Kalman and Adaptive Least Squared Error Filters
Adaptive least squared error filters are used with nonstationary signals and environments, or in applications where a low processing delay is required, such as multi-channel noise reduction, radar signal processing, channel equalisation, echo cancellation, and speech coding. We begin this chapter with a study of state-space Kalman filters. In Kalman theory a state equation models the dynamics of the signal generation process, and an observation equation models the channel distortions and additive noise. We then consider recursive least squared (RLS) error adaptive filters. The RLS filter is a sample adaptive formulation of the Wiener filter, and for stationary signals should converge to the same solution as the Wiener filter. In least squared error filtering, an alternative to using a Wiener-type closed form solution, is an iterative search for the optimal filter coefficients. The steepest descent search is a gradient-based method of searching the least squared error performance curve for the minimum error filter coefficients. We study the steepest descent method, and then consider the LMS method which is a computationally inexpensive gradient search algorithm.
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