Circular convolution: Difference between revisions

Revision as of 01:19, 25 September 2019

Revision as of 01:20, 25 September 2019

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A case of great practical interest is illustrated in the figure. The duration of the '''x''' sequence is '''N''' (or less), and the duration of the '''h''' sequence is significantly less. Then many of the values of the circular convolution are identical to values of '''x∗h''',  which is actually the desired result when the '''h''' sequence is a [[finite impulse response]] (FIR) filter. Furthermore, the circular convolution is very efficient to compute, using a [[fast Fourier transform]] (FFT) algorithm and the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]].

There are also methods for dealing with an '''x''' sequence that is longer than a practical value for '''N'''. The sequence is divided into segments (''blocks'') and processed piecewise. Then the filtered segments are carefully pieced back together. Edge effects are eliminated by <u>''overlapping</u>'' either the input blocks or the output blocks. To help explain and compare the methods, we discuss them both in the context of an '''h''' sequence of length 201 and an FFT size of&nbsp;''N''&nbsp;=&nbsp;1024.

=== Overlapping input blocks ===