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Causal Zero Padding

In practice, a signal $ x\in{\bf R}^N$ is often an $ N$ -sample frame of data taken from some longer signal, and its true starting time can be anything. In such cases, it is common to treat the start-time of the frame as zero, with no negative-time samples. In other words, $ x(n)$ represents an $ N$ -sample signal-segment that is translated in time to start at time 0 . In this case (no negative-time samples in the frame), it is proper to zero-pad by simply appending zeros at the end of the frame. Thus, we define e.g.,

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

Causal zero-padding should not be used on a spectrum of a real signal because, as we will see in §7.4.3 below, the magnitude spectrum of every real signal is symmetric about frequency zero. For the same reason, we cannot simply append zeros in the time domain when the signal frame is considered to include negative-time samples, as in ``zero-centered FFT processing'' (discussed in Book IV [72]). Nevertheless, in practice, appending zeros is perhaps the most common form of zero-padding. It is implemented automatically, for example, by the matlab function fft(x,N) when the FFT size N exceeds the length of the signal vector x.

In summary, we have defined two types of zero-padding that arise in practice, which we may term ``causal'' and ``zero-centered'' (or ``zero-phase'', or even ``periodic''). The zero-centered case is the more natural with respect to the mathematics of the DFT, so it is taken as the ``official'' definition of ZEROPAD(). In both cases, however, when properly used, we will have the basic Fourier theorem7.4.12 below) stating that zero-padding in the time domain corresponds to ideal bandlimited interpolation in the frequency domain, and vice versa.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University