An apodization function (also called a tapering function or window function) is a function used to smoothly bring a sampled signal down to zero at the edges of the sampled region. This suppresses leakage sidelobes which would otherwise be produced upon performing a discrete Fourier transform, but the suppression is at the expense of widening the lines, resulting in a decrease in the resolution.
A number of apodization functions for symmetrical (two-sided) interferograms are summarized below, together with the instrument functions (or apparatus functions) they produce and a blowup of the instrument function sidelobes. The instrument function corresponding to a given apodization function can be computed by taking the finite Fourier cosine transform,
(1)
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type | apodization function | instrument function |
Bartlett | ||
Blackman | ||
Connes | ||
cosine | ||
Gaussian | ||
Hamming | ||
Hanning | ||
uniform | 1 | |
Welch |
where
(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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The following table summarizes the widths, peaks, and peak-sidelobe-to-peak (negative and positive) for common apodization functions.
type | instrument function FWHM | IF peak | ||
Bartlett | 1.77179 | 1 | 0.00000000 | |
Blackman | 2.29880 | 0.00124325 | ||
Connes | 1.90416 | |||
cosine | 1.63941 | |||
Gaussian | -- | 1 | -- | -- |
Hamming | 1.81522 | 0.00734934 | ||
Hanning | 2.00000 | 1 | 0.00843441 | |
uniform | 1.20671 | 2 | ||
Welch | 1.59044 |
A general symmetric apodization function can be written as a Fourier series
(12)
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where the coefficients satisfy
(13)
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The corresponding instrument function is
(14)
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(15)
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To obtain an apodization function with zero at , use
(16)
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Plugging in (14),
(17)
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(18)
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(19)
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(20)
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The Hamming function is close to the requirement that the instrument function goes to 0 at , giving
(21)
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(22)
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The Blackman function is chosen so that the instrument function goes to 0 at and , giving
(23)
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(24)
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(25)
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