An overview of robust compressive sensing of sparse signals in impulsive noise

While compressive sensing (CS) has traditionally relied on L2 as an error norm, a broad spectrum of applications has emerged where robust estimators are required. Among those, applications where the sampling process is performed in the presence of impulsive noise, or where the sampling of the high-dimensional sparse signals requires the preservation of a distance different than L2. This article overviews robust sampling and nonlinear reconstruction strategies for sparse signals based on the Cauchy distribution and the Lorentzian norm for the data fidelity. The derived methods outperform existing compressed sensing techniques in impulsive environ- ments, thus offering a robust framework for CS

Compressed sensing based cyclic feature spectrum sensing for cognitive radios

Spectrum sensing is currently one of the most challenging design problems in cognitive radio. A robust spectrum sensing technique is important in allowing implementation of a practical dynamic spectrum access in noisy and interference uncertain environments. In addition, it is desired to minimize the sensing time, while meeting the stringent cognitive radio application requirements. To cope with this challenge, cyclic spectrum sensing techniques have been proposed. However, such techniques require very high sampling rates in the wideband regime and thus are costly in hardware implementation and power consumption. In this thesis the concept of compressed sensing is applied to circumvent this problem by utilizing the sparsity of the two-dimensional cyclic spectrum. Compressive sampling is used to reduce the sampling rate and a recovery method is developed for re- constructing the sparse cyclic spectrum from the compressed samples. The reconstruction solution used, exploits the sparsity structure in the two-dimensional cyclic spectrum do-main which is different from conventional compressed sensing techniques for vector-form sparse signals. The entire wideband cyclic spectrum is reconstructed from sub-Nyquist-rate samples for simultaneous detection of multiple signal sources. After the cyclic spectrum recovery two methods are proposed to make spectral occupancy decisions from the recovered cyclic spectrum: a band-by-band multi-cycle detector which works for all modulation schemes, and a fast and simple thresholding method that works for Binary Phase Shift Keying (BPSK) signals only. In addition a method for recovering the power spectrum of stationary signals is developed as a special case. Simulation results demonstrate that the proposed spectrum sensing algorithms can significantly reduce sampling rate without sacrifcing performance. The robustness of the algorithms to the noise uncertainty of the wireless channel is also shown

Compressed sensing of monostatic and multistatic SAR

In this paper we study the impact of sparse aperture data collection of a SAR sensor on reconstruction quality of a scene of interest. Different mono and multi-static SAR measurement configurations produce different Fourier sampling patterns. These patterns reflect different spectral and spatial diversity trade-offs that must be made during task planning. Compressed sensing theory argues that the mutual coherence of the measurement probes is related to the reconstruction performance of sparse domains. With this motivation we compare the mutual coherence and corresponding reconstruction behavior of various mono-static and ultra-narrow band multi-static configurations, which trade-off frequency for geometric diversity. We investigate if such simple metrics are related to SAR reconstruction quality in an obvious way

Compressed sensing of monostatic and multistatic SAR

In this letter, we study the impact of compressed data collections from a synthetic aperture radar (SAR) sensor on the reconstruction quality of a scene of interest. Different monostatic and multistatic SAR measurement configurations produce different Fourier sampling patterns. These patterns reflect different spectral and spatial diversity tradeoffs that must be made during task planning. Compressed sensing theory argues that the mutual coherence of the measurement probes is related to the reconstruction performance of sparse domains. With this motivation, we propose a closely related t%-average mutual coherence parameter as a sensing configuration quality parameter and examine its relationship to the reconstruction behavior of various monostatic and ultranarrow-band multistatic configurations. We investigate how this easily computed metric is related to SAR reconstruction quality

Quantization and Compressive Sensing

Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This book chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems. Specifically, we first provide a brief overview of quantization and examine fundamental performance bounds applicable to any quantization approach. Next, we consider several forms of scalar quantizers, namely uniform, non-uniform, and 1-bit. We provide performance bounds and fundamental analysis, as well as practical quantizer designs and reconstruction algorithms that account for quantization. Furthermore, we provide an overview of Sigma-Delta ($\Sigma\Delta$) quantization in the compressed sensing context, and also discuss implementation issues, recovery algorithms and performance bounds. As we demonstrate, proper accounting for quantization and careful quantizer design has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing and Its Applications", 201

Lorentzian Iterative Hard Thresholding: Robust Compressed Sensing with Prior Information

Commonly employed reconstruction algorithms in compressed sensing (CS) use the $L_2$ norm as the metric for the residual error. However, it is well-known that least squares (LS) based estimators are highly sensitive to outliers present in the measurement vector leading to a poor performance when the noise no longer follows the Gaussian assumption but, instead, is better characterized by heavier-than-Gaussian tailed distributions. In this paper, we propose a robust iterative hard Thresholding (IHT) algorithm for reconstructing sparse signals in the presence of impulsive noise. To address this problem, we use a Lorentzian cost function instead of the $L_2$ cost function employed by the traditional IHT algorithm. We also modify the algorithm to incorporate prior signal information in the recovery process. Specifically, we study the case of CS with partially known support. The proposed algorithm is a fast method with computational load comparable to the LS based IHT, whilst having the advantage of robustness against heavy-tailed impulsive noise. Sufficient conditions for stability are studied and a reconstruction error bound is derived. We also derive sufficient conditions for stable sparse signal recovery with partially known support. Theoretical analysis shows that including prior support information relaxes the conditions for successful reconstruction. Simulation results demonstrate that the Lorentzian-based IHT algorithm significantly outperform commonly employed sparse reconstruction techniques in impulsive environments, while providing comparable performance in less demanding, light-tailed environments. Numerical results also demonstrate that the partially known support inclusion improves the performance of the proposed algorithm, thereby requiring fewer samples to yield an approximate reconstruction.Comment: 28 pages, 9 figures, accepted in IEEE Transactions on Signal Processin