Full-Rate, Full-Diversity, Finite Feedback Space-Time Schemes with   Minimum Feedback and Transmission Duration
  In this paper a MIMO quasi static block fading channel with finite N-ary delay-free, noise-free feedback is considered. The transmitter uses a set of N Space-Time Block Codes (STBCs), one corresponding to each of the N possible feedback values, to encode and transmit information. The feedback function used at the receiver and the N component STBCs used at the transmitter together constitute a Finite Feedback Scheme (FFS). Although a number of FFSs are available in the literature that provably achieve full-diversity, there is no known universal criterion to determine whether a given arbitrary FFS achieves full-diversity or not. Further, all known full-diversity FFSs for T<N_t where N_t is the number of transmit antennas, have rate at the most 1. In this paper a universal necessary condition for any FFS to achieve full-diversity is given, using which the notion of Feedback-Transmission duration optimal (FT-Optimal) FFSs - schemes that use minimum amount of feedback N given the transmission duration T, and minimum transmission duration given the amount of feedback to achieve full-diversity - is introduced. When there is no feedback (N=1) an FT-optimal scheme consists of a single STBC with T=N_t, and the universal necessary condition reduces to the well known necessary and sufficient condition for an STBC to achieve full-diversity: every non-zero codeword difference matrix of the STBC must be of rank N_t. Also, a sufficient condition for full-diversity is given for the FFSs in which the component STBC with the largest minimum Euclidean distance is chosen. Using this sufficient condition full-rate (rate N_t) full-diversity FT-Optimal schemes are constructed for all (N_t,T,N) with NT=N_t. These are the first full-rate full-diversity FFSs reported in the literature for T<N_t. Simulation results show that the new schemes have the best error performance among all known FFSs. 
Lakshmi Prasad Natarajan and B. Sundar Rajan
cs.IT math.IT