  Unitary space-time modulation is known to be an efficient means to
communicate over non-coherent Multiple Input Multiple Output (MIMO) channels.
In this letter, differential unitary space-time coding and non-coherent
space-time coding for the training based approach of Kim and Tarokh are
addressed. For this approach, necessary and sufficient conditions for
multi-group decodability are derived in a simple way assuming a Generalized
Likelihood Ratio Test receiver and a unitary codebook. Extending Kim and
Tarokh's approach for colocated MIMO systems, a novel training based approach
to distributed non-coherent space-time coding for wireless relay networks is
proposed. An explicit construction of two-group decodable distributed
non-coherent space-time codes achieving full cooperative diversity for all even
number of relays is provided.
