Constellation shaping for broadcast channels in practical situations

This paper analysis the problem of constellation shaping for degraded broadcast channels using a finite dimension constellation, where a single source communicates simultaneously with two receivers. We are interested in a particular situation where the private message to be sent to each user is limited to unity. This corresponds to a very practical situation, and hierarchical modulation is a special case of this framework. We investigate if higher achievable rates can be obtained by using non uniform constellations following a nonequiprobable distribution. Achievable rates regions are derived for Additive White Gaussian Noise (AWGN) and for finite input Pulse Amplitude Modulation (PAM) constellations. A noticeable shaping gain (compared with the equiprobable case) is observed with 4 PAM symbols, when achievable rates are maximized over the probability distribution of the input signals, the shape of the constellation, and the labeling.