4.11 Summary of Important Results

• Space-time block codes from orthogonal designs provide:

  • simple decoding: maximum-likelihood decoding results in each symbol being decoded separately using only linear processing;

  • maximum diversity: the code satisfies the rank criterion.

• A real orthogonal design of size N is an N N orthogonal matrix _N with real entries x ₁, ? x ₁, x ₂, ? x ₂, , x _N, ? x _N such that

  
  

• A real orthogonal design exists if and only if N = 2, 4, 8.

• A generalized real orthogonal design is a T N matrix with real entries x ₁, ? x ₁, x ₂, ? x ₂, , x _K, ? x _K such that

  
  

  where I _N is the N N identity matrix and ? is a constant. The rate of the code is R = K/ T.

• For any number of transmit antennas, N, there exists a full rate, R = 1, real space-time block code with a block size T = min[2 ^{4 c + d}], where the minimization is over all possible integer values of c and d in the set { c ? 0, d ? 08 c + 2 ^d ? N}.

• A complex orthogonal design of size N is an