Adaptive beamforming using cyclic signals in the presence of cycle frequency error

ADAPTIVE BEAMFORMING USING CYCLIC SIGNALS IN

THE PRESENCE OF CYCLE FREQUENCY ERROR*

Ju-Hong Leet and Yung-Ting Lee

Room 517, 2nd Building, Department of Electrical Engineering

National Taiwan University Taipei, Taiwan,

R.O.C.

Abstract

A

well-known property exhibited by most man- made signals is that they generate spectral lines when they are passed through a nonlinear trans- form[l]. For instance, BPSK signal generates spectral lines a t multiples of the symbol rate and the twice of the carrier frequency due to cyclostationary property. The Self-Coherent REstoral (SCORE) algorithms presented by [2] have shown to give the optimal solution for the signal extraction problem using cyclostationary statistics. However, they suffer performance degradation in the presence of cycle frequency error. This paper considers the behavior of the

SCORE algorithms in the presence of cycle fre-

quency error. Based on the theoretical analy- sis, w e p r e s e n t an efficient m e t h o d to alleviate the performance degradation. Simulation result-

s are presented for showing the effectiveness of

the proposed method.

I.

The SCORE Algorithms

Consider an adaptive beamforming using an antenna array excited by a SO1 (Signal Of Inter- est), J interferers, and background noise. If the ‘This work was supported by the Natioual Science t Author for All Correspondence.

Councd Under Grant NSC85-2213-E002-008

inverse bandwidth of the receiver is small with respect to the electrical distance between the ar- ray elements, then the received data vector x ( t )

is given by

where s d and Sj denote the SO1 and interference aperture vectors, respectively. Assume that s ( t ) is self-coherent at a which is one of the cycle fre- quencies of s ( t ) , and that i(t) is not self-coherent at a and is temporally uncorrelated with s ( t ) . The array output is given by

y ( t ) = WHX(t) (2)

From ( I ) , we note that the extractioti of t he SO1 can be accomplished by constructing a suitable cost function on y ( t ) and optimizing it with re- spect to the weight vector W. In the LS-SCORE algorithm, the following cost function is used[2].

Fsc(w; e )

=<

IY(t)

-

.(t)I* >T

?‘(t)

= C H X ( ’ ) ( t

-

T)t?32aui

(3)

(4)

where the reference signal r ( t ) is given by

and

< .

>i- denotes the time average over [0, TI.

The solution of minimizing (3) is given by

w,,

= R;:Rz7(a) (5)

where

Rr,

and Rz,(a) are the sample autocor- relation matrix of x(t) and the cross-correlation vector of x(t) and r(t) computed over

[O,T],

re- spectively. It is shown in [2] that ( 5 ) converges to the maximum-SINR solution when

T

approaches infinite for any vector of c as long as CHSd

#

0.

Due to the fact that r(t) contains the interfer- ence, the LS-SCORE algorithm converges slow- ly. To alleviate this difficulty, Gardner [2] also proposed t o find the optimal weight vector

WO,,

and the optimal control vector cOpt by maximiz- ing the correlation coefficient between y(t) and

T(t).

where u(t) = x(*I(t

-

T ) e j Z n a f is the control sig- nal. The solutions of maximizing (6) are given by

where

R,,

=

RzUcepl

and

R=,

= RurWopl, It is shown[2] that WO,, also maximizes the output

SINR as

T -9 ca. Simulation results show that

the performance of Cross-SCORE is much better than LS-SCORE, especially in the presence of strong interference.

11. Performance Analysis

From (l), we obtain (by letting

f

= a )

ILL=

=

c

x(t

+

T ) x ~ ( t

-

7)e-32"Jt > m 2 2 J

=

RfSdSdH

+

Ri,SjSjH

+

Ri

j=1 where

R!

=<

a(t

+

T ) a * ( t -

F)e-j2*ff 2 2 "

From the property of Fourier transform, the sample cross-correlation vector can be represent- ed by

where 8 denotes the convolution operation and

S I N C ( f T ) =

w.

Moreover, due to the fact that cycle frequencies are discrete, the cyclic autocorrelation function of the SO1 can be writ- ten as

R!(T)

=

C d n ( T ) h ( f

-

a n ) (11)

n

From (8), the cross-correlation vector can be ex- pressed by

R&)

= R~zc e-J "f7

where k d = SdHce-J"fT, kj

=

SjHce-jfffT and

k,

= R!,(T)ce-jXr'. Substituting (12) into (10) yields (Actually, Rrr(f) includes the cyclic cross correlations between the

SOI,

interference, and noise. However, they are negligible due to large

T .

1

k v ( f )

=

kf{x&(T)SINC((f- 0n)T))Sd

t

~ > { P ~ , ~ ( T ) S I N C ( ( ~

-

Pi,m)T)}Sj

+

ic,,

(3 S I N C ( f T ) (13)

i.m

where non-cyclostationary interference and noise are contained in

k,,,.

Without loss of generality, we set the known cycle frequency a = al. If a and P,,m are well separated and due to the fact that the value of S I N C ( f T ) is small for large

fT

then the effect of interference and noise in (13) is small. When

f =

a!

