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

Abstract

Abstract-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 nonzero 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.

System identification in the presence of outliers and random noises: A compressed sensing approach

“…Though applicable to control applications where FIR models are used, it is not as complete as IIR models from a control point of view. However, identification in the presence of outliers and random noises is not limited to control applications and in fact routinely applied to many other areas, e.g., signal processing and communication [15,22]. Note in those areas, the systems are dominantly and overwhelmingly FIR models and thus the results derived in this paper can be readily applied.…”

“…With the development of compressed sensing theory in recent years, the role of ℓ 1 regularization has been studied in system identification [14,15,[20][21][22]. In these works, system parameters are often assumed to be sparse, and then ℓ 1 regularization can be used to reduce the number of needed samples for system identification.…”

Compressive origin-destination estimation

“…The gap between existing recovery guarantees and actual recovery performance is being narrowed in different applications. In particular, when sparse dynamical systems are involved this gap is usually large and has been investigated in system identification [26]- [30], observability and control of linear systems [31], [32], and identification of interconnected networks [33], [34].…”

An abrupt change detection heuristic with applications to cyber data attacks on power systems

“…Lemma 1: ( [15], [16,Lemma 2]) Let e be a vector in R M whose entries are independent Gaussian random variables with zero mean and ν 2 variance. Then for every ≥ 0,…”