Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals. A great many applications feature smooth or modulated signals that can be modeled as a linear combination of a small number of sinusoids; such signals are sparse in the frequency domain. In practical applications, the standard frequency domain signal representation is the discrete Fourier transform (DFT). Unfortunately, the DFT coefficients of a frequency-sparse signal are themselves sparse only in the contrived case where the sinusoid frequencies are integer multiples of the DFT’s fundamental frequency. As a result, practical DFT-based CS acquisition and recovery of smooth signals does not perform nearly as well as one might expect. In this paper, we develop a new spectral compressive sensing (SCS) theory for general frequency-sparse signals. The key ingredients are an over-sampled DFT frame, a signal model that inhibits closely spaced sinusoids, and classical sinusoid parameter estimation algorithms from the field of spectrum estimation. Using peridogram and eigen-analysis based spectrum estimates (e.g., MUSIC), our new SCS algorithms significantly outperform the current state-of-the-art CS algorithms while providing provable bounds on the number of measurements required for stable recovery.

Abstract—Wideband spectrum sensing (WSS) encompasses a collection of techniques intended to estimate or to decide over the occupancy parameters of a wide frequency band. However, broad bands require expensive acquisition systems, thus moti-vating the use of compressive schemes. In this context, previous works in compressive WSS have already realized that great compression rates can be achieved if only second-order statistics are of interest in spectrum sensing. In this paper, we go a step further by exploiting spectral prior information that is typically available in practice in order to reduce the sampling rate even more. The signal model assumes that the acquisition is done by means of an analog-to-information converter (A2I). The input signal is the linear combination of a number of signals whose second-order statistics are known and the goal is to estimate/de-cide over the coefficients of this combination. The problem is thus a particular instance of the well-known structured covariance estimation problem. Unfortunately, the algorithms used in this area are extremely complex for inexpensive spectrum sensors so that alternative techniques need to be devised. Exploiting the fact that the basis matrices are Toeplitz, we use the asymptotic theory of circulant matrices to propose a dimensionality reduction technique that simplifies existing structured covariance estimation algorithms, achieving a similar performance at a much lower computational cost. Index Terms—Analog-to-information converters, compressed sensing, covariance matching, wideband spectrum sensing.

Abstract—We investigate the problem of a monostatic pulse-Doppler radar transceiver trying to detect targets sparsely populated in the radar’s unambiguous time-frequency region. Several past works employ compressed sensing (CS) algorithms to this type of problem but either do not address sample rate reduction, impose constraints on the radar transmitter, propose CS recovery methods with prohibitive dictionary size, or per-form poorly in noisy conditions. Here, we describe a sub-Nyquist sampling and recovery approach called Doppler focusing, which addresses all of these problems: it performs low rate sampling and digital processing, imposes no restrictions on the transmitter, and uses a CS dictionary with size, which does not increase with increasing number of pulses. Furthermore, in the presence of noise, Doppler focusing enjoys a signal-to-noise ratio (SNR) improvement, which scales linearly with, obtaining good detec-tion performance even at SNR as low as 25 dB. The recovery is based on the Xampling framework, which allows reduction of the number of samples needed to accurately represent the signal, directly in the analog-to-digital conversion process. After sampling, the entire digital recovery process is performed on the low rate samples without having to return to the Nyquist rate. Finally, our approach can be implemented in hardware using a previously suggested Xampling radar prototype. Index Terms—Compressed sensing, rate of innovation, radar, sparse recovery, sub-Nyquist sampling, delay-Doppler estimation. I.

Abstract—We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpass-filtered samples. We show that maximum-likelihood estimation of the unknown parameters amounts to a difficult, even believed NPhard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed. Index Terms—Recovery of Dirac pulses, spike train, finite rate of innovation, super-resolution, spectral estimation, maximum

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 flexibility in the postprocessing 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.

