The Nuttall window coefficients are given by the following formula

$$a(k)= 0.355768 - 0.487396 \, \cos(\frac{2 \pi k}{N-1}) + 0.144232 \, \cos(\frac{4 \pi k}{N-1}) - 0.012604 \, \cos(\frac{6 \pi k}{N-1})$$

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

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

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

## Measures for the Nuttall window

The following is a comparison of the discrete Fourier transform of the Nuttall window and the rectangular window.

The Nuttall window measures are as follows.

Coherent gain | 0.36 |

Equivalent noise bandwidth | 2.03 |

Processing gain | -3.06 dB |

Scalloping loss | -0.81 dB |

Worst case processing loss | -3.87 dB |

Highest sidelobe level | -93.3 dB |

Sidelobe falloff | -23.4 dB / octave, -77.6 dB / decade |

Main lobe is -3 dB | 1.92 bins |

Main lobe is -6 dB | 2.68 bins |

Overlap correlation at 50% overlap | 0.035 |

Amplitude flatness at 50% overlap | 0.423 |

Overlap correlation at 75% overlap | 0.452 |

Amplitude flatness at 75% overlap | 1.000 |

See also:

Window

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