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?
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
