure.

The basic idea behind linearly constrained minimumvariance (LCMV) beamforming i s to constrain the re-sponse of the beamformer so signals from the direction ofinterest are passed with specified gain and phase. Theweights are chosen to minimize output variance or powersubject to the response constraint. This has the effect of


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preserving the desired signal while minimizing con-tributions to the output due to interfering signals andnoise arriving from directions other than the direction ofinterest. The analogous FI R filter has the weights chosento minimize the filter output power subject to the con-straint that the filter response to signals of frequency w obe unity.

In Section II we saw that the beamform er response to asource at angle 8 and temporal frequency w i s given bywHd(8,U ) . Thus, by linearly constraining the weights tosatisfy w Hd (8 ,w ) = g, where g i s a complex constant, weensure that any signal from angle 0 and frequency U ispassed to the output with response g. Minimization ofcontributions to the outpu t from interference (signals notarriving from 6, with frequency w ) i s accomplished bychoosing the weights to minimize the expected value ofoutput power or variance E{jyj2}= wHR,w. The LCMVproblem for choosing the weights i s thus written

"2w"R,w subject to dH(H,w)w = g*. (4.2)The method of Lagrange multi pliers can be used to solve(4.2) resulting i n

R;'d( 8 , U )d"(8, w)R['d(O, w )

w = g* (4.3)Note that in practice the presence of uncorrelated noisewill ensure that R, i s invertible. If g = 1, then (4.3) s oftentermed the minimum variance distortionless response(MVDR) beamformer. It can be shown that (4.3) i s equiva-lent to the maximum SNR soluti on given in (4.1) by substi-tuting cr2d(8,w)dH(6,,w)+ R, for R, in (4.3) and applyingthe matrix inversion lemma.

The single linear constraint in (4.2) is easily generalizedto multiple linear constraints for added control over thebeampattern. For example, if there i s a fixed interfe rencesource at a known direction 6,hen it may be desirable toforce zero gain in that direction in ad dition to maintainingthe response g to th e desired signal. This is expressed as

(4.4)If there are L < N linear constraints on w, we write themin the form CH w = f where the N by L matrix C and Ldimensiona l vector f are termed the constraint matrix andresponse vector. The constraints are assumed to be lin-early independent so C has rank L. The LCMV problem andsolution with this more general constraint equation aregiven in Table 4.1.Constraint Design. Several different philosophies can

be employed for choosing the constraint matrix and re-sponse vector. We discuss point (Kelly and Levin [19641),deriv&ive (Ows ley [1973], Er and Cantoni [1983]), and ei-genvector (Buckley [1987]) constraints below. In manyapplications, a combination of the different types of con-straints i s most effective. Each linear constraint uses onedegree of freedom in the weight vector so with L constraintsthere are only N - L degrees of freedom available form n rnizing variance.

Point constraints fix the beamformer response at pointsof spatial direction and temporal frequency. Equation (4.4)represents an example of two po int constraints on w . Thenumbe r of points at which response can be constrained i slimited to N . I f N constraints are used then there are nodegrees of freedom left for powe r minimiz ation and a dataindependent beamformer i s obtained.

Derivative constraints are employed to influence re-sponse over a region of direction and/or frequency byforcing the derivatives of the beamformer response atsome point of direction and frequency to be zero. Theyare usually employed in conjunction with point con-straints. An example where derivative constraints are use-fu l is when the desired signal direction i s only knownapproximately. If the signal arrives near the direc tion atwhich a poin t constraint i s employed, then application ofa derivative constraint at that point prevents the beam-former from synthesizing a response of zero to the de-sired signal.

