Fast coupled adaptation for sparse impulse responses using a partial haar transform

This work presents a novel scheme for identifying the impulse response of a sparse channel. The scheme consists of two adaptive filters operating sequentially. The first adaptive filter adapts using a partial Haar transform of the input and yields an estimate of the location of the peak of the sparse impulse response. The second adaptive filter is then centered about this estimate. Both filters are short in comparison to the delay uncertainty of the unknown channel. The principle advantage of this scheme is that two short adaptive filters can be used instead of one long adaptive filter, resulting in faster overall convergence and reduced computational complexity and storage. The scheme is analyzed in detail for a least mean squares (LMS) LMS-LMS type of structure, although it can be implemented using any combination of adaptive algorithms. Monte Carlo simulations are shown to be in good agreement with the theoretical model for the behavior of the peak estimating filter as well as for the mean square error (MSE) behavior of the second filter.