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This paper proposes the use of wavelets for the identification of an unknown sparse system whose impulse response (IR) is rich in spectral content. The superior time localization property of wavelets allows for the identification and subsequent adaptation of only the nonzero IR regions, resulting in lower complexity and faster convergence speed. An added advantage of using wavelets is their ability to partially decorrelate the input, thereby further increasing convergence speed. Good time localization of nonzero IR regions requires high temporal resolution while good decorrelation of the input requires high spectral resolution. To this end we also propose the use of biorthogonal wavelets which fulfil both of these two requirements to provide additional gain in performance. The paper begins with the development of the wavelet-basis (WB) algorithm for sparse system identification. The WB algorithm uses the wavelet decomposition at a single scale to identify the nonzero IR regions and subsequently determines the wavelet coefficients of the unknown sparse system at other scale levels that require adaptation as well. A special implementation of the WB algorithm, the successive-selection wavelet-basis (SSWB), is then introduced to further improve performance when certain a priori knowledge of the sparse IR is available. The superior performance of the proposed methods is corroborated through simulations.