Compressed sensing deals with the reconstruction of signals

from sub-Nyquist samples by exploiting the sparsity of their

projections onto known subspaces. In contrast, the present

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.

%B IEEE Signal Processing Magazine
%V 33
%P 78--93
%8 01/2016
%G eng
%N 1
%0 Journal Article
%J IEEE Trans. Information Theory
%D 2015
%T Compression Limits for Random Vectors with Linearly Parameterized Second-Order Statistics
%A Daniel Romero
%A R. López-Valcarce
%A Geert Leus
%K compass
%K compressed sensing
%X The class of complex random vectors whose covariance

matrix is linearly parameterized by a basis of Hermitian

Toeplitz (HT) matrices is considered, and the maximum

compression ratios that preserve all second-order information

are derived — the statistics of the uncompressed vector must

be recoverable from a set of linearly compressed observations.

This kind of vectors arises naturally when sampling widesense

stationary random processes and features a number of

applications in signal and array processing.

Explicit guidelines to design optimal and nearly optimal

schemes operating both in a periodic and non-periodic fashion

are provided by considering two of the most common linear

compression schemes, which we classify as dense or sparse. It

is seen that the maximum compression ratios depend on the

structure of the HT subspace containing the covariance matrix of

the uncompressed observations. Compression patterns attaining

these maximum ratios are found for the case without structure as

well as for the cases with circulant or banded structure. Universal

samplers are also proposed to compress unknown HT subspaces.

%B IEEE Trans. Information Theory
%V 61
%P 1410-1425
%8 03/2015
%G eng
%N 3
%& 1410
%R 10.1109/TIT.2015.2394784