Adaptive Linearly Constrained Inverse QRD-RLS Beamformer for Multiple Jammers Suppression Chung-Yao Chang and Shiunn-Jang Chem Department of Electrical Engineering, National Sun Yat-Sen University Kaohsiung, 80424, Taiwan. Fax(886)7-5254199 E-Mail:
[email protected]
Abstract-In this paper, a new general linearly constrained recursive least squares (RLS) array beamforming algorithm, based on an inverse QR decomposition, is developed for multiple jammers suppression. It is known the LS weight vector can be computed without back substitution in the inverse QRD based algorithms and is suitable to be implemented using the systolic array. Also, the problem of the unacceptable numerical performance in limited precision environments, occurred in the “fast” RLS filtering algorithms, can be avoided. To document the advantage of this new constrained algorithm performance, in terms of convergenceproperty of the learning curve and the capability of jammer’s suppression, is investigated. We show that our proposed algorithm outperforms the LCLMS algorithm [SI and the linearly constraint fast LS algorithm (LCFLS) and its robust version (LCRFSL) algorithm 141. I. INTRODUCTION The linearly constrained minimum-variance (LCMV) beamformer is considered to be one of the most popular approaches for suppressing the undesired interfering signals. The Frost’s beamforming algorithm proposed by Frost [ 5 ] can be viewed as an adaptive implementation of the method of the LCMV beamformer. However, under certain circumstance, the conventional Frosf s beamforming algorithm may have some problem associated with the performance degradation in multiple jammers environment. It is known that, in general, the RLS algorithm offers better convergence rate, steady-state means-square error (MSE), and parameter tracking capability over the adaptive LMS based algorithm. But, the widespread acceptance of recursive LS (RLS) filters has been impeded by a sometime unacceptable numerical instability in limited precision environments due to round off error. This degradation of performance is especially noticeable for the family of techniques collectively known as “fast” RLS filters. To overcome this problem, a well known numerical stable RLS algorithm, which is called the QR-decomposition RLS (QRD-RLS) algorithm was proposed. Basically, it computes the QR
0-7803-6720-0/01/$10.000200 1 IEEE
294
decomposition of the input data matrix using Givens rotation and solving the LS weight vector by the back substitution. This, in turn, causes the numerical dynamic range of the transformed computational problem to be reduced. Also, it has the benefit of using the QR-based approaches, that is, the rotation-based computations are easily mapped onto systolic array structures for a parallel implementation with VLSI technology. However, in some practical applications if the LS weight vector is desired in each of iteration then back substitution steps must be performed accordingly. Due to the fact that back substitution is a costly operation to be performed in array structure, in such circumstance the so-called inverse QRD-RLS (IQRD-RLS) algorithm was proposed, where the LS weight vector can be computed without back substitution. In this paper, a linearly constrained adaptive beamformer based on the IQRD-RLS algorithm IS developed, where an adaptive narrowband array structure is adopted. It is known that in the conventional IQRD-RLS algorithm the adaptation gain or Kalman gain is evaluated using the Givens rotation [1][2], where the LS weight vector can be computed without back substitution. This results in having better numeric a1 accuracy and stable steady-state mean square error (MSI:) and better capability to null the multiple undesircd interferences. In what follows, the development of tile adaptive LC-IQRD-RLS beamforming algorithm is briefly described 11. ADAPTIVE LINEARLY CONSTRAINED INVERSE QRD-RLS BEAMFORMER To start our derivation, let us consider a uniformly linear array (ULA), depicted in Fig. 1, with N element array of sensors. Now, taking the first element in the array as the phase reference and with equal array spacing, d, the relative phase shift of the received signal at the n,fh element can be expressed as:
#nk
21F
= -d(n
A
- 1)sin 8 , Where w (0 < w 5 1) is an exponential weighting. For convenience, let us denote a n x N data matrix, in terms of snapshots, x ( l ) , x(2), x(3), which is defined by X(n)=[x (I), x(2), . ., x(n)IT, with x'(i) =[xI(i), x2(i), .., x N ( i ) ] .In consequence, we may rewrite (4) in a matrix form, i.e.,
..,
...,&,,I] is a diagonal matrix. where ,p(n)=diag r,/T,JT, In the linearly constraint problem, the constraints of the weights are introduced by a linear system: Stmirig vator
Adaptive Algorithm
C H h ( n )= f
Fig.1. Configuration of adaptive linearly constrained array beamformer.
