#### 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 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 an 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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Strong COLA

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Example 1: Low-Pass Filtering by FFT Convolution