The Bohman window coefficients are given by the following formula

$$w(k)=(1-|\frac{k}{M}-1|)\,cos(\pi |\frac{k}{M}-1|)+\frac{1}{\pi}\,sin(\pi |\frac{k}{M}-1|)$$

where N is the length of the filter, M = (N – 1) / 2, and k = 0, 1, …, N – 1.

The Bohman window is the convolution of the sine window with itself.

## An example Bohman window

Take a finite impulse response (FIR) low pass filter of length N = 201. The following is the Bohman 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 Bohman window is as follows.

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

## Measures for the Bohman window

The following is a plot of the discrete Fourier transform of the Bohman window against the discrete Fourier transform of the rectangular window.

The measures of the Bohman window are as follows.

Coherent gain | 0.40 |

Equivalent noise bandwidth | 1.79 |

Processing gain | -2.53 dB |

Scalloping loss | -1.02 dB |

Worst case processing loss | -3.55 dB |

Highest sidelobe level | -46.0 dB |

Sidelobe falloff | -21.4 dB / octave, -71.0 dB / decade |

Main lobe is -3 dB | 1.70 bins |

Main lobe is -6 dB | 2.38 bins |

Overlap correlation at 50% overlap | 0.073 |

Amplitude flatness at 50% overlap | 0.637 |

Overlap correlation at 75% overlap | 0.544 |

Amplitude flatness at 75% overlap | 0.982 |

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