Linearly-Constrained Adaptive Signal Processing Methods

Lloyd J. Griffiths

doi:10.1117/12.942019

21 January 1988 Linearly-Constrained Adaptive Signal Processing Methods

Lloyd J. Griffiths

Proceedings Volume 0826, Advanced Algorithms and Architectures for Signal Processing II; (1988) https://doi.org/10.1117/12.942019
Event: 31st Annual Technical Symposium on Optical and Optoelectronic Applied Sciences and Engineering, 1987, San Diego, CA, United States

Abstract

In adaptive least-squares estimation problems, a desired signal d(n) is estimated using a linear combination of L observation values samples xi (n), x2(n), . . . , xL-1(n) and denoted by the vector X(n). The estimate is formed as the inner product of this vector with a corresponding L-dimensional weight vector W. One particular weight vector of interest is Wopt which minimizes the mean-square between d(n) and the estimate. In this context, the term `mean-square difference' is a quadratic measure such as statistical expectation or time average. The specific value of W which achieves the minimum is given by the prod-uct of the inverse data covariance matrix and the cross-correlation between the data vector and the desired signal. The latter is often referred to as the P-vector. For those cases in which time samples of both the desired and data vector signals are available, a variety of adaptive methods have been proposed which will guarantee that an iterative weight vector Wa(n) converges (in some sense) to the op-timal solution. Two which have been extensively studied are the recursive least-squares (RLS) method and the LMS gradient approximation approach. There are several problems of interest in the communication and radar environment in which the optimal least-squares weight set is of interest and in which time samples of the desired signal are not available. Examples can be found in array processing in which only the direction of arrival of the desired signal is known and in single channel filtering where the spectrum of the desired response is known a priori. One approach to these problems which has been suggested is the P-vector algorithm which is an LMS-like approximate gradient method. Although it is easy to derive the mean and variance of the weights which result with this algorithm, there has never been an identification of the corresponding underlying error surface which the procedure searches. The purpose of this paper is to suggest an alternative approach to providing adaptive solutions to problems in which samples of d(n) are unavailable.

Citation Download Citation

Lloyd J. Griffiths "Linearly-Constrained Adaptive Signal Processing Methods", Proc. SPIE 0826, Advanced Algorithms and Architectures for Signal Processing II, (21 January 1988); https://doi.org/10.1117/12.942019

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