Reconciling Compressive Sampling Systems for Spectrally-sparse Continuous-time Signals

The Random Demodulator (RD) and the Modulated Wideband Converter (MWC) are two recently proposed compressed sensing (CS) techniques for the acquisition of continuous-time spectrally-sparse signals. They extend the standard CS paradigm from sampling discrete, finite dimensional signals to sampling continuous and possibly infinite dimensional ones, and thus establish the ability to capture these signals at sub-Nyquist sampling rates. The RD and the MWC have remarkably similar structures (similar block diagrams), but their reconstruction algorithms and signal models strongly differ. To date, few results exist that compare these systems, and owing to the potential impacts they could have on spectral estimation in applications like electromagnetic scanning and cognitive radio, we more fully investigate their relationship in this paper. We show that the RD and the MWC are both based on the general concept of random filtering, but employ significantly different sampling functions. We also investigate system sensitivities (or robustness) to sparse signal model assumptions. Lastly, we show that "block convolution" is a fundamental aspect of the MWC, allowing it to successfully sample and reconstruct block-sparse (multiband) signals. Based on this concept, we propose a new acquisition system for continuous-time signals whose amplitudes are block sparse. The paper includes detailed time and frequency domain analyses of the RD and the MWC that differ, sometimes substantially, from published results.Comment: Corrected typos, updated Section 4.3, 30 pages, 8 figure

Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples

This paper presents a novel power spectral density estimation technique for band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The technique employs multi-coset sampling and incorporates the advantages of compressed sensing (CS) when the power spectrum is sparse, but applies to sparse and nonsparse power spectra alike. The estimates are consistent piecewise constant approximations whose resolutions (width of the piecewise constant segments) are controlled by the periodicity of the multi-coset sampling. We show that compressive estimates exhibit better tradeoffs among the estimator's resolution, system complexity, and average sampling rate compared to their noncompressive counterparts. For suitable sampling patterns, noncompressive estimates are obtained as least squares solutions. Because of the non-negativity of power spectra, compressive estimates can be computed by seeking non-negative least squares solutions (provided appropriate sampling patterns exist) instead of using standard CS recovery algorithms. This flexibility suggests a reduction in computational overhead for systems estimating both sparse and nonsparse power spectra because one algorithm can be used to compute both compressive and noncompressive estimates.Comment: 26 pages, single spaced, 9 figure

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

Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences

Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements

Fully-passive switched-capacitor techniques for high performance SAR ADC design

In recent years, SAR ADC becomes more and more popular in various low-power applications such as wireless sensors and low energy radios due to its circuit simplicity, high power efficiency, and scaling compatibility. However, its speed is limited by its successive approximation procedures and its power efficiency greatly reduces with the ADC resolution going beyond 10 bit. To address these issues, this thesis proposes to embed two techniques: 1) compressive sensing (CS) and 2) noise shaping (NS) to a conventional SAR ADC. The realization of both techniques are based on fully-passive switched-capacitor techniques. CS is a recently emerging sampling paradigm, stating that the sparsity of a signal can be exploited to reduce the ADC sampling rate below the Nyquist rate. Different from conventional CS frameworks which require dedicated analog CS encoders, this thesis proposes a fully-passive CS-SAR ADC architecture which only requires minor modification to a conventional SAR ADC. Two chips are fabricated in a 0.13 µm process to prove the concept. One chip is a single-channel CS-SAR ADC which can reduce the ADC conversion rate by 4 times, thus reducing the ADC power by 4 times. In many wireless sensing applications, multiple ADCs are commonly required to sense multi-channel signals such as multi-lead ECG sensing and parallel neural recording. Therefore, the other chip is a multi-channel CS-SAR ADC which can simultaneously convert 4-channel signals with a sampling rate of one channel’s Nyquist rate. At 0.8 V and 1 MS/s, both chips achieve an effective Walden FoM of around 5 fJ/conversion-step. This thesis also proposes a novel NS SAR ADC architecture that is simple, robust and low power for high-resolution applications. Compared to conventional ∆Σ ADCs, it replaces the power-hungry active integrator with a passive integrator which only requires one switch and two capacitors. Compared to previous 1st-order NS SAR ADC works, it achieves the best NS performance and can be easily extended to 2nd-order. A 1st-order 10-bit NS SAR ADC is fabricated in a 0.13 µm process. Through NS, SNDR increases by 6 dB with OSR doubled, achieving a 12- bit ENOB at OSR = 8. An improved version of a 2nd-order 9-bit NS SAR ADC is designed and simulated in a 40 nm process. The SNDR increases by 10 dB with OSR doubled, achieving a 14-bit ENOB at OSR = 16. At a bandwidth of 312.5 kHz, the Schreier FoM is 181 dB and the Walden FoM is 12.5 fJ/conversion-step, proving that the proposed NS SAR ADC architecture can achieve high resolution and high power efficiency simultaneously.Electrical and Computer Engineerin

A Cognitive Radio Compressive Sensing Framework

With the proliferation of wireless devices and services, allied with further significant predicted growth, there is an ever increasing demand for higher transmission rates. This is especially challenging given the limited availability of radio spectrum, and is further exacerbated by a rigid licensing regulatory regime. Spectrum however, is largely underutilized and this has prompted regulators to promote the concept of opportunistic spectrum access. This allows unlicensed secondary users to use bands which are licensed to primary users, but are currently unoccupied, so leading to more efficient spectrum utilization. A potentially attractive solution to this spectrum underutilisation problem is cognitive radio (CR) technology, which enables the identification and usage of vacant bands by continuously sensing the radio environment, though CR enforces stringent timing requirements and high sampling rates. Compressive sensing (CS) has emerged as a novel sampling paradigm, which provides the theoretical basis to resolve some of these issues, especially for signals exhibiting sparsity in some domain. For CR-related signals however, existing CS architectures such as the random demodulator and compressive multiplexer have limitations in regard to the signal types used, spectrum estimation methods applied, spectral band classification and a dependence on Fourier domain based sparsity. This thesis presents a new generic CS framework which addresses these issues by specifically embracing three original scientific contributions: i) seamless embedding of the concept of precolouring into existing CS architectures to enhance signal sparsity for CR-related digital modulation schemes; ii) integration of the multitaper spectral estimator to improve sparsity in CR narrowband modulation schemes; and iii) exploiting sparsity in an alternative, non-Fourier (Walsh-Hadamard) domain to expand the applicable CR-related modulation schemes. Critical analysis reveals the new CS framework provides a consistently superior and robust solution for the recovery of an extensive set of currently employed CR-type signals encountered in wireless communication standards. Significantly, the generic and portable nature of the framework affords the opportunity for further extensions into other CS architectures and sparsity domains

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

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