Tukey window

The Tukey window coefficients are given by the following formula

Tukey window 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.

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

The Tukey window approaches the Hann window at alpha equal to zero

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

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