Moreover, we assume that the spacing between array elements is set to be h/2 , the array response vector of this N-antenna ULA can be denoted by
Thus, we choose 8 toward the direction of arrive (DOA) of desired source signal and suitably adjust the weights of adaptive array, the array will pass the desired source signal from direction 8 and steer nulls toward interference sources located at 8 ,for k # 0 . It can be shown that an N element array has N-1 degrees of freedom giving up to N-1 independent pattem nulls. So it has better performance if the array has more antenna elements. It is known that the principle of a LCMV beamformer [ 3 ] is to minimize tbe powers of background noise and the interference at the linear array output, while maintaining a chosen frequency response in the look direction. The vector of sampled signals at the time index n is denoted by x(n)=[x,(n)&), ..,nN(n)]" and the corresponding vector of the weights appearing at each tap is designated as h(n)=[h,(n),h,(n),..,h&)]". Where the superscript H i s denoted as the Hermitian operation and N is the number of array elements. The output signal is given by A n ) =hH(n)x(n)
(3)
In the method of exponentially weighted least-square, we choose the weights at time n, so as to minimize the cost function that consists of the sum of weighted output power:
0-7803-6720-0/01/$10.0002001 EEE
295
(6)
In ( ~ ) , C = [ C , ,,..., C , c,] is a N x K constraint matrix, constructed by array steering vectors, ci '[c, ,c,,,...,c , ~ ] ~ , and f = K , f , , , , f K ] His the K-element response column vector. The constrained optimization problem becomes to minimize the cost function defined in ( 5 ) , subject to the constraints defined in (6). It is known that in the conventional QRD-RLS algorithm, an orthogonal matrix Q ( n ) is employed to do the triangular factorization of the data matrix, , p 2 ( n ) ~ ( n ) , by Givens rotations, i.e., Q (n)A" (n)X ( n) =
r]:
(7)
where R(n) is an N x N upper triangular matrix, and 0 is the ( n - N ) x N null matrix. Since orthogonal matrix is length preserving, using the result of (7), the cost function, based on QR-decomposition, can be rewritten as IIh)h(n)l12 (8)
$wl=
where [I(. )/I denoted the Euclidean norm of ). Now, the constrained optimization problem is stated by (8) and (6), consequently, we may derive the constrained optimal solution of the LS weight vector via Lagrange multiplier method, based on the inverse QRD-RLS notation, that is
h(n)=[RH(n)R(n)]-lC(CH[RH(n)~n)]-'C~f(9) Based on equation (9), a recursive implementation of the optimum linearly constrained LS solution, using the IQRD-RLS algorithms, can be developed. In order to derive the recursive equation of (9), we define a new N x N matrix, S(n), i.e.,
S(n)=R4(n)Ry(n)
of learning curves with 3000 iterations, for case 2, are shown in Fig.2, while the performance of nhlling capability with 200 iterations, is listed in Table 2, for different methods. The results are the averaged of 500 independent runs. As observed from Table 2, the presented method has much better nulling capability than the LCFLS and the LCRFLS algorithms and the conventional Frost's algorithm. Also, it has more stable steady-state mean square error (MSE) than the LCFLS and the LCRFLS algorithms as evident from Fig.2.
(10)
In the Kalman filtering matrix s'(n) is referred to as an "information matrix" of the exponential weighting sensor outputs averaged over n snapshots. Moreover, we define =S(n)C and Q(n) =C"S(n)C, in consequence, (9) can be expressed as
r(n)
h(n) = r(n)W'(n)f
(1 1)
Here (1 1) can be viewed as the MVDR beamformer, based on the inverse Q R decomposition. Finally, the recursive implementation of (1 1) can be developed based on the recursive equations of r ( n ) and @-'(U). After some mathematical manipulation and simplification, we obtain the adaptive linearly constrained inverse QRD-RLS beamforming algorithm, i.e.,
h(n) = h(n - 1) - p (n)e(n,n -1)
Table 2. Comparison ofnulling capabilitydifferent methods (200 iterations, unit of nulling gain is dB)
Case1
Case2
Jammer
(12)
Jammer 30dB 30dB Power
where
p(n>= k(n) - wW>q(n)
I
1
30dB
lOdB 40dB 40dB
LCLMS -35.39 -47.16 -53.55
-12.59 -39.81 -39.07
(13)
and e(n,n-1) = x H(n)h(n-l)
(14)
It is noted that the Kulman gain, k(n) = g(n)/t(n) can be evaluated as the conventional inverse QRD-RLS algorithm [I] and the updated equations of the parameters, r ( n ) and d n ) are derived as
r(n) = w T ( n - 1) - g ( n ) a ( n ) q(n>=
with
-'(n -I)&
(n)
I -w@n)@-'(n -I)olH(n)
I
I
I MVDR I -82.02 I -81.80 I -83.24 I -72.91 I -84.26 1-88.57 I
wloz LCLMS
(15) (16)
LCFLS
'
I"
1 o'
1oo
@n) =g"(n)C and
I
I
0
1000
.
