Virtual Array Processing Using Wideband Cyclostationary Signals

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Virtual Array Processing Using Wideband Cyclostationary Signals

Mari lynn P. Wylie

*

WINLAB

Rutgers University

P.O. Box 909

Piscataway,

N J

08855

Abstract

A new spatio-temporal array signal processing tech-

nique for wideband array data is presented that im-

proves spatial resolution and increases the number of

resolvable sources. T he method is predicated on digi-

tal communication signals that are temporally cyclo-

stationary. We show that th e (frequency-dependent)

array manifold has a

separable

representation in the

directions of arrival (DOAs) and the array geometry

and exploit the structure of the cyclic cross spectral

density matrix in order to obtain virtual array ‘obser-

vations’

wzthout

a-priori knowledge of the DOAs.

1

Introduction

Frequency focusing techniques for direction of ar-

rival estimation (DOA) of temporally wideband sig-

nals is

a

problem of considerable continuing interest

motivated, for example, by developments in mobile

communications. One approach to this problem for

wide-band array data as will be shown) is to design

frequency-dependent transformations which focus the

DOA information to

a

reference frequency and, simul-

taneously, generate virtual. array ‘observations’. Wit h

proper design, the new, virtual array geometry may

be selected

so

as

to

increase the aperture and hence

obtain improved DOA estimates.

An important contribution t o wide-band DOA es-

timation can be traced to the subspace focusing tech-

nique introduced by [l] termed

as

Coherent Signal

Subspace Method (CSSM). In this method, stationary

wide-band array d at a is decomposed i nto narrowband

components followed by focusing of the narrowband

components to a reference frequency. Subsequent to

frequency focusing, the DOA estimates are obtained

using conventional signal subspace methods (such as

MUSIC) for narrowband signals. Some new interpre-

tations and results related to CSSM were proposed

[2]-[3]. Unfortunately, performance o the CSSM is

critically dependent on some a-priori initial estimate

of the DOAs close to the true value, in order to con-

struc t the focusing matrix. An import ant contribu-

tion to resolution of the above dilemna was provided

by [GI, who demonstrated the

separabi l i t y

of the array

manifold into two functions: one of the DOAs and an-

ot,her of the array geometry and temporal frequencies,

*Author for

all

correspondence; e-mail:

mwyIieQ winlab Tu gers edu

Sumit

Roy

Divn.

of

Engineering

University of Texas

San A ntonio , T X 78249

respectively. Th e resulting array manifold interpola-

tion matrix (focusing matrix) is then computed wzth-

out

any a-priori knowledge of the DOAs of the sources.

In this work, we develop a new technique for DOA

estimation of wide-band signals for 2-D (planar) arrays

by combining the separability of the array manifold

with exploitation of th e wide-sense cyclostationarity

of the signals of interest (SOIs). It is shown that the

array manifold has

a

separable representation which

may be exploited in order to focus its information to

a reference frequency bin

without

a-priori knowledge

of the DOAs. We also introduce a spatial extrapo-

lation parameter which, if judiciously selected during

the frequency focusing procedure, generates ‘virtual’

array observations at the reference frequency.

i)

A-priori DOA estimates are not required to gener-

ate

the ‘virtual’ array observations;

ii) Rejection of wide-sense stationary or cyclostation-

ary interference and noise;

iii) Increased resolution by spatial extrapolation.

The salient features of this new method are:

2 Problem Formulation

The signal observed by an M-element planar array

is assumed to be the supe rposition of L a cyclostation-

ary waveforms received in the presence of interference

and noise. Th e signal measured at the output

of

the

mth sensor can be described by

I=1

over the observation interval 0 5 t 5 TOand for

m = 1, . . . ,M .

{ ~ ~ ( t ) } ~ ~re the radiated signals,

and {n,(t)}Z=, is interference. Th e delay T ~ B I ) =

, where 01 denotes the DOA of the

c m sin h + y m c os )

lth radiated waveform

as

measured from the array

broadside and (x m , m) denotes the 2-D coordinates

of the mth array element normalized

relatzve

to the

wavelength,

A 0

= c/fo.

The Fourier coefficient of (1)

at

frequency 6 s de-

fined as Z m f k )

=

?

zm( t )e- l”fk tdt

In vector

f o

1058-6393/96 5.00 1996 EEE

Proceedings ofASILOMAR-29

506



notation, we have

L

I=1

where

.

4:)

2 T a z m

s n ~~+y,,, cos^^

[GIl,,l ( f k ) = e fo

Sl fk)

nd

;V,,,(fk)

are the lcth Fourier coefficients of

the Ith radiated waveform and the noise/interference

present at the mth sensor, respectively.

s / t ) is a cyclostationary signal and therefore ex-

hibits spectral correlation at frequencies separated by

multiples of the cycle frequency, a ,

{fk

+ 5 , k

-

5 )

[7], where Q. typically corresponds to the bit rate, chip

rate, or twice the carrier frequency. Let us define the

cyclic autocorrelation of the I t h waveform (in discrete

time) to be

(where

i

is the discrete time variable).

