I have a sampled DTFT. If we assume that there isn't aliasing in time domain, what is the best way to reconstruct DTFT from its equidistant samples? I though about Dirichlet interpolation. Do you know if there are other ways to do it? There is any difference between a theoretical approach and a practical approach?
1$\begingroup$ This answer (especially Eq. (6)) should answer your question. $\endgroup$– Matt L.Dec 28, 2016 at 17:07
$\begingroup$ Thank you! I read all the theory about it, but I've found a question that confused me. As I write above, the differences between a theoretical approach and a practical approach to the DTFT reconstruction. I think that the theoretical one, is the Dirichlet interpolation; the second one could be IFFT-> zero-padding -> FFT with a high sample density? That should be useful to see a quasi-continuous spectrum? $\endgroup$– Francesco MancusoDec 28, 2016 at 17:48
$\begingroup$ Yes, the second approach is practical if you're OK with a finite and equidistant set of points. See my answer below. $\endgroup$– Matt L.Dec 28, 2016 at 19:14
For finite length time domain signals, the DTFT can be obtained from the DFT via interpolation, as explained in this answer. In this case the DTFT can be evaluated at any desired frequency point because it is expressed as a function of a continuous frequency variable.
If a finite set of points is sufficient, it is much more efficient to apply an inverse DFT (using an FFT algorithm), then zero-pad the time-domain data, and apply a DFT (again, using an FFT algorithm). This will give you an equidistantly sampled DTFT with a sample density determined by the DFT length.