Linear Diophantine Constrained Intra-Fix for Frequency Domain Equalization of Multi-h CPM

This paper considers the problem of constructing a symbol block to generate a cyclic multi-h continuous phase modulated (CPM) signal so as to permit frequency domain equalization (FDE) at the receiver. Like single-h CPM, the memory inherent in a multi-h CPM waveform requires the use of an additional segment, called an intra-fix, in each transmitted block. The intra-fix forces the transmitter state machine to regress to the state it had at the start of the block. The constraints on the symbols of the intra-fix, which ensure above condition, are shown to be a set of Linear Diophantine Equations (LDEs). Three different methods are proposed for finding the symbols of the intra-fix, such that the length of the intra-fix is minimized. The last two methods connect the problem of intra-fix insertion to the problem of trellis termination of the tilted phase. This connection enables us to state and justify that existing bounds established on the length of the terminating sequence are effective for the length of the intra-fix. We observe that multi-h CPM schemes typically require a shorter intra-fix, as compared to single-h CPM schemes having similar modulation parameters. We apply the proposed procedure to a dual-h CPM scheme.