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). If each of the component codes encodes K independent complex symbols and is of transmission duration T, the rate of the FFS is K/T complex symbols per channel use. 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 proposed necessary condition reduces to the well known necessary and sufficient condition for an STBC to achieve full-diversity, viz. 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 those FFSs in which the component STBC yielding 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.