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Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Zero Padding


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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) $ \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) $ \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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Previous: Stretch Operator
Next: Causal (Periodic) Signals
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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