Comparison

A number of simulations were carried out to illustrate the performance of the proposed algorithm. The following parameters were assumed: a BPSK signal was used, the channels were independent Rayleigh flat-fading and the temporal variation of each channel between a transmit and receive antenna pair was based on the Clarke-Gans model [10]. The symbols $\textbf{d}[n]$ were grouped into frames consisting of $N=256$ slots. In the first setup a MIMO system with $N_t = N_r = 2$ antennas was used, in the second setup $N_t = N_r = 4$ , and in the last setup $N_t=2$ and $N_r=4$ . In all cases the normalized Doppler frequencies $f_dT = 5 \cdot 10^{-4}$ and $f_dT = 10^{-3}$ were considered, where $f_d = f_c \cdot v/c$ with receiver velocity $v$ and $c$ is the speed of light. For a GSM symbol period $T = 3.7 \cdot 10^{-6}$ seconds and a carrier frequency $f_c = 900$ MHz, these normalized Doppler frequencies correspond to receivers moving at $162$ km/h and $324$ km/h, respectively. For a carrier frequency $f_c = 1800$ MHz, the normalized Doppler frequencies correspond to $81$ km/h and $162$ km/h, respectively.

The bit error rate (BER) curves of two algorithms were compared. In the first place the proposed spectral clustering method (referred to as SPC) was tested, in which self-tuning spectral clustering was applied with $L=5$ . The number of pilot symbol slots used was fixed as $N_t$ for this method. In the second place, the GDFE algorithm from [2] was applied, with forgetting factor $\lambda=0.95$ .

The BER against $Eb/N_0$ for the $2 \times 2$ , a $4 \times 4$ and a $2 \times 4$ setup are shown in Fig. 4, Fig. 5 and Fig. 6, respectively. Because the GDFE algorithm is essentially a supervised method, it requires transmitting more pilot symbols. Therefore, apart from its BER curves for $N_t$ pilots (both of which coincide, in all figures), a second set of BER curves was also displayed for a higher number of pilots, to achieve the same performance as the spectral clustering algorithm. For the three cases, the GDFE algorithm needs $32$ pilot symbols to achieve similar performance as the presented method when $f_dT = 5 \cdot 10^{-4}$ , and $8$ pilots when $f_dT=1\cdot 10^{-3}$ . Comparing Fig. 4 and Fig. 6 shows that the presented algorithm performs significantly better when receiver antennas are added. This can be explained by observing that the clusters will be more separated in space when dimensions are added to the data points.

In cases where only a few pilot symbols can be sent, the spectral clustering algorithm obtains superior performance for the tested MIMO systems. However, it requires the calculation of the eigenvectors of its affinity matrix, which generally requires $O(N^3)$ operations. In most cases this can be lowered to $O(N^2)$ [13] taking into account that the affinity matrix is symmetric and can be approximated by a tridiagonal matrix.

This analysis suggests that spectral clustering could be used as an initialization for the GDFE or any other supervised algorithm. Specifically, given only $N_t$ pilot symbol slots it can estimate a short symbol vector sequence which can be used as a pilot sequence for a (computationally more efficient) supervised algorithm.

Figure 4: BER curves for a $2 \times 2$ BPSK MIMO system.

Figure 5: BER curves for a $4 \times 4$ BPSK MIMO system.

Figure 6: BER curves for a $2 \times 4$ BPSK MIMO system.