A tutorial on recovery conditions for compressive system identification of sparse channels

In this tutorial, we review some of the recent results concerning Compressive System Identification (CSI) (identification from few measurements) of sparse channels (and in general, Finite Impulse Response (FIR) systems) when it is known a priori that the impulse response of the system under study is sparse (high-dimensional but with few non-zero entries) in an appropriate basis. For the systems under study in this tutorial, the system identification problem boils down to an inverse problem of the form Ax = b, where the vector x ∈ ℝ^N is high-dimensional but with K ≪ N nonzero entries and the matrix A ∈ ℝ^M×N is underdetermined (i.e., M <; N). Over the past few years, several algorithms with corresponding recovery conditions have been proposed to perform such a recovery. These conditions provide the number of measurements sufficient for correct recovery. In this note, we review alternate approaches to derive such recovery conditions concerning CSI of FIR systems whose impulse response is known to be sparse.