A Family of Distributed Space-Time Trellis Codes Achieving Full Diversity for Asynchronous Cooperative Communications

In current cooperative communication schemes, to achieve cooperative diversity, synchronization between terminals is usually assumed, which may not be practical since each terminal has its own local oscillator. In this paper, we first present a necessary and sufficient condition for the space-time trellis codes based on the stack construction proposed by Hammons and El Gamal to possess the full cooperative diversity order without the synchronization assumption. The condition is that the binary generator metrices of the trellis codes are shift full rank (SFR) matrices, i.e., have full row rank no matter the shifts of their row vectors, where a row corresponds to a terminal (or transmit antenna) and the length of a row vector is the memory size of the corresponding trellis code on the corresponding terminal. We then present a simple construction of SFR for any number of rows, whose number of columns, however, grows exponentially with the number of rows. We finally present some systematic SFR matrix constructions including shortest, i.e., square, SFR matrices.