Bandwidth of Gamma-Distribution-Shaped Functions via Lambert W Function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918146535849984 |
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| author | LoPrete, Anthony Burge, Johannes |
| author_facet | LoPrete, Anthony Burge, Johannes |
| contents | The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma distribution probability density function (PDF)-is not widely available. Here, we derive and present just such an expression. To do so, we use the Lambert W function to compute the inverse of the gamma PDF. We use this inverse to derive an exact analytic expression for the width of a gamma distribution at an arbitrary proportion of the maximum, from which the FWHM follows trivially. (An expression for the octave bandwidth of gamma-shaped functions is also provided.) The FWHM is then compared to the Gaussian approximation of gamma-shaped functions. A few other related issues are discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_19307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bandwidth of Gamma-Distribution-Shaped Functions via Lambert W Function LoPrete, Anthony Burge, Johannes Signal Processing Probability 60E05 (primary), 33E20 (secondary) The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma distribution probability density function (PDF)-is not widely available. Here, we derive and present just such an expression. To do so, we use the Lambert W function to compute the inverse of the gamma PDF. We use this inverse to derive an exact analytic expression for the width of a gamma distribution at an arbitrary proportion of the maximum, from which the FWHM follows trivially. (An expression for the octave bandwidth of gamma-shaped functions is also provided.) The FWHM is then compared to the Gaussian approximation of gamma-shaped functions. A few other related issues are discussed. |
| title | Bandwidth of Gamma-Distribution-Shaped Functions via Lambert W Function |
| topic | Signal Processing Probability 60E05 (primary), 33E20 (secondary) |
| url | https://arxiv.org/abs/2509.19307 |