# For a wireless channel, what is the difference between Ergodic capacity and Outage capacity?

What do the terms Ergodic and Outage mean? And how do they relate to Shannon capacity?

Ergodic Capacity: is the same as Shannon Capacity.

Outage Capacity: is the highest rate of communication that occurs given a certain outage probability. A measure for slow fading channels.

This is good place to start: https://en.wikipedia.org/wiki/Channel_capacity

Imagine that you want to transmit a block of symbols over a wireless channel (infinite block if we want to discuss capacity). The wireless channel will vary throughout your transmission due to mobility and multi-path and therefore, your received symbols will undergo fading. Now, one may refer to two different scenarios, one where the fading is varying fast and one where the fading is varying slowly. I will assume that we are aware of the fading realization, i.e., we have CSI.

The fading remain constant over a small number of symbols and hence, we may obtain a good estimate of the capacity by averaging what we receive over the fading realizations. This is called the ergodic regime. Hence, the ergodic capacity is given as $$C_{\mathrm{erg}} = \mathrm{E}_{H}\!\left[ \log(1+\vert H \vert^2\gamma) \right]$$ where $\gamma$ is the SNR and $H$ the fading coefficient.

In this regime, the fading stays constant over a large number of symbols. Therefore, averaging is not possible since we will not have enough fading realizations. Also, since the fading can be arbitrarily small and remain for a very long time, there is no way to guarantee that we can communicate at any rate above zero with arbitrarily small error probability.

Instead, we take another route and define the outage probability as

$$p_\mathrm{out}(R)= \Pr\left( R > C(H) \right)$$

where $C(H) = \log(1+\vert H\vert^2 \gamma)$ is the instantaneous capacity.

We can now define the outage capacity (also called $\epsilon$-capacity as the largest rate that we can communicate with, given that the probability of being in outage is smaller than $\epsilon$. Hence, $$C_{\epsilon} = \sup\! \left \lbrace R:p_\mathrm{out}(R) < \epsilon \right \rbrace.$$