The Lanczos window coefficients are given by the following formula

$$a(k)=\begin{cases} \frac{\sin(\pi (\frac{2k}{N-1} - 1))}{\pi (\frac{2k}{N-1} - 1)}, k \ne \frac{N-1}{2} \\ 1, k=\frac{N-1}{2} \end{cases}$$

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 Lanczos 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 Lanczos window is as follows.

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

## Measures for the Lanczos window

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

The Lanczos window measures are as follows.

Coherent gain | 0.59 |

Equivalent noise bandwidth | 1.30 |

Processing gain | -1.14 dB |

Scalloping loss | -1.88 dB |

Worst case processing loss | -3.03 dB |

Highest sidelobe level | -26.4 dB |

Sidelobe falloff | -11.5 dB / octave, -38.3 dB / decade |

Main lobe is -3 dB | 1.26 bins |

Main lobe is -6 dB | 1.74 bins |

Overlap correlation at 50% overlap | 0.272 |

Amplitude flatness at 50% overlap | 0.785 |

Overlap correlation at 75% overlap | 0.733 |

Amplitude flatness at 75% overlap | 0.947 |

See also:

Window

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