The Cognitive Compressive Sensing Problem

In the Cognitive Compressive Sensing (CCS) problem, a Cognitive Receiver (CR) seeks to optimize the reward obtained by sensing an underlying $N$ dimensional random vector, by collecting at most $K$ arbitrary projections of it. The $N$ components of the latent vector represent sub-channels states, that change dynamically from "busy" to "idle" and vice versa, as a Markov chain that is biased towards producing sparse vectors. To identify the optimal strategy we formulate the Multi-Armed Bandit Compressive Sensing (MAB-CS) problem, generalizing the popular Cognitive Spectrum Sensing model, in which the CR can sense $K$ out of the $N$ sub-channels, as well as the typical static setting of Compressive Sensing, in which the CR observes $K$ linear combinations of the $N$ dimensional sparse vector. The CR opportunistic choice of the sensing matrix should balance the desire of revealing the state of as many dimensions of the latent vector as possible, while not exceeding the limits beyond which the vector support is no longer uniquely identifiable.Comment: 8 pages, 2 figure

Xampling in Ultrasound Imaging

Recent developments of new medical treatment techniques put challenging demands on ultrasound imaging systems in terms of both image quality and raw data size. Traditional sampling methods result in very large amounts of data, thus, increasing demands on processing hardware and limiting the exibility in the post-processing stages. In this paper, we apply Compressed Sensing (CS) techniques to analog ultrasound signals, following the recently developed Xampling framework. The result is a system with significantly reduced sampling rates which, in turn, means significantly reduced data size while maintaining the quality of the resulting images.Comment: 17 pages, 9 Figures. Introduced in SPIE Medical Imaging Conference, Orlando Florida, 201

Statistical reconstruction of pulse shapes from pulse streams

A short sample sequence of a finite-length pulse signal allows for its reconstruction only if the signal has a sparse representation in some basis. The recurrence of the pulse allows for a statistical approach to its reconstruction. We propose a novel method for this task. It is based on the distribution of short sample sequences treated as points which lie along a curve in a low-dimensional Euclidean space. We prove that the probability distribution of the points along this curve determines the underlying pulse signal uniquely. Based on this discovery, we propose an algorithm for pulse estimation from a finite number of short sequences of pulse-stream samples

Multichannel Sampling of Pulse Streams at the Rate of Innovation

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

Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie

Xampling: Signal Acquisition and Processing in Union of Subspaces

We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. Analog compression that narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm then detects the input subspace prior to conventional signal processing. A representative union model of spectrally-sparse signals serves as a test-case to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy and software complexities. We conduct a comprehensive comparison between two sub-Nyquist acquisition strategies for spectrally-sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address lowrate signal processing and develop an algorithm for that purpose that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.Comment: 16 pages, 9 figures, submitted to IEEE for possible publicatio

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Sampling streams of pulses with unknown shapes

This paper extends the class of continuous-time signals that can be perfectly reconstructed by developing a theory for the sampling and exact reconstruction of streams of short pulses with unknown shapes. The single pulse is modelled as the delayed version of a wavelet-sparse signal, which is normally not band limited. As the delay can be an arbitrary real number, it is hard to develop an exact sampling result for this type of signals. We achieve the exact reconstruction of the pulses by using only the knowledge of the Fourier transform of the signal at specific frequencies. We further introduce a multi-channel acquisition system which uses a new family of compact-support sampling kernels for extracting the Fourier information from the samples. The shape of the kernel is independent of the wavelet basis in which the pulse is sparse and hence the same acquisition system can be used with pulses which are sparse on different wavelet bases. By exploiting the fact that pulses have short duration and that the sampling kernels have compact support, we finally propose a local and sequential algorithm to reconstruct streaming pulses from the samples