We assume that the signals of interest are mutually

cyclically uncorrelated with each other and with the

noise, and that the noise is cyclically uncorrelated with

itself at cycle frequency

cr,

i.e.,

~ { s l [ i ] s : , [ ; j e - j ~ ~ ~ ~ }

P S ~ ~ (6)

E {

SI [;In;

[ i ] e - j?=ai} =

0

( 7 )

E {

Invt[i] T a i

} = O

(8)

where

E { . }

s the expecta.tion operator.

3

The

Virtual Array Processing Algo-

rithm

Th e first step in generating the virtual arr ay obser-

vation is to operate on the vector pair, z ( f k + a / 2 )

and

e ( f k

a/2)

in

order t o generate

a

set of obser-

vations that are commensurate with the narrowband

model of signal recept ion, which (in general) , is given

bY

In the sections to follow, it will be shown that is

selected

as a

spatial extrapolation parameter.

In general, t he focusing matrices Tm fk) m

=

1,2)

are functions of the unknown {Oi}f:,. However, we

show that by exploiting the separability of the depen-

dence of

(f)

on the array geometry and frequency

versus the direction of arrival, that we can construct

Tl fk) and Tz fk) without a priori knowlege

of

the

source directions of arrival.

3.1 Separability of (f)

The focusing matrices Tl fk) and T? fk)are cle-

rived by exploiting the separability of (f) using the

Jacobi-Anger expansion of a plane wave in a series of

cylindrical waves [9]

m

r = - m

where J,.(.) s the r%horder Bessel function of the first

kind. Using this expansion, we approximate

z e f ) )

~ ( f ) $ e

(15)

where G f) and

2R

+

1 vector given, respectively, by

are the M x 2R + 1 matrix and

[G f)],, =

j r - R- 1

e

- j r -R- l )Lrt

Jr -

R-

1 (2r f /fO rra 1

n

= l , . . . , M , T = 1 , . . . , 2 R +

1.

T = y, + j ~ , ,

and R is the highest order Bessel function used

in

the

approximation.

Using (15), we define

Tl fk)

and

T? fk) as

the

M

x M matrices which satisfy

Tl fk)G fk + a / 2 )

= G ( ( 1 -

7 ) f o )

(17)

T~ fk)G* fk

a / 2 )

= G* -yfo). (18)

3.2

Virtual Array Observations

(13) at each frequency, we have th at

Applying the transformations indicated in (12) and

=

a'e(f)s(t)

+ n- t).

(9)

L ,

@l(fk) = Czer((l ) f O ) S I ( f k

/ a )

Assuming t,hat the L , cyclostationary arrivals are

characterized by the common cycle frequency

a,

we

I 1

define + Qd f k +ala )

(19)

507



Now, taking the inverse DFT of (19) and (20) , the

nth snapshot

is

given by

l O

+

.i[nl (21)

La-1

7 i i [ 7 l ]

=

4,

-7fo)s;

[ i ] e - ~ ~ *

l O

.i[n]. (22)

Because of the focusin operations (using Tl (f k) and

T ? ( f k ) , the vectors &fi] and

&[i]

can be modeled as

narrowband. The final step is to take the expected

value of the random vectors:

2

= E

{ w l [ i ]

@ w z [ i ] )

(23)

where

@

represents the Kronecker product.

(In this

case, the Kronecker product is just the element-by-

element product of the vectors).

Substituting (21 and (22) into (23

,

assuming un-

correlated SOIs (61 and using the i ntity

AB) @

CD)

=

A ) B

)

[SI, it follows that

La

*

C s s , ( f O ) C P

(24)

1=1

where c y = E {

sr i]sy i ] e - - 3 2 T a r }

nd the M2-vector

( f a ) = G 1 - *Ofo) @ G,(-rfo). (25)

j j ~ ,

f o ) is the M2 x 1 vzrtual array manifold vector

generated by taking the Knonecker product and is the

concantenation of M subvectors each of length M x 1.

The

mth

such subvector of Yo fo) (for m = 1, . ,M )

is given by

(26)

{~,(fo)

thus corresponds to a virtual array with el-

ements places at locations

2, +

y(z1

z m ) , y m +

Y( YI

- ~ m ) )

. .

~ ~ + Y ( ~ M - - ~ ) , Y ~ + Y ( Y M

~ m . ) )

for m =

1, . . . ,

M . We note that for

y

=

0,

the vir-

tual array sensor geometry coincides with the original

array geometry (with redundancy).

