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Beamforming:

Versatile

Approach to Spatial Filtering

Barry

Van

beamformer

a processor used in conjunction with an

array of sensors to provide a versatile form of spatial filtering.

The sensor array collects spatial samples of propagating wave

fields, which are processed by the beamformer. The objective

to estimate the signal arriving from a desired direction in

the presence of noise and interfering signals.

beamformer

performs spatial filtering to separate signals that have over-

lapping frequency content but originate from different spatial

locations. This paper provides an overview of beamforming

from a signal processing perspective, with an emphasis on re-

cent research. Data independent, statistically optimum, adap-

tive, and partially adaptive beamforming are discussed.

INTRODUCTION

The term beamforming derives from the fact that early

spatial filters were designed to form pencil beams (see

polar plot in Fig.

.I)

in order to receive a signal radiating

from

specific location and attenuate signals from other

locations. ”Forming beams” seems to indicate radiation of

energy; however, beamforming

applicable to either

radiation or reception of energy. In this paper we discuss

formation of beams for reception.

Systems designed to receive spatially propagating sig-

nals often encounter the presence of interference signals.

If the desired signal and interferers occupy the same tem-

poral frequency band, then temporal filtering cannot be

used to separate signal from interference. However, the

desired and interfering signals usually originate from dif-

ferent spatial locations. This spatial separation can be ex-

ploited to separate signal from interference using

spatial

filter at the receiver. Implementing

temporal filter requires

processing of data collected over

temporal aperture.

Similarly, implementing

spatial filter requires processing

of data collected over

spatial aperture.

Several applications that employ spatial filtering of data

are listed in Table

.1.

Fig.

illustrates

microwave com-

munications antenna that employs

continuous spatial

aperture to accomplish spatial filtering with a single an-

tenna. Fig.

1.2

depicts a low frequency-towed sonar array

in which the spatial aperture

obtained through

dis-

crete spatial sampling by an array

sensors. When the

spatial sampling

discrete, the processor that performs

the spatial filtering

termed

beamformer. Typically

IEEE

ASSP MAGAZINE APRIL

1988

Veen and Kevin

Buckley

’igure

continuous spatial aperture provides one

nechanism for spatial filtering. Illustrated is a parabolic

nicrowave antenna system. The antenna dish provides

;he spatial aperture over which energy is collected. This

3nergy

reflected to the antenna feed. The dish and feed

iperate as a spatial integrator. The energy from a far field

source located directly in front

the antenna arrives at

;he feed temporarily aligned [i.e., all source-to-feed path

engths are equal1 and

coherently summed. In general,

?nergy from other sources will arrive at the feed via

iariable length paths, and add incoherently.

polar plot

a typical antenna beampattern (i.e., power gain

vs.,

;his case, azimuth angle1

shown for a selected fre-

iuency and for the elevation angle at which the antenna

minted.

0740-746718810400-0004$01

.OOGI9881

EEE

TABLE

1.1

ARRAYS

AN0 BEAMFORMERS PROVIDE AN EFFECTIVE AND VERSATILE MEANS

SPATIAL

FILTERING.

THIS

TABLE LISTS A NUMBER OF APPLICATIONS OF SPATIAL FILTERING,

GIVES

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

Application Description References

RADAR

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

SONAR source localization and classification Knight et al. [19811; Owsley

Communications

directional transmission and Mayhan 119761; Compton

119781; Adams et al. 119801

control; synthetic aperture RADAR

Munson et al. (19831

[I9851

reception; sector broadcast

satellite communications

Imaging ultrasonic; optical; tomographic Macovski [19831; Pratt I19781;

Geophysical earth crust mapping;

oil

exploration Justice [I9851

Exploration

Astrophysical

high

resolution imaging of the Readhead [19821; Yen [I985

Exploration universe

Biomedical fetal heart monitoring; tissue

Kak [I9851

Widrow et al. 119751; Gee et al.

[19841; Peterson

al. [I9871

hyperthermia; hearing aids

beamformer linearly combines the spatially sampled time

series from each sensor to obtain a scalar output time

series in the same manner that an FIR filter linearly

combines temporally sampled data. Two principal advan-

tages of spatial sampling with an array of sensors are dis-

cussed below.

Spatial discrimination capability depends on the size of

the spatial aperture; as the aperture increases, discrimi-

nation improves. The absolute aperture size

not impor-

tant, rather its size in wavelengths

the critical parameter.

single physical antenna (continuous spatial aperture)

capable of providing the requisite discrimination

often

practical for high frequency signals since the wavelength

short. However, when low frequency signals are of in-

terest, an array of sensors can often synthesize a much

larger spatial aperture than that practical with

single

physical antenna.

second very significant advantage of using an array of

sensors, relevant at any wavelength,

the spatial filtering

versatility offered by discrete sampling. In many applica-

tion areas it

necessary to change the spatial filtering

function in real time to maintain effective suppression of

interfering signals. This change

easily implemented in

discretely sampled system by changing the way in which

the beamformer linearly combines the sensor data.

