Performance gains due to improper signals in MIMO broadcast channels with widely linear transceivers

Proper Gaussian signals have been shown to be optimal in multiple-input multiple-output (MIMO) broadcast channels from an information theoretic point of view, i.e., capacity can be achieved with a strategy that transmits circularly symmetric complex Gaussian signals. In this work, we show that optimality of proper Gaussian signals does not necessarily hold if the transmit strategy is restricted to widely linear transceivers. The proof is performed by identifying a rate tuple that is achievable in a certain set of channels with widely linear transceivers and improper Gaussian signals, but lies outside the achievable rate region for widely linear transceivers and proper Gaussian signals.