The processing gain of a filter or a window (or a system) is

$$PG=\frac{1}{ENBW}$$

where ENBW is equivalent noise bandwidth of the filter or window.

ENBW measures the noise power accumulated in the magnitude response of the filter. The term "processing gain" follows, since this noise power is larger than the noise power that would be accumulated by an ideal filter with the same cutoff frequency. Alternatively, since some of the noise power of the filter spills out of the cutoff frequency, we can define the term "processing loss". These two terms are equivalent.

In spectral analysis, when windows are used before the Fourier transform, and since windows decrease the power of the signal before the transform, the processing gain may be defined as

$$PG=\frac{CG}{ENBW}$$

where CG is the coherent gain of the window. Usually though, when measures for windows are listed, the processing gain will be computed assuming the coherent gain is 1 (i.e., the results are adjusted for the coherent gain).

The processing gain can be expressed in decibels, but since it is a measure of power, the computation of the decibel value uses

$$10\,log_{10} (\frac{1}{ENBW})$$

Whether the result is called processing gain or processing loss, it will usually be written as a positive number.

The ENBW of the Hamming window, for example, is 1.36 (bins). This means that the processing gain of the window is 1 / 1.36 = 0.735 = -1.335 dB.

## Processing gain for common windows

The following is the processing gain (in dB) of some common windows (with the window definitions on this site).

Bartlett-Hann | -1.64 |

Blackman Exact Blackman Generalized Blackman α = 0.05 α = 0.20 α = 0.35 |
-2.38 -2.30 -1.93 |

Blackman-Harris | -3.03 |

Blackman-Nuttall | -2.97 |

Bohman | -2.53 |

Dolph-Chebychev ω _{0} = 0.1ω _{0} = 0.2ω _{0} = 0.3 |
-2.56 -3.66 -4.35 |

Flat top | -5.77 |

Gaussian σ = 0.3 σ = 0.5 σ = 0.7 Approximate confined Gaussian σ = 0.3 σ = 0.5 σ = 0.7 Generalized normal α = 2 α = 4 α = 6 |
-2.76 -0.91 -0.32 -2.77 -0.91 |

Hamming | -1.35 |

Hann | -1.77 |

Hann-Poisson α = 0.3 α = 0.5 α = 0.7 |
-1.95 -2.07 -2.20 |

Kaiser α = 0.5 α = 1.0 α = 5.0 |
-0.10 -0.62 -2.44 |

Kaiser-Bessel | -2.55 |

Lanczos | -1.14 |

Nuttall | -3.06 |

Parzen | -0.50 |

Planck-taper ε = 0.2 ε = 0.4 ε = 0.5 |
-0.77 -1.64 -2.11 |

Poisson α = 0.2 α = 0.5 α = 0.8 |
-0.01 -0.09 -0.22 |

Power of cosine α = 1.0 α = 2.0 α = 3.0 |
-0.92 -1.77 -2.40 |

Rectangular | 0.00 |

Sine | -0.92 |

Triangular | -1.26 |

Tukey α = 0.3 α = 0.5 α = 0.7 |
-1.25 -0.88 -0.52 |

Ultraspherical (x_{0} = 1)μ = 2 μ = 3 μ = 4 |
-0.71 -1.42 -1.96 |

Welch | -0.78 |

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