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Example 2: Time Domain Aliasing

Figure 8.7 shows the effect of insufficient zero padding, which can be thought of as undersampling in the frequency domain. We will see aliasing in the time domain results.

The lowpass filter length is $ L= 65$ and the input signal consists of an impulse at times $ 10$ and $ M-(L-1)/4 = 85$ , where the data frame length is $ M=100$ . To avoid time aliasing (i.e., to implement acyclic convolution using an FFT), we must use an FFT size $ N$ at least as large as $ 85+65-1=149$ . In the figure, the FFT sizes $ 116$ , $ 132$ , and $ 165$ are used. Thus, the first case is heavily time aliased, the second only slightly time aliased (involving only some of the filter's ``ringing'' after the second pulse), and the third is free of time aliasing altogether.

Figure 8.7: Illustration of FFT convolution with insufficient zero padding. From the top: (1) Input signal (two impulses) and lowpass-filter impulse response; (2) heavily time-aliased convolution in which the second filter impulse has wrapped around to low times; (3) slightly time-aliased result in which some of the filter ``post-ring'' from the second pulse wraps around; (4) result with no time aliasing.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University