We consider compressed sampling over finite fields and investigate the number of compressed measurements needed for successful L0 recovery. Our results are obtained while the sparseness of the sensing matrices as well as the size of the finite fields are varied. One of interesting conclusions includes that unless the signal is “ultra” sparse, the sensing matrices do not have to be dense. Keywor...

This survey provides a brief introduction to compressed sensing as well as several major algorithms to solve it and its various applications to communications systems. We firstly review linear simultaneous equations as ill-posed inverse problems, since the idea of compressed sensing could be best understood in the context of the linear equations. Then, we consider the problem of compressed sens...

The paper presents a study of Compressed Sensing application in a passive radar, where the range resolution is limited by the bandwidth of signal used. The application of Compressed Sensing allows to obtain superresolution in a presence of a point target, which is useful e.g. when exploiting multipath information for estimating the target elevation. However, in such setup, Compressed Sensing al...

Over the past decade, compressed sensing has delivered significant advances in the theory and application of measuring and compressing data. Consider capturing a 10 mega pixel image with a digital camera. Emailing an image of this size requires an unnecessary amount of storage space and bandwidth. Instead, users employ a standard digital compression scheme, such as JPEG, to represent the image ...

This paper considers the problem of recovering an unknown sparse p× p matrix X from an m ×m matrix Y = AXBT , where A and B are known m × p matrices with m p. The main result shows that there exist constructions of the “sketching” matrices A and B so that even if X has O(p) non-zeros, it can be recovered exactly and efficiently using a convex program as long as these non-zeros are not concentra...

Recall the setup in compressive sensing. There is an unknown signal z ∈ R, and we can only glean information about z through linear measurements. We choose m linear measurements a1, . . . , am ∈ R. “Nature” then chooses a signal z, and we receive the results b1 = 〈a1, z〉, . . . , bm = 〈am, z〉 of our measurements, when applied to z. The goal is then to recover z from b. Last lecture culminated i...

In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted `1 minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted `1 minimization recovery algorithm and analyze its perfor...

In this chapter, we introduce a unjfied rugh-dimensional geometric framework for analyzing the phase transition phenomenon of (1 minimization in compressive sensing. This framework connects srudying the phase transitions of ( 1 minimization with computing the Grassmann angles in high-dimensional convex geometry. We demonstrate the broad applications of this Grassmann angle framework by giving s...