Eigenvector constraints are based on a least squares ap-proxim ation to the desired response and are typically usedto contr ol beam former response over regions of directionand/or frequency. Constraining the beamformer responsein a least squares sense ensures that the mean square errorbetween desired and actual beamformer response over aregion is minimized for a given number of constraints. Inthis sense eigenvector constraints are efficient. Considerdesigning a set of constraints which w ill co ntrol the beam-former response to a source from direction 8, over thefrequency band [w , ,ub ] . The dimension of the span ofd ( O , , w ) over this band i s approximately given by thesource TBWP discussed in Section I I . Eigenvector con-straints are derived fro m (3.1) by choosing P significantlygreater than the TBWP. The w , then oversample [ ua , b land A is ill-conditioned. A rank L approximation of A i sobtained from its singular value decomposition

A L = VZ LUH (4.5)where XL s an L by 1.diagonal matrix containing the largestsingular values of A, and the L columns of V and U arerespectively the left and right singular vectors of A corre-sponding to these singular values. Replacing A in (3.1) byits rank L approximate (4.5) and bri nging UC , to the rightside (the pseudo inverse of U i s U"), yields

Equation (4.6) has the same form as the constraint equa-tion C H w = f . The columns of V correspond to the eigen-vectors of AAH; hence the name eigenvector Constraints.(Note that AA H represents an approximation of R, in (2.9)if S ( w ) = 1.)

Example. Fig. 4.2 depicts the response of an LCMVbeamformer at eight equally spaced frequencies on thein te rva l [ 2~ / 5 , 4 ~ / 51 hen two interferers arrive from-17.5 and -5.75 degrees in the presence of whi te noise.The interferer power levels relative to the wh ite noise are



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icts the L C M V beamformer51 when two inter-

151and have powers o f 40

30 and 40 dB, respectively, and the inte rfere rs have a flatspectrum on [ 2 ~ / 5 , 4 ~ / 5 ] .he array has sixteen linearequi-spaced sensors with five taps per sensor. The tapspacing i s normalized to one second and the sensors arespaced at one-half wavelength corre spondi ng to fre-quency 4 ~ / 5 .he response i s const rained to pass signalsarriving from 18 degrees in the band [ 2 ~ / 5 , 4 ~ / 5 ] ,it hunit gain and linear phase using ten eigenvector con-straints designed from (4.6). The effectiveness of the con-straints i s evident, since all the frequency curves passthrough zero dB at 18 degrees. The response has nulls inthe directions of the interferers with the deeper nu ll corre-spond ing to the stronger interferer. The response as afunction of frequency for the interferer directions i s plot-ted in Fig. 6.2.The array gain i s 50 dB for this example.

Generalized Sidelobe Canceller. The generalized side-lobe canceller (GSC) represents an alternative formulationof the LCMV problem, which provides insight, is useful foranalysis, and can simplify LCMV beamformer implementa-tion. It also illustrates the relationship between the MSCand LCMV beamforming. Essentially, the CSC i s a mecha-nism for changing a constrained minimization probleminto unconstrained form. Perhaps the first reference tothis concept i s in Hanson and Lawson [19691,where a pro-cedure for transforming constrained least squares prob-lems to unconstrained least squares problems i s given.Griffiths and Jim [I9821applied the same concept to LCMVbeamforming and coined the term GSC. Similar tech-niques were discussed in Applebaum and Chapman [1976].

Suppose we decompose the weight vector w into twoorthogonal components w, and -v (w = wo - v) that liein the range and nul l space of C , respectively. The rangeand null space of a matrix span the entire space so thisI 6 I EEE ASSP MAG AZI NE A P R I L 1988

decomposition can be used to represent any w. SinceCH v = 0, we must have

WO = C(CHC) - f (4.7)if w is to satisfy the constraints. (4.7) is the minimum Lnorm solution to the underdeterm ined equivalent of (3.1).The vector v is a linear combination of the columns of anN by N-L mat rix C,(v = C,w,) provided the columns of C,form a basis for the null space of C. C, can be obtainedfrom C using any of several orthogonalization proceduressuch as Gram-Schmidt, QR decomposition, or singularvalue decomposition. The weight vector w = w, - Cnw,,is depicted in block diagram form in Fig. 4.3.The choicefor w, and C, implies that w satisfies the constraints inde-pendent of w n and reduces the LCMV problem to theunconstrained problem

Gnn[wo- CnwnIHRx[wo - Cnwn]. (4.8)The solution i s

w, = (C!R,C,) -C:R,w,. (4.9)

Figure4.3 The generalizedsidelabe canceller (GSCI rep-beamformer inned. I t consistsmatrix, C,,and

The primary implementation advantages of this alter-nate but equivalent formulation stem from the facts thatthe weights w, are unconstrained and a data independentbeamformer w, is implemented as an integral part of theadaptive beamformer. The unconstrained nature of theadaptive weights permits much simpler adaptive algo-rithms to be employed and the data independent beam-former i s useful in situations where adaptive signalcancellation occurs (see subsection E).

