# Solved – How is the impulse-response function of a given system related to the autocorrelation function

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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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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# Solved – How is the impulse-response function of a given system related to the autocorrelation function

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?