Multicellular Alamouti scheme performance in Rayleigh and shadow fading

Abstract

In this paper, we study the performance of two downlink multicellular systems: a multiple inputs single output (MISO) system using the Alamouti code and a multiple inputs multiple outputs (MIMO) system using the Alamouti code at the transmitter side and a maximum ratio combining (MRC) as a receiver, in terms of outage probability. The channel model includes path-loss, shadowing, and fast fading, and the system is considered interference-limited. Two cases are distinguished: constant shadowing and log-normally distributed shadowing. In the first case, closed form expressions of the outage probability are proposed. For a log-normally distributed shadowing, we derive easily computable expressions of the outage probability. The proposed expressions allow for fast and simple performance evaluation for the two multicellular wireless systems: MISO Alamouti and MIMO Alamouti with MRC receiver. We use a fluid model approach to provide simpler outage probability expressions depending only on the distance between the considered user and its serving base station.

Appendix: Independence result

In this appendix, we recall an independence result presented in [9]. Consider zero mean complex Gaussian vectors $\textbf{h}_{0}=[h_{1,0},h_{2,0},...,h_{N,0}]^{H}$ and $\textbf{h}_{j}=[h_{1,j},h_{2,j},...,h_{N,j}]^{H}$ and let g _j be a random variable given by

$$ g_{j}=\frac{\textbf{h}^{H}_{0}\textbf{h}_{j}}{\left\|\textbf{h}_{0}\right\|}. $$

(58)

Since the elements of $\textbf{h}_{j}$ are i.i.d zero mean complex Gaussian, g _j conditioned on $\textbf{h}_{0}$ is also zero mean complex Gaussian. The mean and the variance of g _j can be calculated as follows:

$$ \textrm{E}[g_{j}|\textbf{h}_{0}]=\frac{\textbf{h}^{H}_{0}}{\left\|\textbf{h}_{0}\right\|}\textrm{E}[\textbf{h}_{j}]=0, $$

(59)

$$ \textrm{E}[|g_{j}|^{2}|\textbf{h}_{0}] = \frac{\textbf{h}^{H}_{0}\textrm{E}[\textbf{h}_{j}\textbf{h}_{j}^{H}]\textbf{h}_{0}}{\left\|\textbf{h}_{0}\right\|^{2}}, $$

(60)

$$ =\frac{\textbf{h}^{H}_{0}\textbf{I}_{N}\textbf{h}_{0}}{\left\|\textbf{h}_{0}\right\|^{2}}, $$

(61)

$$ = 1, $$

(62)

$\textbf{I}_{N}$ being the identity matrix of dimension N.

The pdf of g _j conditioned on $\textbf{h}_{0}$ can thus be written as

$$ f_{g_{j}}(g_{j}/\textbf{h}_{0})=\frac{1}{\pi}\exp\left(-|g_{j}|^{2}\right). $$

(63)

From the expression of the pdf, it can be clearly stated that g _j is independent of $\textbf{h}_{0}$.