pseudocovariance

Pseudocovariance, also called pseudo-covariance or complementary covariance, is a second-order statistic used for complex random variables. For a complex random vector Z with mean μ, the pseudocovariance is defined as E[(Z − μ)(Z − μ)ᵀ], while the ordinary covariance is E[(Z − μ)(Z − μ)ᴴ], where ᵀ denotes transpose and ᴴ denotes conjugate transpose. The pseudocovariance captures the correlation between a complex variable and its own complex conjugate, and it is a key descriptor of noncircular (improper) complex signals.

The relationship between the two statistics determines circularity. If Z is proper or circular, its pseudocovariance is zero; a nonzero pseudocovariance indicates noncircularity, meaning the distribution does not remain invariant under multiplication by a unit-magnitude complex scalar. In the scalar case, pseudocovariance is a single complex number; in the vector case, it is a matrix called the complementary covariance matrix.

Estimation and interpretation are straightforward in practice. The sample pseudocovariance matrix is formed as (1/n) Σ (Zᵢ − μ̂)(Zᵢ − μ̂)ᵀ, where μ̂ is the sample mean. Together with the covariance matrix, it provides a complete second-order description for Gaussian complex vectors and improves modeling of non-Gaussian or noncircular signals. In signal processing and communications, pseudocovariance is important for processing improper complex signals, such as certain modulation schemes or polarized waves. Algorithms that account for pseudocovariance often use augmented representations that pair Z with its complex conjugate Z*, enhancing performance in the presence of improperness.