# Autocorrelation of sinc function

I'm having trouble on computing the autocorrelation of the sinc function
I want to compute $$R_{hh}(\tau)=\int_{-\infty}^{\infty}\operatorname{sinc}(t) \ \operatorname{sinc}(t-\tau) \ \mathrm{d}t$$
where $$\operatorname{sinc}(t) \triangleq \begin{cases} \dfrac{\sin(\pi t)}{\pi t} \qquad & t \ne 0\\ \\ \quad \ \ 1 & t = 0 \end{cases}$$

Is the result another sinc function?

• Hint: Since sinc is symmetrical, convolution and correlation will yield the same result. Second hint : What is the fourier transform of a sinc? Third hint : Convolution (or cross-correlation in this case) is equivalent to multiplication in the frequency domain.
– Ben
Apr 1 '19 at 2:26
• The fourier transform of a sinc is a rectangular pulse in frequency domain. So multiplying two rect functions i get another rectangular pulse, and then its antitransform is sinc? Is that right? Apr 1 '19 at 2:32
• yes. and directly integrating the above leads only to headache and sorrow. when converting autocorrelation to power spectrum or energy spectrum, be careful to dot your t's and cross your i's. Apr 1 '19 at 3:34
• Yes, that's right. The Fourier transform of $x(t) = \operatorname{sinc}(t)$ is $X(f)=\operatorname{rect}(f)$ and thus, $|X(f)|^2 = \operatorname{rect}(f)$. Apr 1 '19 at 3:37

Because of the symmetry of the $$\operatorname{sinc}(t)$$ we have
$$R_{hh}(\tau)=\int_{-\infty}^{\infty}\operatorname{sinc}(t) \ \operatorname{sinc}(t-\tau) \ \mathrm{d}t=\int_{-\infty}^{\infty}\operatorname{sinc}(t) \ \operatorname{sinc}(\tau-t) \ \mathrm{d}t = \operatorname{sinc}(\tau) \ast \operatorname{sinc}(\tau)$$
Analyzing in the frequency domain we have $$\mathcal{F}(R_{hh}(\tau)) = \mathcal{F}(\operatorname{sinc}(\tau))^2=\operatorname{rect}(f)^2=\operatorname{rect}(f)$$
$$\Rightarrow R_{hh}(\tau)=\mathcal{F}^{-1}(\mathcal{F}(R_{hh}(\tau))) = \mathcal{F}^{-1}(\operatorname{rect}(f))=\operatorname{sinc}(\tau)$$
where $$\operatorname{rect}(f)$$ denotes the rectangular function in the frequency domain
• Good job! Now try finding the autocorrelation function of $\displaystyle \operatorname{sinc}\left(\frac tT\right)$ Apr 2 '19 at 15:32