Abstract—Finding sparse solutions of underdetermined sys-tems of linear equations is a fundamental problem in signalprocessing and statistics which has become a subject of interestin recent years. In general, these systems have infinitely ma nysolutions. However, it may be shown that sufficiently sparsesolutions may be identified uniquely. In other words, the cor -responding linear transformation will be invertible if we restrictits domain to sufficiently sparse vectors. This property may beused, for example, to solve the underdetermined Blind SourceSeparation (BSS) problem, or to find sparse representation o fa signal in an ‘overcomplete’ dictionary of primitive elements(i.e., the so-called atomic decomposition). The main drawback ofcurrent methods of finding sparse solutions is their computa tionalcomplexity. In this paper, we will show that by detecting ‘active’components of the (potential) solution, i.e., those componentshaving a considerable value, a framework for fast solution ofthe problem may be devised. The idea leads to a family ofalgorithms, called ‘Iterative Detection-Estimation (IDE)’, whichconverge to the solution by successive detection and estimation ofits active part. Comparing the performance of IDE(s) with oneof the most successful method to date, which is based on LinearProgramming (LP), an improvement in speed of about two tothree orders of magnitude is observed.