#### Relation to Stretch Theorem

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

*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 ,

*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.*,

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Bandlimited Interpolation of Time-Limited Signals

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Illustration of the Downsampling/Aliasing Theorem in Matlab