A New Finite Series Expansion of Continuous Phase Modulated Waveforms

The Laurent Decomposition expresses any binary single-h continuous phase modulated (CPM) signal as the summation of a finite number of pulse amplitude modulated (PAM) waveforms, and the resulting signal space is so constructed that the waveform can usually be synthesized with a reasonable degree of accuracy by using only the ldquomainrdquo 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 show that by expressing a CPM signal in its equivalent binary multi- form, we can derive a generalization of Laurent´s result for the general class of such waveforms that use noninteger modulation indices. In this new signal representation, there is a clearly identifiable data-dependent ldquomainrdquo expansion pulse during each symbol interval which carries most of the signal energy. As in the Laurent Decomposition, the number of terms in this decomposition is only dependent on the CPM signal memory.