Continuous phase modulation (CPM) is a very attractive digital modulation scheme, with constant envelope feature and high efficiency in meeting the power and bandwidth requirements. CPM signals with pairs of input sequences that differ in an infinite number of positions and map into pairs of transmitted signals with finite Euclidean distance (ED) are called catastrophic. In the CPM scheme, data sequences that have the catastrophic property are called the catastrophic sequences; they are periodic difference data patterns. The catastrophic sequences are usually with shorter length of the merger. The corresponding minimum normalized squared ED (MNSED) is smaller and below the distance bound. Two important CPM schemes, viz., LREC and LRC schemes, are known to be catastrophic for most cases; they have poor overall power and bandwidth performance. In the literatures, it has been shown that the probability of generating such catastrophic sequences are negligible, therefore, the asymptotic error performance (AEP) of those well-known catastrophic CPM schemes evaluated with the corresponding MNSED, over AWGN channels, might be too negative or pessimistic. To deal with this problem in AWGN channel, this paper presents a new split-merged MNSED and provide criteria to explore which conventional catastrophic CPM scheme could increase the length of mergers with split-merged non-periodic events, effectively. For comparison, we investigate the exact power and bandwidth performance for LREC and LRC CPM for the same bandwidth occupancy. Computer simulation results verify that the AEP evaluating with the split-merged MNSED could achieve up to 3dB gain over the conventional approach.

Conventional CPM signals employ information sequence with time-unlimited phase shaping pulse (PSP) to achieve power and bandwidth efficient transmission. On the contrary, information sequence using time-limited PSP was believed to produce power-wasting data-independent discrete spectral lines in CPM spectra, and was suggested to be avoided. In this paper, we revisit this problem and adopt the time-limited PSP to replace the one with time-unlimited, it turns out to have an alternative solution to the CPM scheme. We first modify the spectral computing formula for the CPM with time-limited PSP (or CPM-TL) from conventional CPM formula and show that the discrete spectral lines appeared in the power density spectrum of CPM-TL signals can be diminished or become negligible by appropriately choosing PSP. We also show that this class of CPM can use any real number modulation index (h) and the resultant trellis structure of CPM guarantees the maximum constraint length allowed by the number of states in the MLSD receiver. Finally, the energy-bandwidth performance of CPM using time-limited PSP is investigated and compared with conventional CPM with time-unlimited PSP. From numerical results we show that, under the same number of states in the MLSD receiver and bandwidth occupancy, this subclass of CPM could outperform the conventional CPM up to 6dB coding gain, for h<1, in many cases.