Problem formulation

MIMO systems are used in wireless communications to enhance signal diversity, spectral efficiency, or both. In a typical MIMO flat-fading system with $N_t$ transmit and $N_r$ receive antennas, the $N_r \times 1$ received vector $\textbf{x}[n]$ at time $n$ is expressed as

$$\textbf{x}[n] = \textbf{H}[n] \textbf{d}[n] + \textbf{v}[n]$$ (1)

where $\textbf{H}[n]$ is the $N_r \times N_t$ channel matrix whose elements represent independent flat-fading SISO channels, $\textbf{d}[n]$ contains the $N_t$ (in general, complex) symbols transmitted by the $N_t$ antennas at time $n$ , and $\textbf{v}[n]$ represents both spatially and temporally white complex zero-mean Gaussian noise.

In MIMO systems with block fading channels, variations of the channel during the transmission of one block of symbols are so small that they can be ignored. Hence the channel matrix $\textbf{H}[n] = \textbf{H}$ is considered constant during transmission of one block of symbols. This is not the case for MIMO systems with fast time-varying channels, where the channel matrix changes from symbol to symbol due to the Doppler spread caused by the movement of the transmitter and/or receiver. In time-varying MIMO systems, depending on the Doppler spread, the channel matrices $\textbf{H}[n]$ are temporally correlated. The variations can be modeled for instance by the Clarke-Gans model [10] which states that if a vertical $\lambda/4$ antenna with uniform power distribution is used to transmit a single tone, the received spectrum is

$$S_{E_z}(f) = \frac{1.5}{\pi f_m \sqrt{1-\left(\frac{f-f_c}{f_m}\right) ^2}},$$ (2)

where $f_c$ and $f_m$ are the carrier frequency and the maximum Doppler shift, respectively.

Figure 1: (a) and (b): Scatter plots of the data received by the two antennas of a $2 \times 2$ BPSK MIMO system with fast time-varying channels. The points corresponding to the symbols $[+1,+1]^T$ and $[-1,-1]^T$ are represented by circles and the points corresponding to $[+1,-1]^T$ and $[-1,+1]^T$ by black dots, emphasizing the symmetry of the used constellation. Note that no such information is available at receiver side.

(a)	(b)

The proposed method aims to estimate the symbols $\textbf{d}[n]$ given the received data points $\textbf{x}[n]$ . This problem is illustrated in Fig. 1, which shows typical scatter plots of the complex data $\textbf{x}_1[n]$ and $\textbf{x}_2[n]$ received by the two antennas in a time-varying $2 \times 2$ MIMO system with binary phase-shift keying (BPSK) modulation, for which the basic constellation points are $d \in \{+1,-1\}$ . Classical clustering algorithms that operate directly on the data of these scatter plots will fail due to overlapping of the clusters. In the next section we propose a solution to these problems that combines a spectral clustering approach with the incorporation of the temporal dimension into the clustering process.