Differential phase shift keying of the quaternion carrier

Proposed a method of spatial differential phase shift keying of a carrier in 2n-dimensional state space, n =1,2,…, formed by the  dynamic equation using state transition matrices in the form of a real extension of the hypercomplex numbers obtained by the doubling procedure. Using the quaternion carrier as an example, the method of forming a reference matrix from a received signal with phase shift keying with a random phase shift is shown. It is shown that the reference matrix is orthogonal and represents the system matrix of the MIMO channel. Decoding matrices are obtained, which allow us to calculate the transmitted information symbols from the received information symbols. As a result, we get a gain in using the signal power for each channel, in general, 2n times.