Zero Padding
Zero padding consists of extending a
signal (or
spectrum)
with zeros. It maps a length

signal to a length

signal, but

need not divide

.

Definition:
![$\displaystyle \hbox{\sc ZeroPad}_{M,m}(x) \isdef \left\{\begin{array}{ll} x(m),...
...ert m\vert < N/2 \\ [5pt] 0, & \mbox{otherwise} \\ \end{array} \right. \protect$](http://www.dsprelated.com/josimages_new/mdft/img1223.png) |
(7.4) |
where

, with

for

odd,
and

for

even.
For example,
In this example, the first sample corresponds to time 0, and five
zeros have been inserted between the samples corresponding to times

and

.
Figure
7.7 illustrates zero padding from length

out to length

. Note that

and

could be replaced by

and

in the
figure caption.
Figure 7.7:
Illustration of zero padding:
a) Original signal (or spectrum)
plotted over the
domain
where
(i.e., as the samples would
normally be held in a computer array).
b)
.
c) The same signal
plotted over the domain
which
is more natural for interpreting negative times (frequencies).
d)
plotted over the zero-centered domain.
![\includegraphics[width=\twidth]{eps/zpad}](http://www.dsprelated.com/josimages_new/mdft/img1232.png) |
Note that we have unified the time-domain and
frequency-domain
definitions of zero-padding by interpreting the original time axis
![$ [0,1,\dots,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1233.png)
as indexing positive-time samples from 0 to

(for

even), and negative times in the interval
![$ n\in[N-N/2+1,N-1]\equiv[-N/2+1,-1]$](http://www.dsprelated.com/josimages_new/mdft/img1235.png)
.
7.8 Furthermore, we require

when

is even, while odd

requires no such
restriction. In practice, we often prefer to interpret time-domain
samples as extending from 0 to

,
i.e., with no negative-time
samples. For this case, we define ``
causal zero padding'' as
described below.
Next Section: Causal (Periodic) SignalsPrevious Section: Stretch Operator