If I have the autocorrelation function of an observed system output, how does this relate to the impulse-response function of that system, if I don't have information about the input?

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#### Best Answer

The autocorrelation function $R_y$ of the system output $y(t)$ is related to the *autocorrelation* function $R_h$ of the system impulse response $h(t)$ and the *autocorrelation* function $R_x$ of the system input $x(t)$ as $$R_y = R_h star R_x tag{1}$$ where $star$ denotes the convolution operation. But note that because $R_h$ and $R_x$ are (real-valued) *even* functions of time with a peak at the origin (as indeed is $R_y$), we can also think of the result as saying that $R_y$ is the *cross-correlation* function of $R_h$ and $R_x$. Unfortunately, all this does not help you very much if all you know is $R_y$ and have no idea what $R_x$ is (or possibly even what $x(t)$ might have been). If $x(t)$ is indeed white noise with autocorrelation function $R_x(tau) = Kdelta(tau)$ where $delta(cdot)$ is the Dirac impulse, then you can estimate $R_h(tau) = K^{-1}R_y(tau)$ which is progress of sorts, but it is still not possible to say *exactly* what the impulse response $h(t)$ might be. Quite *different* signals can have the *same* autocorrelation function: that is, knowledge of $R_h(tau)$ does not tell us what $h(t)$ is, though it does reduce the possibilities considerably.

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