This paper considers the capacity of sub-sampled analog channels when the sampler is designed to operate independent of instantaneous channel realizations. A compound multiband Gaussian channel with unknown subband occupancy is considered, with perfect channel state information available at both the receiver and the transmitter. We restrict our attention to a general class of periodic sub-Nyquist samplers, which subsumes as special cases sampling with periodic modulation and filter banks. We evaluate the loss due to channel-independent (universal) sub-Nyquist design through a sampled capacity loss metric, that is, the gap between the undersampled channel capacity and the Nyquist-rate capacity. We investigate sampling methods that minimize the worst-case (minimax) capacity loss over all channel states. A fundamental lower bound on the minimax capacity loss is first developed, which depends only on the band sparsity ratio and the undersampling factor, modulo a residual term that vanishes at high signal-to-noise ratio. We then quantify the capacity loss under Landau-rate sampling with periodic modulation and low-pass filters, when the Fourier coefficients of the modulation waveforms are randomly generated and independent (resp. i.i.d. Gaussian-distributed), termed independent random sampling (resp. Gaussian sampling). Our results indicate that with exponentially high probability, independent random sampling and Gaussian sampling achieve minimax sampled capacity loss in the Landau-rate and super-Landau-

In light of the ever-increasing demand for new spectral bands and the underutilization of those already allocated, the concept of Cognitive Radio (CR) has emerged. Opportunistic users could exploit temporarily vacant bands after detecting the absence of activity of their owners. One of the crucial tasks in the CR cycle is therefore spectrum sensing and detection which has to be precise and efficient. Yet, CRs typically deal with wideband signals whose Nyquist rates are very high. In this paper, we propose to reconstruct the power spectrum of such signals from sub-Nyquist samples, rather than the signal itself as done in previous work, in order to perform detection. We consider both sparse and non sparse signals as well as blind and non blind detection in the sparse case. For each one of those scenarios, we derive the minimal sampling rate allowing perfect reconstruction of the signal’s power spectrum in a noise-free environment and provide power spectrum recovery techniques that achieve those rates. The analysis is per-formed for two different signal models considered in the literature, which we refer to as the analog and digital models, and shows that both lead to similar results. Simulations demonstrate power spectrum recovery at the minimal rate in noise-free settings and the impact of several parameters on the detector performance, including signal-to-noise ratio, sensing time and sampling rate.

Abstract—This paper investigates the effect of sub-Nyquist sampling upon the capacity of an analog channel. The channel is assumed to be a linear time-invariant Gaussian channel, where perfect channel knowledge is available at both the transmitter and the receiver. We consider a general class of right-invertible time-preserving sampling methods which include irregular nonuniform sampling, and characterize in closed form the channel capacity achievable by this class of sampling methods, under a sampling rate and power constraint. Our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal-to-noise ratio among all spectral sets of measure equal to the sampling rate. This can be attained through filterbank sampling with uniform sampling at each branch with possibly different rates, or through a single branch of modulation and filtering followed by uniform sampling. These results reveal that for a large class of channels, employing irregular nonuniform sampling sets, while typically complicated to realize, does not provide capacity gain over uniform sampling sets with appropriate preprocessing. Our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain, which is in contrast to the benefits obtained from random mixing in spectrum-blind compressive sampling schemes. Index Terms—nonuniform sampling, irregular sampling, sam-pled analog channels, sub-Nyquist sampling, channel capacity, Beurling density, time-preserving sampling systems I.

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 flexibility 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.

This paper considers the capacity of sub-sampled analog channels when the sampler is designed to operate independent of instantaneous channel realizations. A compound multiband Gaussian channel with unknown subband occupancy is considered, with perfect channel state information available at both the receiver and the transmitter. We restrict our attention to a general class of periodic sub-Nyquist samplers, which subsumes as special cases sampling with periodic modulation and filter banks. We evaluate the loss due to channel-independent (universal) sub-Nyquist design through a sampled capacity loss metric, that is, the gap between the undersampled channel capacity and the Nyquist-rate capacity. We investigate sampling methods that minimize the worst-case (minimax) capacity loss over all channel states. A fundamental lower bound on the minimax capacity loss is first developed, which depends only on the band sparsity ratio and the undersampling factor, modulo a residual term that vanishes at high signal-to-noise ratio. We then quantify the capacity loss under Landau-rate sampling with periodic modulation and low-pass filters, when the Fourier coefficients of the modulation waveforms are randomly generated and independent (resp. i.i.d. Gaussian-distributed), termed independent random sampling (resp. Gaussian sampling). Our results indicate that with exponentially high probability, independent random sampling and Gaussian sampling achieve minimax sampled capacity loss in the Landau-rate and super-Landau-