Compressed Sensing of Analog Signals

A traditional assumption underlying most data converters i s that the signal should be sampled at a rate which exceeds twice the highest frequency. This statement is base d on a worst-case scenario in which the signal occupies the entire available bandwidth. In practice, many signals p o ses a sparse structure so that a large part of the bandwidth is not exploited. In this paper, we consider a fram ework for utilizing this sparsity in order to sample such analog signals at a low rate. More specifically, we consi der continuous-time signals that lie in a shift-invariant (SI) space generated by m kernels, so that any signal in the space can be expressed as an infinite linear combination of the shifted kernels. If the period of the underlying SI spa ce is equal toT , then such signals can be perfectly reconstructed from samples at a rate of m/T . Here we treat the case in which only k out of them generators are active, meaning that the signal actually lies in a lower dime nsional space spanned by k generators. However, we do not know whichk are chosen. By relying on results developed in the context of compressed sensing (CS) of finite-length vectors, we develop a general framework for sa mpling such signals at a rate much lower than m/T . The distinguishing feature of our results is that in contras t to the problems treated in the context of CS, here we consider sampling of analog-signals for which no underly ing finite-dimensional model exists. Our approach combines ideas from analog sampling in a subspace with a rece ntly developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Usin g these two components we formulate our problem within the framework of finite CS and then rely on efficient and stable algorithms developed in that context. Our results allow to extend much of the recent literature on CS to the truly analog domain. Department of Electrical Engineering, Technion—Israel In stitute of Technology, Haifa 32000, Israel. Phone: +972-48293256, fax: +9724-8295757, E-mail: yonina@ee.technion.ac.il. This work w as supported in part by the Israel Science Foundation under G rant no. 1081/07 and by the European Commission in the framework of the FP7 Net work of Excellence in Wireless COMmunications NEWCOM++ (co ntract no. 216715). 2