Interpolating a DFT

Starting with a sampled spectrum , , typically obtained from a DFT, we can interpolate by taking the DTFT of the IDFT which is not periodically extended, but instead zero-padded [248]:^3.8

(The aliased sinc function, , was defined in Eq.(4.4) and repeated in Eq.(4.7).) Thus, zero-padding in the time domain interpolates a spectrum consisting of samples around the unit circle by means of `` interpolation.'' This is ideal, time-limited interpolation in the frequency domain using the aliased sinc function as an interpolation kernel. We can almost rewrite the last line above as , but such an expression would normally be defined only for , where is some integer.

Figure G.1 lists a matlab function for performing ideal spectral interpolation directly in the frequency domain. Such an approach is normally only used when non-uniform sampling of the frequency axis is needed. For uniform spectral upsampling, it is more typical to take an inverse FFT, zero pad, and a longer FFT, as discussed further in the next section.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.