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GeezerVison is showing mathworld.wolfram.com — Instrument Function -- from Wolfram MathWorld

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Instrument Function

The finite Fourier cosine transform of an apodization function, also known as an apparatus function. The instrument function I(k) corresponding to a given apodization function A(x) is

I(k)=int_(-a)^ae^(2piikx)A(x)dx
(1)

which, upon expanding the complex exponential,

I(k)=int_(-a)^ae^(2piikx)A(x)dx
(2)
=int_(-a)^acos(2pikx)A(x)dx+iint_(-a)^asin(2pikx)A(x)dx.
(3)

For A(x) an even function, the left integrand is even (and hence is equal to twice its value over half its interval) and the right integrand is odd (and hence equal to 0), so

I(k)=2int_0^acos(2pikx)A(x)dx.
(4)

See also

Apodization Function, Fourier Cosine Transform

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Cite this as:

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

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