Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

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

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 lowpass 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, realtime 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.Comment: 17 pages, 12 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

Feedback Acquisition and Reconstruction of Spectrum-Sparse Signals by Predictive Level Comparisons

In this letter, we propose a sparsity promoting feedback acquisition and reconstruction scheme for sensing, encoding and subsequent reconstruction of spectrally sparse signals. In the proposed scheme, the spectral components are estimated utilizing a sparsity-promoting, sliding-window algorithm in a feedback loop. Utilizing the estimated spectral components, a level signal is predicted and sign measurements of the prediction error are acquired. The sparsity promoting algorithm can then estimate the spectral components iteratively from the sign measurements. Unlike many batch-based Compressive Sensing (CS) algorithms, our proposed algorithm gradually estimates and follows slow changes in the sparse components utilizing a sliding-window technique. We also consider the scenario in which possible flipping errors in the sign bits propagate along iterations (due to the feedback loop) during reconstruction. We propose an iterative error correction algorithm to cope with this error propagation phenomenon considering a binary-sparse occurrence model on the error sequence. Simulation results show effective performance of the proposed scheme in comparison with the literature

Non-uniform sampling and reconstruction of multi-band signals and its application in wideband spectrum sensing of cognitive radio

Sampling theories lie at the heart of signal processing devices and communication systems. To accommodate high operating rates while retaining low computational cost, efficient analog-to digital (ADC) converters must be developed. Many of limitations encountered in current converters are due to a traditional assumption that the sampling state needs to acquire the data at the Nyquist rate, corresponding to twice the signal bandwidth. In this thesis a method of sampling far below the Nyquist rate for sparse spectrum multiband signals is investigated. The method is called periodic non-uniform sampling, and it is useful in a variety of applications such as data converters, sensor array imaging and image compression. Firstly, a model for the sampling system in the frequency domain is prepared. It relates the Fourier transform of observed compressed samples with the unknown spectrum of the signal. Next, the reconstruction process based on the topic of compressed sensing is provided. We show that the sampling parameters play an important role on the average sample ratio and the quality of the reconstructed signal. The concept of condition number and its effect on the reconstructed signal in the presence of noise is introduced, and a feasible approach for choosing a sample pattern with a low condition number is given. We distinguish between the cases of known spectrum and unknown spectrum signals respectively. One of the model parameters is determined by the signal band locations that in case of unknown spectrum signals should be estimated from sampled data. Therefore, we applied both subspace methods and non-linear least square methods for estimation of this parameter. We also used the information theoretic criteria (Akaike and MDL) and the exponential fitting test techniques for model order selection in this case

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

Advancements of MultiRate Signal processing for Wireless Communication Networks: Current State Of the Art

With the hasty growth of internet contact and voice and information centric communications, many contact technologies have been urbanized to meet the stringent insist of high speed information transmission and viaduct the wide bandwidth gap among ever-increasing high-data-rate core system and bandwidth-hungry end-user complex. To make efficient consumption of the limited bandwidth of obtainable access routes and cope with the difficult channel environment, several standards have been projected for a variety of broadband access scheme over different access situation (twisted pairs, coaxial cables, optical fibers, and unchanging or mobile wireless admittance). These access situations may create dissimilar channel impairments and utter unique sets of signal dispensation algorithms and techniques to combat precise impairments. In the intended and implementation sphere of those systems, many research issues arise. In this paper we present advancements of multi-rate indication processing methodologies that are aggravated by this design trend. The thesis covers the contemporary confirmation of the current literature on intrusion suppression using multi-rate indication in wireless communiquE9; networks

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