Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals

By Moshe Mishali, Yonina C. Eldar. IEEE Transactions on Signal Processing, 2007

858 citations recorded by the source corpus.

We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. To date, recovery methods for this sampling strategy ensure perfect reconstruction either when the band locations are known, or under strict restrictions on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finite-dimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate, and provides a first systematic study of compressed sensing in a truly analog setting. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/b17f2a7961f98d86b8780395761e82e51b425dd7

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Citations in the corpus

Understanding training and generalization in deep learning by Fourier analysis (2018, result): “This discussion is consistent with other studies (Poggio et al. (2018), Nagarajan & Kolter (2017)).”
Convex Optimization in Signal Processing and Communications (2009, methodology): “This model can be used, for example, to describe multiband signals whose total number of active bands is small compared to the Nyquist rate [22, 23].”
Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation (2009, background): “I NTRODUCTION Recovery of sparse signals from a small number of measurements is a fundamental problem in many different signal processing tasks such as image denoising [8], analog-to-di gital conversion [31], [19], [32], radar, compression, inpainti ng, and many more.”
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals (2009, background): “ere known in advance [44]. See also the subsequent work [45]. Afterward, Mishali and Eldar showed that related methods can be used to acquire multiband signals without knowledge of the band locations [46]. Researchers have also considered generalizations of the multiband signal model. These approach”
Rank Awareness in Joint Sparse Recovery (2010, background): “These ideas have also been recently expanded to include sub-Nyquist sampling of structured ana log signals [19], [20], [21], [22], [23].”
Structured Compressed Sensing: From Theory to Applications (2011, methodology): “Theorem 27 [108]: Consider a multiband model with maximal frequency and total support .”
From Theory to Practice: Sub-Nyquist Sampling of Sparse Wideband Analog Signals (2009, methodology): “This translates to an average sampling rate of 2NB, which is the lowest possible for x(t) ∈ M [5].”
Xampling: Analog to digital at sub-Nyquist rates (2009, background): “We note that the theory of multiband sampling necessitates a minimal sampling rate of 2NB = 228 MHz, regardless of the sampling architecture [19].”
Robust Recovery of Signals From a Structured Union of Subspaces (2008, methodology): “By formulating this problem within the framework of CS, explic it sub-Nyquist sampling and reconstruction schemes were developed in [15], [16] that ensure perfect-recovery a t the minimal possible rate.”
Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (2009, methodology): “ing in blocks (or clusters) rather than being arbitrarily spread throughout the vector. Speciﬁc examples include signals that lie in unions of subspaces [25], [24], [14], [26], and multi-band signals [16], [17], [18]. The recovery of block-sparse vectors x from measurements y = Dx is the focus of th”
Training behavior of deep neural network in frequency domain (2018, background): “For a finite training set, there exists an effective frequency range [17,23,12] beyond which the information of the signal is lost.”
Wideband spectrum sensing for cognitive radio networks: a survey (2013, methodology): “ted by design imperfections or model mismatches. 2) Multi-channel Sub-Nyquist Wideband Sensing:To circumvent model mismatches, Mishali and Eldar proposed a modulated wideband converter (MWC) model in [12] by modifying the AIC model. The main difference between MWC and AIC is that MWC has multiple sa”
Spectral compressive sensing (2013, methodology): “Additionally, several alternative frameworks to CS have been proposed for the modeling and acquisition of frequency-sparse signals from a small number of samples, including finite rate of innovation (FROI) sampling [29, 30, 31, 32] and Xampling [9, 10, 33].”
Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors (2008, methodology): “As described in [15], in this application can be computed by evaluating correlations between the sampling sequences in the time domain.”