Symbol decoding

Once the symbol clusters have been successfully retrieved, the original time-varying problem has been reduced to a simpler decoding problem, which is the only supervised part of the proposed algorithm. Symbols need to be assigned to each cluster, and to this end a small number of pilot symbol slots $\textbf{d}[i]$ is transmitted at the start of the symbol block, specifically $N_t$ . Not that these pilot symbol slots are not needed for the clustering process.

Defining the matrix of pilot symbols $\textbf{D}_p = [\textbf{d}[1],\textbf{d}[2],\dots,\textbf{d}[N_t]]$ and the matrix of corresponding received data $\textbf{X}_p = [\textbf{x}[1], \textbf{x}[2], \dots, \textbf{x}[N_t]]$ , an approximation of the initial channel matrix $\textbf{H}$ can be obtained as

$$\hat{\textbf{H}} = \textbf{X}_p \textbf{D}_p^{-1}.$$ (8)

The algorithm concludes by assigning the symbol slot $\textbf{d}$ to the cluster whose first data point in time is closest to the vector $\hat{\textbf{H}} \textbf{d}$ .