ABSTRACT

Sparse signal reconstruction from undersampled measurements has been well studied for some time. The goal is to nd the sparsest signal that satises the constraint imposed by the measurements, that is, min‖β‖0 st. y = Aβ (here, β is the unknown signal, y is the undersampled measurements, and A is a fat matrix and ‖⋅‖0 counts the number of nonzero elements in x). A direct solution to this has exponential complexity and hence is not practical. Two classes of practical approaches have been proposed in literature-convex relaxation methods, which replace the ℓ0 norm by the ℓ1 norm, for example [6-8,10], and greedy methods, for example [21]. Recent work on compressed sensing (CS) gives conditions for its exact reconstruction [6-8] and bounds the error when this is not possible [6,10]. The terms compressed sensing and sparse reconstruction are used interchangeably these days, and henceforth, we will mostly use the term compressed sensing. In this work, we develop recursive algorithms for reconstructing a time sequence of sparse signals using

2.1 Introduction ....................................................................................................27 2.1.1 Our Contributions ...............................................................................30 2.1.2 Other Related Work ............................................................................ 32

2.2 Problem Denition .......................................................................................... 32 2.2.1 Notation .............................................................................................. 32 2.2.2 Problem Formulation .......................................................................... 35

2.3 Modied-CS-Residual ....................................................................................36 2.4 Experimental Results ......................................................................................40