Diversity-multiplexing tradeoff

Introduction

For slow-fading channels, outage probability is one of the performance measures. There is a fundamental trade off between the rate and the outage probability. The exact expression of outage probability is often difficult to obtain. One way to simplify the problem is to consider high signal-to-noise ratio scenario to obtain a trade-off between the diversity order, which is the exponent of the outage probability, and the multiplexing gain, which is the pre-log factor of the rate.

Example

Consider a single-input single-output (SISO) slow-fading channel: \begin{equation} y = h x + n \end{equation}

where $y$ is the output, $h$ is the fading coefficient, which is a circularly-complex Gaussian random variable with variance 1, $x$ is the input, $n$ is circularly-complex Gaussian noise with variance $N$. The channel coefficient is random, but once picked, it remains fixed for the rest of the transmission. Define the outage event $\mathcal{O}$ to the event that the mutual information between the input and the output less than the information rate $R$, i.e. $\mathcal{O} = \{I(X;Y)<R\}$. This case is simple enough to compute the exact outage probability. \begin{equation} \mathbb{P}(\mathcal{O}) = \mathbb{P}\left( \log(1+|h|^2 \text{SNR}) < R \right) = \mathbb{P}\left( |h|^2 < \frac{e^R-1}{\text{SNR}} \right) = 1 - \exp\left( \frac{e^R-1}{\text{SNR}} \right) \end{equation} Consider the rate of codes use grows with $\text{SNR}$ such that $R(\text{SNR}) = r \log(\text{SNR})$, then in the high-SNR regime \begin{equation} \mathbb{P}(\mathcal{O}) \approx \frac{\text{SNR}^r-1}{\text{SNR}} \approx \text{SNR}^{-(1-r)} \end{equation} the diversity is $d(r) = 1-r$ for $r \geq 0$.

Multiple-Input Multiple-Output Point-to-Point Channel