- Email: [email protected]

Adaptive H∞ robust beamforming for imperfect antenna array Ann-Chen Changa , Ching-Tai Chiangb; ∗ a Department

b Department

of Electrical Engineering, Chung-Cheng Institute of Technology, Taoyuang 335, Taiwan, ROC of Electrical Engineering, I-SHOU University, 1, Section 1, Hsueh-Cheng Rd., Ta-Hsu Hsiang, Kaohsiung County 84008, Taiwan, ROC Received 7 December 2000; received in revised form 12 December 2001

Abstract The inherent robustness of the H∞ algorithm in dealing with modeling disturbances is used to improve the performance of imperfect antenna array with the generalized sidelobe canceller structure. The modeling disturbances include sensor gain perturbation, initial condition error, and colored noise contamination. The in,uence of modeling disturbances, which is not involved in the recursive least squares (RLS) algorithm, is lumped into the estimation error minimization of the adaptive H∞ algorithm. Thus, an H∞ -based beamformer converges faster and is more robust to imperfect antenna array than an RLS-based beamformer. Computer simulation results have been provided to show the robustness feature of the proposed technique when applied to an imperfect antenna array. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Imperfect antenna array; H∞ algorithm; Generalized sidelobe canceller; Beamforming

1. Introduction Recently, the adaptive array beamforming techniques have been widely adopted for mobile radio communication systems. Beamformers can adaptively form radiation patterns in a dynamic radio propagation environment to suppress interference [3]. The Wiener and the recursive least square (RLS) 7ltering approaches [1] are frequently applied in array beamforming. However, both the Wiener and the RLS 7lters may not be robust to the disturbances of the array signal model. Array structure imperfection and colored noise contamination may deteriorate the SINR performance signi7cantly [2,4]. Unless a ∗

Corresponding author. E-mail address: [email protected] (C.-T. Chiang).

robust 7ltering algorithm is used, the small array sensor gain perturbations (amplitude and phase) may result in important performance degradation. The H∞ minimum estimation error criterion [5] has been developed and implemented to generate a new class of optimal 7lters. The H∞ algorithm ensures that the operator relating modeling disturbances to the resultant estimation error has an H∞ norm less than a prescribed positive value. Since the H∞ estimator can minimize the worst possible ampli7cation of the modeling disturbances, it can be seen as a dynamic, two-person, and zero sum game [6]. In the game, the H∞ 7lter is a player preparing for the worst scenario provided by the other player (i.e. modeling disturbances). In other words, the goal of the H∞ 7lter is to achieve a uniformly small estimation error for any process and measurement noise as well as initial condition error [6].

0165-1684/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 2 ) 0 0 2 7 6 - 1

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Nomenclature H∞

an adaptive algorithm minimax the in7nity norm of the system error transfer function array response vector impinging angle sensor gain perturbation amplitude perturbation phase perturbation variance of sensor gain perturbation

ai i Bgm Bam Bpm g2

In this paper, the application of H∞ algorithm to imperfect array beamforming with the generalized sidelobe canceller (GSC) structure [1] is investigated. The state and the measurement equations to describe the GSC structure are written for a stationary channel. The sensor gain perturbation for the imperfect antenna array and the colored noise contamination are lumped together into the measurement noise. Besides the measurement noise, the initial condition error is accounted for the modeling disturbances. By minimizing the eDect of the worst case of modeling disturbances on the estimation error, the H∞ algorithm can perform robust beamforming for an imperfect array, even in a colored noise environment. Simulation results show that the H∞ algorithm is superior to the RLS counterpart in dealing with imperfect array beamforming.

