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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "This worksheet allows us t
o study aliasing." }}{PARA 0 "" 0 "" {TEXT -1 26 "We begin with the fu
nction" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f0 := (x,a)->" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 32 "2*((sin(x)/x)+(cos(x)-1)/(x*x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#f0Gf*6$%\"xG%\"aG6\"6$%)operatorG%&arrowG
F),&*&-%$sinG6#9$\"\"\"F2!\"\"\"\"#*(F5F3,&-%$cosGF1F3F3F4F3F2!\"#F3F)
F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Here's what it looks like
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f0(x),x=-10..10);" }}
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}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "This function has its Fourier \+
transform supported in the interval [-1,1]." }}{PARA 0 "" 0 "" {TEXT 
-1 72 "To reconstruct it perfectly, Nyquist's theorem say we need to s
ample it " }}{PARA 0 "" 0 "" {TEXT -1 73 "at equally spaced points pla
ced a distance Pi apart. If instead we sample" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "at points that are 2Pi/(1+a) for 0<a<1 we get aliasing as
 illustrated below." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f := (x,a)->
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "2*((sin(x)/x)+(cos(x)-1)/(x*x)+
(1-a)*(sin(x))/x+(cos(x)-cos(a*x))/(x*x" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"aG6\"6
$%)operatorG%&arrowGF),**&-%$sinG6#9$\"\"\"F2!\"\"\"\"#*(F5F3,&-%$cosG
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2F3F4F3F2F:F3F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Using anim
ation we can see how aliasing effects the reconstruction of our" }}
{PARA 0 "" 0 "" {TEXT -1 9 "function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
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58 "Here we consider a more realistic example. Our function is" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "g0 := x -> exp(-abs(x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#g0Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$
expG6#,$-%$absG6#9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 
"Its Fourier transform is 2Pi/(1+w^2). It's NOT bandlimited!" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "plot(g0(x),x=-10..10);" }}{PARA 13 "" 1 "
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