Intrinsic Secrecy of EGT and MRT Precoders for Proper and Improper Modulations

Abstract

This paper makes an information theoretical analysis of the intrinsic secrecy level of M-QAM and M-PSK modulation schemes considering the use of equal gain transmission and maximum ratio transmission precoding techniques. In addition to the analysis of the conventional proper M-QAM and M-PSK constellations, a recently proposed family of improper versions of the M-QAM and M-PSK modulation schemes is also evaluated. With the exception of proper M-PSK, which verifies always full secrecy for the considered precoders, the main results show that for low order constellations, the amount of intrinsic secrecy provided by the combination of the precoder and modulation scheme is significant and, therefore, can be exploited in the design of a full secrecy solution. The theoretical derivations provided in this paper can be directly applied to quantify the minimal entropy that a secret key must have to fully secure the exchange of information for these transmission schemes.