Statistical mechanical analysis of the Kronecker channel model for multiple-input multiple-output wireless communication

The Kronecker channel model of wireless communication is analyzed using statistical mechanics methods. In the model, spatial proximities among transmission/reception antennas are taken into account as certain correlation matrices, which generally yield nontrivial dependence among symbols to be estimated. This prevents accurate assessment of the communication performance by naively using a previously developed analytical scheme based on a matrix integration formula. In order to resolve this difficulty, we develop a formalism that can formally handle the correlations in Kronecker models based on the known scheme. Unfortunately, direct application of the developed scheme is, in general, practically difficult. However, the formalism is still useful, indicating that the effect of the correlations generally increase after the fourth order with respect to correlation strength. Therefore, the known analytical scheme offers a good approximation in performance evaluation when the correlation strength is sufficiently small. For a class of specific correlation, we show that the performance analysis can be mapped to the problem of one-dimensional spin systems in random fields, which can be investigated without approximation by the belief propagation algorithm.

Figure 1

(a) Schematic picture of the MIMO system. (b) The MIMO system with correlation. Correlation is propagated through adjacent transmission antennas. (c) The correlated system with unitary or orthogonal matrix multiplication. Correlations are smeared by the multiplication of unitary or orthogonal matrix.

Figure 2

Conditional entropy $h_{\mathit{I}+\rho \mathit{R}}\left(\mathit{b}\mid \mathit{r}\right)$ for chainlike system [Eq. (1)]. Entropy vs SNR [(a) $\rho =0.2$ and (b) $\rho =0.5$]. The solid lines show the results obtained by exact analysis, the broken lines show the results obtained by matrix integration, and the dotted lines show the results for iid channel.

Figure 3

Conditional entropy $h_{\mathit{I}+\rho \mathit{R}}\left(\mathit{b}\mid \mathit{r}\right)$ for chainlike system [Eq. (2)]. Entropy vs correlation parameter $\rho$ [(a) $\text{SNR}=2\text{dB}$ and (b) $\text{SNR}=6\text{dB}$]. The solid lines show the results obtained by exact analysis, the broken lines show the results obtained by matrix integration with correction from the fourth order, and the dotted lines show the results obtained by matrix integration without correction.

Figure 4

Performance by the replica analysis and the result of demodulation experiment [Eq. (1)]. BER vs SNR [(a) $\rho =0.2$ and (b) $\rho =0.5$]. We set the parameters $K=4400$ and $L=4000$ and take the average of the results over 128 samples with various input signals $\mathit{b}$, matrices $\mathbf{\Xi }$, and noises $\mathit{\eta }$. We also varied the orthogonal matrix $\mathit{O}$ for the model with orthogonal-matrix multiplication. The lines depict the results of the replica analysis for two models, the chainlike model ${\mathit{R}}_{\text{t}}=\mathit{I}+\rho \mathit{R}$ with tridiagonal $\mathit{R}$ (solid) and the model with orthogonal-matrix multiplication ${\mathit{R}}_{\text{t}}=\mathit{I}+\rho {\mathit{O}}^T\mathit{RO}$ (dotted). The symbols in the figure denote the results of demodulations, namely, demodulation for the chainlike model $(\ast )$, demodulation for the model with orthogonal-matrix multiplication $(\times )$, inappropriate choice of the demodulation algorithm, i.e., the demodulation algorithm for the orthogonal-matrix multiplication model applied to the chainlike model $(+)$.

Figure 5

Performance by the replica analysis and the results of demodulation experiment [Eq. (2)]. BER vs correlation parameter $\rho$ [(a) $\text{SNR}=5\text{dB}$ and (b) $\text{SNR}=7\text{dB}$’. We set the parameter $K=4400$ and $L=4000$ and take the average over 1024 samples for various values of $\mathit{b}$, $\mathbf{\Xi }$, $\mathit{\eta }$, and $\mathit{O}$. The lines depict the results of the replica analysis for two models, the chainlike model ${\mathit{R}}_{\text{t}}=\mathit{I}+\rho \mathit{R}$ with tridiagonal $\mathit{R}$ (solid) and the model with orthogonal-matrix multiplication ${\mathit{R}}_{\text{t}}=\mathit{I}+\rho {\mathit{O}}^T\mathit{RO}$ (dotted). The symbols in the figure denote the results of demodulations, namely, demodulation for the chainlike model $(\ast )$, demodulation for the model with orthogonal-matrix multiplication $(\times )$, inappropriate choice of the demodulation algorithm, i.e., the demodulation algorithm for the orthogonal-matrix multiplication model applied to the chain-like model $(+)$.