A COMPARATIVE STUDY OF ADAPTIVE ALGORITHMS WITH APPLICATIONS TO BEAMFORMING

Abstract

Adaptive arrays are currently the subject of extensive investigations, as a means for reducing the vulnerability of the reception of desired signals to the presence of interference signals in radar, sonar, seismic and communication systems. The principal reason behind this widespread interest lies in their ability to sense automatically the presence of interference noise sources and to suppress them, while simultaneously enhancing the desired signal reception without the prior knowledge of the signal/interference environment. The interference signals may not only consist of deliberate electronic counter measures, nonhostile RF interferences, clutter scatter returns and natural noise sources but also coherent interferences. Coherent interferences can arise when multipath propagation is present or when "smart" jammers deliberately introduce coherent jamming by retrodirecting the signal energy to the receiver. Also, the signal environment may consist of either narrowband or broadband signal and interferences. An adaptive array can be best described as a collection of sensors, feeding a weighting and summing network, with automatic signal dependent weight adjustment to reduce unwanted signals and/or emphasize the desired signal. In the case of broadband adaptive arrays, a tapped delay line is connected behind each sensor to compensate for the inter-element phase shift. The weight coefficients are adjusted recursively using suitable algorithms. In an adaptive array, the interference suppression is obtained by appropriately steering beam pattern nulls in the direction of interference sources, while signal reception is maintained by preserving desirable main lobe features. Therefore, an adaptive array system relies heavily on spatial characteristics to improve the output signal-to-noise ratio (SNR). A wide range of algorithms have been reported in the signal processing literature which can be used for adjusting the weights of an adaptive array. These include the conventional least-squares (LS) solution by direct matrix inversion or by Cholesky factorization, the classical least-mean-square (LMS) algorithm, the recursive least-square (RLS) algorithm, the fast RLS algorithms, QR decomposition algorithms based on Givens, Householders and Modified Gram-Schmidt techniques, and the rotation based fast RLS algorithms. Some of these algorithms are suitable for implementation using VLSI technology. Moreover, due to the recent advances in parallel computing architectures and VLSI technology, various computational, numerical and architectural concepts have merged. Consequently, it is becoming increasingly difficult to comprehend the interrelationships and tradeoffs among these concepts and approaches. A few of the above techniques, viz, the direct matrix inversion and the LMS algorithm, have been widely studied in the context of adaptive arrays. The difficulties in obtaining the inverse of the correlation matrix, when the matrix is ill conditioned, and the slow convergence and dependence of the time constant on the eigenvalues in LMS algorithm, make these techniques less attractive for application to adaptive arrays. The QRD-LS algorithm based on Givens rotations has been recommended in the literature for narrowband adaptive array applications. This algorithm has fast convergence and is numerically

stable but, unfortunately, it is computationally expensive because of the square-root operations involved. Some of the other techniques, viz, the RLS algorithm in the case of narrowband beamformers and multichannel fast transversal filters (MFTF) and QRD-multichannel lattice algorithms for broadband arrays have been discussed only briefly in the literature and detailed investigations have not been carried out so far. The recursive modified Gram-Schmidt (RMGS) and the multichannel least-square Lattice (MLSL) algorithms have not been studied at all in the context of adaptive arrays. The adaptive arrays based on above mentioned algorithms are effective in suppressing the interferences and enhancing the desired signal reception in a noncoherent signal environment. However, these techniques fail to suppress the coherent interferences. To overcome this problem, methods such as the structured correlation matrix method (redundancy averaging) and the spatial smoothing preprocessing scheme have been proposed in the literature. Of the two, the spatial smoothing scheme is more attractive and has received relatively wider attention. Several modifications of this scheme have also been proposed in the literature. Of these, the modified or forward/backward spatial smoothing scheme is important. However, the adaptive implementation in various algorithm based arrays has not received much attention so far. This work encompasses the study of adaptive arrays covering the above aspects. A comparative study of the structured correlation matrix method and the spatial smoothing scheme using an optimum beamformer revealed that the structured correlation matrix method introduces a bias while placing nulls in the direction of i i i interferences. Also, the method is not suitable for broadband adaptive arrays as in this case the correlation matrix is nontoeplitz even in noncoherent situation. Moreover, the adaptive implementation of this method in various algorithm based processor is not possible, where as the spatial smoothing scheme is a practical method to suppress coherent interferences in an adaptive array. We next consider the study of adaptive arrays based on recursive least-square algorithms having a computational complexity of

the 0(P ), where 'P' is the number of sensors in the array. It has been found that the conventional RLS algorithm based array suffers from numerical instability and fails to produce nulls in the direction of interferences arriving from endfire directions. Though the QRD-LS array based on Givens rotations has excellent numerical properties and superior nulling performance, it is computationally expensive because of the involvement of square-root operations. As an alternative, we have proposed the use of RMGS algorithm and its error feedback version for adaptive beamformers. These algorithms can also be implemented using systolic structures. The arrays based on these algorithms have numerical properties and nulling performance that are comparable with Givens rotation based QRD-LS array and at the same time, they are computationally less expensive. Therefore, the proposed RMGS algorithms based arrays represent a good compromise between the numerical stability and computational cost in the adaptive beamforming problems. For broadband arrays, however, these algorithms turn out to be computationally expensive with a complexity of the 0(P n), where 'M' is the number of taps in each delay line.

Next, we consider the arrays based on Fast-RLS algorithms for realizing broadband arrays. We have proposed the multichannel least-square Lattice [MLSL] algorithm for broadband adaptive array which has a computational complexity of 0(P M). Using MLSL algorithm as the basis, we formulate the Givens rotation based QRD-MLSL algorithm and apply it to the adaptive beamforming problem. We then derive the MFTF algorithm and study the adaptive arrays based on these algorithms. The algebraic approach has been used to derive these algorithms. Of the three broadband arrays realized, the MFTF algorithm has the least computational complexity. Finally, we have considered, the spatial smoothing scheme and the forward/backward spatial smoothing scheme as an effective means to suppress coherent interferences. Our studies have revealed that, in optimum beamformers, both the methods are effective in placing nulls in the direction of coherent interferences. The adaptive implementation of spatial smoothing scheme on the QR decomposition algorithms, such as QRD-LS and RMGS algorithm based arrays, has received little attention so far. We have proposed a method of implementing spatial smoothing scheme on the arrays based on these algorithms. In this method, the elements of the upper triangular matrix are smoothed first and after a fixed number of snapshots, this smoothed upper triangular matrix Is used to compute the optimum weights of the beamformer. The proposed method has been tested through computer simulations and produces deep nulls in the direction of coherent interferences. In the forward/backward spatial smoothing scheme, the signals of the respective forward and complex conjugated backward subarrays are first averaged. Then, the resultant signals are used to smooth the weights or the elements of upper triangular matrix. Our numerical experiments show that the conventional spatial smoothing scheme has a much superior nulling performance as compared to the forward/backward spatial smoothing scheme. The performance of various algorithms has been evaluated for the narrowband and broadband arrays in noncoherent as well as coherent interference environment, using computer simulations. Convergence characteristics of various beamformers have been tested by computing the residual power as afunction of the number of adaptation samples. The comparative study has revealed that, QRD-LS array based on Givens rotations has the fastest convergence and the least residual power. The RMGS algorithm based array with error feedback has characteristics comparable with those of the QRD-LS array. The nulling performance of various arrays has been studied with the help of voltage patterns. In the case of broadband arrays, the output waveforms has also been extracted to demonstrate the ability of the beamformers to track the desired signal.