### 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: **

where , with for odd, and for even. For example,

Figure 7.7 illustrates zero padding from length out to length . Note that and could be replaced by and in the figure caption.

Note that we have unified the time-domain and frequency-domain
definitions of zero-padding by interpreting the original time axis
as indexing positive-time 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 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) Signals

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Stretch Operator