Uniform Apodization Function

Author

Eric W. Weisstein
October 7, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/UniformApodizationFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/UniformApodizationFunction.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Apodization Function

Uniform[x]

Maximum

Limit[Uniform[x],x->0]

Instrument Function

FullSimplify[FullSimplify[InstrumentFunction[Uniform[x],{x,a},k,Assumptions->k>0&&a>0]]/.Sin[x_]:>Sinc[x]x]

i[k_]:=((2a Sin[2 a k π])/((2 a k π)))

Maximum

Limit[i[k],k->0]

Plots

ApodizationPlot[{Uniform[x],((2a Sin[2 a k π])/((2 a k π)))},{x,k},a]

FWHM

Apodization Function

N/A

Instrument Function

2NFWHM[i[k]/a/.a->x/k,x]/L

Sidelobes

Numerator[FullSimplify[Together[D[i[k],k]]]]/.a->x/k

f[x_]:=2 π x Cos[2 π x]-Sin[2 π x]

roots=x/.PlotRoot[f[x],{x,0,5},
PlotStyle->Red]

extrema={#,i[#]/.a->1}&/@roots

Plot[Evaluate[i[k]/.a->1],{k,0,5},PlotStyle->Red,
Epilog->{PointSize[.02],Red,Point/@extrema}]