Multidimensional Analog Modulation and Optimal Mappings

We consider nonlinear n-dimensional modulations for transmission of uniformly distributed analog source. We show that under certain conditions, a class of constant stretch modulations yields purchase of n dB in resulting SNR for every dB of transmitted power while keeping the probability of threshold effect constant. We also derive the optimal mappings for a class of modulations with two-dimensional spiral trajectory and show that constant stretch modulations are close to optimal in this case. Receiver design for two-dimensional case is proposed.