#### 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