E[ ] diag[ ] n2 wopt W0 Rxx wˆ sup

mean operator diagonal matrix noise power optimum weight vector weighting matrix correlation matrix weight vector forgetting factor supremum prescribed level of noise attenuation

where d is the spacing between two adjacent sensors, is the wavelength of the carrier, i is the impinging angle with respect to the array broadside, and the superscript T denotes the transposition operator. In an ideal array, all sensor gains are set to unity. But amplitude and phase perturbations of array sensors may occur due to imperfect array structure. Thus, the assumption of the nominal unity sensor gain is no longer valid. In such environment, the gain gm of the mth sensor can be modeled as gm = (1 + Bam )ejBpm ;

m = 1; 2; : : : ; M;

where Bam and Bpm are zero mean random amplitude and phase perturbations of the mth sensor, respectively. Under the assumption of the small amplitude and phase perturbations, the sensor gain gm can be approximated as gm ≈ 1 + Bgm ;

2. Array data model and GSC-based array beamforming 2.1. Array data model Consider a desired source and J narrowband uncorrelated sources impinging on a uniform linear array with M sensors. Then, the direction vector of the ith source is given by d ai = 1 exp j2 sin i · · ·

d exp j2(M − 1) sin i

T ;

(1)

(2)

(3)

where Bgm = Bam + jBpm represents a zero mean complex gain perturbation of the mth sensor. The complex gain perturbation is assumed to remain constant during the adaptation period. It is also assumed that the random gain disturbances are independent among sensors with the same variance given by [2] g2 , E[|Bgm |2 ];

m = 1; 2; : : : ; M;

(4)

where E denotes the mean operator. From the above description of array imperfections, the received array data at the kth snapshot can be represented as an M ×1 vector form x(k) =

J +1 i=1

si (k)Gai + n(k);

(5)

A.-C. Chang, C.-T. Chiang / Signal Processing 82 (2002) 1183 – 1188

x(k)

y(k) +

Quiescent Weight

wq

P(k + 1) = −1 [P(k) − K(k + 1)uH (k + 1)P(k)];

Σ

(9)

_

z(k) Blocking Matrix B

u(k)

w

3. Adaptive H∞ beamforming algorithm for the imperfect array with GSC structure

where the ith source si is assumed to be a complex Gaussian process with power equal to i2 ; G , diag[g1 ; g2 ; : : : ; gM ] represents the complex sensor gain matrix, and the noise vector n(k) is spatially and temporally Gaussian noise with power n2 . 2.2. The GSC-based array beamforming In this subsection, the generalized sidelobe canceller [1] is described in Fig. 1. The nonadaptive weight vector, wq = C(CH C)−1 f, in the upper branch is data independent, where C is the constraint matrix and f is the constraint vector. The lower branch weight vector is free to be adapted to suppress the interference, and a blocking matrix B is inserted to prevent the signal cancellation. The matrix B of rank M − 1 is chosen to satisfy BH C = 0. Accordingly, the optimal solution wopt is given by [1] wopt = (B Rxx B)

−1

where y(k + 1) = wqH x(k + 1) is the required reference signal, the superscript ∗ denotes the complex conjugate, and denotes the forgetting factor (0 ¡ 6 1).

Adaptive Weight

Fig. 1. The GSC-based adaptive beamformer.

H

1185

H

B Rxx wq ;

(6)

where Rxx is the correlation matrix of x. Let the lower branch data vector which passes the signal blocking matrix B be denoted as an (M − 1) × 1 vector u(k) = BH x(k). It is easy to show that the weight vector wˆ can be recursively computed based on RLS algorithm as follows [1]:

In this section, the adaptive H∞ 7ltering algorithm is applied to adapt the weighting vector in an imperfect array with the GSC structure. For a stationary channel, the channel impulse response is time invariant over a number of data snapshots. As a result, the optimal weighting vector can be assumed to be unchanged over the adapted time interval. The array data model can be described by the following state and measurement equations: wopt (k + 1) = wopt (k);

(10)

H y(k) = wopt (k)u(k) + v(k);

(11)

where the measurement noise v(k) accounts for the input additive noise and the modeling error caused by the array imperfections. No statistical assumptions on the disturbances v(k) are made. The estimation error at the kth snapshot is de7ned as H e(k) = wopt (k)u(k) − wˆ H (k)u(k):

(12)

The performance measure for the H∞ algorithm is de7ned as the transfer operator which transforms the measurement noise v(k) and the initial condition error ˆ wopt (0) − w(0) into the estimation error e(k) [5] K−1 2 k=0 |e(k)|Q(k) F= ; (13) K−1 2 ˆ |wopt (0) − w(0)| + k=0 |v(k)|2V −1 (k) W−1 0