As an example, assume the constraints are as given in(4.2). (4.7) implies w, = g*d(H, w)/[dH(8,w)d(O, w ) ] . C, sat-isfies dH(8,w)C,, = 0 so each column [ C , , ] , ;1 5 i 5 N-L,can be viewed as a data independent beamformer with anull in direction 0 at frequency w : dH(8,w ) C,,],= 0. Thus,a signal of frequency w and direction 8 arriving at the a r-ray will be blocked or nul led by the matrix c , , . In general,if the constraints are designed to present a specified re-sponse to signals from a set of directions and frequencies,then the columns of C, will block those directions andfrequencies. This characteristic has led to the term block -ing matrix for C,. These signals are only processed by wc ,and since w, satisfies the constraints, they are presented


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with the desired response independent of w,,. Signalsfrom directions and frequencies over whic h the responsei s not constrained will pass through the upper branch inFig. 4.3 with some response determined by w,. The lowerbranch chooses w n to estimate the signals at the output ofw0 as a linear combination of the data at the output of theblock ing matrix. This is similar to the operation of the MSC,in which weights are applied to the output of auxiliarysensors in order to estimate the primary channel output(see Fig. 4.1).E . Signal Cancellation in S tatistically Optimu mBeam orming.

Optimum beamforming requires some knowledge ofthe desired signal characteristics, either its statistics (formaximum SNR or reference signal methods), its direction(for the MSC), or its response vector d(0,w ) for the LCMVbeamformer). If the required knowledge i s inaccurate, theopt imu m beamformer will attenuate the desired signal asi f it were interterence. Cancellation of the desired signal i soften significant, especially if the SNR of the desired signalis large (Cox [1973]).Several approaches have been sug-gested to reduce this degradation (e.g., Jablon [19861,Coxet al. 119871).

A second cause of signal cancellation i s correlation be-tween the desired signal and one or more interferencesignals. This can result either fr om multipa th propagationof a desired signal or from smart (correlated) jamming.When interference and desired signals are uncorrelatedthe beamformer attenuates interferers to minimize outputpower. However, with a correlated interferer the beam-former minimizes output power by processing the inter-fering signal in such a way as to cancel the desired signal.If the interferer i s partially correlated with the desiredsignal, then the beamformer wi ll cancel the por tion of thedesired signal that i s correlated with the interferer. Meth -ods for reducing signal cancellation due to correlatedinter ference have been suggested (e.g., Wid row et al.[19821, Shan and Kailath [19851, Yang and Kaveh [19871).

, ? - I : I 1Figure 5.1 The standard adaptive filter configuration.

V. ADAPTIVE ALGO RITHM S FOR BEAM FORM INGThe optimum beamformer weight vector equations

listed in Table 4.1 require knowledge of second order sta-tistics. These statistics are usually not known, but with theassumption of ergodicity, they (and therefore the opti-mum weights) can be estimated from available data. Sta-tistics may also change over time (e.g., due to movinginter ferers ). To solve these prob lems, weights are typicallydetermined by adaptive algorithms.

There are two basic adaptive approaches: 1) block adap-tation, where statistics are estimated from a temporalblock of array data and used in an op tim um weight equa-tion; and 2) continuous adaptation, where the weights areadjusted as the data i s sampled such that the resultingweight vector sequence converges to the optimum solu-tion. If a nonstationary environment is anticipated, blockadaptation can be used, provided that the weights arerecomputed periodically. Adams, et al. [I9801and others

have described applications of block data processing.Continuous adaptation is usually preferred when statisticsare time-varying or (for computational reasons) when thenumber of adaptive weights M s moderate to large (valuesof M > 50 are not uncommon).

