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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 50 "Animation of the Fourie
r Analysis of a Square Wave" }}{PARA 0 "" 0 "" {TEXT -1 76 "   The fuc
tion to study is F(t) = \{ -1/2 if  -1<t<0   and +1/2 if 0<t<+1   \}" 
}}{PARA 0 "" 0 "" {TEXT -1 92 "   The coefficients for the sine compon
ents of the Square wave are :  2/(nPi), for all odd n" }}{PARA 0 "" 0 
"" {TEXT -1 85 "     m=number of terms to keep - then we create a \"se
quence\" of the sums (myseqsum) :" }}{PARA 0 "" 0 "" {TEXT -1 68 "    \+
       first term of the sequence is just the first fourier term" }}
{PARA 0 "" 0 "" {TEXT -1 119 "           second term of the sequence i
s the sum of the first two terms, third in sequence is sum of first th
ree terms" }}{PARA 0 "" 0 "" {TEXT -1 69 "   Thus, we can watch the fi
nal waveform evolve into the Square Wave." }}{PARA 0 "" 0 "" {TEXT -1 
135 "   Also, we could create a sequence of just the fourier terms (my
seq2) - we can see the amplitude of each successive term gets smaller.
" }}{PARA 0 "" 0 "" {TEXT -1 139 "                              [ Once
 the \"graphs\" are created, click once on the graph to highlight it, \+
and you will see \"player controls\" " }}{PARA 0 "" 0 "" {TEXT -1 91 "
                                 appear in a toolbar above - to let yo
u run the animation.]" }}{PARA 0 "" 0 "" {TEXT -1 147 "   Notice that \+
as the Square wave evolves .. there is some \"ringing\" at the discont
inuities .. this is called Gibb's Phenomena - no matter how many" }}
{PARA 0 "" 0 "" {TEXT -1 141 "       terms you take, you can't get rid
 of it (at least not the over-ringing .. you can reduce the oscillatio
ns after that with more terms)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 74 "m:=40: # Number of terms to keep  (this will be the non-zero coe
fficients)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "mycoeffs := s
eq( (2/((2*n-1)*Pi)),n=1..m);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 87 "myseq := seq( (2/((2*n-1)*Pi))*sin((2*n-1)*Pi*x),n=1..m):   # (2
n-1) selects only odd n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "
myseqsum := seq(sum(myseq[nn],nn=1..n),n=1..m):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 154 "display(seq(plot(myseqsum[i], x=-0.2..2.2,col
or=blue, thickness=2),i=1..m), insequence=true,          title=`Evolut
ion of Fourier series to Square Wave`);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "                                                         \+
                      " }{TEXT 257 67 "  Note the \"ringing\" in the c
orners of the picture as it evolves!!!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "myseq2 := seq(myseq[n],n=1..m):  # just the individua
l terms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "display(seq(plo
t(myseq2[i], x=-0.2..2.2,color=blue, thickness=2),i=1..m), insequence=
true,          title=`Individual terms of the Fourier series solution \+
to the Square Wave`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
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