On Space-Time Trellis Codes Achieving Optimal Diversity Multiplexing Tradeoff

Multiple antennas can be used for increasing the amount of diversity (diversity gain) or increasing the data rate (the number of degrees of freedom or spatial multiplexing gain) in wireless communication. As quantified by Zheng and Tse [1], given a Multiple Input Multiple Output (MIMO) channel, both gains can, in fact, be simultaneously obtained, but there is a fundamental tradeoff (called the Diversity-Multiplexing Gain (DM-G) tradeoff) between how much of each type of gain, any coding scheme can extract. Space-time codes (STC's) can be employed to make use of these advantages offered by multiple antennas. STC's can be broadly classified in two categories; namely space-time block codes (STBC) and space-time trellis codes (STTC). STTCs are known to have better bit error rate performance than STBCs, but with a penalty in decoding complexity. Also, for STTCs, the frame length is assumed to be finite and hence zeros are forced towards the end of the frame (called the trailing zeros), inducing rate loss. In this paper, we derive an upper bound on the DM-G tradeoff of full-rate STTCs with non-vanishing determinant (NVD). Also, we show that the full-rate STTCs with NVD are optimal under the DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros. Next we give a explicit generalized full-rate STTC construction for any number of states of the trellis, which achieves the optimal DM-G tradeoff for any number of transmit and receive antennas, neglecting the rate loss due to trailing zeros.