+

Aa,

where

Aa

is

the amount of deviation, (13) can be written by

R2,(

f)

=

.kd{ d l

(

T ) S I N C ( AaT)

+

dn(r)SfNC(&T)}Sd

n f l

+

k , { ~ J , , ~ ( . ) S I N c ( f i J , ~ , T ) } s j

J F

+

kn

C3 S I N C ( ( a

+

A a ) T )

(14) where 6, = a -0,

+

Acu and

fiJ,,,,

=

cy

-

&m

+

Aa. Due to the fact that S I N C ( A a T ) = 0

when

T

=

T,

=

2,

n = 1 ,2

,...,

Rsr(j)

~ ~ ~ ~ ( T ) S I N C ( A L Y T ) S ~

as

T

is far from

T,.

However, when T is near to

T,,

the compo- nents of Sj in (14) are not negligible and thus

the performance of the SCORE algorithms be- comes poor. From (7), we note that the differ- ence between LS-SCORE and Cross-SCORE is that Cross-SCORE adjusts adaptively the con- trol vector c and thus the behavior of the Cross- SCORE algorithm is similar t o the LS-SCORE algorithm when cycle frequency is error.

111.

An Efficient

Method

Consider the following new reference sigual in the LS-SCORE algorithm

? [ t )

= {y

+

SINC(r;t)}cHxi’)[t

-

T)e12nc.f (15)

where both y and fi are two constants. Ac-

cordingly, we also obtain the new control signal G ( t ) = {y

+

S I N C ( n t ) } x ( * ) ( t

-

r)eJ2aof in the Cross-SCORE algorithm. From (15), we obtain

Rzi(f) =

rRSr(f)

+

;lsect(;) 1 f @

R z , ( f )

where red(:) equals 1 when

-:

5

f

5

:

and equals 0, elsewhere. The corresponding sample

cross-correlation vector when f = a+

Aa

is also given by

%i(f)

= k d { d l ( T ) n ( n a T )

+

dn(T)n(&T)}Sd i- Cli,{pJ,n,(.)n(~j,mT)}sj

+

k,

c3Q ( ( a

t

A a ) T )

(17) n#l Jlm where

n ( f T )

= y S I N C ( f T )

+

?;red(:) @

S I N C ( f T ) . Under the conditions that Acu =

0 and Aa

#

0, R&) is proportional to

sd.

The result is due to the fact that from ( 1 6 ) ,r e c t( i)

8

R,’ =

Cd,(r)rect(*) is not equal to 0 as long as Acu

5 f.

Moreover, owing to red(?) @ S I N C ( A a T ) and SINC(AqT) in (17) are not equal to 0 at the same time,

R,?

is not seriously effected by Sj when

T

is near to T, =

2.

As a result, using ? ( t ) in the LS- SCORE algorithm and C ( t ) in the Cross-SCORE algorithm make these algorithms robust against the effect of cycle frequency error. We note that

r e c t ( i ) @ S f N C ( & , , T ) in (17) enhances the components of Sj in R,;(f) and leads to per- formance degradation. However, choosing larger y can alleviate this difficulty. Furthermore, the co~iipone~its of Sj iu (17) becoiiie larger as K or

y increases. Therefore, by choosing the values of y and n adequately, we can alleviate the effect of cycle frequency error 011 the performance of the SCORE algorithms.

IV.

Simulation

and Observation

Consider an adaptive beamfornier using a 21 u-

niform linear array with interelement spacing = half wavelength of the desired signal. All of the inputs are BPSK. The carrier frequency of the desired signal is set to 1. The noise is spatial- ly white. Moreover, we set y = 0.3, f i = 0.02,

and the sampling interval T, = 0.2. Two cases in which J equals 0 and 2 are considered. For each case, f = 2 and

f =

2.01 (i.e.. cycle fre- quency error = 0.01) are used. The vector c in the LS-SCORE algorithm is fixed and given by

s d is proportional to

SINC(AcrT)

when

J

= 0. Owing to

SINC(AcrT)

= 0 when

T

=

2,

n = 1,2,

....

there are nulls a t about 500n snap- shots which are shown in Figure l(b) when using the LS-SCORE and Cross-SCORE algorithms. In the case of J = 2, nulls still appear at about

500n snapshots which are shown in Figure 2(b). For both cases, the proposed method improves the performance of the SCORE algorithms i n the presence of cycle frequency error. Moreover, we note that an acceptable performance can be achieved by choosing appropriate y and K .

c = [1,0,0,

....

OIT. From (14), the coefficient of

References

[I] W. A. Gardner, "Exploitation of spectral re- dundancy in cyclostationary signals," IEEE

SP MAGAZINE, pp. 14-36. Apr. 1991.

[2] B. G. Agee, S. V. Schell and

N.

A. Gardner. "Spectral self-coherence restoral: A new ap- proach to blind adaptive signal extraction us-

ing antenna arrays," Proc. IEEE, Vol. 78, pp.

753-767, Apr. 1990.

!I

,"

:.

... ... i.. ....

" V k d S n p h a , lb)

Fig. 1: J=O. (a): 1=2, (b): 1=2.01.

'-' line: LS-SCORE using i ( t ) . 'x' line: Cross-SCORE using i ( f ) . '- -' line: LS-SCORE. '+' line: CrosrSCORE.

I5

...

...i...*.*...

...

I O .

. . .

I ,

't

to

. . . .

'":"

...

_:

...

;

...

Fig. 2: J=Z. (a): f=Z, (b): f=2.01. I-' line: LS-SCORE using i ( f ) . 'x' line: Cross-SCORE using 6(t).

'- -' line: LS-SCORE. '+' line: Cross-SCORE.