We know that the discrete Fourier transform (DFT) of a discrete rectangular function is related to Dirichlet kernel: $D_n$(x)=$\frac{sin[(n+1/2)x]}{sin(x/2)}$, and the Fourier transform of a continuous rectangular function is Sinc function. But now I wonder what is the (inverse) discrete Fourier transform of a discretely sampled Sinc function? In my current work, I have two finite discrete signal $f_1$ and $f_2$ both in frequency domain, and now I want to simplify the equation:
$f_1[x]$$f_2[x]$$\frac{sin[kx]}{kx}$.
I want to first fix $f_1[x]$ and use the convolution theorem, getting the result:
=$f_1[x]$$\hat{f_2}[x]$,
where $\hat{f_2}[x]$ is the DFT of average smoothed inverse DFT of $$f_2[x]$. But I notice the difference between Dirichlet kernel and Sinc function. I wonder whether there is a method to bridge them together.
Thank you for your help!