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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 55 "Animation of the Fourie
r Analysis of a \"Saw-Tooth\" Wave" }}{PARA 0 "" 0 "" {TEXT -1 55 "   \+
The fuction to study is F(t) =  t    for -1 < t < 1 " }}{PARA 0 "" 0 "
" {TEXT -1 118 "   The coefficients for the sine components of the \"S
aw tooth\" wave are complicated (switching signs and magnitudets)." }}
{PARA 0 "" 0 "" {TEXT -1 85 "     m=number of terms to keep - then we \+
create a \"sequence\" of the sums (myseqsum) :" }}{PARA 0 "" 0 "" 
{TEXT -1 68 "           first term of the sequence is just the first f
ourier term" }}{PARA 0 "" 0 "" {TEXT -1 119 "           second term of
 the sequence is the sum of the first two terms, third in sequence is \+
sum of first three terms" }}{PARA 0 "" 0 "" {TEXT -1 74 "   Thus, we c
an watch the final waveform evolve into the \"Saw Tooth\" Wave." }}
{PARA 0 "" 0 "" {TEXT -1 135 "   Also, we could create a sequence of j
ust the fourier terms (myseq2) - we can see the amplitude of each succ
essive term gets smaller." }}{PARA 0 "" 0 "" {TEXT -1 139 "           \+
                   [ Once the \"graphs\" are created, click once on th
e graph to highlight it, and you will see \"player controls\" " }}
{PARA 0 "" 0 "" {TEXT -1 91 "                                 appear i
n a toolbar above - to let you run the animation.]" }}{PARA 0 "" 0 "" 
{TEXT -1 151 "   Notice that as the \"SawTooth\" wave evolves .. there
 is some \"ringing\" at the discontinuities .. this is called Gibb's P
henomena - 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 oscillations after that with more terms)." }}
{PARA 0 "" 0 "" {TEXT -1 104 "     Also, we can see here that the \"ov
ershoot\" is more pronounced when there is a larger discontinuity " }}
{PARA 0 "" 0 "" {TEXT -1 100 "                      (note the ringing \+
near the 2 unit drops vs the ringing 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:=20: # Number of terms to \+
keep" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "mycoeff := seq( (-8
*(-sin(n*Pi)+n*Pi*cos(n*Pi))/(n^2*Pi^2)),n=1..m);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "myseq
 := seq( (-8*(-sin(n*Pi)+n*Pi*cos(n*Pi))/(n^2*Pi^2))*sin(n*Pi*x/2),n=1
..m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "myseqsum := seq(su
m(myseq[nn],nn=1..n),n=1..m):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 147 "display(seq(plot(myseqsum[i], x=-5..5,color=blue, thickness=2),
i=1..m), insequence=true,          title=`Evolution of Fourier series \+
to SawTooth`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 268 "#  Note:
  Watch the discontinuities .. the series will never fully match the S
awTooth because of the discontinuity ... that \"ringing\" at the corne
rs is called the Gibb's Phenomena!  And, the ringing is more pronounce
d 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 0 161 "display(seq(plot(myseq2[i],
 x=-5..5,color=blue, thickness=2),i=1..m), insequence=true,          t
itle=`Individual terms of the Fourier series for the SawTooth`);" }}}
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