Example 2: Time Domain Aliasing

Figure 8.8 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 and the input signal consists of an impulse at times and , where the data frame length is . To avoid time aliasing (i.e., to implement acyclic convolution using the FFT), we must use an FFT size at least as large as . In the figure, the FFT sizes , , and 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.

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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.