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.

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 $2L-1$.