Wideband analog signals push contemporary analog-to-digital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system’s performance that supports the empirical observations.

We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. Prior recovery methods for this sampling strategy either require knowledge of band locations or impose strict limitations on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finitedimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed ℓ2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.

A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to over-sampling with similar reconstruction algorithms. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and Error Locator Polynomials in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method. Such narrowband spectral estimation is then related to multi-source location and direction of arrival estimation in array processing. The notions of sparse array beamforming and sparse sensor networks are also introduced. Sparsity in unobservable source signals is also shown to facilitate source separation in Sparse Component Analysis (SCA); the algorithms developed in this area are also widely used in compressed sensing. Finally, the nature of the multipath channel estimation problem is shown to have a sparse formulation; algorithms similar to sampling and coding are used to estimate typical multicarrier communication channels.

We consider the recovery of jointly sparse multichannel signals from incomplete measurements using convex relaxation methods. Worst case analysis is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Therefore, we analyze an average case by imposing a probability model on the measured signals. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. 1.

Digital applications have developed rapidly over the last few decades. Since many sources of information are of analog or continuous-time nature, discrete-time signal processing (DSP) inherently relies on sampling a continuous-time signal to obtain a discrete-time representation. Consequently, sampling theories lie at the heart of signal processing devices and communication systems. Examples include sampling rate conversion for software radio [1] and between audio formats [2], biomedical imaging [3], lens distortion correction and the formation of image mosaics [4], and super-resolution of image sequences [5]. To accommodate high operating rates while retaining low

We consider the recovery of jointly sparse multichannel signals from incomplete measurements using convex relaxation methods. Worst case analysis is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Therefore, we analyze an average case by imposing a probability model on the measured signals. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. 1.

Abstract. Relationships that exists between the classical, Shannon-type, and geometric-based approach to sampling are investigated. Some aspects of coding and communication through a Gaussian channel prob-lem are considered. In particular, a constructive method to determine the quantizing dimension in Zador’s theorem is provided. A geomet-ric version of Shannon’s Second Theorem is introduced. Applications to Pulse Code Modulation and Vector Quantization of Images are pro-vided. In addition, we sketch the geometerization of wavelets for Image Processing purposes. We also discuss the implications of the Uncertainty Principle on sampling and reconstruction of images. An extension of our sampling scheme to a certain class of infinite dimensional manifolds is considered. 1. General background 1.1. Introduction. We consider a geometric approach to Shannon’s sam-pling theorem, i.e. one based on sampling the graph of the signal, considered

insight into how sparse a signal x can be represented in an over-complete dictionary consisting of ~ and '11. It also sheds light on

I thank my adviser Dr. José C. Príncipe, for giving me the opportunity and funds to pursue my PhD. I specially thank my committee members, who have been a great inspiration. Dr. Murali Rao, helped me through my first steps in sampling theory. Dr. Entezari formally introduced me to the problems in interpolation, sampling and reconstruction. He has been instrumental in my learning of B-splines and multivariate splines. Dr. Banerjee who was always optimistic, provided great insight into the connections between this work and the results in computational neuroscience. I would always leave their offices motivated and ready to tackle the next problem. Finally I thank Dr. Harris, who guided a great part of this work and would keep me grounded to the application. I would also like to thank Dr. Hans G. Feichtinger, who gave me the opportunity to spend a summer with his group in Vienna during the summer of 2009. This was a great opportunity to meet some of the leading researchers in the sampling field. Furthermore, I had the chance to work with his group specifically Jose Luis Romero, Gino Velasco and Saptarshi Das to whom part of this work is in debt to. During this visit I also met one of