The vector of virtual array 'observations' given in

(24) may be used in a conventional narrowband direc-

tion finding algorithm in order t o estimate the source

directions,

{6}fg1.

Note the attenuation of noise and

interference since they are assumed to be cyclically un-

correlated a t cycle frequency

CY.

Also, the parameter

may be used in order to increase the virtual array

aperture vis-a-vis the true (physical) aperture.

e32 T f o

k m -r zi l m

I sinel+[ym+-r Y1 - Y ) ] co s

0

f o

2 T f o =m+-r =M - = m I

s i n

@ I + [ Y ~ + ~ Y M - Y ~ ) ] ol 8

f0

In practice, we estimate

R

using N snapshots to

form the statistic

The conventional beamforming algorithm employed in

the next section operates on the vector in (27) (assum-

ing the virtual array geometry) in order to achieve

enhanced resolution vis-a-vis the original array geom-

etry.

4

Simulatioiis

The algorithm described above was implemented

using simulated array da ta and assuming uniform cir-

cular arrays of various radii.

M =

15 uniform circular

array with a (normalized) radius of unity. The SOIs

and cyclostationary interferers were all direct sequence

spread spectrum observed over Nb observation inter-

vals. The P h transmitted waveform is characterized

by the chip rate, CY. The frequency band of interest

in each simulation, (fo

5

: fo + .5)/fo, with fo nor-

malized to unity. In each example, the cutoff for the

approximation in (16) is chosen

as

R was chosen to

equal the largest integer not exceeding ( M 1)/2. In

all examples, a maximal length code was used for the

chipping sequence.

4.1 Simulation

Examples

Example 1: The simulation indicates the increased

resolution obtained using the virtual array process-

ing algorithm compared to results obtained using the

narrowband focusing technique discussed in [3]. The

array has

M

= 15 elements, with y = -0.5. There are

7

SOIs present spaced uniformly every 30 from 0 to

180'.

Colored noise with variance (a t each element) of

unity and correlation of

0.5

with adjacent sensors was

used. 50 bit intervals were observed and the number

of chips per bit was 6. A 15 chip maximal length code

was used. Th e cycle frequency,

CY

=

116.

Example 2: Thi s simulation demonstrates the ability

the anti-aliasing feature of th e virtual array processing

algorithm. A

9

element array was used with

=

-0.5

and one source present at 81 = 100'. Length 7 chip-

ping code was generated. Whi te noise was generated

and th e signal to noise ratio was 10 dB.

As

is evident

from Figure 2, the virtual array processing algorithm

is capable of estimating the source location although

the information is aliased using the original array (and

narrowband focusing).

Example

3:

This simulation demonstrates the sup-

pressio n of cyclostationary interference using a 15el-

ement array with 7 = -0.5. The

SO1

has unit energy

and arrives at l oo , while t he cyclostationary interfer-

ers are present at

02

= 120°,

6

= 150 and

64 = 90°.

30 bit intervals of the SO1 are observed. The SIR is 10

dB for each interferer and the noise is white with unit

variance. Th e cycle frequency of the SO1 is CY

=

1/6

while the interferers share the common cycle frequency

Y

=

1/7.

As

is demonstrated in Figure 3 , the virtual

array processing algorithm suppresses the undesired

interference.

508



5 Conclusion

In this paper, we have

a

described

a

new method

for obtaining a virtual array geometry for wide-band

array signal processing. Th e struc ture of the array

manifold was exploited in order to refocus

a

vector

pair of frequency shifted measurements without a-

priori knowledge of the DOAs and to generate new,

virtual array ’observations’

at

the final operating fre-

quency, fo We have shown the improved resolution

after application of the virtual array processing algo-

rithm vis-a-vis that of the original array geometry. I[n

addition, this algorihm is useful for the cancellation

of

unwanted cyclostationary and wide-sense stationary

interference (noise) present

at

the array.

References

[l]

H. Hung and M. Kaveh, ‘Coherent Signal-

Subspace Processing for the Detection and

Es-

timation of Angles of Arrival of Multiple Wide-

Band Sources,’ IEEE Trans. on Sign. Proc., Aug.

[2]

Ii Buckley and L. J . Griffiths, ‘Broad-band

signal- subspace spatial spectrum BASS-ALE)

Estimation,’

IEEE

Trans.

Acoust .

eech Sign(u1

Processing, July

1988,

pp.

953-964.

[3]

M.

A .

Doron and A. J . Weiss, ‘On Focusing Ma-

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IEEE

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June

1992,

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1295-130:2.

[4]

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111

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E.

Doron, and A. J. Weiss, ‘CO-

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J . W .

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509



Actual Array (M=9)

90,

270

27

Simulation 1

901

Simulation

0

80

270

70

510

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Virtual Array Processing Using Wideband Cyclostationary Signals