Compressive power spectral density estimation

Abstract

In this paper, we consider power spectral density estimation of bandlimited, wide-sense stationary signals from sub-Nyquist sampled data. This problem has recently received attention from within the emerging field of cognitive radio for example, and solutions have been proposed that use ideas from compressed sensing and the theory of digital alias-free signal processing. Here we develop a compressed sensing based technique that employs multi-coset sampling and produces multi-resolution power spectral estimates at arbitrarily low average sampling rates. The technique applies to spectrally sparse and nonsparse signals alike, but we show that when the widesense stationary signal is spectrally sparse, compressed sensing is able to enhance the estimator. The estimator does not require signal reconstruction and can be directly obtained from a straightforward application of nonnegative least squares.

Universal lower bounds on sampling rates for covariance estimation

“…More recently, Tarczynski [15], using multicoset (or periodic non uniform) sampling shows that "in the case of PSD [power spectral density] estimation, the average sampling rate can be arbitrarily low". In [11], where multicoset sampling is considered as well, "the noncompressive estimates can theoretically be computed at arbitrarily low sampling rates". In the context of co-prime sampling [18], the authors show that "the sampling rate can be made arbitrarily small".…”

“…This is owing to the fact that, while signal recovery from compressive measurements is an underdetermined problem, compressive covariance recovery has been shown to lead to overdetermined systems in certain settings [11], [1]. In such cases, the sparsity constraint required for signal recovery can be dropped [1].…”

“…In the latter case, CS techniques are used, which exploit the signal sparsity, whereas the former setting does not assume any sparsity. In [11], the authors assume that the sampling pattern is such that the system they obtain has a unique solution but no specific sampling pattern or rate satisfying this condition is discussed. In [16], sampling patterns generated uniformly at random and the Golomb ruler are considered in simulations but no analysis of the required rate is performed.…”

“…The proposed estimator is shown to be consistent for all positive values of the average sampling rate. As opposed to random sampling, a deterministic approach, multicoset, or periodic nonuniform sampling, has been investigated in [15], [1], [11], [16]. In [15], an unbiased power spectrum estimator from multicoset samples is proposed with arbitrarily low average sampling rate.…”

“…In the second overdetermined case, they show that the so-called minimal sparse and minimal circular sparse ruler patterns [17] provide optimal solutions for sub-Nyquist sampling, within the class of multicoset samplers, without any prior sparsity assumption. In [11], [16], the authors propose a method to estimate finite resolution approximations to the true power spectrum exploiting multicoset sampling. That is, they estimate the average power within subbands rather than the power spectrum for each frequency.…”