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:

$\displaystyle \hbox{\sc ZeroPad}_{M,m}(x) \isdef \left\{\begin{array}{ll} x(m),... ...ert m\vert < N/2 \\ [5pt] 0, & \mbox{otherwise} \\ \end{array} \right. \protect$

(7.4)

where $m=0,\pm1,\pm2,\dots,\pm M_h$ , with $M_h\isdef (M-1)/2$ for

odd, and

for

even. For example,

$\displaystyle \hbox{\sc ZeroPad}_{10}([1,2,3,4,5]) = [1,2,3,0,0,0,0,0,4,5].$

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 $n\in [0,N-1]$ where (*i.e.*, as the samples would normally be held in a computer array). b) $\protect\hbox{\sc ZeroPad}_{11}(x)$ . c) The same signal plotted over the domain $n\in [-(N-1)/2, (N-1)/2]$ which is more natural for interpreting negative times (frequencies). d) $\protect\hbox{\sc ZeroPad}_{11}(x)$ plotted over the zero-centered domain.
$\includegraphics[width=\twidth]{eps/zpad}$

Note that we have unified the time-domain and frequency-domain definitions of zero-padding by interpreting the original time axis $[0,1,\dots,N-1]$ as indexing positive-time samples from 0 to (for even), and negative times in the interval $n\in[N-N/2+1,N-1]\equiv[-N/2+1,-1]$ .^7.8 Furthermore, we require $x(N/2)\equiv x(-N/2)=0$ 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.