Compressed sensing deals with the reconstruction of signals from sub-Nyquist samples by exploiting the sparsity of their projections onto known subspaces. In contrast, this article is concerned with the reconstruction of second-order statistics, such as covariance and power spectrum, even in the absence of sparsity priors. The framework described here leverages the statistical structure of random processes to enable signal compression and offers an alternative perspective at sparsity-agnostic inference. Capitalizing on parsimonious representations, we illustrate how compression and reconstruction tasks can be addressed in popular applications such as power-spectrum estimation, incoherent imaging, direction-of-arrival estimation, frequency estimation, and wideband spectrum sensing.@en

This paper considers multiple wireless sensors that cooperatively estimate the power spectrum of the signals received from several sources. We extend our previous work on cooperative compressive power spectrum estimation to accommodate the scenario where the statistics of the fading channels experienced by different sensors are different. The signals received from the sources are assumed to be time-domain wide-sense stationary processes. Multiple sensors are organized into several groups, where each group estimates a different subset of lags of the temporal correlation. A fusion centre (FC) combines these estimates to obtain the power spectrum. As each sensor group computes correlation estimates only at a subset of lags, the sampling rate per sensor can be less than the Nyquist rate. The conditions required for uniqueness of the least-squares estimator are derived based on our previous results. The sensors are combined into clusters in such a way that all sensors within the same cluster experience approximately the same fading statistics. We find that, as long as the number of sensors of each group is the same across clusters, the resulting power spectrum estimate computed by the FC converges to the power spectrum of the transmitted signal scaled by the averaged fading statistics. In a simulation study, we also investigate the performance of our approach when the aforementioned assumption is not true, i.e., when the number of sensors of each group is not the same across clusters. The simulation study shows degradation in the performance of our approach for this case.@en

In this paper, two problems that show great similarities are examined. The first problem is the reconstruction of the angular-domain periodogram from spatial-domain signals received at different time indices. The second one is the reconstruction of the frequency-domain periodogram from time-domain signals received at different wireless sensors. We split the entire angular or frequency band into uniform bins. The bin size is set such that the received spectra at two frequencies or angles, whose distance is equal to or larger than the size of a bin, are uncorrelated. These problems in the two different domains lead to a similar circulant structure in the so-called coset correlation matrix. This circulant structure allows for a strong compression and a simple least-squares reconstruction method. The latter is possible under the full column rank condition of the system matrix, which can be achieved by designing the spatial or temporal sampling patterns based on a circular sparse ruler. We analyze the statistical performance of the compressively reconstructed periodogram, including bias and variance. We further consider the case when the bins are so small that the received spectra at two frequencies or angles, with a spacing between them larger than the size of the bin, can still be correlated. In this case, the resulting coset correlation matrix is generally not circulant and thus a special approach is required.@en

This paper focuses on the design of a Fourier dictionary matrix formed by selecting specific rows of the inverse discrete Fourier transform matrix based on coherence-related metrics. While maximum coherence is a popular metric in compressive sampling, we also consider rms LN-coherence, which focuses on the largest LN (instead of one) inner products between different columns of the dictionary matrix. Finding a dictionary matrix optimizing either the maximum or the rms LN-coherence lead to a complicated optimization problem. Hence, we introduce a new metric called coherence deviation (CD), which gives a measure on the variation of all the inner products between different columns of the dictionary matrix, and motivate its use as an amenable alternative for both the maximum and rms LN-coherence. While finding a dictionary matrix optimizing the CD leads to a simplified optimization problem, the resulting cost function is a quartic function of a binary vector variable. Hence, we propose Greedy-β algorithm to provide sub-optimal solutions.@en