ˆ + 1) w(k H

∗

ˆ = w(k)+K(k+1)[y(k+1) − wˆ (k)u(k+1)] ; K(k + 1) =

+

P(k)u(k + 1) ; + 1)P(k)u(k + 1)

uH (k

(7) (8)

where Q(k) ¿ 0 and V (k) ¿ 0 are weighting variables, W0 ¿ 0 is a weighting matrix, and |x|2G , xH Gx. The weighting variables and matrix are determined by the performance requirement. The optimal H estimate of wopt (k)u(k) among all possible estimates H wˆ (k)u(k) (i.e. the worst-case performance measure)

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should satisfy 1 F∞ = sup F 6 ;

ˆ (v(k);w(0))

(14)

where “sup” stands for supremum and ¿ 0 is a prescribed level of noise attenuation [6]. The design objective of the H∞ algorithm is to process the received signal samples to produce an ˆ estimate of the weighting vector w(k). Moreover, the in7nity norm of the system error transfer function F∞ is guaranteed to be less than or equal to a prescribed positive value 1= . The discrete H∞ algorithm can be interpreted as a minimax problem where the estimator strategy wˆ H (k)u(k) plays against the exogenous inputs v (k) and the initial condition ˆ error wopt (0) − w(0). Then, the performance criterion can be equivalently represented as 1 2 ˆ F = − |wopt (0) − w(0)| min max W0−1 e(k) (v(k);w(0)) ˆ 2 K−1 1 1 |e(k)|2Q(k) − |v(k)|2V −1 (k) ; (15) + 2

k=0

where “min” and “max” stand for minimization and maximization, respectively. The minimax problem can be solved by the game theory approach [6]. Following the result given in [6], it can be proven that for a given

¿ 0 value, there exists an H∞ 7lter for optimal estiH mate wopt (k)u(k) if and only if there exists a Hermition solution, P(k) ¿ 0, to the following discrete-time Riccati equation: P(k + 1) = P(k){I − [ Q(k) −V −1 (k)]u(k)uH (k)P(k)}−1 ; P(0) = W0 :

(16) (17)

Furthermore, the estimated weight vector can be described as ˆ + 1) = w(k) ˆ w(k + K(k)[y(k) − wˆ H (k)u(k)]∗ ;

(18)

where K(k) = V −1 (k)P(k){I − [ Q(k) −V −1 (k)]u(k)uH (k)P(k)}−1 u(k) (19) and the initial condition of the estimated weight ˆ vector can be selected as w(0) = 0. Based on the performance requirement, a designer can choose the

appropriate weighting variables and matrix, which the RLS algorithm does not support. 4. Simulation results Several simulation examples for demonstration and comparison are presented under additive white Gaussian and colored noise assumptions. A 16-element linear uniform array with half wavelength separation was used for simulation. Four uncorrelated signals, each of them contaminated with background noise of n2 = 1, were fed into the system. Let the ideal steering angle ◦ be d = 0 and the input SNR be 10 dB. Three interfering signals with INR =30; 20, and 30 dB arrived at ◦ ◦ ◦ −30 ; −10 and 20 oD array broadside, respectively. The weighting variables Q(k) and V (k) were set to 1 for all the snapshots. The weighting matrix W0 was strengthened as 1000I to reduce the eDect of initial condition error on the convergence rate. The forgetting factor of the RLS algorithm was set to 1 (i.e. a stationary channel). Moreover, each simulation result presented is the average of 100 independent runs with uncorrelated samples for each run. Example 1. This example demonstrates the robustness of the H∞ algorithm for an imperfect array in presence of white Gaussian noise. Fig. 2(a) shows the resultant SINR versus the number of snapshots for an ideal array and an imperfect array with gain perturbation of g2 = 0:0002. Because the weight matrix W0 reduces the eDect of initial condition uncertainty, the H∞ algorithm leads to a convergence rate faster than the RLS algorithm. For the ideal array, the RLS-based beamformer can also converge to maximum output SINR performance. The output beampatterns of the RLS algorithm and of the H∞ algorithm after 1000 snapshots are shown in Fig. 2(b). It is obvious that H∞ -based imperfect array beamforming still suppresses the interference properly. With the same snapshots and INR values as Fig. 2(b), Fig. 2(c) shows the sensitivity curves in terms of the output SINR versus the variance of the gain perturbation. The responsive curves for several values of the input SNR are plotted. These curves show that the sensitivity to errors is higher for larger SNR values, because the modeling errors cause the desired signal to pass through the block matrix in the lower branch of the

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Fig. 2. The results of Example 1 with additive white Gaussian noise. (a) Output SINR versus the number of snapshots of the ideal and the imperfect array. (b) Output beampatterns under gain perturbation of g2 = 0:0002. (c) Output SINR versus variance of the gain perturbation for various input SNR’s.

