Interpolating a DFT

Starting with a sampled spectrum $X(\omega_k)$ , $k=0,1,\ldots,N-1$ , we may interpolate ideally by taking the DTFT of the zero-padded IDFT:

$$\begin{eqnarray*} X(\omega) &=& \hbox{\sc DTFT}_{\omega}(\hbox{\sc ZeroPad}_{\infty}(\hbox{\sc IDFT}_N(X)))\\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \sum_{n=-N/2}^{N/2-1}\left[\frac{1}{N}\sum_{k=0}^{N-1}X(\omega_k) e^{j\omega_k n}\right]e^{-j\omega n}\\ &=& \sum_{k=0}^{N-1}X(\omega_k) \left[\frac{1}{N}\sum_{n=-N/2}^{N/2-1} e^{j(\omega_k-\omega) n}\right]\\ &=& \sum_{k=0}^{N-1}X(\omega_k)\hbox{asinc}_N(\omega-\omega_k)\\ &=& \left<X,\hbox{\sc Sample}_{\Omega_N}(\hbox{\sc Shift}_{\omega}(\hbox{asinc}_N))\right>\\ &=& (X\circledast \hbox{asinc}_N)_\omega, \end{eqnarray*}$$

where $\circledast$ denotes convolution between a discrete ($X$ ) and continuous ( $\hbox{asinc}$ ) signal. (If math operators adapt to their argument types like perl functions, we can simply use $\ast$ as usual.)

• Zero-padding in the time domain corresponds to `` $\hbox{asinc}_N$ interpolation'' in the frequency domain
• This is ``ideal time-limited spectral interpolation''