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Welch Apodization Function

Welch

The apodization function

A(x)=1-(x^2)/(a^2).

(1)

Its full width at half maximum is sqrt(2)a.

Its instrument function is

where J_nu(z) is a Bessel function of the first kind. This function has a maximum of 4a/3. To investigate the instrument function, define the dimensionless parameter u=2pika and rewrite the instrument function as

I(u)=4a(sinu-ucosu)/(u^3).

(4)

Finding the full width at half maximum then amounts to solving

u^3+6ucosu-6sinu=0,

(5)

which gives u_(1/2)=2pik_(1/2)=2.498255533736..., so for L=2a, the full width at half maximum is

FWHM=2k_(1/2)=(0.795219)/a=(1.59043886)/L.

(6)

The maximum negative sidelobe of -0.0861713 times the peak, and maximum positive sidelobe of 0.356044 times the peak.

See also

Apodization Function, Instrument Function, Parabola

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References

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. 554-556, 1992.

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Welch Apodization Function

Cite this as:

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

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