Determining the Dimension of the Improper Signal Subspace in Complex-Valued Data

Abstract

A complex-valued signal is improper if it is correlated with its complex conjugate. The dimension of the improper signal subspace, i.e., the number of improper components in a complex-valued measurement, is an important parameter and is unknown in most applications. In this letter, we introduce two approaches to estimate this dimension, one based on an informationtheoretic criterion and one based on hypothesis testing. We also present reduced-rank versions of these approaches that work for scenarios where the number of observations is comparable to or even smaller than the dimension of the data. Unlike other techniques for determining model orders, our techniques also work in the presence of additive colored noise.

Asymptotic regime for impropriety tests of complex random vectors

“…where χ 2 N (N +1) denotes the chi-squared distribution with N (N + 1) degrees of freedom. Note that this approximation was used recently in [26], in a high-dimensional setting in the absence of better approximation.…”