Tukey window

The Tukey window coefficients are given by the following formula

$$a(k)=\begin{cases} 0.5 (1+\cos(\frac{\pi(|k-M|-\alpha M)}{(1-\alpha)M})), \,\,\, |k-M| \ge \alpha M \\ 1, \,\,\, |k-M| \lt \alpha M \end{cases}$$

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.

Tukey window

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.

Impulse response of a low pass filter with and without the Tukey window

The magnitude response of the same filter is shown on the graph below.

Magnitude response of a low pass filter with and without the Tukey window

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

$$a(k)=0.5 (1+\cos(\frac{2\pi|k-\frac{N-1}{2}|}{N-1}))=0.5 (1+\cos(\frac{2 \pi k}{N-1}-\pi))=0.5 (1+\cos(\frac{2 \pi k}{N-1}))$$

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).

Tukey window at three different values for alpha

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.

Magnitude response of the Tukey window with different values of alpha

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.

Discrete Fourier transform of the Tukey window

The Tukey window measures are as follows.

α 0.3 0.5 0.7
Coherent gain 0.65 0.75 0.85
Equivalent noise bandwidth 1.33 1.22 1.13
Processing gain -1.25 dB -0.88 dB -0.52 dB
Scalloping loss -1.81 dB -2.23 dB -2.79 dB
Worst case processing loss -3.06 dB -3.11 dB -3.31 dB
Highest sidelobe level -18.2 dB -15.1 dB -13.8 dB
Sidelobe falloff -16.3 dB / octave, -54.3 dB / decade -15.8 dB / octave, -52.6 dB / decade -15.3 dB / octave, -50.9 dB / decade
Main lobe is -3 dB 1.28 bins 1.16 bins 1.04 bins
Main lobe is -6 dB 1.76 bins 1.58 bins 1.42 bins
Overlap correlation at 50% overlap 0.272 0.362 0.430
Amplitude flatness at 50% overlap 0.616 0.500 0.500
Overlap correlation at 75% overlap 0.710 0.727 0.738
Amplitude flatness at 75% overlap 0.978 1.000 0.776

See also:
Window

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