It is instructive to interpret the periodic interpolation theorem in terms of the stretch theorem, . To do this, it is convenient to define a ``zero-centered rectangular window'' operator:
Definition: For any and any odd integer we define the length even rectangular windowing operation by
Thus, this ``zero-phase rectangular window,'' when applied to a spectrum , sets the spectrum to zero everywhere outside a zero-centered interval of samples. Note that is the ideal lowpass filtering operation in the frequency domain. The ``cut-off frequency'' is radians per sample. For even , we allow to be ``passed'' by the window, but in our usage (below), this sample should always be zero anyway. With this notation defined we can efficiently restate periodic interpolation in terms of the operator:
Theorem: When consists of one or more periods from a periodic signal ,
In other words, ideal periodic interpolation of one period of by the integer factor may be carried out by first stretching by the factor (inserting zeros between adjacent samples of ), taking the DFT, applying the ideal lowpass filter as an -point rectangular window in the frequency domain, and performing the inverse DFT.
Proof: First, recall that . That is, stretching a signal by the factor gives a new signal which has a spectrum consisting of copies of repeated around the unit circle. The ``baseband copy'' of in can be defined as the -sample sequence centered about frequency zero. Therefore, we can use an ``ideal filter'' to ``pass'' the baseband spectral copy and zero out all others, thereby converting to . I.e.,
The last step is provided by the zero-padding theorem (§7.4.12).