# Why do we use sinc interpolation rather than just multiply the frequency domain signal by rect?

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?

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.
• The only way to avoid that using a $\mbox{sinc}$ is to use an infinitely long one... – Peter K. Oct 9 '15 at 22:24