628 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 3, MARCH 2003 Rapid Identification of a Sparse Impulse Response Using an Adaptive Algorithm in the Haar Domain K. C. Ho, Senior Member, IEEE, and Shannon D. Blunt, Member, IEEE Abstract—This paper proposes a fast convergence adaptive al- gorithm 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 un- necessary adaptation of zero coefficients. The proposed method, which is called the Haar–Basis algorithm, employs a transform ap- proach by modeling the sparse impulse response in the Haar do- main. 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 dra- matic improvement in either convergence speed or steady state ex- cess mean-square error (EMSE), while requiring no a priori knowl- edge such as the number of nonzero coefficients in the unknown sparse impulse response. The proposed algorithm has been tested with a variety of un- known 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. Index Terms—Adaptive algorithm, Haar transform, sparse im- pulse response, system identification. I. INTRODUCTION A DAPTIVE filtering algorithms are finding much wide- spread use nowadays when the exact nature of a system is unknown or is time-varying [1], [2]. Some applications of adap- tive filters, such as echo cancellation and time delay estimation, require the identification of a sparse impulse response. A sparse impulse response is defined here as an impulse response that contains a large number of zero coefficients. When carefully excluding the zero coefficients from adaptation, it is possible to increase the convergence speed, decrease the additional Manuscript received July 28, 2000; revised October 25, 2002. The associate editor coordinating the review of this paper and approving it for publication was Dr. Steven T. Smith. K. C. Ho is with the Department of Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO 65211 USA (e-mail: hod@missouri.edu). S. D. Blunt was with the Department of Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO 65211 USA. He is now with the Radar Division, U.S. Naval Research Laboratory, Washington, DC 20375 USA. Digital Object Identifier 10.1109/TSP.2002.808077 error introduced by coefficient adaptation, i.e., the excess mean-square error (EMSE) [1], and reduce the computational complexity. Several previous techniques in literature exploited the sparse nature of a system in the time domain to improve performance. The scrub taps waiting in a queue (STWQ) algorithm proposed by Kawamura and Hatori [3] adapts a fraction of the filter co- efficients that have significant magnitude. The coefficient with the smallest magnitude is placed in a first-in first-out queue and is not adapted until it reaches the front of the queue. Etter et al. [4]–[8] have proposed a method that adaptively identifies the delay regions where significant nonzero coefficients exist and then adapts a preselected amount of coefficients around each delay region. Both of these methods require a priori knowledge of the number of nonzero coefficients and possibly their loca- tions in the time domain. More recently, Gay [9] has developed a sparse version of the normalized least-mean-square (NLMS) algorithm that achieves faster convergence than NLMS yet re- quires slightly more computational complexity. The algorithm developed in this paper is based on a trans- form domain approach. The idea of using the transform domain for system identification is not new. Hosur and Tewfik [10] de- veloped an algorithm based on the wavelet transform to reduce computation and speed up convergence. This method is devel- oped for a general impulse response that is not sparse. In [11], Doroslovacki and Fan proposed an algorithm that capitalizes on the nature of sparse systems by modeling them with a small number of Haar domain coefficients. This method is limited to systems where the impulse response matches the expected model and does not adaptively determine which coefficients are needed to model an unknown impulse response. The algorithm we propose transforms the system into the Haar domain where the coefficients of the sparse system can be decomposed into many subsets that contain different character- istics of the sparse impulse response. The important innovation of the proposed Haar domain approach is that, given that the sparse impulse response has rich spectral content, the nonzero coefficients in one subset are used to identify the nonzero co- efficients in other subsets. While this is intuitively similar to a filterbank approach, the important difference lies in the fact that the basis vectors within each basis set have finite span and therefore provide some degree of temporal localization. It is this temporal localization property that we exploit in order to deter- mine the temporal location of the nonzero regions in the impulse response. The proposed algorithm, which is called the Haar–Basis al- gorithm, adapts the Haar domain filter coefficients in a single subset, applies statistical detection of nonzero magnitude of the 1053-587X/03$17.00 © 2003 IEEE