# Practical implementation of Expected Value?

If i compute the average power of my input signal random variable $$X(t)$$ as $$R = E[X^2(t)]$$ i.e. as the expected value of a random process, is this really just an estimate of average power? More generally, is it true that we don't have (cannot compute) the Expected Values of actual input signals, because we have only one instance of a random signal to work with? And so we are stuck with just estimates of Expected Values, instead of the real, actual Expected Values?

thanks, js.

The answer is yes. Even if we could generate several realizations of the random process by conducting the random experiment a number of times. We could only have a limited number of realizations and for a limited period of time practically. Now to take expectation of the random process we will have to take freeze time and look at the distribution of the values across the several realizations we generated. But that would be an estimated expectation because we can only estimate the distribution, because although we have conducted the experiment several times, we cannot practically be sure to exhaust the sample space.

• I agree with everything you said but note that the assumption of ergodicity ( which can't be tested as far as I know ) means that having one loooooooooooooooooooooooooooooooooooong series is equivalent to have many realizations of shorter series. – mark leeds Jul 1 '20 at 3:46
• @markleeds Agreed, but that would be the case only if the random process was assumed to be ergodic. Generally speaking every outcome $\zeta$ of the random experiment gets mapped to a possibly different function of time. So, having a long series of one realization will tell nothing about another realization. We could however study the question under different assumptions on stationarity of the random process. – DSP Rookie Jul 1 '20 at 9:22
• AFAIK, all random processes are assumed to be ergodic because there's no practical way of testing for it. But, if you can show me that I'm wrong, I'm opened to learning. Thanks. Oh, also, if you want to just agree to disagree, that's okay too :). I shouldn't assume you want to do the extra work especially since ergodicity is kind of an esoteric topic. – mark leeds Jul 2 '20 at 2:15
• @markleeds Well, I am not an expert on this matter. But, practically speaking I can agree with what you are saying. Except for a few carefully constructed counter-examples, I think we can make that assumption for real life random processes. In essence of OP's original question, your statement only makes OP's claim more concrete that everything is estimated because that's the best we can do. – DSP Rookie Jul 2 '20 at 7:26

I would say that $$R$$ is an estimate of the instantaneous power in $$x(t)$$, as opposed to the average power. But this assumes that you implement the expectation by averaging over multiple realizations of $$x(t)$$. In this case, the average is still a function of time, $$R(t) = E \left\{ x^2(t) \right\},$$ and is an estimate of the instantaneous power in $$x(t)$$ at time $$t$$.

If you are computing a time average, i.e. $$R = \frac{1}{T} \int_{t_0}^{t_0+T} x^2(t) \, dt,$$ then I agree with the comments above that this converges to the average power in $$x(t)$$ only if $$x(t)$$ is an ergodic random process. By definition, ergodic processes have equal time and ensemble averages.

It sounds as if what you really want to do is compute the autocorrelation function for the random process. If $$x(t)$$ is wide-sense stationary, then the autocorrelation is

$$R(\tau) = E \left\{ x(t) x(t + \tau) \right\},$$

where the expectation is an ensemble average, and the average power is $$R(0)$$. Unfortunately, this won't help if you have only one realization, and you cannot assume that it is ergodic, which would allow you to create many realizations as segments of one long realization and average over the realizations.

• Yes, I believe my original expression is equivalent to R(0) - autocorrelation function for lag 0 - and that is defined as "average power" in my Leon-Garcia Random Processes textbook, which also says that X^2(t) is the instantaneous power (developed across a 1-ohm resistor). Thank you to all who responded to my question - very helpful as I haven't looked at this subject for over 25 years :) – j03y_ Jul 9 '20 at 2:54