Low-Delay, High-Rate Nonsquare Complex Orthogonal Designs
Smarajit Das, Balaji Sundar Rajan · IEEE Transactions on Information Theory · 2012
The maximal rate of a nonsquare complex orthogonal design forntransmit antennas is [1/2]+[1/(n)] ifnis even and [1/2]+[1/(n+1)] ifnis odd and the codes have been constructed for allnby Liang (2003) and Lu (2005) to achieve this rate. A lower bound on the decoding delay of maximal-rate complex orthogonal designs has been obtained by Adams (2007) and it is observed that Liang's construction achieves the bound on delay fornequal to 1 and 3 modulo 4 while Lu et al.'s construction achieves the bound forn=0 , 1, 3 mod 4. Forn=2 mod 4, Adams (2010) have shown that the minimal decoding delay is twice the lower bound, in which case, both Liang's and Lu et al.'s construction achieve the minimum decoding delay. For large value ofn, it is observed that the rate is close to half and the decoding delay is very large. A class of rate-[1/2] codes with low decoding delay for allnhas been constructed by Tarokh (1999). In this paper, another class of rate-[1/2] codes is constructed for allnin which case the decoding delay is half the decoding delay of the rate-[1/2] codes given by Tarokh This is achieved by giving first a general construction of square real orthogonal designs which includes as special cases the well-known constructions of Adams, Lax, and Phillips and the construction of Geramita and Pullman, and then making use of it to obtain the desired rate-[1/2] codes. For the case of nine transmit antennas, the proposed rate-[1/2] code is shown to be of minimal delay. The proposed construction results in designs with zero entries which may have high peak-to-average power ratio and it is shown that by appropriate postmultiplication, a design with no zero entry can be obtained with no change in the code parameters.