The Shannon/Nyquist sampling theorem tells us that in order to not lose information when uniformly sampling a signal we must sample at least two times faster than its bandwidth. In many applications, including digital image and video cameras, the Nyquist rate can be so high that we end up with too many samples and must compress in order to store or transmit them. In other applications, including imaging systems (medical scanners, radars) and high-speed analog-to-digital converters, increasing the sampling rate or density beyond the current state-of-the-art is very expensive. In this lecture, we will learn about a new technique that tackles these issues using compressive sensing [1, 2]. We will replace the conventional sampling and reconstruction operations with a more general linear measurement scheme coupled with an optimization in order to acquire certain kinds of signals at a rate significantly below Nyquist. 2

Non-parametric Bayesian techniques are considered for learning dictionaries for sparse image representations, with applications in denoising, inpainting and compressive sensing (CS). The beta process is employed as a prior for learning the dictionary, and this non-parametric method naturally infers an appropriate dictionary size. The Dirichlet process and a probit stick-breaking process are also considered to exploit structure within an image. The proposed method can learn a sparse dictionary in situ; training images may be exploited if available, but they are not required. Further, the noise variance need not be known, and can be nonstationary. Another virtue of the proposed method is that sequential inference can be readily employed, thereby allowing scaling to large images. Several example results are presented, using both Gibbs and variational Bayesian inference, with comparisons to other state-of-the-art approaches.

We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block k-sparse signals in no more than k steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed ℓ2/ℓ1-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.

Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analogto-digital converters and digital imagers in certain applications. A key hallmark of CS is that it enables sub-Nyquist sampling for signals, images, and other data. In this paper, we explore and exploit another heretofore relatively unexplored hallmark, the fact that certain CS measurement systems are democractic, which means that each measurement carries roughly the same amount of information about the signal being acquired. Using the democracy property, we re-think how to quantize the compressive measurements in practical CS systems. If we were to apply the conventional wisdom gained from conventional Shannon-Nyquist uniform sampling, then we would scale down the analog signal amplitude (and therefore increase the quantization error) to avoid the gross saturation errors that occur when the signal amplitude exceeds the quantizer’s dynamic range. In stark contrast, we demonstrate that a CS system achieves the best performance when it operates at a significantly nonzero saturation rate. We develop two methods to recover signals from saturated CS measurements. The first directly exploits the democracy property by simply discarding the saturated measurements. The second integrates saturated measurements as constraints into standard linear programming and greedy recovery techniques. Finally, we develop a simple automatic gain control system that uses the saturation rate to optimize the input gain.

Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In this paper, we develop a unified approach to time delay estimation from low rate samples. This problem can be formulated in the broader context of sampling over an infinite union of subspaces. Although sampling over unions of subspaces has been receiving growing interest, previous results either focus on unions of finite-dimensional subspaces, or finite unions. The framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses, and allows for a variety of different sampling methods. The sampling rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. This result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces. By properly manipulating the low-rate samples, we show that the time delays can be recovered using the well-known ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation, we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate.

We consider the reconstruction of sparse signals in the multiple measurement vector (MMV) model, in which the signal, represented as a matrix, consists of a set of jointly sparse vectors. MMV is an extension of the single measurement vector (SMV) model employed in standard compressive sensing (CS). Recent theoretical studies focus on the convex relaxation of the MMV problem based on the (2, 1)-norm minimization, which is an extension of the well-known 1-norm minimization employed in SMV. However, the resulting convex optimization problem in MMV is significantly much more difficult to solve than the one in SMV. Existing algorithms reformulate it as a second-order cone programming (SOCP) or semidefinite programming (SDP) problem, which is computationally expensive to solve for problems of moderate size. In this paper, we propose a new (dual) reformulation of the convex optimization problem in MMV and develop an efficient algorithm based on the prox-method. Interestingly, our theoretical analysis reveals the close connection between the proposed reformulation and multiple kernel learning. Our simulation studies demonstrate the scalability of the proposed algorithm.

We consider minimal-rate sampling schemes for infinite streams of delayed and weighted versions of a known pulse shape. The minimal sampling rate for these parametric signals is referred to as the rate of innovation and is equal to the number of degrees of freedom per unit time. Although sampling of infinite pulse streams was treated in previous works, either the rate of innovation was not achieved, or the pulse shape was limited to Diracs. In this paper we propose a multichannel architecture for sampling pulse streams with arbitrary shape, operating at the rate of innovation. Our approach is based on modulating the input signal with a set of properly chosen waveforms, followed by a bank of integrators. This architecture is motivated by recent work on sub-Nyquist sampling of multiband signals. We show that the pulse stream can be recovered from the proposed minimal-rate samples using standard tools taken from spectral estimation in a stable way even at high rates of innovation. In addition, we address practical implementation issues, such as reduction of hardware complexity and immunity to failure in the sampling channels. The resulting scheme is flexible and exhibits better noise robustness than previous approaches.

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Abstract—Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Instead of taking N periodic samples, we measure M ≪ N inner products with random vectors and then recover the signal via a sparsity-seeking optimization or greedy algorithm. A new framework for CS based on unions of subspaces can improve signal recovery by including dependencies between values and locations of the signal’s significant coefficients. In this paper, we extend this framework to the acquisition of signal ensembles under a common sparse supports model. The new framework provides recovery algorithms with theoretical performance guarantees. Additionally, the framework scales naturally to large sensor networks: the number of measurements needed for each signal does not increase as the network becomes larger. Furthermore, the complexity of the recovery algorithm is only linear in the size of the network. We provide experimental results using synthetic and real-world signals that confirm these benefits. I.

Abstract—Multipath interference is an ubiquitous phenomenon in modern communication systems. The conventional way to compensate for this effect is to equalize the channel by estimating its impulse response by transmitting a set of training symbols. The primary drawback to this type of approach is that it can be unreliable if the channel is changing rapidly. In this paper, we show that randomly encoding the signal can protect it against channel uncertainty when the channel is sparse. Before transmission, the signal is mapped into a slightly longer codeword using a random matrix. From the received signal, we are able to simultaneously estimate the channel and recover the transmitted signal. We discuss two schemes for the recovery. Both of them exploit the sparsity of the underlying channel. We show that if the channel impulse response is sufficiently sparse, the transmitted signal can be recovered reliably. I.