Changing the spatial filtering function of a continuous ap-

erture antenna

impractical.

The purpose of this paper

to describe beamforming

from

signal processing perspective, provide an overview

of beamformer design, and briefly discuss perform-

ance and implementation issues with an emphasis on re-

cent research. The paper begins with a section devoted to

defining basic terminology, notation, and concepts. Suc-

ceeding sections cover data-independent, statistically op-

timum, adaptive, and partially adaptive beamforming. We

then provide

brief discussion of implementation issues

and conclude with a summary.

Throughout the paper we use familiar methods and

techniques from FIR filtering to provide insight into vari-

ous aspects of spatial filtering with a beamformer. How-

ever, in some ways beamforming differs significantly from

FIR filtering. For example, in beamforming a source of

energy has several parameters that can be of interest:

range, azimuth and elevation angles, polarization, and

temporal frequency content. Different signals are often

mutually correlated as a result of multipath propagation.

The spatial sampling

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 problem than FIR filtering and as

result, more general design procedures and processing

structures are common.

Rather than making

futile attempt at attributing devel-

opments due to many different researchers in beam-

forming, we refer the reader to the following references:

books-J.

Griffiths et al., ed. [19731, Hudson [19811,

Monzingo and Miller [19801, Haykin, ed. [19851, Compton

[19881; special issues

-/€€E

Transactions on Antennas

and

Propagation [19761, [19861, journal

Ocean Engineering

[19871; tutorial

Gabriel [1976]; and bibliography- Marr

[19861. Papers devoted to beamforming are often found in

the

IEEE

Transactions on: Antennas

and

Propagation,

APRIL

1988

IEEE

ASSP MAGAZINE

have any necessary receiver electronics and an

con-

verter if beamforming

performed digitally.

The second beamformer in Fig. 2.1 samples the propa-

gating wave field in both space and time and

often used

when signals of significant frequency extent (broadband)

are of interest. The output in this case can be expressed as

(2.2)

where

the number of delays in each of the

sensor

channels. If the signal at each sensor

viewed as an input,

then a beamformer represents a multi-input single out-

put system.

convenient to develop notation which permits us to

treat both beamformers in Fig.

2.1

simultaneously. Note

that (2.1) and (2.2) can be written as

y(k)

wHx(k).

(2.3)

by appropriately defining a weight vector

and data vec-

tor x(k). We use lower and upper case boldface to denote

vector and matrix quantities respectively, and let super-

script H represent Hermitian (complex conjugate) trans-

pose. Vectors are assumed to be column vectors. Assume

that

and x(k) are

dimensional; this implies that

when referring to (2.2) and

when referring to (2.1).

Except for Section V on adaptive algorithms, we will drop

the time index and assume that

its

presence

understood

throughout the remainder of the paper. Thus (2.3)

writ-

ten as y

wHx.

Many of the techniques described in this

paper are applicable to continuous time as well

discrete

time beamforming.

The frequency response of an

FIR

filter with tap weights

w;,

and

tap delay of T seconds

given by

K-1

y(k)

WtpXl(k

I=1

p=o

p=l

,.(@)

Wp+e-~~T(~-l)

(2.4a)

Alternatively

r(w)

wHd(w)

(2.4b)

where

[w? w;.

. .

WT] and

d(w)

[I eJwTe12wT

...

e~il-l)wT

r(o) represents the response of the filter to

complex sinusoid of frequency

and

d(w)

a vector de-

scribing the phase of the complex sinusoid at each tap in

the

FIR

filter relative to the tap associated with wl.

Similarly, beamformer response

defined as the ampli-

tude and phase presented to a complex plane wave as a

function of location and frequency. Location

in general

a 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 not

consider range. Fig. 2.2 illustrates the manner in which an

array of sensors samples a spatially propagating signal.

Assume that the signal

complex plane wave with

DOA

and frequency

For convenience let the phase

be zero at the first sensor. This implies xl(k)

elwk and

xdk)

eJw[k-AI'o)l,

Al(8)

represents the time delay

due to propagation from the first to the

Ith

sensor. Substi-

tution into (2.2) results in the beamformer output

Sensor

Reference

Y--l

y(k)

'2'

Wtpe-lWIAl(o)+Pl

eIwkr(8,w) (2.5)

I=1

p=o

where

A,(@

r(0,

the beamformer response and

can be expressed in vector form as

r(8,

wHd(8,

0).

