Hamming Function

Author

Eric W. Weisstein
October 7, 2004

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

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

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

Apparatus Function

Hamming[x,a]

Maximum

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

Instrument Function

v4.2

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

v5.0

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

i[k_]:=((a (-1.08+0.64 (a^2) (k^2)) Sin[2 a k π])/((-1+4 (a^2) (k^2))(2a k π)))

Maximum

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

Plots

ApodizationPlot[{Hamming[x,a],((a (-1.08+0.64 (a^2) (k^2)) Sin[2 a k π])/((-1+4 (a^2) (k^2))(2a k π)))},{x,k},a]

FWHM

Apodization Function

FWHM[Hamming[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_]:=-(r (6.785840131753954`-31.16459912361075` (r^2)+16.084954386379742` (r^4)) Cos[2 π r]+(-1.08`+12.320000000000002` (r^2)-2.5599999999999996` (r^4)) 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}]