Real-Time Algorithm For Adaptive Beamforming Using Cyclic
Signals
Shiann-Jeng Yu * and Ju-Hong Lee Department of Electrical Engineering
National Taiwan University Taipei, Taiwan, R.O.C.
Abstract
Adaptive beamforming using signal cyclostalioiiarity ca.n preserve the desired signal and cancel the interferers without prior information of the steering vector. In this paper, we consider the Cross-SCORE processor which is one of this class or beam formers and uses time-consuming eigen value decomposition (EVD) to compute weight vectors. 'rhus, this processor is not suitable for real-time processing. We here apply modular Gram-Schmidt Orthogonalization (GSO) structure in
conjunction with power norinalization scheine to the Cross-SCORE processor and propose a LMS based adaptive algorithm to update weight vectors. Due to the pipeline and parallel properties of the modular GSO structure, our approach is very suitable for real-time processing and the required computing time for the array to process an output is O ( N ) , where N is the number of array elements.
I.
Problem Formulation
Consider a narrowband array with
N
isotropic receiving sensor elements. Let the signal data vector A'(!) received by the seiisors and / I ( ! ) = ,Y*( 1+
~)exp(j2atv!), where T is a tiine delay and the cyclic frequency t v is equal to doubled symbol rate or inultiple baud rates or both. The Cross-SCORE processor computes the control vector C: and weight vector bV by solving the following problem
In (l),
IZ,,
= < A'(l)X"(l)>m,
Itx,, =<
X ( t ) U l ' ( L ) >m, andH,,
=<
U ( t ) / / I l ( l )><=,
where <.
denotes infinite time average. Thus, the opl.imal weight vectors for W and C can be found as the eigenvectors with the largest eigenvalues ofrespectively. Using the EVD, the Cross-SCORE requires 0 ( N 3 ) complex multiplications. 'This work was supported hy the National Science Council Under Cimnt NSC84-2213-E002-07 I
435
II. The Proposed Technique
In this paper, we notice that finding the eigenvectors of (2) is equivalent to finding the eigen- vectors of
I,;' I<,,, /,;Ii ancl 1,;' I(,,, I(;,' I<, , , I ~ ; " , ( 3 )
whcre I<, = I,, 1,;' and I<,,,, = I,,, I,:,'
.
~ iI,, eant1 I,,, arc Iowcr-tri;ingiilar initti ices ant1 their inverses exist. Therefore, we can express the optimal weight vectors as 1/11 = 1,;" I&'/, and ( ' =L t N CI, with W L and CL being the largest left and right singular vectors of I I ' L ~ ~ ~ = l<zt,L;/',
respective1 y
.
We here employ the inodular GSO
structure. Let the equivalent transformation matrix of the GSO be expressed as a upper triangular matrix V,. Forming the correlation matrix
'
:
I
= diag{ Pzl, P:2, 3 ..,
P,,, } = V,"IZ,,
I/,,
we canfind
L;"
= V,I];'.
In practice, V, and I: are updated recursively, we may rewrite L ; " ( L ) =There exists a number of methods to factorize I<,, =
K(
t ) Ii-I(1).Given L, and L,, we rewrite the cost of (1) as
If we restrict that the weight vector W L and C L are unit-norm, p = I~/V~JZL,,~CLI' becomes a siinple quadratic cost liinction. 'I'hcrefore, usiiig tlic instantaneous gradicnt, we can develop thc LMS based algorithm as follows.
W L ( f ) = W,(t
-
1 )+
/ L , X I L ( f ) U ~ ' ( l ) C L ( t - l ) , kV,(C) +- l ~ l ' L ( t ) / l l k l ' / , ( ~ ) ~ ~ , (:/,(I) = ('/,(I - 1)+
~L,~//,(/),~~,/(/)I/l;/,(/ -11,
( ' / , ( 1 ) +-~ ' / , ( ~ ) / l l ~ ~ / , ( ~ ) l l ,
W ( 1 ) = rt,, 12;" ( I )W/,(
1,
( ' ( I ) = 7, /,;//(/)('/,(/),( 5 )
where X L ( ~ ) = K ' f ( t ) X ( f ) , U/,(/) =
X;(/
+ T ) & ~ ~ ~ ' , and /ic are step sizes, and T?,, and r, aretwo scalars. The configuration of the proposed approach is shown in Figure 1.
We have noticed that the proposed algorithm is affected by the initial guesses of the weight vectors WL and CL. For some applications,
we
may roughly estimate the direction of the desired signal by using the beamspa?e preprocessor and then use the estimated directional steering vector as the initial guess. If we pick the initial weightsbV~(0)
and C,(O) such that L;H(0)kl/~(O) = y,$l and L ; " ( O ) C L ( O ) =rc.?i,
where ,?cl is the initial estiniate of the desired signal direction, y,,, and yc are constants such that I.V/,(O) and (:/,(U) are unit-norm.111. Siniulation Examples
We here provide an example to show the performance of the proposed algorithm. A uniform linear array with N = 6 and half-wavelength element spacing is employed. The scenario of
the signal sources is shown in Table 1. The initial /,,(O) = I and /ttt(O) = for t = I , 2,
..
.,
6. The cyclic frqiency I Y = 0.2. We choose p,,, = 0 . 0 S / ( , ~ ~ ~ ( / ) , ~ / , ( / ) ) ;aid p , =U.US/(//A/(
/ ) / I , , ( / )1.
Wic initials IV,,(U) ancl (,',,(U) iiie the norriiaIizerI Iiaiiiiaiiig wiidow fiiiiclioiiswith iiiaxiniiiiii gain i n tlic broadside. Figure 2 prcsciits tlic output sigii;il-(o-iiilcrf'c~cticc plus noise ratio (SINK) versus iiuinhcr o f s ~ ~ a p ~ l ~ o l s . ' I ~ I C ~csril~s ;iiic ;ivciagctl by 100 iritlcpciitlctil ~ I I I I S .
Fig. 2(a) uses T = 0 and Fig. 2(b) uses T = A'/', wliere A'/' = 1 is the sainpling inlerval.
They show that the Cross-SCORE [2] has the fastest convergence speed at the expense of' large computation load. However, the proposed algorithm can be iinpleinented in real time and achieve better performance when it converges for different values of' T .
References
[ 11
W.
A. Gardner, STATISTICAL SPECTRAL ANALYSIS. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.121 R. G . Agcc, S. V. Sclicll, W. A. Ciartliicr,"Sl)ccll-al slclf-colicrciice rcstorail: a ncw ;ipproach to blind adaptive signal extraction using aiiteniia arrays," Prvc. IEEE, Vol. 78, pp. 753-766,
Apr. 1990.
Table 1 The scenario of Figure 2
Signal
SNR
(dB) Symbol Rate Baud Rate DOA Pulse Shape0 cosine
BPSK 20 35" cosine
QAM 20 0.15 117 20"
QAM 20 0.25 1/8 -2.5"
* is the desired
signal0
- -
J
Figure
1.
The Proposed Configuration
5 9 0 2 -j 0 0 200 600 800 1000 I200 I400
Nuiiihcr ol' Sitapsho~s
Fig. 2(a) The Output SINR versus Number of Snapshots. Solid line: The Proposed Approach. Dash line: The EVD based Cross-SCORE. T = 0.
12,
I 200 400 600 800 loo0 1200 1400
Number of Snapshots
Fig. 2(b) The Output SINR versus Number of Snapshots. Solid line: The Proposed Approach. Dash line: The EVD based Cross-SCORE. T = A'/'.
