Empirical Model for Spectrally Efficient Continuous Phase Modulation

The continuous phase property of Continuous Phase Modulation (CPM) makes it possible to define spectrally efficient digital modulation schemes with a narrow spectral main lobe and small spectral side lobes by using a small modulation index, a non-binary alphabet size and a smooth phase pulse [1]. The constant envelope property of CPM also allows the use of non-linear amplifiers, which have lower cost and significantly higher power efficiency than linear amplifiers, since they can work in a saturated mode. In previous work [2], we searched for optimum phase pulses and modulation indices for spectrally efficient CPM. All the resulting CPM schemes provide the same information bit rate and satisfy the same spectrum requirement. In this paper we analyze some properties of these numerically optimized CPM schemes. It is found that the optimized schemes behave similarly for all alphabet sizes and that they follow a simple relation with the normalized one-sided effective bandwidth of CPM. We derive the expression for the normalized one-sided effective bandwidth of M -ary CPM and derive an approximate expression for the minimum Euclidean distance. Then, we define some phase pulse parameters that are approximate invariants. Using these approximate invariants as empirical constants for optimum phase pulses, we derive an expression for the minimum Euclidean distance as a function of the alphabet size. This empirical model predicts the minimum Euclidean distance of optimized CPM with even-integer alphabet sizes in the range M = 4 − 32 and that satisfy the same information bit rate and spectral requirement. The empirical model can be used for predicting the behavior of optimized spectrally efficient CPM schemes with requirements on both small spectral main lobe and low spectral side-lobes. Motivated by the results for the optimized CPM schemes in [2], we investigate properties of optimized CPM schemes for two other less spectrally efficient cases by scaling the frequency axis with the factor 1.5 and 2, respectively. We find that the empirical constants are still approximate invariants, but the relation with the normalized one-sided effective bandwidth becomes more complicated. Finally, we investigate under what circumstances the LRC and the GMSK phase pulses satisfy the empirical constants and find that the LRC phase pulse is close for L = 3, which is the so-called saturation length of the optimized CPM schemes with large M and that the GMSK phase pulse can be made close by adjusting the BT parameter for lengths L ≥ 2. We define the saturation length Ls and show that the gain in minimum Euclidean distance of the CPM phase code is small when increasing the phase pulse length L beyond Ls. For saturated schemes, i.e. L = Ls, using jointly optimized phase pulse and modulation index, the numerically optimized