# How to find the impulse response of filter that outputs the autocorrelation of the input?

I am trying to solve question 9.61 in the attached image.

I guess, by evaluating the autocorrelation for a specific instance, say $$t = 0$$ , that the impulse response, $$h(t)$$, should be : $$h (t) = x(-t)$$.

But can I get this from equating the convolution integral to the given output integral, i.e.

$$y(t) = \int_{-\infty}^{\infty} x(\tau)x(t + \tau) d\tau = \int_{-\infty}^{\infty} x(\tau)h(t - \tau) d\tau$$ .

Is it then right to equate as below?

$$x(t + \tau) = h (t - \tau)$$

Thank you!

• Read the answers to “Understanding the matched filter” on this website – Dilip Sarwate Jan 1 at 5:00

$$\int_{-\infty}^{\infty}x(\tau)x(t+\tau)d\tau=\int_{-\infty}^{\infty}x(\tau-t)x(\tau)d\tau$$