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)

where

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)

where the coefficients satisfy

(13)

The corresponding instrument function is

To obtain an apodization function with zero at , use

(16)

Plugging in (14 ),

(17)

(18)

The Hamming function is close to the requirement that the instrument function goes to 0 at
,
giving

The Blackman function is chosen so that the instrument function goes to 0 at and , giving

See also Bartlett Function ,

Blackman Function ,

Connes Function ,

Cosine
Apodization Function ,

Full Width at Half
Maximum ,

Gaussian Function ,

Hamming
Function ,

Hanning Function ,

Leakage ,

Mertz Apodization Function ,

Parzen
Apodization Function ,

Uniform Apodization
Function ,

Welch Apodization Function
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References Ball, J. A. "The Spectral Resolution in a Correlator System" §4.3.5 in Astrophysics,
Part C: Radio Observations (Ed. M. L. Meeks). New York: Academic Press,
pp. 55-57, 1976. Blackman, R. B. and Tukey, J. W. "Particular
Pairs of Windows." In The
Measurement of Power Spectra, From the Point of View of Communications Engineering.
New York: Dover, pp. 95-101, 1959. Brault, J. W. "Fourier
Transform Spectrometry." In High
Resolution in Astronomy: 15th Advanced Course of the Swiss Society of Astronomy and
Astrophysics (Ed. A. Benz, M. Huber, and M. Mayor). Geneva
Observatory, Sauverny, Switzerland, pp. 31-32, 1985. Harris, F. J.
"On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform."
Proc. IEEE 66 , 51-83, 1978. Norton, R. H. and Beer,
R. "New Apodizing Functions for Fourier Spectroscopy." J. Opt. Soc.
Amer. 66 , 259-264, 1976. Press, W. H.; Flannery, B. P.;
Teukolsky, S. A.; and Vetterling, W. T. Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 547-548, 1992. Schnopper, H. W.
and Thompson, R. I. "Fourier Spectrometers." In Astrophysics,
Part A: Optical and Infrared (Ed. N. P. Carleton). New York: Academic
Press, pp. 491-529, 1974. Referenced on Wolfram|Alpha Apodization Function
Cite this as:
Weisstein, Eric W. "Apodization Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ApodizationFunction.html

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