On orthogonal designs and space-time codes

Two aspects of orthogonal designs are explored in this paper. The first relates to the existence of restricted-alphabet orthogonal designs. While orthogonal designs yield efficient space-time codes, real orthogonal designs exist only for sizes n=2, 4, 8. This raises the question of existence of orthogonal designs of other sizes when the alphabet is restricted to be a finite or infinite proper subset of the real numbers. We answer this question in the negative by showing that the only exception is when the signal alphabet is BPSK. We provide an example construction that yields orthogonal designs with alphabet {±1} whenever the size of the alphabet n is such that an (n×n) binary Hadamard matrix exists. Binary Hadamard matrices are conjectured to exist whenever n is a multiple of 4. Our second observation concerns the recent interesting differential-detection space-time coding scheme of Jafarkhani and Tarokh (see IEEE Trans. Inform. Theory, vol.47, p.2626-31, Sept. 2001) that is based on orthogonal designs and that possesses a very efficient decoding algorithm. We provide a simpler description of this differential detection scheme that also makes a connection with the differential modulation scheme of Hughes (see IEEE Trans. Inform. Theory, vol.46, p.2567-78, Nov. 2000).