(2.6)

The elements of

d(8,w)

correspond to the complex

exponentials e-lw[Al(o)+P'

In general it can be expressed

d(8,

eJWT2(o)

elWT3(8)

elwTN(@I]H

(2.7)

where the

7,(8),

N, are the time delays due to

propagation and any tap delays from the zero phase refer-

ence to the point at which the

ith

weight

applied. We

refer to

d(0,w)

as the array response vector. It

also

known as the steering vector or direction vector. Non-

ideal sensor characteristics can be incorporated into

d(8,w)

by multiplying each phase shift by

function

ado,

w),

which describes the associated sensor response

a function of frequency and direction.

APRIL

1988

IEEE

ASSP MAGAZINE

The beampattern

defined

the magnitude squared

of r(0,w). Note that each weight in w affects both the

temporal and spatial response of the beamformer. His-

torically, use of FIR filters has been viewed

providing

frequency dependent weights in each channel. This inter-

pretation

accurate but somewhat incomplete since the

coefficients in each filter also influence the spatial filtering

characteristics of the beamformer.

a multi-input single

output system, the spatial and temporal filtering that oc-

curs

a result of mutual interaction between spatial and

temporal sampling.

The correspondence between FIR filtering and beam-

forming

closest when the beamformer operates at

single

temporal frequency

w,)

and the array geometry

linear

and equi-spaced as illustrated in Fig.

2.3.

Letting the sen-

sor spacing be d, propagation velocity be c, and

repre-

sent

DOA

relative to broadside (perpendicular to the array)

we have

T,(@)

(d/c) sin

In this case we identify

the relationship between temporal frequency

d(w)

(FIR filter) and direction

d(O,wO)

(beamformer)

wo(d/c) sin

Thus, temporal frequency in an

FIR

fil-

Figure

2.3

The analogy between an equi-spaced omni-

directional narrowband line array and a single-channel

FIR

Filter

illustrated in this figure.

ter corresponds to the sine of direction in

narrowband,

linear equi-spaced beamformer. Complete interchange of

beamforming and

FIR

filtering methods

possible for this

special case provided the mapping between frequency

and direction

accounted for.

The vector notation introduced in (2.6) suggests

vector

space interpretation of beamforming. This point of view

useful both in beamformer design and analysis. We use it

here in consideration of spatial sampling and array geome-

try. The weight vector

and the array response vectors

d(0,

are vectors in an

dimensional vector space. The

angles between w and

d(0,

determine the response

r(0,

w).

For example, if for some

(0,

the angle between

wand

d(0,

90" (i.e., if w

orthogonal to

d(0,

w)),

then

the response

zero.

the angle

close to

o",

then the

response magnitude will be relatively large. The ability to

discriminate between sources at different locations and/or

frequencies, say

(01,

wl)

and

(02,

w2),

determined by the

angle between their array response vectors,

d(0,

w,)

and

The general effects

spatial sampling are similar to

temporal sampling. Spatial aliasing corresponds to an am-

biguity in source locations. The implication

that sources

at different locations have the same array response vector,

e.g., for narrowband sources

d(Ol,w0)

d(02,wo).

This

can occur if the sensors are spaced too far apart. If the

sensors are too close together, spatial discrimination suf-

fers as a result of the smaller than necessary aperture;

array response vectors are not well dispersed in the

dimensional vector space. Another type of ambiguity oc-

curs with broadband signals when

source at one location

and frequency cannot be distinguished from a source at a

different location and frequency, i.e.,

d(&,

w,)

d(02,

wJ.

For example, this occurs in

linear equi-spaced array

whenever

sin

02.

(The addition of temporal

samples at one sample prevents this particular ambiguity.)

primary focus of this paper

on designing response

via weight selection; however, (2.6) indicates that response

also a function of array geometry (and sensor character-

istics

the ideal omnidirectional sensor model

invalid).

In contrast with single channel filtering where

AID

con-

verters provide a uniform sampling in time, there

compelling reason to space sensors regularly. Sensor loca-

tions provide additional degrees of freedom in designing

desired response and can be selected

that over the

range of

(0,~)

of interest the array response vectors are

unambiguous and well dispersed in the N dimensional

vector space. Utilization of these degrees of freedom can

become very complicated due to the multidimensional

nature of spatial sampling and the nonlinear relationship

between r(0,

and sensor locations. References discus-

sing array geometry design for response synthesis include

Unz [19561, Harrington [19611, lshimaru [19621, Lo [19631,

and Skolnik et al. [19641.

d(02,

~2)

(COX [19731).

Second Order Statistics

Evaluation of beamformer performance usually involves

power

variance,

the second order statistics of the

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