The expected value of the product of a random variable or signal realization with a time-shifted version of itself
With a simple calculation and analysis of the autocorrelation function, we can discover a few important characteristics about our random process. These include:
1)How quickly our random signal or processes changes with respect to the time function
2)Whether our process has a periodic component and what the expected frequency might be
As was mentioned above, the autocorrelation function is simply the expected value of a product. Assume we have a pair of random variables from the same process, X 1 =X (t 1)and X 2 =X (t 2) , then the autocorrelation is often written as
The above equation is valid for stationary and nonstationary random processes. For stationary processes, we can generalize this expression a little further. Given a wide-sense stationary processes, it can be proven that the expected values from our random process will be independent of the origin of our time function. Therefore, we can say that our autocorrelation function will depend on the time difference and not some absolute time. For this discussion, we will let τ= t 2 - t 1 , and thus we generalize our autocorrelation expression as
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