Among notable adaptive algorithms proposed forbeamforming are the Howells-Applebaum adaptive loopdeveloped in the late 1950's and reported by Howells[1966,1976]and Applebaum [1966], and the Frost LCMValgorithm [19721. Rather than recapi tulating adaptive algo-rithms for each optimum beamformer listed in Table 4.1(for this see texts by Monzi ngo and Mille r [1980],Hudson[I9811 and Compton [1988]),we take a unifying approachusing the standard adaptive filter problem illustrated inFig.5.1.I I

1 K -1K k= UR, = - c u(k)u"(k)

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and sample cross-covariance vector

aV w M t k ) = - / (WM )dWM WM wM(kl

adaptive algorithm. The L M S algorithm replac:s Ywu(klwith the instantaneous gradient estimate Vw, , (k , =- 2 [ u ( k ) y d ( k ) - u ( k ) u ( k ) w M ( k ) l .Denoting y ( k ) = yd(k ) -w % k ) u ( k ) ,we have

WM(k + 1) = WM(k) + p u ( k ) y * ( k ) . (5.4)The gain constant p controls convergence characteristicsof the random vector sequence w d k ) . Table 5.1 providesguidelines for its selection.

The primary virtue of the L M S algorithm i s its simplicity.Its performance i s acceptable in many applications; how-ever, its convergence characteristics depend on the shapeof the error surface and therefore the eigenstructure of R,.When the eigenvalues are widely spread, convergencecan be slow and other adaptive algorithms with betterconvergence characteristics should be considered. Alter-native procedures for searching the error surface havebeen proposed in addition to algorithms based on least-squares and Kalman filtering. Roughly speaking, thesealgorithms trade-off computational requirements withspeed of convergence to wept. We refer you to texts onadaptive filtering for detailed descriptions and analysis(Widrow and Stearns 119851, Haykin [19861, Alexander[1986], Treichler et al. [1987], and others).

One alternative to LMS i s the exponentially weightedrecursive least squares (RLS) algorithm. At the K th time

AlgorithmInitialization

UpdateEquations

Multipliesper update

PerformanceCharacteristics

TABLE 5.1COMPAAJSON OF THE LMS AND RLS WEIGHT ADAPTATtON ALGORITHMS

2M

RLSWM(0) = 0P(0) = rll

6 small, I identity matrixv(k) = P(k - I)u(k)

k(k) = 1 f h-uH(k)v(k)X-v(k)

a( k ) = y&) - wZ(k - )u(k)WM ( ~ )= w M ( ~- I) + k(k)a*(k)

P(k) = h-P(k - 1) - A-k(k)vH(k)4M* 4M + 2

The w d k ) represents the least-squares solution at each instantk and are optimum in a deter-ministic sense. Convergence tothe statistically o ptim um weightvector wept is often faster thanthat obtained using the LMSalgorithm because it is inde-pendent of the eigenvaluespread of R,.

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step, w M( K)s chosen to minimize a weighted sum of pastsquared errors

Kmi n AK-k Iyd(k)- w%K)u(k)/ ' . (5.5)W M ( K ) k 0

A i s a positive constant less than one which determineshow quickly previous data are deemphasized. The RLSalgorithm is obtained from (5.5) by expanding the mag-nitude squared and applying the matrix invers ion lemma.Table 5. 1 summarizes both the L M S and RLS algorithms.VI. INTERFERENCE CANCE LLATION A ND PARTIALLYADAPTIVE BEAMFORMING

The computational requirements of each update inadaptive algorithms are proporti onal t o either the weightvector dimension M or dimension squared (M'). If M i slarge, this requirement i s quit e severe and for practical realtime implementation it i s often necessary to reduce M.

The expression degrees o f freedom refers to the num-ber of unconstrained or "fre e" weights i n an imple -mentation. For example, an LCMV beamformer with Lconstraints on N weights has N -L degrees of freedom; theGSC implementation separates these as the unconstra inedweight vector w n .There are M degrees of freedom in thestructure of Fig. 5.1. A fu l ly adaptive beamformer uses al lavailable degrees of freedom and apartiallyadaptive beam-former uses a reduced set of degrees of freedom. Re-ducing degrees of f reedom lowers computat ionalrequirements and often improves adaptive responsetime.' However, there is a performance penalty associatedwith reducing degrees of freedom. A partially adaptivebeamformer cannot converge to the same opti mum solu-tion as the full y adaptive beamformer. The goal of partiallyadaptive beamformer design i s to reduce degrees of free-dom wi thou t significant degradation in performance.

