Abstract
In this work, an estimate of the power spectrum of a real-valued wide-sense stationary autoregressive signal is computed from sub-Nyquist or compressed measurements in additive white Gaussian noise. The problem is formulated using the concepts of compressive covariance sensing and Blackman-Tukey nonparametric spectrum estimation. Only the second-order statistics of the original signal, rather than the signal itself, need to be recovered from the compressed signal. This is achieved by solving the resulting overdetermined system of equations by application of least squares, thereby circumventing the need for applying the complicated ℓ 1 -minimization otherwise required for the reconstruction of the original signal. Moreover, the signal need not be spectrally sparse. A study of the performance of the power spectral estimator is conducted taking into account the properties of the different bases of the covariance subspace needed for compressive covariance sensing, as well as different linear sparse rulers by which compression is achieved. A method is proposed to benefit from the possible computational efficiency resulting from the use of the Fourier basis of the covariance subspace without considerably affecting the spectrum estimation performance.
Highlights
The efficiency of signal acquisition systems has been greatly improved by the introduction of the concept of compressive sensing (CS) [1]
This information enables the reconstruction of wide-sense stationary (WSS) signal statistics by covariance sensing (CCS), and the technique has been extended to nonstationary signals as well using online CCS [3]
Compression is assumed to be performed using a linear sparse ruler (LSR) [2, 3] that enables subsequent recovery of the Toeplitz covariance matrix by sampling the delays at least once, and the development of the CCS problem leads to an overdetermined system of equations that can be solved by least squares to obtain the covariance values of the original signal
Summary
The efficiency of signal acquisition systems has been greatly improved by the introduction of the concept of compressive sensing (CS) [1]. Reconstruction of second-order statistics from the compressed measurements is possible without the need for the original signal reconstruction and without the condition of spectral sparsity. Such a technique is referred to as compressive covariance sensing (CCS) [2]. Structure information present in the statistical domain is captured during compression such as the common positive semidefinite Hermitian Toeplitz (HT) structure of the signal covariance matrix. This information enables the reconstruction of wide-sense stationary (WSS) signal statistics by CCS, and the technique has been extended to nonstationary signals as well using online CCS [3]. Compression is assumed to be performed using a linear sparse ruler (LSR) [2, 3] that enables subsequent recovery of the Toeplitz covariance matrix by sampling the delays at least once, and the development of the CCS problem leads to an overdetermined system of equations that can be solved by least squares to obtain the covariance values of the original signal
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