Least squares linear phase filters and their applications

Abstract

The main objective of this dissertation work is to develop and analyze the efficient algorithms for adaptive FIR digital filters with linear phase. Linear phase filters with constant phase delay and constant group delay are both examined. We focus on the LS optimization for every point in time adn also reduction of the computational complexity to the same order as the suboptimum gradient-type algorithms. First, we investigate unwindowed fast RLS algorithms with linear phase constraint on the filter impluse response. It is shown that the reduction of the computational complexity of the fast RLS algorithm to the same order as the number of filter coefficients is possible due to the fact that the algorithm takes advantage of the near-to Toepltz and the centrosymmetry of the normal equation matrix. Second, we present a recursive modified Gram-Schmidt (RMGS) algorithm for LS linear phase filters to allow for the tracking of time-varying parameters. Both exponentially windowed and sliding window convariance cases are examined, including the prewindowed case as a special case of the exponentially windowed one. We describe quantitatively the performance characteristics of the RMGS filters for the linear phase system identification problem when the unknown system parameters vary with time. Also, we implement the RMGS algorithm for the linear phase filters using a triangular array with a highly modular structure. Third, we design a frequency-domain LS adaptive filter with the linear phase constraint directly on the phase of the filter frequency response. The pseudo-orthogonality property of the discrete Fourier transform that affects the convergence behaviors of the adaptive filter is discussed. Finally, we apply the adaptive RLS-type updating of the linear phase filter to adaptive notch filtering. The convergent adaptive notch filter is modelled as an IIR linear phase filter. We show the performance behavior of adaptive notch filters with the fast RLS linear phase a...