The discussion in this section i s general, applying todifferent types of beamformers although we borrow muchof the notation fro m the GSC. We assume the beamformeri s described by the adaptive structure of Fig. 5.1 where thedesired signal yd i s obtained as yd = w,"x and the datavector U as U = T H x . Thus, the beamformer output i sy = w H xwhere w = wo - TwM. n order to dis tinguish be-tween fully and partially adaptive implementations wedecompose T into a product of two matrices C,TM. Thedefinition of C, depends on the particular beamformerand TM represents the mapping which reduces degrees offreedom. The MSC and GSC are obtained as special casesof this representation. In the M S C wo i s an N vector thatselects the primary sensor, C, is an N by N- I matrix thatselects the N-I possible auxiliary sensors from the com-plete set of N sensors, and TM i s an N- I by M matrix thatselects the M auxiliary sensors actually utilized. I n terms of

'Adaptive algori thm convergence characteristics have not been dis-cussed in this paper. Generally, mo re data are requi red to derive anaccurate estimate of a larger opti mum weight vector with block adap-tive processing or RL S [Reed et al., 19741. With LMS, convergence isgoverned by the covariance matrix eigenvalue spread, which tends tobe larger for larger dimensional problems.

the GSC, w, and C, are defined as in Section I V and TM i san N-L by M matrix that reduces degrees of freedom

This section begins by considering the interferencecancellation process in these general beamformer imple-mentations. This develops the intuiti on required for under-standing why and how the number of adaptive weights canbe reduced. We conclude this section by surveying differ-ent partially adaptive beamformer design philosophies.A. Interference Can cellation Vs Degrees of Freedom.The results in this subsection depend on T and are inde-pendent of the individual terms C, and TM. We assumethat the beamformer does not cancel the desired signal(see Section 1V.E.) and that the optimum weights affectonly interferers and uncorrelated noise. This simplifiesthe analysis by permitting us to exclude consideration ofthe desired signal.

Suppose a narrowband inter fering source of frequencyw o arrives at the array from direction 0, . The response ofthe w, branch i s g, = w,Hd(O,,wo ) . Perfect cancellation ofthis source requires w Hd( Ol ,w , )= 0 so we must chooseWM to satisfy

WETHd(O,, C1)= g i . (6.1)If we asume that THd(O,, ,) i s nonzero, (6.1) represents asystem of one equation in M unknowns (elements of wM )for which a solution always exists. To simultaneously can-cel a second interferer located at 0 2 , w M must satisfy

(M < N-L).

where g 2 = w,Hd(O,, w o ) . Assuming T H d ( B , ,w o ) andTHd(f12, ,) are linearly independent and nonzero, andprovided M 2 2, then at least one whl exists that satisfies(6.2). Continuing this reasoning, we see that wM can bechosen to cancel M narrowband interferers (assuming theTHd(O,, ,) are linearly independent and nonzero), inde-pendent of T. Total cancellation occurs i f wM s chosen sothe response of Tw Mperfectly matches the wo branch re-sponse to the interferers.So far we have only considered narrowband point inter-ferers. Uncorrelated noise will be present in any real sys-tem and contributes to the output power. In an optimumbeamformer w M s chosen to m inimize the overall outputpower. Recall that the output power due to uncorrelatednoise i s proportional to the L2 nor m squared of the overallweight vector w (white noise gain). The norm of w canbecome large when wM s chosen to provide total inter-ference cancellation, depending on the choice for T andthe interferer locations. Thus, although in princip le pointsources of energy in direc tion and frequency can be totallycanceled with one weight per interferer independent of T,the presence of uncorre lated noise results in the degree ofcancellation being dependent o n the mapping describedby T.Now consider inter ferers that are spatial point sourcesbut emit broadband energy on the band w a 5 w 5 wh.The response of the w, branch to an interferer at 0 , isw,Hd(e,,w) = g l (w) . To achieve total cancellation w Mmust

