This lecture note presents a new method to capture and represent compressible signals at a rate significantly below the Nyquist rate. This method, called compressive sensing, employs nonadaptive linear projections that preserve the structure of the signal; the signal is then reconstructed from these projections using an optimization process.

Compressive Sensing [Lecture Notes]

Cooperative spectrum sensing in cognitive radio networks: A survey

This paper overviews the recent work on compressive sensing, a new approach to data acquisition in which analog signals are digitized for processing not via uniform sampling but via measurements using more general, even random, test functions. In stark contrast with conventional wisdom, the new theory asserts that one can combine "low-rate sampling" with digital computational power for efficient and accurate signal acquisition. Compressive sensing systems directly translate analog data into a compressed digital form; all we need to do is "decompress" the measured data through an optimization on a digital computer. The implications of compressive sensing are promising for many applications and enable the design of new kinds of analog-to-digital converters, cameras, and imaging systems.

http://ieeexplore.ieee.org/iel5/4555640/4558476/04558479.pdf

Compressive sensing

We propose simple and extremely efficient methods for solving the basis pursuit problem $\min\{\|u\|_1 : Au = f, u\in\mathbb{R}^n\},$ which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem $\min_{u\in\mathbb{R}^n} \mu\|u\|_1+\frac{1}{2}\|Au-f^k\|_2^2$ for given matrix $A$ and vector $f^k$. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving $A$ and $A^\top$ can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.

https://epubs.siam.org/doi/pdf/10.1137/070703983

Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing

Conventional sub-Nyquist sampling methods for analog signals exploit prior information about the spectral support. In this paper, we consider the challenging problem of blind sub-Nyquist sampling of multiband signals, whose unknown frequency support occupies only a small portion of a wide spectrum. Our primary design goals are efficient hardware implementation and low computational load on the supporting digital processing. We propose a system, named the modulated wideband converter, which first multiplies the analog signal by a bank of periodic waveforms. The product is then low-pass filtered and sampled uniformly at a low rate, which is orders of magnitude smaller than Nyquist. Perfect recovery from the proposed samples is achieved under certain necessary and sufficient conditions. We also develop a digital architecture, which allows either reconstruction of the analog input, or processing of any band of interest at a low rate, that is, without interpolating to the high Nyquist rate. Numerical simulations demonstrate many engineering aspects: robustness to noise and mismodeling, potential hardware simplifications, real-time performance for signals with time-varying support and stability to quantization effects. We compare our system with two previous approaches: periodic nonuniform sampling, which is bandwidth limited by existing hardware devices, and the random demodulator, which is restricted to discrete multitone signals and has a high computational load. In the broader context of Nyquist sampling, our scheme has the potential to break through the bandwidth barrier of state-of-the-art analog conversion technologies such as interleaved converters.

/pdf/from-theory-to-practice-sub-nyquist-sampling-of-sparse-4swzjs0wes.pdf

From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals

The new theory of compressive sensing enables direct analog-to-information conversion of compressible signals at sub-Nyquist acquisition rates. The authors develop new theory, algorithms, performance bounds, and a prototype implementation for an analog-to-information converter based on random demodulation. The architecture is particularly apropos for wideband signals that are sparse in the time-frequency plane. End-to-end simulations of a complete transistor-level implementation prove the concept under the effect of circuit nonidealities.

/pdf/theory-and-implementation-of-an-analog-to-information-4gq0kr2lqy.pdf

Theory and Implementation of an Analog-to-Information Converter using Random Demodulation

Many problems in radar and communication signal processing involve radio frequency (RF) signals of very high bandwidth. This presents a serious challenge to systems that might attempt to use a high-rate analog-to-digital converter (ADC) to sample these signals, as prescribed by the Shannon/Nyquist sampling theorem. In these situations, however, the information level of the signal is often far lower than the actual bandwidth, which prompts the question of whether more efficient schemes can be developed for measuring such signals. In this paper we propose a system that uses modulation, filtering, and sampling to produce a low-rate set of digital measurements. Our "analog-to-information converter" (AIC) is inspired by the recent theory of Compressive Sensing (CS), which states that a discrete signal having a sparse representation in some dictionary can be recovered from a small number of linear projections of that signal. We generalize the CS theory to continuous-time sparse signals, explain our proposed AIC system in the CS context, and discuss practical issues regarding implementation.

/pdf/analog-to-information-conversion-via-random-demodulation-bfn6hw1u7n.pdf

Analog-to-Information Conversion via Random Demodulation

In this paper, we utilize recent advances in compressive sensing theory to enable signal acquisition beyond Nyquist sampling constraints. We successfully recover signals sampled at sub-Nyquist sampling rates by exploiting additional structure other than bandlimitedness. We present a working prototype of compressive analog-to-digital converter (CADC) based on a random demodulation architecture. The architecture is particularly suitable for wideband signals that are sparse in the time-frequency plane. CADC has the advantage of enhancing the performance of communication and multimedia systems by increasing the transmission rate for the same bandwidth. We report successful reconstruction of AM modulated signals at sampling rates down to 1/8 of the Nyquist-rate, which represents an up to 87.5% savings in the bandwidth and the storage memory.

A prototype hardware for random demodulation based compressive analog-to-digital conversion

A typical data acquisition system takes periodic samples of a signal, image, or other data, often at the so-called Nyquist/Shannon sampling rate of two times the data bandwidth in order to ensure that no information is lost. In applications involving wideband signals, the Nyquist/Shannon sampling rate is very high, even though the signals may have a simple underlying structure. Recent developments in mathematics and signal processing have uncovered a solution to this Nyquist/Shannon sampling rate bottlenck for signals that are sparse or compressible in some representation. We demonstrate and reduce to practice methods to extract information directly from an analog or digital signal based on altering our notion of sampling to replace uniform time samples with more general linear functionals. One embodiment of our invention is a low-rate analog-to-information converter that can replace the high-rate analog-to-digital converter in certain applications involving wideband signals. Another embodiment is an encoding scheme for wideband discrete-time signals that condenses their information content.

Method and apparatus for on-line compressed sensing

The stability and programmability of digital signal processing systems has motivated engineers to move the analog-to-digital conversion (ADC) process closer and closer to the front end of many signal processing systems in order to perform as much processing as possible in the digital domain. Unfortunately, many important applications, including radar and communication systems, involve wideband signals that seriously stress modern ADCs; sampling these signals above the Nyquist rate is in some cases challenging and in others impossible. While wideband signals by definition have a large bandwidth, often the amount of information they carry per second is much lower; that is, they are compressible in some sense. The first contribution of this paper is a new framework for wideband signal acquisition purpose-built for compressible signals that enables sub-Nyquist data acquisition via an analog-to-information converter (AIC). The framework is based on the recently developed theory of compressive sensitng in which a small number of non-adaptive, randomized measurements are sufficient to reconstruct compressible signals. The second contribution of this paper is an AIC implementation design and study of the tradeoffs and nonidealities introduced by real hardware. The goal is to identify and optimize the parameters that dominate the overall system performance.

T. Ragheb

Papers

Theory and Implementation of an Analog-to-Information Converter using Random Demodulation

Analog-to-Information Conversion via Random Demodulation

A prototype hardware for random demodulation based compressive analog-to-digital conversion

Method and apparatus for on-line compressed sensing

Practical Issues in Implementing Analog-to-Information Converters