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:

(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 zerocentered domain.

Note that we have unified the timedomain and
frequencydomain
definitions of zeropadding by interpreting the original time axis
as indexing positivetime samples from 0 to
(for
even), and negative times in the interval
.
^{7.8} Furthermore, we require
when
is even, while odd
requires no such
restriction. In practice, we often prefer to interpret timedomain
samples as extending from 0 to
,
i.e., with no negativetime
samples. For this case, we define ``
causal zero padding'' as
described below.
Next Section: Causal (Periodic) SignalsPrevious Section: Stretch Operator