MMSE Optimal Space-Time Block Codes from Crossed Product Algebras

Design of Space-Time Block codes (STBCs) for Maximum Likelihood (ML) reception has been predominantly the main focus of researchers. However the ML decoding complexity of STBCs become prohibitive large as the number of transmit and receive antennas increase. To solve this problem, orthogonal designs, single symbol ML decodable and double symbol ML decodable STBCs have been proposed in the literature. But the rate of these STBCs fall exponentially with increase in the number of transmit antennas. Hence it is natural to resort to suboptimal reception techniques like linear Minimum Mean Squared Error (MMSE) receiver. Barbarossa et al and Liu et al have independently derived necessary and sufficient conditions for a full rate linear STBC to be MMSE optimal, i.e achieve least Symbol Error Rate (SER). Motivated by this problem, we construct a new class of full rate MMSE optimal STBCs for any number of transmit antennas using crossed product algebras. It is also shown that codes from cyclic division algebras which are special cases of crossed product algebras are also MMSE optimal. Hence these STBCs achieve least SER when MMSE reception is employed and are fully diverse when ML reception is employed.