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 Spectral Clustering

Spectral clustering has become a popular technique due to its high performance in many contexts. It comprises three main steps: create a similarity graph between N objects to cluster, compute the first k eigenvectors of its Laplacian matrix to define a feature vector for each object, and run k-means on these features to separate objects into k classes. Each of these three steps becomes computationally intensive for large N and/or k. We propose to speed up the last two steps based on recent results in the emerging field of graph signal processing: graph filtering of random signals, and random sampling of bandlimited graph signals. We prove that our method, with a gain in computation time that can reach several orders of magnitude, is in fact an approximation of spectral clustering, for which we are able to control the error. We test the performance of our method on artificial and real-world network data.Comment: 12 pages, 2 figure

Compressive Raman imaging with spatial frequency modulated illumination

We report a line scanning imaging modality of compressive Raman technology with spatial frequency modulated illumination using a single pixel detector. We demonstrate the imaging and classification of three different chemical species at line scan rates of 40 Hz

A sub-Nyquist co-prime sampling music spectral approach for natural frequency identification of white-noise excited structures

Motivated by practical needs to reduce data transmission payloads in wireless sensors for vibration-based monitoring of civil engineering structures, this paper proposes a novel approach for identifying resonant frequencies of white-noise excited structures using acceleration measurements acquired at rates significantly below the Nyquist rate. The approach adopts the deterministic co-prime sub-Nyquist sampling scheme, originally developed to facilitate telecommunication applications, to estimate the autocorrelation function of response acceleration time-histories of low-amplitude white-noise excited structures treated as realizations of a stationary stochastic process. This is achieved without posing any sparsity conditions to the signals. Next, the standard MUSIC algorithm is applied to the estimated autocorrelation function to derive a denoised super-resolution pseudo-spectrum in which natural frequencies are marked by prominent spikes. The accuracy and applicability of the proposed approach is numerically assessed using computer-generated noise-corrupted acceleration time-history data obtained by a simulation-based framework pertaining to a white-noise excited structural system with two closely-spaced modes of vibration carrying the same amount of energy, and a third isolated weakly excited vibrating mode. All three natural frequencies are accurately identified by sampling at as low as 78% below Nyquist rate for signal to noise ratio as low as 0dB (i.e., energy of additive white noise equal to the signal energy), suggesting that the proposed approach is robust and noise-immune while it can reduce data transmission requirements in acceleration wireless sensors for natural frequency identification of engineering structures

The Turbulence Spectrum of Molecular Clouds in the Galactic Ring Survey: A Density-Dependent PCA Calibration

Turbulence plays a major role in the formation and evolution of molecular clouds. The problem is that turbulent velocities are convolved with the density of an observed region. To correct for this convolution, we investigate the relation between the turbulence spectrum of model clouds, and the statistics of their synthetic observations obtained from Principal Component Analysis (PCA). We apply PCA to spectral maps generated from simulated density and velocity fields, obtained from hydrodynamic simulations of supersonic turbulence, and from fractional Brownian motion fields with varying velocity, density spectra, and density dispersion. We examine the dependence of the slope of the PCA structure function, alpha_PCA, on intermittency, on the turbulence velocity (beta_v) and density (beta_n) spectral indexes, and on density dispersion. We find that PCA is insensitive to beta_n and to the log-density dispersion sigma_s, provided sigma_s 2, alpha_PCA increases with sigma_s due to the intermittent sampling of the velocity field by the density field. The PCA calibration also depends on intermittency. We derive a PCA calibration based on fBms with sigma_s<2 and apply it to 367 CO spectral maps of molecular clouds in the Galactic Ring Survey. The average slope of the PCA structure function, =0.62\pm0.2, is consistent with the hydrodynamic simulations and leads to a turbulence velocity exponent =2.06\pm0.6 for a non-intermittent, low density dispersion flow. Accounting for intermittency and density dispersion, the coincidence between the PCA slope of the GRS clouds and the hydrodynamic simulations suggests beta_v~1.9, consistent with both Burgers and compressible intermittent turbulence