Cosine Apodization Function

Author

Eric W. Weisstein
October 7, 2004

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

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

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

Apparatus Function

Cosine[x,a]

Maximum

Limit[Cosine[x,a],x->0]

Instrument Function

FullSimplify[InstrumentFunction[Cosine[x,a],{x,a},k,Assumptions->k>0&&a>0]]

i[k_]:=((4 a Cos[2 a k π])/(π-16 (a^2) (k^2) π))

Maximum

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

Plots

V5

ApodizationPlot[{Cosine[x,a],((4 a Cos[2 a k π])/(π-16 (a^2) (k^2) π))},{x,k},a]

V7

ApodizationPlot[{Cosine[x,a],((4 a Cos[2 a k π])/(π-16 (a^2) (k^2) π))},{x,k},a,PlotStyle->{Thickness[.01],Red}]

FWHM

Apodization Function

FWHM[Cosine[x,a]/.a->1,x]a

Instrument Function

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

Sidelobes

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

f[r_]:=8 (r^2) (16 r Cos[2 π r]+π (-1+16 (r^2)) Sin[2 π r])

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

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

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