Rapid identification of a sparse impulse response using an adaptive algorithm in the Haar domain

This paper proposes a fast convergence adaptive algorithm for identifying a sparse impulse response that is rich in spectral content. A sparse impulse response is referred here as a discrete time impulse response that has a large number of zero or near zero coefficients. The basic idea for rapid identification is to automatically determine the locations of the nonzero impulse response coefficients for their adaptation and eliminate the unnecessary adaptation of zero coefficients. The proposed method, which is called the Haar-Basis algorithm, employs a transform approach by modeling the sparse impulse response in the Haar domain. The Haar transform has many basis sets and each of them contains basis vectors that span the entire time domain range. This special nature of the Haar transform allows for the selection of one small subset of adaptive filter coefficients whose basis vectors span the entire range of the impulse response. These coefficients are adapted at the beginning and are then used subsequently to identify, from the hierarchical structure of the Haar transform, the rest of the filter coefficients that must be adapted to correctly model the unknown sparse impulse response. The consequence is avoiding adaptation of many zero coefficients, leading to a dramatic improvement in either convergence speed or steady state excess mean-square error (EASE), while requiring no a priori knowledge such as the number of nonzero coefficients in the unknown sparse impulse response. The proposed algorithm has been tested with a variety of unknown sparse systems using both white noise input and colored input whose spectrum closely resembles that of speech. Simulation results show that the new approach provides promising results.