Frequency-Domain Implementation of the Blackman-Harris Family

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

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

1. Start with a length rectangular window.
2. Take an -point DFT.
3. Convolve the DFT data with the 3-point smoother .

Note that the frequency-domain implementation of the Hann window requires no multiplies in linear fixed-point data formats.

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

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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.