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







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).
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




![$ \omega_c = 2\pi[(M-1)/2]/N$](http://www.dsprelated.com/josimages_new/mdft/img1463.png)



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.,

Bandlimited Interpolation of Time-Limited Signals
The previous result can be extended toward bandlimited interpolation
of
which includes all nonzero samples from an
arbitrary time-limited signal
(i.e., going beyond the interpolation of only periodic bandlimited
signals given one or more periods
) by
- replacing the rectangular window
with a smoother spectral window
, and
- using extra zero-padding in the time domain to convert the
cyclic convolution between
and
into an acyclic convolution between them (recall §7.2.4).


















The approximation symbol `


![$ \hbox{\sc Chop}_{N_x}([1,\dots,1])$](http://www.dsprelated.com/josimages_new/mdft/img1484.png)

Equation (7.8) can provide the basis for a high-quality
sampling-rate conversion algorithm. Arbitrarily long signals can be
accommodated by breaking them into segments of length , applying
the above algorithm to each block, and summing the up-sampled blocks using
overlap-add. That is, the lowpass filter
``rings''
into the next block and possibly beyond (or even into both adjacent
time blocks when
is not causal), and this ringing must be summed
into all affected adjacent blocks. Finally, the filter
can
``window away'' more than the top
copies of
in
, thereby
preparing the time-domain signal for downsampling, say by
:






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Why a DFT is usually called an FFT in practice
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Zero Padding Theorem (Spectral Interpolation)