We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known to be NP-hard, many single-measurement suboptimal algorithms have been formulated that have found utility in many different applications. Here, we consider in depth the extension of two classes of algorithms-Matching Pursuit (MP) and FOCal Underdetermined System Solver (FOCUSS)-to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed. Cost functions appropriate to the multiple measurement problem are developed, and algorithms are derived based on their minimization. A simulation study is conducted on a test-case dictionary to show how the utilization of more than one measurement vector improves the performance of the MP and FOCUSS classes of algorithm, and their performances are compared.

Channels with a sparse impulse response arise in a number of communication applications. Exploiting the sparsity of the channel, we show how an estimate of the channel may be obtained using a matching pursuit (MP) algorithm. This estimate is compared to thresholded variants of the least squares (LS) channel estimate. Among these sparse channel estimates, the MP estimate is computationally much simpler to implement and a shorter training sequence is required to form an accurate channel estimate leading to greater information throughput.

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Recently, the problem of signal representation in terms of basis vectors from a large, "overcomplete", spanning dictionary has been the focus of much research. Achieving a succinct, or "sparse", representation is known as the problem of best basis representation. We consider methods which seek to solve this problem by sequentially building up a basis set for the signal. Three distinct algorithm types have appeared in the literature which we term Basic Matching Pursuit (BMP), Order Recursive Matching Pursuit (ORMP) and Modified Matching Pursuit (MMP). The algorithms are first described and then their computation is closely examined. Modifications are made to each of the procedures which improve their computational efficiency. Each algorithm's complexity is considered in two contexts: one where the dictionary is variable (time dependent), and the other where the dictionary is fixed (time independent). Experimental results are presented which demonstrate that the ORMP method is the best procedure in terms of its ability to give the most compact signal representation, followed by

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