**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

*and is temporally uncorrelated with*

**a***The array output is given by*

**s ( t ) .***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, **

*relation matrix of*

**and Rz,(a) are the sample autocor-****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 *converges to the maximum-SINR solution when*

**( 5 )***T *

approaches
*CHSd*

**infinite for any vector of c as long as**### #

**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 **

### -

*is the control sig- nal. The solutions of maximizing*

**T ) e j Z n a f****(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 **

= **f**

*a )*

**ILL= **

= **c **

**x(t**

### +

**T ) x ~ ( t **

**T ) x ~ ( t**

### -

7)e-32"Jt*2 2*

**> m**

**J**### =

**RfSdSdH**### +

**Ri,SjSjH**

### +

**Ri **

*where*

**j=1****R! **

**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 **ic,,**

*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***and P,,m are well separated and due to the fact*

**al.**If a**that the value of**

*S I N C ( f T )*

**is small for large**

**fT **

**fT**

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

**f = **

**f =**

**a!**### +

**Aa, **

where **Aa,**

**Aa **

is
**Aa**

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

**R2,( **

**f) **

**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,,,, **

**fiJ,,,,**

### =

**cy**### -

**&m**

### +

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

when

*T *

= *T, *

= **2, **

n = 1 ,2 **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 samplecross-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, **

c3*Q ( ( 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 **

**8**

*R,’ = *

Cd,(r)rect(*) is not
equal to 0 as long as Acu **5 f. **

Moreover, owing
to red(?) @ **5 f.**

*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-

*algorithm make these algorithms robust against the effect of cycle frequency error. We note that*

**SCORE algorithm and C ( t ) in the Cross-SCORE***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 **f =**

**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. **