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Here it goes. I was writing up some elementary course material for an instrumentation course, and wanted to quickly introduce "white noise".

Now, the usual definition of white noise is something like a stationary random process such that E( X(t) ) = 0 for all t and a flat power spectral density.

On the other hand, the autocorrelation function is defined as R(tau) = E (X(t) X(t+tau) ) (independent of t).

But here's the problem. The Wiener-Khinchine theorem states that the power spectral density equals the Fourier transform of the autocorrelation function, so a flat power spectral density comes down to a Dirac for the autocorrelation. And for example on Wiki, you find that as a defining property of white noise.

But the autocorrelation of white noise E (X(t) X(t) ) is nothing else but sigma-squared.

So it would seem that the autocorrelation function is everywhere 0, except in 0, where it is a finite number.

What am I missing here ?