Recursive space-time trellis codes using differential encoding

Differential space-time modulation (DSTM) has been recently proposed by Hughes, and Hochwald and Sweldens when the channel information is not known at the receiver, where the demodulation is in fact the same as the coherent demodulation of space-time block coding by replacing the channel matrix with the previously received signal matrix. On the other hand, the DSTM also needs a recursive memory of a matrix block at the encoder and therefore provides a trellis structure when the channel information is known at the receiver, which is the interest of this paper. This recursive structure of the DSTM has been adopted lately by Schlegel and Grant in joint with a conventional binary code and joint iterative decoding/demodulation with a superior performance. The number of states of the trellis from the recursive structure depends on both the memory size, which is fixed in this case, and the unitary space-time code (USTC). When a USTC for the DSTM forms a group, the number of states is the same as the size of the USTC, otherwise the number of the states is the size of the semi-group generated by the USTC from all the multiplications of the matrices in the USTC. It is well known in the conventional convolutional coding (CC) or the trellis coded modulation (TCM), the free (Hamming or Euclidean) distance (or the performance) increases when the number of states increases by adding more memory with a properly designed CC or TCM. In this paper, we systematically study and design the USTC/DSTM for the recursive space-time trellis modulation and show that the diversity product increases when the number of states increases, which is not because of the memory size but because of the different USTC designs that generate different sizes of semi-groups. We propose a new USTC design criterion to ensure that the trellis structure improves the diversity product over the USTC as a block code. Based on the new criterion, we propose a new class of USTC design for an arbitrary number of transmit antennas that has an analytical diversity product formula for two transmit antennas. We then follow Schlegel and Grant's approach for joint encoding and iterative decoding of a binary coded DSTM (turbo space-time coding) and numerically show that our new USTC designs for the recursive space-time trellis modulation outperforms the group USTC used by Schlegel and Grant.