Frequency-Domain Implementation

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. Example:

1. Hann Window = 3-Point $\hbox{\sc DFT}_M$ Smoother:
   • Start with a length $M$ rectangular window
   • Take an $M$ -point DFT
   • Convolve the DFT data with the 3-point smoother $[1/4,1/2,1/4]$ to implement a Hann window
   • Note that the Hann window requires no multiplies in linear fixed-point data formats
2. Any Blackman window is a 5-point smoother for a Length $M$ (critically sampled) DFT