Zero padding consists of extending a signal (or spectrum)
with zeros to extend its time (or frequency band) limits. It maps a
length signal to a length signal, but need not be an
integer multiple of . To unify the time-domain and
frequency-domain definitions of zero-padding, it is necessary to
regard the original time axis
as indexing
positive-time samples from 0 to (for even), and negative
times in the interval
.
Furthermore, we require
when is even,
while odd requires no such restriction.

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 zero-centered domain.

When a signal is causal, that is, for all
negative-time samples (
), then zero-padding can be
carried out by simply appending zeros to the original signal.
For example,

(causal case)

The causal definition is natural when represents a signal
starting at time 0 and extending for samples. On the other
hand, when we are zero-padding a spectrum, or we have a time-domain
signal including both positive- and negative-time samples, then the
zeros should be inserted ``outside the nonzero interval'' of the
signal or spectrum, as defined in Eq. (7.4).