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 going wrong in one step.
With the definition of autocorrelation, the LHS is $$\int_{-\infty}^{\infty}{f(\tau)f^*(x-\tau)dx}$$
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. In general, what happens to $f(x-k)$ when you are time-shifting after time-reversal?