The Blackman-Nuttall window coefficients are given by the following formula

$$a(k)=0.3635819-0.4891775 \, cos(\frac{2\pi k}{N-1})$$ $$+0.1365995 \, cos(\frac{4\pi k}{N-1})-0.0106411 \, cos(\frac{6\pi k}{N-1})$$

where N is the length of the filter and k = 0, 1, …, N – 1.

The Blackman-Nuttall window is a generalized cosine window (see Hamming window).

## An example Blackman-Nuttall window

Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Blackman-Nuttall 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 Blackman-Nuttall window is as follows.

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

## Measures for the Blackman-Nuttall window

The following graph compares the discrete Fourier transform of the Blackman-Nuttall window and the rectangular window.

The Blackman-Nuttall window measures are as follows.

Coherent gain | 0.36 |

Equivalent noise bandwidth | 1.98 |

Processing gain | -2.97 dB |

Scalloping loss | -0.85 dB |

Worst case processing loss | -3.81 dB |

Highest sidelobe level | -98.3 dB |

Sidelobe falloff | -12.7 dB / octave, -42.2 dB / decade |

Main lobe is -3 dB | 1.88 bins |

Main lobe is -6 dB | 2.62 bins |

Overlap correlation at 50% overlap | 0.041 |

Amplitude flatness at 50% overlap | 0.454 |

Overlap correlation at 75% overlap | 0.469 |

Amplitude flatness at 75% overlap | 1.000 |

See also:

Window

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