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be chosen to satisfyWzTHd(OT,W)= g i ( w ) w a 5 w 5 w b . (6.3a)

Define the response of each column of T asf,(w) = [T]?d(O,,w) 1 5 i 5 M (6.3b)

where [TI, denotes the ith column of T. (6.3a) requiresg l ( w ) to be expressed as a linear combina tion of the f,(w),1 5 i 5 M, on w , 5 w 5 wb. In general, this cannot beaccomplished and we conclude that total cancellation ofbroadband interference cannot be obtained. The outputpower due t o the broadband interferer can be expressedas the integral over frequency of the magni tude squared ofthe difference between the wo branch and adaptivebranch responses weighted by the interferer power spec-trum. The degree of cancellation can vary dramatically andi s critically dependent on the interferer direction, fre-quency content, and choice for T. Good cancellation canbe obtained in some situations when M = 1, while in oth-ers even large values of M result in poor cancellation.These conclusions are also valid for narrowband sourcesthat are broad in direct ion (spatially distributed radiation).B . Partially Adaptive Beam former Design.

The preceding discussion indicates that the degree ofinterference cancellation i s critically dependent on theability of the adaptive channel (TWM) to match the mainbeam response over the interferer frequency extent. Thisprovides a means by which to evaluate partially adaptivebeamformers. The majority of work reported on partiallyadaptive beamforming has been concerned with narrow-band environments. We begin with a discussion of severalnarrowband approaches and briefly discuss their ex-tension to broadband situations. We then considertechniques that are directly applicable to broadband situ-ations. Some techniques select Th l for a given C , whileothers select T directly.2

Several approaches to reducing degrees of freedom arebased on processing a subset of the outpu ts of the matrixC , . This implies that TM i s a sparse matrix of zeros andones. The outputs of C , in the MSC are simply auxiliarysensor outputs. Morgan [I9781 evaluated partially adaptivebeamformer performance when TM selected various sub-sets of the auxiliary outputs. This i s termed an elementspace approach, since a subset of t he sensor element out -puts i s utilized. Several investigators, including Vural[19771, Adams, et al. [19801, and Gabriel [1986al have con-sidered choosing the columns of T to form beams. This istradit ionall y termed a beam space approach. The columnsof C , , are designed as data independent beamformers,each steered to a differen t direction, and TM can be usedto select a subset of the beam outputs. The objective i s todirect a beam at each interfering source so that it can besubtracted from the outpu t of thew, branch. One way to

'In principl e one can generate auxiliary constraints in an LCMVbeamformer to reduce the number of adaptive weights in a GSCimplementation (Griffiths (19871). Here we assume all constraints arealready specified in partially adaptive LCMV beamforming.

accomplish this i s by selecting enough beams to cover al lpossible directions from which interferers might arrive.Another i s to utilize source direction finding techniques toselect which beams correspond to estimated interfererdirections. The biggest advantage of the element spaceapproach i s the simplicity of implementation. Improvedperformance obtained using beam space processing i s es-pecially evident for interference due to either spatiallydistributed sources or sources with appreciable temporalbandwidth. However, this improvement i s obtained at theexpense of implementing the required beams.

Chapman 119761 and Owsley [I9781 have consideredchoosing the columns of T to select subarrays, i.e., eachcolumn involves only a subset of the sensors in the array.The weightings applied to each subarray (elements of T)can be chosen in various ways, one of which i s to use thesubarray to form a beam. Performance depends on thenumber of sensors in each subarray, which sensors areused in each subarray, and the weightings used to com-bine the sensor outputs i n each subarray. Note that eachcolumn of T wil l have zeros in locations corresponding tosensors excluded from that subarray, so the overall T i s ofsparse structure.

Owsley [I9851 suggests a narrowband method for theGSC in which the columns of TM are chosen as a basis forthe space spanned by the fully adaptive weight vectors.The dimension of this space i s given by the rank of thespatially correlated component of the interference co-variance matrix. Van Veen [1988a] extended this approachto the broadband case. The dimension of the fully adap-tive space can be large in this case since it i s given by therank of the correlated components of the broadband in-terference covariance matrix.

