38 research outputs found

    Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples

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    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

    Group Iterative Spectrum Thresholding for Super-Resolution Sparse Spectral Selection

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    Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency, thereby resulting in a coherent design. The popular convex compressed sensing methods break down in presence of high coherence and large noise. We propose a new regularization approach to handle model collinearity and obtain parsimonious frequency selection simultaneously. It takes advantage of the pairing structure of sine and cosine atoms in the frequency dictionary. A probabilistic spectrum screening is also developed for fast computation in high dimensions. A data-resampling version of high-dimensional Bayesian Information Criterion is used to determine the regularization parameters. Experiments show the efficacy and efficiency of the proposed algorithms in challenging situations with small sample size, high frequency resolution, and low signal-to-noise ratio

    Computational Methods for Sparse Solution of Linear Inverse Problems

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    The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications

    Dictionary Learning-Based Speech Enhancement

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    Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions

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    High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and 1\ell^1 minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best ss-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.Comment: 33 pages, 9 figure

    Global testing under sparse alternatives: ANOVA, multiple comparisons and the higher criticism

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    Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings. Suppose we have pp covariates and that under the alternative, the response only depends upon the order of p1αp^{1-\alpha} of those, 0α10\le\alpha\le1. Under moderate sparsity levels, that is, 0α1/20\le\alpha\le1/2, we show that ANOVA is essentially optimal under some conditions on the design. This is no longer the case under strong sparsity constraints, that is, α>1/2\alpha>1/2. In such settings, a multiple comparison procedure is often preferred and we establish its optimality when α3/4\alpha\geq3/4. However, these two very popular methods are suboptimal, and sometimes powerless, under moderately strong sparsity where 1/2<α<3/41/2<\alpha<3/4. We suggest a method based on the higher criticism that is powerful in the whole range α>1/2\alpha>1/2. This optimality property is true for a variety of designs, including the classical (balanced) multi-way designs and more modern "p>np>n" designs arising in genetics and signal processing. In addition to the standard fixed effects model, we establish similar results for a random effects model where the nonzero coefficients of the regression vector are normally distributed.Comment: Published in at http://dx.doi.org/10.1214/11-AOS910 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Compressive Sensing for Speech Signals in Mobile Systems

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    Compressive sensing is an emerging and revolutionary technology that strongly relies on the sparsity of the signal. In compressive sensing the signal is sparsely compressively sampled by taking a small number of random projections of the signal, which contain most of the salient information. Compressive sensing has been previously applied in areas like: image processing, radar systems and sonar systems. This research work will discuss the potential implementation of compressive sensing in mobile communication systems and how it will influence their data rates. In a typical mobile communication system, the signal of interest is sampled at least at the Nyquist rate. The Nyquist sampling theorem states that the frequency used to sample a signal should be at least twice the maximum frequency contained within the signal. However, this is not the most efficient way to compress the signal, as it places a lot of burden in sampling the entire signal while only a small percentage of the transform coefficients are needed to represent it. The recent results in compressive sampling (also known as compressive sensing) provide a new way to reconstruct the original signal with a minimal number of observations. In compressive sensing the significant information about the signal/image is directly acquired, rather than acquiring the significant information that will be eventually thrown away. The goal of this research is to propose a new mobile communication system which employs compressive sampling to compress the speech signal at the transmitter and decompress it at the receiver. The expected results from the proposed system will be an increment in the data rates of these systems. In order to simulate how compressive sensing could be applied, a small speech signal was recorded in MATLAB. The signal at the transmitter is then multiplied by the measurement matrix which in this case is composed of randomly generated numbers. The measurement matrix is chosen in such a way that the sparse signal can be exactly recovered at the receiver using one of the different optimization techniques available. Once the signal has gone through the process of compressive sampling, it is ready to be transmitted through the mobile system. The transmitted signal is then reconstructed by the receiver from a significantly small number of samples by using any of the multiple optimization techniques available. The algorithm is simulated in MATLAB. The results show that if a threshold window is applied to the transmitted speech signal and the length of the signal is kept constant, the compression rate of the speech signal is increased
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