Complex Parameter Rao, Wald, Gradient, and Durbin Tests for Multichannel Signal Detection

In the problem of multichannel signal detection, when it comes to the detector design criteria apart from the generalized likelihood ratio test, the traditional method is to cascade the real and imaginary parts of the parameters, and then substitute them into the real parameter statistics. This method is not succinct, and sometimes may be cumbersome and difficult to handle. Recently, a complex parameter Rao test was introduced by Kay and Zhu without the need of cascading the real and imaginary parts of the complex parameters when there is no nuisance parameter. Inspired by this work, we move a further step toward the complex parameter statistics of the Rao, Wald, gradient, and Durbin tests both with and without nuisance parameters, and derive the relationships between their real and complex parameter statistics. Moreover, for a special Fisher information matrix which often holds in practice, we derive a series of simple forms of the complex parameter statistics for the above four criteria, and discuss their application conditions in linear multivariate complex circular Gaussian distribution. Finally, several application examples are given to confirm the proposed schemes.