Properties of the FIR-ALE Structure

Abstract

In this paper, results from the literature on sub-space methods are used to establish some properties of the FIR-ALE structure when the input consists of p real sinusoids in white noise. The low rank property of the covariance matrix of the sinusoids is used to show that the Wiener-Hopf solution is only dependent on the 2p dominant eigenvalues of the covariance matrix. This in turn results in the convergence of the LMS algorithm for the zero initial condition to depend on the dominant 2p eigenvalues, implying that the convergence is improved compared to the general case. Also it is shown that when a filter of length L much larger than 2p is used, the parameter sensitivity is fairly low. The low parameter sensitivity is traced to the minimum norm characterization of the solution in the noise free case. The minimum-norm criteria also turns out to be a useful in selecting an optimum value for the decorrelation delay Δ.