Unconstrained Beamforming : A Versatile Approach.

Transcription

1 Unconstrained Beamforming : A Versatile Approach. Monika Agrawal Centre for Applied Research in Electronics, IIT Delhi October 11, 2005 Abstract Adaptive beamforming is minimizing the output power in constrained manner. The number of available degrees of freedom for this optimization is often limited by the size of the array. Whereas, in many practical applications the number of constraints required to be fulfilled are numerous, much more than the available degrees of freedom. Therefore, the beamformer may not perform satisfactorily. In this paper to address this problem, all the constraints are merged with the cost function and the beamformer is designed by unconstrained minimization. It turns out that the outperforms the standard beamformer in almost all the scenarios. The suggested beamformer reconstructs the signal reasonably well even when the array has few sensors. These observations hold even for signals having temporal bandwidth, spatial bandwidth as well as moving targets. 1 Introduction Since the pioneering work of Capon [1] and Widrow [2] adaptive optimal beamforming has attracted the attention of many researchers [3]. The output of the sensor array is combined by the weight vector to reconstruct the desired (look direction) signal without distortion i.e. having minimum contribution of interferences and noise. This is only possible by rejecting the interferences and minimizing the noise to the extent possible. The designing of beamformer can be seen as a constrained optimization problem where the weight vector is found by minimizing the total output power constraining the look direction gain to unity. Also many a times the look direction is not known exactly, therefore, we need to put additional constraints to handle this mismatch [4]. It is well known that a constrained optimization will perform satisfactorily only when the degrees of freedom are more than the number of constraints. This finding puts a limitation on the size of the array. The length of the array to reject M narrowband interferences has to be more than M. In case of broadband, spread, moving etc. sources the available degrees of freedom are much less than the number of constraints [4], [7],[5], [6]. In this paper, we address this problem of designing the beamformer when the available degrees of freedom are less than the number of constraints required. In this scenario the constrained optimization can not result into satisfactory solution as all the constraints can not be satisfied, therefore, we loosen the requirement to fulfill the constraints exactly. A modified unconstrained optimization problem, by including all the constraints into the cost function, is suggested to design the beamformer. When the available degrees of freedom are less, the weight vector so calculated performs better than the because it uses the available degrees of freedom in the best possible way, unlike a [1] which uses fixed number of degrees of freedom in satisfying

2 the constraints. When we have sufficient degrees of freedom the behaves at par with the. The simulation results show that the outperforms than the. 2 Constrained Beamforming Let us consider the simplest case, where an array of M sensors is receiveing narrowband signals from a desired source and P 1 interfering signals. The received signal at the mth sensor, at tth time instant is y m (t), given by y m (t) = P 1 p=0 e jω 0τ m(θ p) s p (t) + n m (t) (1) where s 0 ( ) is the desired source signal, s p ( ) is the signal from the pth interference, n m ( ) is the noise signal picked by the mth sensor, ω 0 is the operating frequency, τ m (θ p ) is the time delay corresponding to the mth sensor for a plane wavefront coming from direction θ p. The noise is assumed to be spatially uncorrelated, i.e. its covariance is given by E[n m (t)n k (t) H ] = σn 2 δ(m k) (2) where σn 2 is the noise variance. We can write the above equation (1) in vector form as y(t) = a 0 s 0 (t) + As(t) + n(t), (3) where y(t) = [y 1 (t), y 2 (t),, y M (t)] T, n(t)) = [n 1 (t), n 2 (t),, n M (t)] T, s(t) = [s 1 (t), s 2 (t),, s P 1 (t)] T and a p = [1, e jω 0τ 1 (θ p), e jω 0τ 2 (θ p),, e jω 0τ M 1 (θ p) ] T (4) a 0 is the array steering vector corresponding to the desired source and A is the M P 1 dimensional array steering matrix for interfering sources, A = [a 1, a 2,, a P 1 ] (5) The problem of beamforming is to find a weight vector w, such that the beamformer output w H y(t) replicates the desired signal as closely as possible. The weight vector is such that minimizes the output power in all directions, and maintain a unity gain in the look direction, mathematically, w c = arg(min w w H Rw) The solution to this problem is s.t. w H a 0 = 1. (6) w c = R 1 a 0 a H 0 R 1 a 0. (7) If the direction of desired signal is not exactly known, or the source is moving or the source has spatial spread, additional derivative constraints are required in the look direction. These derivative constraints flatten the response in the look direction to allow the look direction signal to pass without distortion. Mathematically the beamfroming problem becomes, Where D k = dk ( ) dθ k w c = arg(min w w H Rw) s.t. w H a 0 = 1 w H D k a 0 = 0 k = 1, 2, (8) is the derivative operator. Let B = [a 0 Da 0 D K a 0 ] f = [100 0] T. (9) Then the solution of above stated beamforming problem is given by w c = R 1 B(B H R 1 B) 1 f (10) Similarly in the case of a broadband beamformer the number of constraints increases because every frequency component of the desired source is constrained to pass. Moreover in case of moving broadband sources and spatial spreaded broadband sources the number of constraints to be satisfied are much larger. These problems will 2

