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Register a new userCompressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples
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Michael Lexa
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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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 estimators 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.
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Covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples, which is of extreme importance in many applications. Classical results have shown that $O(n)$ samples are sufficient to accurately estimate the covariance matrix from $n$-dimensional independent Gaussian samples. However, in many practical applications, the received signal samples might be correlated, which makes the classical analysis inapplicable. In this paper, we develop a non-asymptotic analysis for the covariance matrix estimation from correlated Gaussian samples. Our theoretical results show that the error bounds are determined by the signal dimension $n$, the sample size $m$, and the shape parameter of the distribution of the correlated sample covariance matrix. Particularly, when the shape parameter is a class of Toeplitz matrices (which is of great practical interest), $O(n)$ samples are also sufficient to faithfully estimate the covariance matrix from correlated samples. Simulations are provided to verify the correctness of the theoretical results.
In this paper, we study the prediction of a circularly symmetric zero-mean stationary Gaussian process from a window of observations consisting of finitely many samples. This is a prevalent problem in a wide range of applications in communication theory and signal processing. Due to stationarity, when the autocorrelation function or equivalently the power spectral density (PSD) of the process is available, the Minimum Mean Squared Error (MMSE) predictor is readily obtained. In particular, it is given by a linear operator that depends on autocorrelation of the process as well as the noise power in the observed samples. The prediction becomes, however, quite challenging when the PSD of the process is unknown. In this paper, we propose a blind predictor that does not require the a priori knowledge of the PSD of the process and compare its performance with that of an MMSE predictor that has a full knowledge of the PSD. To design such a blind predictor, we use the random spectral representation of a stationary Gaussian process. We apply the well-known atomic-norm minimization technique to the observed samples to obtain a discrete quantization of the underlying random spectrum, which we use to predict the process. Our simulation results show that this estimator has a good performance comparable with that of the MMSE estimator.
The multipath radio channel is considered to have a non-bandlimited channel impulse response. Therefore, it is challenging to achieve high resolution time-delay (TD) estimation of multipath components (MPCs) from bandlimited observations of communication signals. It this paper, we consider the problem of multiband channel sampling and TD estimation of MPCs. We assume that the nonideal multibranch receiver is used for multiband sampling, where the noise is nonuniform across the receiver branches. The resulting data model of Hankel matrices formed from acquired samples has multiple shift-invariance structures, and we propose an algorithm for TD estimation using weighted subspace fitting. The subspace fitting is formulated as a separable nonlinear least squares (NLS) problem, and it is solved using a variable projection method. The proposed algorithm supports high resolution TD estimation from an arbitrary number of bands, and it allows for nonuniform noise across the bands. Numerical simulations show that the algorithm almost attains the Cramer Rao Lower Bound, and it outperforms previously proposed methods such as multiresolution TOA, MI-MUSIC, and ESPRIT.
Fourier Transform Interferometry (FTI) is an appealing Hyperspectral (HS) imaging modality for many applications demanding high spectral resolution, e.g., in fluorescence microscopy. However, the effective resolution of FTI is limited by the durability of biological elements when exposed to illuminating light. Overexposed elements are subject to photo-bleaching and become unable to fluoresce. In this context, the acquisition of biological HS volumes based on sampling the Optical Path Difference (OPD) axis at Nyquist rate leads to unpleasant trade-offs between spectral resolution, quality of the HS volume, and light exposure intensity. We propose two variants of the FTI imager, i.e., Coded Illumination-FTI (CI-FTI) and Structured Illumination FTI (SI-FTI), based on the theory of compressive sensing (CS). These schemes efficiently modulate light exposure temporally (in CI-FTI) or spatiotemporally (in SI-FTI). Leveraging a variable density sampling strategy recently introduced in CS, we provide near-optimal illumination strategies, so that the light exposure imposed on a biological specimen is minimized while the spectral resolution is preserved. Our analysis focuses on two criteria: (i) a trade-off between exposure intensity and the quality of the reconstructed HS volume for a given spectral resolution; (ii) maximizing HS volume quality for a fixed spectral resolution and constrained exposure budget. Our contributions can be adapted to an FTI imager without hardware modifications. The reconstruction of HS volumes from CS-FTI measurements relies on an $l_1$-norm minimization problem promoting a spatiospectral sparsity prior. Numerically, we support the proposed methods on synthetic data and simulated CS measurements (from actual FTI measurements) under various scenarios. In particular, the biological HS volumes can be reconstructed with a three-to-ten-fold reduction in the light exposure.
In this paper, we consider the problem of compressive sensing (CS) recovery with a prior support and the prior support quality information available. Different from classical works which exploit prior support blindly, we shall propose novel CS recovery algorithms to exploit the prior support adaptively based on the quality information. We analyze the distortion bound of the recovered signal from the proposed algorithm and we show that a better quality prior support can lead to better CS recovery performance. We also show that the proposed algorithm would converge in $mathcal{O}left(logmbox{SNR}right)$ steps. To tolerate possible model mismatch, we further propose some robustness designs to combat incorrect prior support quality information. Finally, we apply the proposed framework to sparse channel estimation in massive MIMO systems with temporal correlation to further reduce the required pilot training overhead.
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