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Apodization Function

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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 I(k) corresponding to a given apodization function A(x) can be computed by taking the finite Fourier cosine transform,

I(k)=int_(-a)^acos(2pikx)A(x)dx.
(1)
InstrumentFunctions
typeapodization functioninstrument function
Bartlett1-(|x|)/aasinc^2(pika)
BlackmanB_A(x)B_I(k)
Connes(1-(x^2)/(a^2))^28asqrt(2pi)(J_(5/2)(2pika))/((2pika)^(5/2))
cosinecos((pix)/(2a))(4acos(2piak))/(pi(1-16a^2k^2))
Gaussiane^(-x^2/(2sigma^2))2int_0^acos(2pikx)e^(-x^2/(2sigma^2))dx
HammingHm_A(x)Hm_I(k)
HanningHn_A(x)Hn_I(k)
uniform12asinc(2pika)
Welch1-(x^2)/(a^2)W_I(k)

where

B_A(x)=(21)/(50)+1/2cos((pix)/a)+2/(25)cos((2pix)/a)
(2)
B_I(k)=(a((21)/(25)-9/(25)a^2k^2)sinc(2piak))/((1-a^2k^2)(1-4a^2k^2))
(3)
Hm_A(x)=(27)/(50)+(23)/(50)cos((pix)/a)
(4)
Hm_I(k)=(a((27)/(25)-(16)/(25)a^2k^2)sinc(2piak))/(1-4a^2k^2)
(5)
Hn_A(x)=cos^2((pix)/(2a))
(6)
=1/2[1+cos((pix)/a)]
(7)
Hn_I(k)=(asinc(2piak))/(1-4a^2k^2)
(8)
=a[sinc(2pika)+1/2sinc(2pika-pi)+1/2sinc(2pika+pi)]
(9)
W_I(k)=a2sqrt(2pi)(J_(3/2)(2pika))/((2pika)^(3/2))
(10)
=a(sin(2pika)-2piakcos(2piak))/(2a^3k^3pi^3).
(11)

The following table summarizes the widths, peaks, and peak-sidelobe-to-peak (negative and positive) for common apodization functions.

typeinstrument function FWHMIF peak(peak (-) sidelobe)/(peak)(peak (+) sidelobe)/(peak)
Bartlett1.7717910.000000000.0471904
Blackman2.29880(21)/(25)-0.001067240.00124325
Connes1.90416(16)/(15)-0.04110490.0128926
cosine1.639414/pi-0.07080480.0292720
Gaussian--1----
Hamming1.81522(27)/(25)-0.006891320.00734934
Hanning2.000001-0.02670760.00843441
uniform1.206712-0.2172340.128375
Welch1.590444/3-0.08617130.0356044

A general symmetric apodization function A(x) can be written as a Fourier series

A(x)=a_0+2sum_(n=1)^inftya_ncos((npix)/b),
(12)

where the coefficients satisfy

a_0+2sum_(n=1)^inftya_n=1.
(13)

The corresponding instrument function is

I(t)=int_(-b)^bA(x)e^(-2piikx)dx
(14)
=2b{a_0sinc(2pikb)+sum_(n=1)^(infty)[sinc(2pikb+npi)+sinc(2pikb-npi)]}.
(15)

To obtain an apodization function with zero at ka=3/4, use

a_0sinc(3/2pi)+a_1[sinc(5/2pi)+sinc(1/2pi)]=0.
(16)

Plugging in (14),

-(1-2a_1)2/(3pi)+a_1(2/(5pi)+2/pi)=-1/3(1-2a_1)+a_1(1/5+1)=0
(17)
a_1(6/5+2/3)=1/3
(18)
a_1=(1/3)/(6/5+2/3)=5/(6·3+2·5)=5/(28)
(19)
a_0=1-2a_1=(28-2·5)/(28)=(18)/(28)=9/(14).
(20)

The Hamming function is close to the requirement that the instrument function goes to 0 at ka=5/4, giving

a_0=(25)/(46) approx 0.5435
(21)
a_1=(21)/(92) approx 0.2283.
(22)

The Blackman function is chosen so that the instrument function goes to 0 at ka=5/4 and ka=9/4, giving

a_0=(3969)/(9304) approx 0.42659
(23)
a_1=(1155)/(4652) approx 0.24828
(24)
a_2=(715)/(18608) approx 0.38424,
(25)

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.

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Apodization Function

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Weisstein, Eric W. "Apodization Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ApodizationFunction.html

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