I have trouble distinguishing between these two concepts. This is my understanding so far.
A stationary process is a stochastic process whose statistical properties do not change with time. For a strict-sense stationary process, this means that its joint probability distribution is constant; for a wide-sense stationary process, this means that its 1st and 2nd moments are constant.
An ergodic process is one where its statistical properties, like variance, can be deduced from a sufficiently long sample. E.g., the sample mean converges to the true mean of the signal, if you average long enough.
Now, it seems to me that a signal would have to be stationary, in order to be ergodic. And what kinds of signals could be stationary, but not ergodic? If a signal has the same variance for all time, for example, how could the time-averaged variance not converge to the true value?
So, what is the real distinction between these two concepts? Can you give me an example of a process that is stationary without being ergodic, or ergodic without being stationary?