  Finding sparse solutions of underdetermined systems of linear equations is a
fundamental problem in signal processing and statistics which has become a
subject of interest in recent years. In general, these systems have infinitely
many solutions. However, it may be shown that sufficiently sparse solutions may
be identified uniquely. In other words, the corresponding linear transformation
will be invertible if we restrict its domain to sufficiently sparse vectors.
This property may be used, for example, to solve the underdetermined Blind
Source Separation (BSS) problem, or to find sparse representation of a signal
in an `overcomplete' dictionary of primitive elements (i.e., the so-called
atomic decomposition). The main drawback of current methods of finding sparse
solutions is their computational complexity. In this paper, we will show that
by detecting `active' components of the (potential) solution, i.e., those
components having a considerable value, a framework for fast solution of the
problem may be devised. The idea leads to a family of algorithms, called
`Iterative Detection-Estimation (IDE)', which converge to the solution by
successive detection and estimation of its active part. Comparing the
performance of IDE(s) with one of the most successful method to date, which is
based on Linear Programming (LP), an improvement in speed of about two to three
orders of magnitude is observed.
