Democracy in action: Quantization, saturation, and compressive sensing

We explore and exploit a heretofore relatively unexplored hallmark of compressive sensing (CS), the fact that certain CS measurement systems are democratic, which means that each measurement carries roughly the same amount of information about the signal being acquired. Using this property, we re-think how to quantize the compressive measurements. In Shannon-Nyquist sampling, we scale down the analog signal amplitude (and therefore increase the quantization error) to avoid the gross saturation errors. In stark contrast, we demonstrate a CS system achieves the best performance when we operate 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

Regime Change: Sampling Rate vs. Bit-Depth in Compressive Sensing

The compressive sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by exploiting inherent structure in natural and man-made signals. It has been demonstrated that structured signals can be acquired with just a small number of linear measurements, on the order of the signal complexity. In practice, this enables lower sampling rates that can be more easily achieved by current hardware designs. The primary bottleneck that limits ADC sampling rates is quantization, i.e., higher bit-depths impose lower sampling rates. Thus, the decreased sampling rates of CS ADCs accommodate the otherwise limiting quantizer of conventional ADCs. In this thesis, we consider a different approach to CS ADC by shifting towards lower quantizer bit-depths rather than lower sampling rates. We explore the extreme case where each measurement is quantized to just one bit, representing its sign. We develop a new theoretical framework to analyze this extreme case and develop new algorithms for signal reconstruction from such coarsely quantized measurements. The 1-bit CS framework leads us to scenarios where it may be more appropriate to reduce bit-depth instead of sampling rate. We find that there exist two distinct regimes of operation that correspond to high/low signal-to-noise ratio (SNR). In the measurement compression (MC) regime, a high SNR favors acquiring fewer measurements with more bits per measurement (as in conventional CS); in the quantization compression (QC) regime, a low SNR favors acquiring more measurements with fewer bits per measurement (as in this thesis). A surprise from our analysis and experiments is that in many practical applications it is better to operate in the QC regime, even acquiring as few as 1 bit per measurement. The above philosophy extends further to practical CS ADC system designs. We propose two new CS architectures, one of which takes advantage of the fact that the sampling and quantization operations are performed by two different hardware components. The former can be employed at high rates with minimal costs while the latter cannot. Thus, we develop a system that discretizes in time, performs CS preconditioning techniques, and then quantizes at a low rate

The Pros and Cons of Compressive Sensing for Wideband Signal Acquisition: Noise Folding vs. Dynamic Range

Compressive sensing (CS) exploits the sparsity present in many signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption, and/or monetary cost of both signal sensors and any associated communication links. This paper examines the use of CS in the design of a wideband radio receiver in a noisy environment. We formulate the problem statement for such a receiver and establish a reasonable set of requirements that a receiver should meet to be practically useful. We then evaluate the performance of a CS-based receiver in two ways: via a theoretical analysis of its expected performance, with a particular emphasis on noise and dynamic range, and via simulations that compare the CS receiver against the performance expected from a conventional implementation. On the one hand, we show that CS-based systems that aim to reduce the number of acquired measurements are somewhat sensitive to signal noise, exhibiting a 3dB SNR loss per octave of subsampling, which parallels the classic noise-folding phenomenon. On the other hand, we demonstrate that since they sample at a lower rate, CS-based systems can potentially attain a significantly larger dynamic range. Hence, we conclude that while a CS-based system has inherent limitations that do impose some restrictions on its potential applications, it also has attributes that make it highly desirable in a number of important practical settings

Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals

Wideband analog signals push contemporary analog-to-digital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations.Comment: 24 pages, 8 figure

Exact Signal Recovery from Sparsely Corrupted Measurements through the Pursuit of Justice

Abstract—Compressive sensing provides a framework for recovering sparse signals of length N from M ≪ N measurements. If the measurements contain noise bounded by ɛ, then standard algorithms recover sparse signals with error at most Cɛ. However, these algorithms perform suboptimally when the measurement noise is also sparse. This can occur in practice due to shot noise, malfunctioning hardware, transmission errors, or narrowband interference. We demonstrate that a simple algorithm, which we dub Justice Pursuit (JP), can achieve exact recovery from measurements corrupted with sparse noise. The algorithm handles unbounded errors, has no input parameters, and is easily implemented via standard recovery techniques. I

Trust, but verify: Fast and accurate signal recovery from 1-bit compressive measurements

Abstract—The recently emerged compressive sensing (CS) framework aims to acquire signals at reduced sample rates compared to the classical Shannon-Nyquist rate. To date, the CS theory has assumed primarily real-valued measurements; it has recently been demonstrated that accurate and stable signal acquisition is still possible even when each measurement is quantized to just a single bit. This property enables the design of simplified CS acquisition hardware based around a simple sign comparator rather than a more complex analog-to-digital converter; moreover, it ensures robustness to gross non-linearities applied to the measurements. In this paper we introduce a new algorithm — restricted-step shrinkage (RSS) — to recover sparse signals from 1-bit CS measurements. In contrast to previous algorithms for 1-bit CS, RSS has provable convergence guarantees, is about an order of magnitude faster, and achieves higher average recovery signal-to-noise ratio. RSS is similar in spirit to trust-region methods for non-convex optimization on the unit sphere, which are relatively unexplored in signal processing and hence of independent interest. Index Terms—1-bit compressive sensing, quantization, consistent reconstruction, trust-region algorithms I