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The multiple-input multiple-output (MIMO) wiretap channel (WTC), which has atransmitter, a legitimate user and an eavesdropper, is a classic model forstudying information theoretic secrecy. In this paper, the fundamental problemfor the complex WTC is whether the proper signal is optimal has yet to be givenexplicit proof, though previous work implicitly assumed the complex signal wasproper. Thus, a determinant inequality is proposed to prove that the secrecyrate of a complex Gaussian signal with a fixed covariance matrix in a degradedcomplex WTC is maximized if and only if the signal is proper, i.e., thepseudo-covariance matrix is a zero matrix. Moreover, based on the result of thedegraded complex WTC and the min-max reformulation of the secrecy capacity, theoptimality of the proper signal in the general complex WTC is also revealed.The results of this research complement the current research on complex WTC. Tobe more specific, we have shown it is sufficient to focus on the proper signalwhen studying the secrecy capacity of the complex WTC.

Optimality of the Proper Gaussian Signal in Complex MIMO Wiretap Channels