A Four Transmit Antenna Orthogonal Space-Time Block Code with Full Diversity and Rate Lori A. Dalton and Costas N. Georghiades Department of Electrical Engineering Texas A&M University College Station, Texas 77843 {ldalton, georghia}@ee.tamu.edu 1 Introduction The Alamouti code [1], remarkable for having an elegant linear receiver, is now a paradigm in space-time block coding. Orthogonal designs [2] generalize Alamouti’s scheme to use more antennas. Unfortunately, the Hurwitz-Radon theorem shows that complex orthogo- nal designs can not achieve full diversity and rate simultaneously for all symbol constella- tions, except in the two transmit antenna case [2]. Other codes show promise, including the STTD-OTD code [3], which is orthogonal but not full diversity, and constellation rotating codes [4], which achieve full rate and diversity but are not orthogonal. In this work we use carefully tailored rotated PSK constellations to design a full rate, full diversity complex orthogonal space-time block code for 4 transmit antennas. 2 The New Code We assume the standard slow, flat Rayleigh fading channel model. The goal is to design a N t = 4 transmit antenna orthogonal space-time block code with M -PSK constellations. The new code has the same form as the orthogonal STTD-OTD code [3] defined by, S =     s 1 s 2 s 3 s 4 −s * 2 s * 1 −s * 4 s * 3 s 1 s 2 −s 3 −s 4 −s * 2 s * 1 s * 4 −s * 3     =  A B A −B  , (1) where A and B are Alamouti blocks. The symbols of the data matrix S are redefined to be linear combinations of complex “base” symbols, d 1 , d 2 , d 3 , and d 4 . We use the following encoding scheme which normalizes the energy transmitted. s 1 = d 1 + d 2 √ 2 , s 2 = d 3 + d 4 √ 2 , s 3 = d 1 − d 2 √ 2 , s 4 = d 3 − d 4 √ 2 . (2) A measure of the quality of a square space-time code is the diversity product [5], ζ v = 1 2 min S 1 =S 2 ∈V |det (S 1 − S 2 )| 1 N t , where V is the set of all data matrices S. We observe that 0 ≤ ζ v ≤ 1 and any square code with ζ v > 0 is said to achieve full diversity. To find ζ v for the new code, notice that det (S 1 − S 2 ) = 4 det (A 1 − A 2 ) det (B 1 − B 2 ), where S 1 and S 2 are matrices of the same form as in(1) and A i and B i are Alamouti