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I know if we have a continuous-time bandlimited signal $x_{ct}(t)$ that is sampled frequently enough we get a signal that we can call $x_s(t)$.

If we take the Fourier transform of $x_s(t)$ to get $X_s(j\Omega)$, we end up with a Fourier transform of the sampled input that resembles $X_{ct}(j\Omega)$ at $\Omega = 0$ and repeating at $2\pi/T$ and $-2\pi/T$. If we want to get $X_c(j\Omega)$, we can multiply by a $\mbox{rect}()$ function. It seems to me that we could then reconstruct the original signal by taking the inverse Fourier transform to get $x_{ct}(t)$

However, when reading on signal processing theory, we instead use a $\mbox{sinc}()$ function to perform reconstruction in the time domain since it is the "ideal interpolator". Why can't we instead use the method described in the frequency domain? It seems like it would be easier?

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The "two" operations you talk about are one and the same: low pass filtering using the $\mbox{rect}()$ function in the frequency domain and $\mbox{sinc}()$ interpolation in the time domain are the same.

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  • $\begingroup$ From what I've read so far, it appears that if you try to use the rect() function in the frequency domain and just zero out all other frequency bins, you'll get ringing? So it is better to just use sinc functions to interpolate instead? $\endgroup$ – Veridian Oct 9 '15 at 22:12
  • $\begingroup$ If you choose the FFT lengths properly, you won't get time aliasing. Not quite sure what "ringing" you're talking about. $\endgroup$ – Peter K. Oct 9 '15 at 22:18
  • $\begingroup$ Gibbs phenomenon $\endgroup$ – Veridian Oct 9 '15 at 22:19
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    $\begingroup$ The only way to avoid that using a $\mbox{sinc}$ is to use an infinitely long one... $\endgroup$ – Peter K. Oct 9 '15 at 22:24
  • $\begingroup$ I believe you have to zero pad the before taking the inverse fft however, correct? That is like using the windowing function? $\endgroup$ – Veridian Oct 10 '15 at 5:29

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