2000
3000
I 0
iteratban LCRFLS
This completes the procedure of the adaptive LC IQRDRLS algorithm-given in Table 1.
1000
2000
I 3000
iteration LCIQRD
111. COMPUTER SIMULATION
To investigate the performance of the presented method, the 'capability of interference rejection, two cases with different eigenvalue spread are considered. In general, since the averaged power of jammer is much larger than the desired source signal, the signal-to-noise ratio (SNR) is set to 0 dB. In the first case, three jammers with jammer-to-noise ratio (JNR) to be JNRl=JNR,=JNR,=30 dB, are chosen, and in the second case, JNR,=10 dB and JN&=JNR3=40 dB are considered. The results, in terms
0-7803-6720-0/0 1/$ 10.0002001 IEEE
1 0
1000
2000
Iteratban
I
1
3000
0
I 1000
2000
iteration
Fig.2. Leaming curves for case 2 after 3000 iterations (using 500 runs)
296
3000
IV. CONCLUSIONS In this paper, a generalized adaptive linearly constrained beamformer based on the inverse QRD-RES algorithm has been derived for multiple jammers suppression. To document the advantage of our proposed method the performance comparison of the learning curves and nulling capability for different methods are evaluated. The computer results have verified the merit of the proposed method. Thus, we can conclude that the overall performance, in term of numerical stability, convergence property and nulling capability, of the presented method did perform over the LCLMS algorithm, the linearly constrained FLS and its modified version.
Table 1. Summary of the LCIQRD beamforming algorithm 1.Initialization:
R-'(0) =#I ,' &small positive constant r(0)= R-' (0)R-H(0)C h (0) = r(o)[CH r(o)r'f 2.Evaluate the Kalman gain using Givens rotation z(n) =
R-H(n-l)x(n)
&
ACKNOWLEDGEMENTS The financial support of this study by the National Science Council, Republic of China, under contract is greatly number NSC-89-2213-E-110-055, acknowledged.
3.Update the intermediate parameter
r(n) = d r ( n - 1) - g(n)a(n)
REFERENCES d n )=
[l] S. T. Alexander and A. L. Ghimikar, " A method for recursive least squares filtering based upon an inverse QR decomposition, " 'IEEE Trans. On Signal Processing, vol. 41, no. 1, pp. 20-30, Jan. 1993. [2] C. T. Pan and R. J. Plemmons, " Least squares modifications with factorization: parallel implications, " Journal of Computational and Applied Mathematics, vol. 27, pp. 109-127, 1989. [3] S. Haykin, Adaptive Filter Theory, 2nd Eds. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1991, Chapter 14. [4] L. S. Resende, J. M. T. Romano and M. G. Bellanger, " A robust algorithm for LCMV adaptive broad band beamformer," 1996 IEEE ICASSP vo1.3, pp. 18261829. [5] 0. L. Frost 111, " An algorithm for linearly constraint adaptive array processing, " Proc. IEEE, vol. 60, no. 8, ~ ~ 9 2 6 - 9 3Aug. 5 , 1972
0-7803-6720-0/01/$10.0002001IEEE
Ah
- 1 ) d (n) l-w~n)@-'(n-l)d(n)
Q-yn) =
4+&(n)ajn)JW(n-
1)
4.Update LS weights vector with LCIQRD algorithm e(n,n-1) = xH(n)h(n-l)
p(n) = Wn)q(n) h(n) = h(n-1) -p(n)e(n, n-1)
297
