The information-theoretic optimality of QPSK

Quadrature Phase Shift Keying (QPSK) finds ubiquitous use in coherent communication systems. In this paper we show that in the low-power wideband regime, QPSK is a capacity-maximizing strategy for additive-noise Gaussian complex-valued channels subject to fading where the receiver has full information about the channel realization. The traditional asymptotic optimality criterion used since Shannon (1948) is that the derivative at zero SNR be optimal. However, this optimality criterion is not strong enough to withstand the test of spectrally efficient finite-bandwidth communication. In contrast, in this paper we propose a new asymptotic optimality criterion where both the first and second derivatives at zero SNR are required to be optimal. The setting of this paper is general enough to encompass multiantenna channels, frequency selectivity, and multiaccess channels. We also illustrate the robustness of QPSK in the absence of full receiver knowledge of the channel.