Compression Limits for Random Vectors with Linearly Parameterized Second-Order Statistics

Abstract

Abstract-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. I. PRELIMINARIES Consider the problem of estimating the second-order statistics of a zero-mean random vector x ∈ C L from a set of K linear observations collected in the vector y ∈ C K given bywhereΦ ∈ C K×L is a known matrix and several realizations of y may be available. This matrix may be referred to as the compression matrix, measurement matrix or sampler, where compression is achieved by setting K < L (typically K L). The covariance matrix Σ = E xx H contains the second-order statistics of x and is assumed to be a linear combination of the Hermitian Toeplitz (HT) matrices in a given set This problem arises in inference operations over the secondorder statistics of a random vector with a Toeplitz covariance matrix. Operating on the compressed observations y entails multiple advantages due to their smaller dimension. In fact, many research efforts in the last decades have been pointed towards designing compression methods and reconstruction algorithms that allow for sampling rate reductions. While most efforts have been focused on reconstructing x, there were also important advances when only the second-order statistics of this vector are of interest. This paper is concerned with problems of the second kind.The compression ratio ρ = L/K measures how much x is compressed. The maximum compression ratio remains an open problem in many cases of interest; and most existing results rely on the usage of specific reconstruction algorithms (see Sec. I-D). This paper presents a general and unifying framework built on abstract criteria where the maximum compression ratio is defined and computed for most relevant settings. The proofs involved in this theory are constructive, resulting in several methods for designing optimal compression matrices. A. Covariance Matching FormulationThe prior information restricts the structure of Σ, thus determining how much x can be compressed. When no information at all is a...

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