A New Finite Series Representation for Continuous Phase Modulation

The Laurent decomposition expresses any binary single-h continuous phase modulated signal as the finite summation of pulse amplitude modulated (PAM) waveforms, and the resulting signal space is so constructed that the CPM signal can usually be synthesized with a reasonable degree of accuracy by using only the "main" PAM component pulse. This derivation has been very useful for reduced complexity demodulation of binary CPM signals. Subsequent to Laurent's work, it was shown that commensurate expressions could be obtained for multilevel and multi-h CPM, but with an exponential increase in the total number of PAM component pulses in the signal representation. In this paper, we derive a generalization of Laurent's result which can be universally applied to all variants of CPM which use a non-integer modulation index. Most notably, the component pulses are naturally ranked in order of decreasing signal energy, so that over each symbol interval there is a single "main pulse" that can be used in a good first-order finite series approximation of the CPM signal