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I need to prove the property $$f(x)⊕f(x)=f(x)⊗f^*(-x)$$ That is the autocorrelation of a function is the convolution with its time-reversed complex conjugate. I have constructed most of the proof but I think I am missing a simple step.

With the definition of autocorrelation, the LHS is $$\int_{-\infty}^{\infty}{f(x)f^*(x-\tau)dx}$$

and with the definition of convolution, the RHS is $$\int_{-\infty}^{\infty}f(x)f^*(-x-\tau)dx$$

Letting $-x-\tau=k$, we get $x=k-\tau$ and the reversal of limits of the integral are compensated for by the negative sign in the differential $dk$ element. So the integral becomes $$\int_{-\infty}^{\infty}f^*(k-\tau)f(k)dk$$

which is in the form I want.

However I am confused at the step where I write down the definition of convolution. Since the signal is time-reversed, do I take $f(-(x-\tau))$ inside the integral or the $f(-x-\tau)$, as I have done? I don't understand what the sign of $\tau$ should be when the signal is time reversed.

Any help would be appreciated!

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    $\begingroup$ Please review your definition of autocorrelation. $\endgroup$
    – Matt L.
    Oct 3, 2023 at 20:54
  • $\begingroup$ Whoops, fixed. Thanks $\endgroup$
    – requiemman
    Oct 3, 2023 at 22:20
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    $\begingroup$ Nope, not fixed yet. $\endgroup$
    – Matt L.
    Oct 4, 2023 at 7:49
  • $\begingroup$ Changed the variable of integration. $\endgroup$
    – requiemman
    Oct 4, 2023 at 13:22
  • $\begingroup$ Ok, finally we're talking. Now you should be able to complete the proof. Some necessary details are given in my answer. $\endgroup$
    – Matt L.
    Oct 4, 2023 at 14:27

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Starting from the convolution of $f(t)$ with $f^*(-t)=\tilde{f}(t)$, you get the following:

\begin{align} (f\star \tilde{f})(t)&=\int_{\infty}^{\infty}\tilde{f}(\tau)f(t-\tau)d\tau\\&=\int_{\infty}^{\infty}f^*(-\tau)f(t-\tau)d\tau\\&=\int_{\infty}^{\infty}f^*(\tau)f(t+\tau)d\tau \end{align}

In order to finish the proof, you should make your definition of autocorrelation somehow resemble this result.

[The following part of my answer referred to previous versions of the question (before the definition of autocorrelation was corrected).]


You won't be able to prove what you'd like to prove before you haven't figured out the correct definition of the autocorrelation of a deterministic signal. The current definition in your question is

\begin{align} \int_{-\infty}^{\infty}{f(\tau)f^*(x-\tau)dx}&=f(\tau)\int_{-\infty}^{\infty}f^*(x-\tau)dx\\&=f(\tau)\int_{-\infty}^{\infty}f^*(x)dx \end{align}

which doesn't make much sense as an autocorrelation function, because it is just a scaled version of the function itself, assuming that the integral is finite (and non-zero).


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