Polar Decomposition of Mutual Information Over Complex-Valued Channels

A polar decomposition of mutual information between a complex-valued channel's input and output is proposed for an input whose amplitude and phase are independent of each other. The mutual information is symmetrically decomposed into three terms: 1) an amplitude term; 2) a phase term; and 3) a cross term, where the cross term is negligible. Theoretical bounds of the amplitude and phase terms are derived for additive white Gaussian noise channels with Gaussian inputs. This decomposition is applied to amplitude phase shift keying with product constellation (product-APSK) design and analysis. It facilitates the product-APSK to achieve a considerable shaping gain at some interesting rates over conventional quadrature amplitude modulation with a similar low complexity.