Trellis Termination of Multi-h CPM and the Diophantine Frobenius Problem

This paper presents a method for terminating the phase trellis in multi-h continuous phase modulation (CPM). It is shown that tilted phase trellis for multi-h CPM signals is periodic and enables the representation of the multi-h CPM modulator as a cascade of a periodic recursive continuous phase encoder (CPE) and a phase modulator (PM). We then pose the problem of finding the minimum number of symbols that will terminate the phase trellis in a known state of the modulator and establish a connection of its solution to the Diophantine Frobenius Problem (FP). Analytical results are established for the existence of a terminating sequence and upper and lower bounds are derived on its length for arbitrary modulation parameters. We show that the trellis terminating symbols and their quantity depends upon the initial state of the encoder, the set of modulation indices, the CPE memory order, and the number of bits per modulated symbol. It is shown that required overhead for trellis termination can be significant for schemes with integer modulation indices.