Power and bandwidth efficiency of shaped modulation by decomposition

In this paper, it is shown that shaped modulations can be decomposed into a shape encoder (a finite state machine) and a memoryless modulator (MM). This decomposition allows the spectrum of shaped modulations to be computed numerically. Although the numerical technique used is not new, the decomposition that allows its implementation has not been recognized previously. In addition, decomposition allows the analytic computation of a union bound for the error probability, whereas previous error probability results have been based on simulation or measurement. The decomposition method shows that shaped modulation can be nonlinear, depending on the vector rotation algorithm (VRA) used; hence the error probability is data sequence dependent. Decomposition also suggests modifications to the VRA which makes for somewhat less dependency of the error probabilities on the data sequences. Finally, simulation results are given to show the performance of shaped binary phase-shift keying (BPSK) in Doppler spread fading channels.