PAM decomposition of M-ary multi-h CPM

It is known that any multilevel continuous phase-modulated (CPM) signal with a single modulation index can be exactly represented by a sum of pulse-amplitude modulated (PAM) waveforms. In this paper, we show how multi-h CPM signals can also be represented in this manner. The decomposition is presented in general terms as a function of the alphabet size, modulation indexes, and phase pulse of the CPM scheme. The number of pulses required to exactly construct the signal is shown to increase over that previously given for single-h schemes; this increase is in proportion to the number of modulation indexes. We propose an approximation which significantly reduces the number of signal pulses and which minimizes the mean-squared error for an arbitrary set of modulation indexes. We show that this approximation can have two objectives: 1) to reduce the number of pulses in the same manner as has been proposed for single-h schemes; and/or 2) to reduce the number of multi-h pulses; we also show the conditions where this latter objective is most practical. We compare this minimum mean-squared error approximation with another method which was recently proposed for CPM. We also give numerical results on detection performance which demonstrate the practicality of the proposed approximation.