Effect of discrete constellations on duality between the Gaussian Multiple Access and the Gaussian Broadcast Channel

The Gaussian Multiple Access Channel (G-MAC) and the Gaussian Broadcast Channel (G-BC) are known to be duals of each other for Gaussian alphabets and their capacity regions are closely related. In this paper, we investigate the duality between the G-MAC and the G-BC when the sources use discrete constellations. Considering the two-user G-MAC and the G-BC and assuming uncoded Pulse Amplitude Modulation (PAM), we show that a rate pair achieved in the G-MAC can be translated to a rate pair in the dual G-BC, such that the equal sum power constraint be satisfied. Due to the similarity of the rate expressions to Shannon´s capacity formula, for an appropriate choice of Signal to Noise Ratio (SNR) gap, we show that when finite constellations are used for transmission, rate regions of these two channels also have a dual relationship (the known for Gaussian alphabets). Thus the rate region of the G-BC can be characterized from the rate region of the dual G-MAC and vice versa.