In one of the numerical, I came across the following power spectral density (PSD):
$S_{Y}(f) = 2; |f| < 2$ and $S_{Y}(f) = 0; \text{otherwise}$.
Its autocorrelation is given as $R_{Y}(\tau) = 8 \text{Sinc}(4\tau)$ ---(1)
I know that autocorrelation is inverse Fourier transform of PSD, but I am not getting how the autocorrelation in equation (1) is obtained.
Any help in this regard will be highly appreciated.
Because $$S_{Y}(f) =\begin{cases} 2&|f|<2\\ 0&f\le -2 \:or\: f \ge 2 \end{cases}$$ we have $$R_{Y} (\tau) = \int_{-\infty}^{+\infty}S_{Y}(f) e^{j2\pi f \tau} \text{d}f = \int_{-2}^{2}2 e^{j2\pi f \tau} \text{d}f=\left[\frac{e^{j2\pi f \tau}}{j\pi\tau}\right]^2_{-2}\\=\frac{e^{j4\pi\tau}-e^{-j4\pi\tau}}{j\pi\tau}=\frac{2\sin4\pi\tau}{\pi\tau}=\frac{8\sin4\pi\tau}{4\pi\tau}=8\;\text{sinc}\:(4\tau)$$