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Zero Padding

Zero padding consists of extending a signal (or spectrum) with zeros. It maps a length $ N$ signal to a length $ M>N$ signal, but $ N$ need not divide $ M$.


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 $ M$ odd, and $ M/2 - 1$ for $ M$ 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 $ n=2$ and $ n=-2$.

Figure 7.7 illustrates zero padding from length $ N=5$ out to length $ M=11$. Note that $ x$ and $ n$ could be replaced by $ X$ and $ k$ in the figure caption.

Figure 7.7: Illustration of zero padding: a) Original signal (or spectrum) $ x=[3,2,1,1,2]$ plotted over the domain $ n\in [0,N-1]$ where $ N=5$ (i.e., as the samples would normally be held in a computer array). b) $ \protect\hbox{\sc ZeroPad}_{11}(x)$. c) The same signal $ x$ 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 $ N/2-1$ (for $ N$ 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 $ N$ is even, while odd $ N$ requires no such restriction. In practice, we often prefer to interpret time-domain samples as extending from 0 to $ N-1$, i.e., with no negative-time samples. For this case, we define ``causal zero padding'' as described below.

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Causal (Periodic) Signals
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Stretch Operator