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Proposes two novel algorithms employing the Haar wavelet for adaptive sparse system identification. A previous study exploited the dyadic, hierarchical structure of the Haar transform to identify the non-zero impulse response coefficients for adaptation. This paper extends this previous work to some common cases in which further performance gains can be realized. We examine the situation where the sparse impulse response has impulse-like non-zero regions that are highly dispersed over the temporal domain. This sparse impulse response allows us to develop an algorithm that seeks to accurately model the impulse response at successively finer levels of temporal resolution and in doing so further reduces the number of adapting coefficients to speed up convergence. The other situation we examine is when the non-zero impulse response coefficients are highly localized in a single temporal region. For this case we can modify the method of wavelet decomposition to focus on better spectral resolution, as fine temporal resolution is unnecessary. This leads to improved decorrelation ability and therefore faster convergence.