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Frequency-Domain Implementation of the
Blackman-Harris Family

The Blackman-Harris window family can be very efficiently implemented in the frequency domain as a $ (2L-1)$ -point convolution with the spectrum of the unwindowed data.

For example, to implement a zero-phase Hann window,

  1. Start with a length $ M$ rectangular window
  2. Take an $ M$ -point DFT
  3. Convolve the DFT data with the 3-point smoother $ W=[1/4,1/2,1/4]$
Note that the frequency-domain implementation of the Hann window requires no multiplies in linear fixed-point data formats [188].

Similarly, any Blackman window may be implemented as a 5-point smoother in the frequency domain. More generally, any $ L$ -term Blackman-Harris window requires convolution of the critically sampled spectrum with a smoother of length $ 2L-1$ .


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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