DFT time-domain interpolation

The paper puts into perspective two computational approaches to discrete-time interpolation. The exact interpolation kernel for the so-called `FFT method´ is derived and compared with that for the `zero-interlace´ method associated with `unsampling´. Both yield precisely the same result, but the FFT method produces it using a finite-length sum, whereas the sum for the other method is infinite-length. The identity responsible for this characteristic is derived. Truncation of the sinc sum in attempts to emulate the efficiency of the FFT method can lead to significant error in reconstruction, especially at the end where the magnitudes of the omitted terms are largest. The FFT method can be used to reconstruct periodic, bandlimited functions without error (excepting roundoff), provided the window contains an integral number of periods and the sampling rate exceeds the Nyquist rate. If it does not, there will be erroneous end effects in the reconstruction. If used in a downsampling-upsampling scheme, one must ensure sufficient oversampling to avoid aliasing, as is the case for the zero-interlace method. Numerical examples illustrate the conclusions