A Quadratic Form Approach to ML Decoding Complexity of STBCs

Abstract— A linear space-time block code (STBC) is a vector space spanned by its defining weight matrices over the real number field. We define a Quadratic Form (QF), called the Hurwitz-Radon QF (HRQF), on this vector space and give a QF interpretation of the ML decoding complexity of a STBC. It is shown that the ML decoding complexity is only a function of the weight matrices defining the code and their ordering, and not of the channel realization (even though the equivalent channel when sphere decoding is used depends on the channel realization) or the number of receive antennas. It is shown that the ML decoding complexity is completely captured into a single matrix obtained from the HRQF. Also, given a set of weight matrices, an algorithm to obtain the best ordering of them leading to the least ML decoding complexity is presented. The well known classes of low ML decoding complexity codes (multi-group decodable codes, fast decodable codes and fast group decodable codes) are presented in the framework of HRQF.