Welch Apodization Function

Author

Eric W. Weisstein
September 30, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Plots

ApodizationPlot[{Welch[x,a],2a Sqrt[2π]BesselJ[3/2,2π k a]/(2π k a)^(3/2)},{x,k},a]

Apodization Function

Welch[x,a]

Maximum

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

FWHM

Solve[Welch[x,a]==Welch[0,a]/2,x]

Instrument Function

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

FullSimplify[FunctionExpand[2a Sqrt[2π]BesselJ[3/2,2π k a]/(2π k a)^(3/2)]]

i[k_]:=2a Sqrt[2π]BesselJ[3/2,2π k a]/(2π k a)^(3/2)

Maximum

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

FWHM

FullSimplify[i[k]/.a->1/.k->u/(2π)]

Solve[i[k]==Limit[i[k],k->0]/2/.a->1/.k->u/(2π),u]

Reduce[((4 (-Cos[u]+((Sin[u])/u)))/(u^2))==(2/3)&&0<u<5,u,Reals]

FullSimplify[((4 (-Cos[u]+((Sin[u])/u)))/(u^2))]

Root[{-6 Sin[#1]+6 Cos[#1] #1+((#1)^3)&,2.4982555337363949996653281067298185744780969455287552299877`30.102999566398122}]/π//N[#,10]&

2Root[{-6 Sin[#1]+6 Cos[#1] #1+((#1)^3)&,2.4982555337363949996653281067298185744780969455287552299877`30.102999566398122}]/π//N[#,10]&

Sidelobes

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

f[x_]:=6 π x Cos[2 π x]+(-3+4 (π^2) (x^2)) 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}]

((0.04747250702347316`)/(4/3))