Hanning Function

Author

Eric W. Weisstein
September 30, 2007

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

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

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

Plots

ApodizationPlot[{Hanning[x,a],((a Sinc[2 a k π])/((1-4 (a^2) (k^2))))},{x,k},a]

Apodization Function

Hanning[x,a]

Maximum

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

FWHM

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

Instrument Function

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

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

i2[k_]:=a(Sinc[2π k a]+(1/2)Sinc[2π k a-π]+(1/2)Sinc[2π k a+π])

FullSimplify[i2[k]]

FullSimplify[i[k]-i2[k]]

Maximum

i[0]

FWHM

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

Reduce[Sin[u]==(1/2)(1-((u^2)/(π^2)))u&&0<u<4,u]

Reduce[Sinc[x]==(1/2)(1-((x^2)/(π^2)))&&0<x<4,x]

Reduce[Sinc[x]==(1/2)(1-((x^2)/(π^2)))&&0<x<4,x,Reals]

FullSimplify[%]

Sinc[x]==(1/2)(1-((x^2)/(π^2)))/.x->π

Reduce[Sinc[2 π u]==(1/2)(1-4 (u^2))&&0<u<1,u,Reals]

u==(1/2)

FindRoot[((Sinc[2 π u])/(1-4 (u^2)))==(1/2),{u,.4}]

{u->0.5000000000273569`}

FindRoot[((Sinc[2 π u])/(1-4 (u^2)))==(1/2),{u,.4,.6}]

{u->0.49999999630069414`}

2k/.Flatten[Solve[2 a k π==π,k]]/.a->L/2

Sidelobes

Plot[i[k]/.a->1,{k,-3,3},PlotRange->{-.03,.02}]

i[k]

i[k]/.a->1/.k->u/(2π)

((Sinc[u])/(1-((u^2)/(π^2))))

Plot[((Sinc[u])/(1-((u^2)/(π^2)))),{u,0,12}]

Minimize[{((Sinc[u])/(1-((u^2)/(π^2)))),0<u<10},u,Reals]

Minimize[{((Sinc[u])/(1-((u^2)/(π^2)))),0<u<10},u,Reals]

Numerator[FullSimplify[D[((Sinc[u])/(1-((u^2)/(π^2)))),u]]]

(π^2) (((π^2)-(u^2)) (u Cos[u]-Sin[u])+2 (u^2) Sin[u])

Plot[ (((π^2)-(u^2)) (u Cos[u]-Sin[u])+2 (u^2) Sin[u]),{u,0,11},PlotRange->{-100,100}]

FindRoot[((π^2)-(u^2)) (u Cos[u]-Sin[u])+2 (u^2) Sin[u]==0,{u,7}]

Reduce[((π^2)-(u^2)) (u Cos[u]-Sin[u])+2 (u^2) Sin[u]==0&&6<u<11,u,Reals]

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

<<Utilities`Plot`

PlotRoots[(((π^2)-(u^2)) (u Cos[u]-Sin[u])+2 (u^2) Sin[u]),{u,0,30},PlotRange->All]