Catastrophic continuous phase modulation schemes and their noncatastrophic equivalents

Continuous phase modulation (CPM) schemes are bandwidth and energy efficient constant-envelope modulation schemes that can be viewed as a continuous-phase encoder (CPE) followed by a memoryless modulator (MM), where the CPE is of convolutional type. It is observed that CPM schemes can be catastrophic in the sense that pairs of input sequences that differ in an infinite number of positions can be mapped into pairs of signals with finite Euclidean distance. This can happen in spite of the fact that the CPE is never catastrophic when considered as a stand alone convolutional encoder. The necessary and sufficient condition for a general CPM scheme to be catastrophic is given. Each member of the two major families of CPM schemes, namely the LREC and the LRC, has been classified as a catastrophic or noncatastrophic scheme. For the catastrophic schemes, the probability that a catastrophic event occurs is determined. A canonical precoder which transforms each scheme of both families into an equivalent noncatastrophic scheme is derived. The equivalent noncatastrophic scheme has the same number of states as the original one. Moreover, it has the property that if two input sequences differ in the ith position, the corresponding output signals have nonzero Euclidean distance in the ith interval