The above approaches are capable of satisfactory per-formance with narrowband interference since cancella-tion requires about one degree of freedom per interfererto match the main beam response at each interfere r direc-tion. However, with broadband interference the mainbeam response must be matched over a range of fre-quency at each interferer direction. Al though the narrow-band approaches can be extended, it is difficult to do thisand keep the number of adaptive weights M small. Forexample, several banks of beams could be designed tospan the range of di rections, w ith each bank operating atdifferent frequency. The problem is that the number ofbeams required can become large as the frequency band-width increases. We seek aT that i s efficient, i .e., providesgood cancellation with a minimum of columns.

Van Veen and Roberts [1987a] have considered an opti-mization based approach for choosing the columns of T inthe context of LCMV beamforming. The matrix C , , i s de -signed to meet the constraints, reducing the problem toan unconstrained optimization over the elements of TU .TM i s chosen to minimize the average interference outputpower over a range of likely interference environments.Let a vector (Y parameterize the interference environmentof interest. In general a can represent interferer loca-tions, power levels, spectral distributions, numbers of in-



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terferers, etc. Defining PI(@)as the interference outputpower, TM is chosen according to

mi n1 Pl(a) d a (6.4)where [a,,Yb ] denotes the set of interference scenarios ofinterest. Since output powe r corresponds to the error be-tween the w, branch and adaptive channel responses, ineffect (6.4) selects TM to provide the best response matchpossible for interference environments in [ a a ,b ] .

Equation (6.4) represents a design problem that i snonlinear in the design parameters (elements of TM). Asuboptimal approach t o solving (6.4), in which TM i s se-quentiallydesigned one column at a time, has been shownto be effective in achieving near fully adaptive interferencecancellation using a small adaptive dimension. The prob-lem i s still nonlinear; however an effective approximatesolution i s obtained by solving a linear least squares prob-lem. An interpretation of this approximate solution i sgiven in Van Veen [1988b].

Fig. 6.1 illustrates the magnitude response at severalfrequencies for a partially adaptive beamformer designedusing this approach for the same interference environ-ment and array geometry as the fully adaptive examplein Section IV (see Fig. 4.2). The partially adaptive beam-former has 28 adaptive weights compared to 70 for thefully adaptive beamformer. Fig. 6.2 compares the magni-tude response of the fully and partially adaptive beam-formers as a function of frequency at the interferer DOA's.TM i s designed assuming (Y i s a two-dimensional vectorwith each element corresponding to the DOA of an inter-ferer. Both interferers have white spectra on 2 ~ / 5 o s4 ~ / 5nd power levels of 40 dB relative to white noise. The

TM a

WA 1n degreeso t depicts the magnitude response ofbeamformer designed to minimize av-r a t eight frequencies on the normal-Val [2n/5,4s/51. The beamformerights and was designed assumingpower 40 dB relative to white noise

-45 and 45 degrees. The actual inter-nt, constraints, and array geometryig. 4.2.Note tha t the beamformer of

Figure 6.2 This plot gnitude response ofthe partially and fully formers correspond-on of frequency a t theIregion [a , ,ab l includes all DOA's between -45 and45 degrees. Partially and fully adaptive array gain wasevaluated at 190 points in this region; the maximum dif-ference between partially adaptive and fully adaptivearray gain was 2.4 dB and the average .7 dB for this par-ticular design.VII. BEAMFORMER IMPLEMENTATIONS

In its simplest form, a beamformer represents a linearcombination of the sensor data. However, there are manydifferent ways of implementing the weighted sum, eachwith its own performance and complexity characteristics.The section begins by introducing a general descriptionfor implementations and then briefly overviews severalbeamformer implementations. For additional detail thereader i s encouraged to consult the references given inthis section.

In essence, most implementations can be representedby decomposing w into a product o f matrices and a vector

w = nv,w,. (7.1[iy,

V, , 1 5 i 5 v are a series of matrix transformations ofconformable dimensions and w v i s a vector. As a generalrule, the matrix transformations are chosen to enhanceperformance and/or lower computational complexity. TheFFT implementation of t he DFT i s analogous to (7.1), sincethe DFT matrix can be expressed as a product o f permuta-tio n and butter fly matrices, representing a series of verysimple computations. The GSC is an example of (7.1) wherev = 1, V, = [w, /C,] , and w, = [1,w,HIH. VI simplifiesadaptive algorithm implementation by permitting uncon-strained adaptation of wn. In order to simplify multiplica-tion of the data by w, and C several researchers havestudied decomposition of [w, IC,] into products of simplematrices (e.g., Kalson and Yao [I9851 and Ward, et al. [19861).



