Incorporating the temporal dimension into the clustering problem

The received data $\textbf{x}[n]$ in a fast time-varying MIMO system can be preprocessed for spectral clustering by simply adding the temporal dimension. If the temporal index is $t[n]$ , the combined vector of data points and temporal indices, $\textbf{x}^{+}[n] = [\textbf{x}[n]^T, t[n]^T ]^T$ , is an $(N_r+1) \times 1$ complex vector. When this extra dimension is added to the scatter plots of Fig. 1, threads appear due to the temporal correlation between consecutive channel matrices (see Fig. 3). Given the non-convex clustering capabilities of spectral clustering algorithms, they should be capable of retrieving the different threads from $\textbf{x}^{+}[n]$ .

Figure 3: Scatter plots of the data of Fig. 1 to which the temporal dimension was added. Threads of data points are now distinguishable in both figures.

(a)	(b)

The performance of a suitable spectral clustering algorithm depends mainly on two factors. In the first place, the number of data points $N$ in one block must be larger than the number of clusters (a rule of thumb is to have at least $10$ samples per cluster). Since spectral clustering is a computationally costly procedure, the number of clusters to detect should be limited. For constellations with alphabet size $M$ (the cardinality) this number of clusters is $M^{N_t}$ , which is exponential in $N_t$ . Taking into account that most commercial MIMO systems use up to $N_t = 4$ transmit antennas, we will only treat BPSK systems ($M=2$ ) in this work. Extensions to more complex modulations will be considered for future investigation.

In the second place, clusters should be well connected, i.e., the distance between neighboring points of the same thread should not be larger than the distance between points of different threads. This requires a rescaling of the temporal dimension to match the scale of the spatial dimensions, for instance $t[n] = n/256$ , $n=0,\dots,255$ for blocks of $256$ symbols. Moreover, this means that if a symbol is not sent during a considerable time, a thread might be incorrectly identified as two separate threads. However, as will be shown in the next section, both difficulties can be reduced by using information derived from the geometric properties of the constellation.