3 also have similar solution, here B will be the corresponding constraint matrix [4]. The available degrees of freedom are limited by the array size, therefore, all the constraints can not be fully satisfied by a limited size array. Hence the desired signal can not be reconstructed from the received signal. 3 Unconstrained Beamforming As stated earlier all the constraints can not be fully met. Therefore, we loosen the requirement of fulfillment of constraints and include them in the cost function. The modified cost function is defined as, w un = arg(min w w H Rw + α w H B f 2 ) (11) Here, The method for estimating the optimum value of α is beyond the scope of this paper. Here we only use optimum value of α to show that unconstrained beamformer behaves better than constrained beamformer. This optimum value of α is obtained by carrying out search over various values. As the purpose of the beamformer is to reconstruct the look direction signal from the observed signal, therefore, two beamformers are compared using metric that computes the difference between the reconstructed signal and the original signal. Let ŝ 0 ( ) is the reconstructed desired signal, then normalized error is defined as, = 1 K K s 0 (t) ŝ 0 (t) 2 (14) t=1 Here s 0 (t) and ŝ 0 (t) are normailzed. w H Rw + α w H B f 2 = (w (R + αbb H ) 1 αbf) H (R + αbb H ) (w (R + αbb H ) 1 αbf) + const (12) Here const is not the function of w. Therefore, 2 w un = (R/α + BB H ) 1 Df, (13) w un is the required beamformer which uses the available degrees of freedom in the best possible way. The constraint set is fulfilled (may be partial if the available degrees of freedom are not sufficient) and also the total output power is minimized. The value of α controls the relative contribution of the constraints and the output power in the optimization problem. A large value of α implies that more degrees of freedom are used in satisfying the constraints and relatively less degrees of freedom are utilized in minimizing the output power and vice versa. The choice of an appropriate value of α depends upon the requirement. For every scenario there exists an optimum value of α for which an unconstrained beamformer w un performs better than or same as a constrained beamformer w c SNR Figure 1: vs. SNR for wideband source. Temporal bandwidth [.9, 1] desired source direction 0 0, interference direction 30 0, M = 3, N = 100, K = Simulation Studies In this section, we present some simulation results to study the performance of the suggested beamformer. An ULA of three sensors (M = 3), is used to observe 100 independent snapshots (N = 100) of array data. The desired source is assumed to be present at the broadside of the 3

4 No of sensors Number of Sensors Figure 2: vs. number of sensors for wideband source. Temporal bandwidth [.9, 1], desired source direction 0 0, interference direction 30 0, SNR = 10dB, N = 100, K = Figure 4: vs. number of sensors for spread source. Spatial bandwidth5 0, desired source direction 0 0, interference direction 30 0, SNR = 10dB, N = 100, K = SNR Figure 3: vs. SNR for spread source. Spatial bandwidth 5 0, desired source direction 0 0, interference 30 0, M = 3, N = 100, K = array (0 degrees), along with the interference of same strength present at 30 degree in White Gaussian Noise (WGN). SNR is defined as SNR = 10log 10 σ 2 p σ 2 n (15) where σp 2 and σ2 n are the desired signal and noise power respectively. First we study the scenario of broadband sources. A Tap-delay line beamformer [3] is designed to pass the desired signal having a bandwidth of [.9 1]. Derivative constraints are also put in the look direction to flatten the beam so that beamformer becomes robust to the error in look direction. Figure 1 and Figure 2 show the as a function of SNR and number of sensors both for constrained and unconstrained beamformers. In all these simulation results the optimum value of α is used. In Figure 2 SNR is assumed to be 10dB. It is clear from the figure that the works better than the constrained beamformer. At low SNR both the beamformers behave similarly. Actually the noise level is high and the number of sensors are limited, therefore, it is very hard to reconstruct the signal. But at high SNR suggested beamformer behave 8dB better than standard beamformer. Next we take up the spread source scenario. Each source is assumed to have uniform spread of 5 0 around its mean position. We put two derivative constraints in the look direction to allow the signal to pass without distortion. Figure 3 plots the vs. SNR for three sensors array and Figure 4 plots vs. number of sensors at 10dB SNR. Again the works better than the. At high SNR there is an improvement of 6dB. Similar observations are made for the case of broadband sources having spatial bandwidth. The plots are not shown here because of the space constraints. The performance of the suggested beamformer is much better than the standard beamformer. 4

5 5 Conclusion A new method of beamforming to use the available degrees of freedom in the best possible way has been suggested. The available degrees of freedom are always limited and when the number of constraints increases, the performance of the beamformer deteriorates. A method is suggested to combat this problem. An unconstrained beamformer is suggested which not only satisfies the constraints but also minimizes the total output power. Simulation results show that the performs much better than the for all kinds of source. Sources, IEEE Transactions on Signal Processing, pp , August [7] M.Agrawal, S.Prasad, Robust Adaptive Beamforming for Wideband, Moving and Coherent Jammer via Uniform Linear Arrays, IEEE Transactions on Antennas and Propagation, pp , August References [1] J. Capon, High Resolution frequencywavenumber spectrum analysis, Proc. IEEE, Vol. 57, pp , Aug [2] B. Widrow, P. E. Mantey, L. J. Griffiths and B. B. Goode, Adaptive Antenna Systems, Proc. IEEE, Vol. 55, pp , Dec [3] B. D. Vanveen and K. M. Buckley, Beamforming: A Versatile Approach to Spatial Filtering, IEEE ASSP Magazine, pp. 4-24, April [4] Er MH, Cantoni A., Derivative constraints for broad-band element space antenna array processors, IEEE Transactions on Acoustics Speech and Signal Processing, pp , Dec [5] J. Ringelstein, A.B. Gershman, J.F. Böhme, Direction Finding in Random Inhomogeneous Media in the Presence of Multiplicative Noise, IEEE Signal Processing Letters, pp , October [6] M. Bengtsson, B. Ottersten, Low- Complexity Estimators for Distributed 5