Periodic Interpolation (Spectral Zero Padding)

The *dual* of the zero-padding theorem states formally that
*zero padding in the frequency domain* corresponds to *periodic
interpolation* in the time domain:

**Definition: **For all
and any integer
,

where zero padding is defined in §7.2.7 and illustrated in Figure 7.7. In other words, zero-padding a DFT by the factor in the frequency domain (by inserting zeros at bin number corresponding to the

Periodic interpolation is ideal for signals that are *periodic*
in
samples, where
is the DFT length. For non-periodic
signals, which is almost always the case in practice, bandlimited
interpolation should be used instead (Appendix D).

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

**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).

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