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.

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

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

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).

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.

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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