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As discussed in Section II, broadband beamforming canbe performed in the frequency domain or time domain.The weights used to obtain the outputs at each frequencyfor the frequency domain beamformer depicted i n Fig. 2.4are easily represented in terms of (7.1) by employing thematrix representation of the DFT. Frequency domain opti -mum beamformers usually choose the weights at eachfrequency based solely on the data at the frequency. Thispartitioning of the data influences both performance andcomputational complexity. Discussion and evaluation offrequency domain beamforming s given in Owsley [1985],Vural [1977], and Gabriel [1986bl.

Systolic implementations of optimum beamformershave been studied by a number of investigators. They areusually designed to both compute and implement theadaptive weights. In general, systolic implementationscan be described in terms of (7.1) where each V, has astructure amenable to parallel computation and local com-munication. McWhirter [I9831 (see also Haykin [I9861ch. I O ) has developed a systolic array that computes thebeamformer output wi thout explicit computation of theadaptive weight vector. Additional references to systolicimplementations include Schreiber and Kuekes [1985],Ward, et al . [19861, Owsley 119871, and Van Veen andRoberts [1987bl.

The study of beamformer implementations i s an evolv-ing research area. Future developments w il l result fromadvances in VLSl and parallel computing technologies.VI I I . S U M M A R Y

A beamformer forms a scalar outp ut signal as a weightedcombination of the data received at an array of sensors.The weights determine the spatial filtering characteristicsof the beamformer and enable separation of signals havingoverlapping frequency content if they originate fromdifferent locations. The weights in a data independentbeamformer are chosen to provide a fixed response inde-pendent of the received data. Statistically optimum beam-formers select the weights to optimize the beamformerresponse based on the statistics of the data. The data sta-tistics are often unknown and may change with time soadaptive algorithms are used to obtain weights that con-verge to the statistically optimum solution. Computationalconsiderations dictate the use of partially adaptive beam-formers with arrays cornposed of large numbers of sen-sors. Many different approaches have been proposed forimplementing optimum beamformers. Future work wi lllike ly address signal cancellation problems, fu rther reduc-tions in computational load for large arrays and improvedstructures for implementation. Beamforming truly repre-sents a versatile approach to spatial filtering.REFERENCESAdams, R. N., Horowitz, L. L. and Senne, K. D. [1980],

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Kevin M. Buckley (M'81)was born in Washing-ton, DC, on December 4, 1954 He receivedthe B.S. and M.S degrees in electrical engi-neering from Villanova University, Philadel-phia, PA, in 1976 and 1980, respectively, andthe Ph.D. degree in electrical engineeringfrom the University of Southern California,Lo s Angeles, in 1986.From 1980 to 1982, he worked at Sonic SciencesInc., Warminister, PA, on communication andsonar applications of digit al signal processing. In 1982-1983 heworked as a full time Instructor at Villanova University. He i s cur-rently an Assistant Professor in the Department of Electrical Engineer-ing, University of Minnesota, Min neapolis. His primary research andteaching interests are in the theory and application of sensor arrayprocessing, spectrum estimation, and adaptive algorithms.

Barry D. Van Veen was born in Green Bay,WI,on June20,1961. He received th e B.S. degreefrom Michigan Technological University in1983 and the Ph.D. deg ree from the Un iversit yof Colorado in 1986, both in electrical engi-neering. He was an ONR Fellow while work-ing on the Ph.D. degree. In the spring of 1987he was with the Department of Electricaland Computer Engineering at the Universityof Colorado-Boulde r. Since August of 1987he has been with the Department of Elec-trical and Computer Engineering a t the Uni-versity of Wisconsin-Mad ison as an Assistant Professor. His researchinterests include array processing, spectral estimation, and adaptivealgorithms.

beam forming a versatile

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