GSC-based beamformer. When the input SNR increases, the cancellation of the desired signal in the output will be augmented. Thus, the output SINR of the RLS-based beamformer for an imperfect array decreases faster as the input SNR increases. On the other hand, the adaptive H∞ -based beamformer is robust to the modeling errors caused by the sensor gain perturbation. From Fig. 2(c), the tolerance to the sensor gain perturbation in the GSC-based beamformer can be estimated and it is a function of input SNR. Example 2. The performances of the RLS and the H∞ algorithms are compared for a colored noise

scenario. The colored noise has been generated by passing a white Gaussian noise signal through a 7lter with transfer function H (z) = (1 − 0:95z)−1 . Fig. 3(a) shows the array output beampatterns after 1000 snapshots. Compared to Fig. 2(b), the beamforming function based on the RLS algorithm in Fig. 3(a) is almost disabled. The colored noise augments the in,uence of the modeling disturbances. However, the H∞ -based beamformer achieves a promising performance, even in a colored noise environment. This advantage is determined by its inherent robustness. Fig. 3(b) shows the output SINR versus the variance of gain error of various input SNR after

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(b)

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Fig. 3. The results of Example 2 with additive colored noise. (a) Output beampatterns under gain perturbation of g2 = 0:0002. (b) Output SINR versus variance of gain error of various input SNR’s.

1000 snapshots. Similar to Fig. 2(c), the large SNR values increase the sensitivity to errors. Moreover, the colored noise decreases the tolerance of the sensor gain perturbation in the GSC-based beamformer. Again, the H∞ algorithm is more robust than the RLS algorithm in an imperfect array beamforming.

5. Conclusion We have developed the H∞ -based beamformer for an imperfect antenna array with the GSC structure. The sensor gain perturbations and the colored noise are lumped into the measurement equation. Moreover, the initial condition error is considered in the modeling disturbance. Since the design criterion is based on estimation error minimization in the worst case of disturbances, the H∞ algorithm is less sensitive then the RLS algorithm to modeling disturbances (exogenous signal statistics, initial condition error and sensor gain perturbation). Accordingly, an imperfect array based on H∞ algorithm can achieve a fast convergence rate and a satisfactory SINR performance.

Acknowledgements We thank the anonymous referees for their constructive suggestions, which help improve the clarity of this paper. References [1] S. Haykin, Adaptive Filter Theory, 3rd Edition, PrenticeHall, Englewood CliDs, NJ, 1996, pp. 227–255 (Chapter 5); S. Haykin, Adaptive Filter Theory, 3rd Edition, PrenticeHall, Englewood CliDs, NJ, 1996, pp. 562–587 (Chapter 10). [2] N.K. Jablon, Adaptive beamforming with the generalized canceller in the presence of array imperfections, IEEE Trans. Antennas Propagation AP-34 (1986) 96–1012. [3] Y. Karasawa, Expectation to software antennas in mobile radio communications, Proceedings of the ’96 Microwave Workshop Digest, 1996, pp. 318–323. [4] R. Rajagopal, K.A. Kumar, P.R. Rao, Signal subspace beamforming in the presence of coherent interferences and unknown correlated noise 7elds, Signal Process. 63 (1997) 17–33. [5] U. Shaked, Y. Theodor, H∞ -optimal estimation: a tutorial, Proceedings of the 31st IEEE CDC, Tucson, AZ, 1992, pp. 2278–2286. [6] X. Shen, L. Deng, Game theory approach to discrete 7lter design, IEEE Trans. Signal Process. 45 (1997) 1092–1095.