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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 60 "Animation of the Fourie
r Analysis of a \"Parital-Square\" Wave" }}{PARA 0 "" 0 "" {TEXT -1 
107 "   The fuction to study is F(t) = \{ 0  if  -2<t<-1    -1 if -1<t
<0     +1 if 0<t<+1  and  0 if  +1<t<+2   \}" }}{PARA 0 "" 0 "" {TEXT 
-1 159 "   The coefficients for the sine components of the \"Partial-S
quare\" wave are :  -2/(nPi)[1-cos(nPi/2)],   this means that n=4, 8, \+
12, 16 have zero coefficients" }}{PARA 0 "" 0 "" {TEXT -1 85 "     m=n
umber of terms to keep - then we create a \"sequence\" of the sums (my
seqsum) :" }}{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 is the sum of the first tw
o terms, third in sequence is sum of first three terms" }}{PARA 0 "" 
0 "" {TEXT -1 79 "   Thus, we can watch the final waveform evolve into
 the \"Partial-Square\" Wave." }}{PARA 0 "" 0 "" {TEXT -1 135 "   Also
, we could create a sequence of just the fourier terms (myseq2) - we c
an see the amplitude of each successive term gets smaller." }}{PARA 0 
"" 0 "" {TEXT -1 139 "                              [ Once the \"graph
s\" 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 you run the an
imation.]" }}{PARA 0 "" 0 "" {TEXT -1 157 "   Notice that as the \"Par
tial-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)." }}{PARA 0 "" 0 "" {TEXT -1 104 "     \+
Also, we can see here that the \"overshoot\" is more pronounced when t
here is a larger discontinuity " }}{PARA 0 "" 0 "" {TEXT -1 100 "     \+
                 (note the ringing near the 2 unit drops vs the ringin
g near the 1 unit drops!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "m:=50: # Number of terms to keep" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "mycoeffs := seq( (-2/(n*Pi))*(1-cos(n*Pi/2)),n=1..m);
  # just the coefficients!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "myseq := seq( (-2/(n*Pi))*(1-cos(n*Pi/2))*sin(n*Pi*x/2),n=1..m):  \+
# n=4,8,12,16 = missing!" }}}{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 0 165 "display(seq(plot(myseqsum[i], x=-6..6,color=
blue, thickness=2),i=1..m), insequence=true,          title=`Evolution
 of Fourier series to Partial Period Square Wave`);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 271 "#  Note:  Watch the discontinuities .. t
he series will never fully match the square wave because of the discon
tinuity ... that \"ringing\" at the corners is called the Gibb's Pheno
mena!  And, the ringing is more pronounced depending on how strong the
 discontinuity step is!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "
myseq2 := seq(myseq[n],n=1..m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 178 "display(seq(plot(myseq2[i], x=-6..6,color=blue, thickness=2),
i=1..m), insequence=true,          title=`Individual terms of the Four
ier series of the Partial Period Square Wave`);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "11" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }