“Adaptive Linearly-Constrained Filtering: Principles and Implementations”

by Ching-Yih Tseng

May 1990

Incorporating linear constraints in adaptive filters has arisen as a new trend to provide robust filtering operations in signal processing. The mechanism of this class of adaptive filter is to minimize the output power while constraining the weight values by a set of linear equations. The constraints are deliberately selected to confine the weights so that the signals of interest will not be canceled when minimizing the filter output power. Through this constrained power minimization process, the noise power is reduced without distorting the signals of interest and therefore it improves the signal to noise power ratio at the output.

The basic requirement to utilize the adaptive linearly-constrained filter is to design the constraints. Appropriately setting up constraints requires the knowledge of the features regarding the particular signal of interest. Such features can be identified either through empirical methods, such as using a large set of training signals, or through theoretical methods, such as using the statistical information about the signals. After all, the resulting constraints have to effectively prevent the cancellation of signals of interest when the filter is applied. The second requirement is an adaptive algorithm to update the weights. To ensure implementation efficiency, it is desirable to develop implementation structures which are both computationally efficient and numerically stable.

This dissertation presents some results from the study of the above two requirements in adaptive linearly-constrained filters. To develop procedures for constraint specification and performance evaluation, the understanding of the effect of the constraints on the steady-state behaviors of the filter is an essential. Such effect is studied here through examining the optimal weights, the signal-to-noise power ratios, and the addition of constraints. On the other hand, a model for implementation is established based on decoupling the weights into fixed and adaptive portions. By this decoupling, any unconstrained adaptive algorithm can be used to implement the adaptive-weight portion. Thus, only the fixed-weight implementation is studied in detailed in which the addressing issues include implementation structures and quantization effects. In the final part of this dissertation, a generalization of the adaptive linearly-constrained filters which allows the constraints to be adjustable is also discussed.