Regularized sampling of multiband signals

This paper presents a regularized sampling method for multiband signals, that makes it possible to approach the Landau limit, while keeping the sensitivity to noise at a low level. The method is based on band-limited windowing, followed by trigonometric approximation in consecutive time intervals. The key point is that the trigonometric approximation "inherits" the multiband property, that is, its coefficients are formed by bursts of non-zero elements corresponding to the multiband components. It is shown that this method can be well combined with the recently proposed synchronous multi-rate sampling (SMRS) scheme, given that the resulting linear system is sparse and formed by ones and zeroes. The proposed method allows one to trade sampling efficiency for noise sensitivity, and is specially well suited for bounded signals with unbounded energy like those in communications, navigation, audio systems, etc. Besides, it is also applicable to finite energy signals and periodic band-limited signals (trigonometric polynomials). The paper includes a subspace method for blindly estimating the support of the multiband signal as well as its components, and the results are validated through several numerical examples.Comment: The title and introduction have changed. Submitted to the IEEE Transactions on Signal Processin

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

Multiband Spectrum Access: Great Promises for Future Cognitive Radio Networks

Cognitive radio has been widely considered as one of the prominent solutions to tackle the spectrum scarcity. While the majority of existing research has focused on single-band cognitive radio, multiband cognitive radio represents great promises towards implementing efficient cognitive networks compared to single-based networks. Multiband cognitive radio networks (MB-CRNs) are expected to significantly enhance the network's throughput and provide better channel maintenance by reducing handoff frequency. Nevertheless, the wideband front-end and the multiband spectrum access impose a number of challenges yet to overcome. This paper provides an in-depth analysis on the recent advancements in multiband spectrum sensing techniques, their limitations, and possible future directions to improve them. We study cooperative communications for MB-CRNs to tackle a fundamental limit on diversity and sampling. We also investigate several limits and tradeoffs of various design parameters for MB-CRNs. In addition, we explore the key MB-CRNs performance metrics that differ from the conventional metrics used for single-band based networks.Comment: 22 pages, 13 figures; published in the Proceedings of the IEEE Journal, Special Issue on Future Radio Spectrum Access, March 201

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

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

Compressed Sensing of Analog Signals in Shift-Invariant Spaces

A traditional assumption underlying most data converters is that the signal should be sampled at a rate exceeding twice the highest frequency. This statement is based on a worst-case scenario in which the signal occupies the entire available bandwidth. In practice, many signals are sparse so that only part of the bandwidth is used. In this paper, we develop methods for low-rate sampling of continuous-time sparse signals in shift-invariant (SI) spaces, generated by m kernels with period T. We model sparsity by treating the case in which only k out of the m generators are active, however, we do not know which k are chosen. We show how to sample such signals at a rate much lower than m/T, which is the minimal sampling rate without exploiting sparsity. Our approach combines ideas from analog sampling in a subspace with a recently developed block diagram that converts an infinite set of sparse equations to a finite counterpart. Using these two components we formulate our problem within the framework of finite compressed sensing (CS) and then rely on algorithms developed in that context. The distinguishing feature of our results is that in contrast to standard CS, which treats finite-length vectors, we consider sampling of analog signals for which no underlying finite-dimensional model exists. The proposed framework allows to extend much of the recent literature on CS to the analog domain.Comment: to appear in IEEE Trans. on Signal Processin