Adjusting the generalized likelihood ratio test of circularity robust to non-normality

Recent research have elucidated that significant performance gains can be achieved by exploiting the circularity/non-circularity property of the complex-valued signals. The generalized likelihood ratio test (GLRT) of circularity assuming complex normal (Gaussian) sample has an asymptotic chi-squared distribution under the null hypothesis, but suffers from its sensitivity to Gaussianity assumption. With a slight adjustment, by diving the test statistic with an estimated scaled standardized 4th-order moment, the GLRT can be made asymptotically robust with respect to departures from Gaussianity within the wide-class of complex elliptically symmetric (CES) distributions while adhering to the same asymptotic chi-squared distribution. Our simulations demonstrate the validity of the chi² approximation even at small sample lengths. A practical communications example is provided to illustrate its applicability. In passing, we derive the connection with the kurtosis of a complex random variable with a CES distribution with the kurtosis of its real and imaginary part.