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{PARA 0 "" 0 "" {TEXT -1 318 "#In this worksheet we further investigat
e the effects of using a smoother window and oversampling on the inter
polation function used in the Shannon-Whitaker formula. First we defin
e the sinc-function, which corresponds to a rectangular window . After
wards we consider windows defined by a once differentiable function." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sinc0 := x-> evalf(1+sum(
(x^(2*m))/((2*m+1)!)*(-1)^m, m=1..infinity));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&sinc0Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%&evalfG6#
,&\"\"\"F0-%$sumG6$*()9$,$%\"mG\"\"#F0-%*factorialG6#,&F8F9F0F0!\"\")F
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1 0 57 "sinc := x->(piecewise(abs(x)<.5, sinc0(x), sin(x)/x))/Pi;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sincGf*6#%\"xG6\"6$%)operatorG%&arr
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\"39`_$Q(>&4W$FH7$$\"3h******>5s#y*F1$!3q2$\\w[_&z8F-7$$\"3U+++l2/P)*F
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car-%+AXESLABELSG6$Q\"x6\"Q!Fhar-%%VIEWG6$;F(F[ar%(DEFAULTG" 1 2 0 1 
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{PARA 0 "" 0 "" {TEXT -1 248 "#The next several commands define the cu
bic polynomial needed to interpolate between 1 and 0. Then we put the \+
pieces together to define the once differentiable window function f.  \+
f(x,b,c) is 1 between [-b,b] and vanishes outside of [-(c+b),(c+b)]." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g0 := (x,b,c) -> x^3/3-(2
*b+c)*x^2/2+b*(b+c)*x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "g1 := (x,
b,c)-> g0(x,b,c)-g0(b+c,b,c);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g \+
:= (x,b,c) -> g1(x,b,c)/g1(b,b,c);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "g(2,1,4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := (x,b,c)->piece
wise(abs(x)<= b,1,b<abs(x) and abs(x)<=b+c, g(x,b,c),0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#g0Gf*6%%\"xG%\"bG%\"cG6\"6$%)operatorG%&arrow
GF*,(*$)9$\"\"$\"\"\"#F3F2*&#F3\"\"#F3*&,&9%F79&F3F3)F1F7F3F3!\"\"*(F:
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6%%\"xG%\"bG%\"cG6\"6$%)operatorG%&arrowGF*,&-%#g0G6%9$9%9&\"\"\"-F06%
,&F3F5F4F5F3F4!\"\"F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6
%%\"xG%\"bG%\"cG6\"6$%)operatorG%&arrowGF**&-%#g1G6%9$9%9&\"\"\"-F06%F
3F3F4!\"\"F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#F\"#K" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"bG%\"cG6\"6$%)operat
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plot([f(x,1,.1),f(x,1,.25), f(x,1,.5),c(x)],x=0..1.75);" }}{PARA 13 "
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hy$\"3Q^=klsx)>#F67$$\"3rTgxtMYm8Fbq$\"3%G#yj#yW)e<F67$F[z$\"3^9Qo%>G^
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3uMXA#*R(y&>F/7$$\"3!=zp)4&=xY\"Fbq$\"3Ss5@&\\!p'>\"F/7$$\"3'o\"HdDv*p
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F^\\lF_\\lF_\\lF(-F$6$7hnF'F,F0F3F7F:F=F@FCFFFIFLFOFRFUFXFenFhnF[oF^oF
aoFdoFgoFjoF]pF`pFcpFfpFip7$$\"3\")fk.aoks)*F6F*F\\q7$$\"3IZ'yZh7H%**F
6F*7$$\"3QUg-NXLm**F6F*7$$\"3/T(\\^\\X!y**F6F*7$$\"3eQMFbkv*)**F6F*7$$
\"3h8(R:uY,+\"FbqF(7$F`qF(7$FfqF(7$F[rF(7$F_sF(7$FctF(7$FavF(FbxFhxF[y
F^yFayFdyFgyFjyF]zF`zFczFfzFizF\\[lF_[lFb[lFe[lFh[l-F\\\\l6&F^\\lF(F(F
_\\l-%+AXESLABELSG6$Q\"x6\"Q!Fg[m-%%VIEWG6$;F($\"$v\"!\"#%(DEFAULTG" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "#Thi
s function is the inverse Fourier transform of the smoothed window fun
ction:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "s := (x,b)-> eval
f(1/(2*Pi)*int(f(y,1,b)*cos(x*y),y=0..1+b));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sGf*6$%\"xG%\"bG6\"6$%)operatorG%&arrowGF)-%&evalfG
6#,$*&%#PiG!\"\"-%$intG6$*&-%\"fG6%%\"yG\"\"\"9%F<-%$cosG6#*&9$F<F;F<F
</F;;\"\"!,&F<F<F=F<F<#F<\"\"#F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 82 "#Here we examine the different interpolating functions near to \+
their peaks at x=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot
([sinc(x),s(x,.1),s(x,.25),s(x,.5)],x=0.001..20,discont=true);" }}
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