TCM with Differential Encoding: Set Partitioning, Trellis Designs, and Distance Analysis

Differential encoding (DE) is a classical technique at the transmitter that allows simple noncoherent detection at the receiver. On the other hand, trellis-coded modulation (TCM) is a bandwidth efficient technique which offers reliable data transmission. In this paper, we aim to find the best concatenation order of TCM and DE for channel phase coherence over $N=2$ symbols. Besides the well-known TCM followed by DE (called TCM-DE), we propose a new trellis coding extension of DE which extends our earlier work, called differential trellis coded modulation (DTCM). DTCM is TCM with DE defined in states where distinct states may have distinct DE functions. We propose design methods of DE functions for noncoherently non-catastrophic DTCM. For both TCM-DE and DTCM, we propose additive distance measures and set partitioning. Based on the proposed set partitioning, trellis codes are designed or searched for both TCM-DE and DTCM. Both minimum distances and simulation results show that TCM-DE outperforms DTCM and the obtained new codes are better than the original codes.