The Tukey window coefficients are given by the following formula
where N is the length of the filter, M = (N – 1) / 2, k = 0, 1, …, N – 1, and α is a constant between zero and one.
Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Tukey window with α = 0.5.
Given a sampling frequency of 2000 Hz and a filter cutoff frequency of 40 Hz, the impulse response of the filter with a rectangular window (with no window) and with the Tukey window above is as follows.
The magnitude response of the same filter is shown on the graph below.
A larger α implies a Tukey window with a "flatter" top, smaller transition band, and worse stop-band attenuation. As α approaches 1, the Tukey window itself approaches a rectangular window. A smaller α creates a Tukey window with a steeper top, larger transition band, and better stop-band attenuation. As α approaches 0, we have
аnd the Tukey window approaches the Hann window.
The following is the Tukey window with three different values for α (0.3, 0.5, and 0.7).
The magnitude response of these same windows given the sampling frequency of 2000 Hz, cutoff frequency of 40 Hz, and a filter of length N = 201 is as follows.
Measures for the Tukey window
The following is a comparison of the discrete Fourier transform of the Tukey window (α = 0.5) and the rectangular window.
The Tukey window measures are as follows.