Noncircularity

Noncircularity is a term used in statistics and signal processing to describe departure from circular (rotational) symmetry in complex-valued data. In the context of complex random variables or vectors, a zero-mean variable z is said to be circular (proper) if its distribution is invariant under multiplication by any unit phasor e^{jθ}, that is, z and e^{jθ}z have the same distribution for all θ. If this invariance fails, z is noncircular (improper).

Second-order statistics provide a concise description of circularity. The covariance is C = E[z z^H] and the pseudo-covariance (or relation) is P = E[z z^T]. Circularity is equivalent to P = 0. A nonzero pseudo-covariance indicates noncircularity, and the degree of noncircularity is often quantified by the noncircularity coefficient κ = |P| / E[z^H z] (for a zero-mean scalar z, κ = |E[z^2]| / E[|z|^2]).

Noncircularity has practical consequences for data analysis and signal processing. Many standard estimators and detectors assume circularity; when signals are noncircular, these methods can be suboptimal. Techniques that use augmented statistics, incorporating both z and its conjugate z*, can exploit noncircularity to improve performance in tasks such as parameter estimation, direction finding, and blind source separation.

In directional statistics, noncircularity can also describe departures from uniform rotational symmetry on the circle, indicating a concentration of data toward preferred angles rather than a uniform spread.