Blackman Function

Author

Eric W. Weisstein
October 7, 2004

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

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

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

Apodization Function

Blackman[x,a]

Maximum

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

Instrument Function

InstrumentFunction[.42,{x,a},k,Assumptions->k>0&&a>0]

InstrumentFunction[.5Cos[π x/a],{x,a},k,Assumptions->k>0&&a>0]

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

FullSimplify[InstrumentFunction[Blackman[x,a],{x,a},k,Assumptions->k>0&&a>0]/.Sin[x_]Cos[x_]y_:>y Sin[2x]/2/.Sin[x_]:>Sinc[x]x]

Factor[1-5 (a^2) (k^2)+4 (a^4) (k^4)]//.(-1+x_)(1+x_)->x^2-1

i[k_]:=((a (0.84-0.36 (a^2) (k^2)) Sin[2 a k π])/((1-(a^2) (k^2)) (1-4 (a^2) (k^2))(2 a k π)))

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

Plot

ApodizationPlot[{Blackman[x,a],((a (0.84-0.36 (a^2) (k^2)) Sin[2 a k π])/((1-(a^2) (k^2)) (1-4 (a^2) (k^2))(2 a k π)))},{x,k},a]

Apodization Function

NFWHM[Blackman[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 (5.277875658030852`-28.651325000738915` (r^2)+32.421236185046666` (r^4)-9.047786842338605` (r^6)) Cos[2 π r]+(-0.84`+12.24` (r^2)-18.599999999999998` (r^4)+4.319999999999999` (r^6)) 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}]