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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 50 "Fourier, Windowed Fourier
, and Wavelet transforms " }}{PARA 0 "" 0 "" {TEXT -1 132 "Here we wan
t to compare the Continuous versions of the  Fourier Transform  the Wi
ndowed Fourier Transform and the Wavelet transform." }}{PARA 0 "" 0 "
" {TEXT -1 139 "We are using Maple and concocted function to get exact
 formulas for good plotting.  Each of these has a finite discrete vers
ion as follows:" }}{PARA 15 "" 0 "" {TEXT -1 28 " Fourier Transform  <
--> FFT" }}{PARA 15 "" 0 "" {TEXT -1 94 "Windowed Fourier Transform  <
---> Finite version using short time FFT and windowing functions." }}
{PARA 15 "" 0 "" {TEXT -1 40 "Wavelet Transform  <--->  Filter banks. \+
" }}{PARA 0 "" 0 "" {TEXT -1 111 "In this work sheet we will just talk
 about the first two, the Wavelet transform will be in the next work s
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{TEXT -1 13 "Load packages" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(
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aph and the formula it would appear that there is a single  frequency \+
of " }{XPPEDIT 18 0 "5/(2*Pi)" "*&\"\"&\"\"\"*&\"\"#F$%#PiGF$!\"\"" }
{TEXT -1 40 "  cylces per second localized near t=1. " }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 32 "The Continuous Fourier transform" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "General definition" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "FTh := Int(h(t)*exp(-I*w*t),t=-infi
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*&-%\"hG6#%\"tG\"\"\"-%$expG6#,$*(%\"IGF-%\"wGF-F,F-!\"\"F-/F,;,$%)inf
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ion and plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "FTf := fouri
er(f(t),t,w);\nFTf := evalc(Re(FTf));" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%$FTfG,&*.-%$expG6##!\"\"\"\"&\"\"\"-F(6#,$%\"IG!\"&F-%#PiG#F-\"\"
#F,F4-F(6#*&,&#\"#DF5F-F1F+F-%\"wGF-F--F(6#,(#!$@'\"#?F-F1F,*$F<F5#F2
\"\"%F-F4*.F'F--F(6#,$F1F,F-F3F4F,F4-F(6#*&,&#!#DF5F-F1F+F-F<F-F--F(6#
,(F@F-F1F2FCFDF-F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$FTfG,**(,&*.-
%$expG6##!\"\"\"\"&\"\"\"-%$cosG6#F.F/%#PiG#F/\"\"#F.F4-F*6#,$%\"wG#\"
#DF5F/-F16#F9F/F/*.F)F/-%$sinGF2F/F3F4F.F4F6F/-F@F=F/F-F/-F*6#,&#!$@'
\"#?F/*$F9F5#!\"&\"\"%F/F0F/F4*(,&*.F)F/F?F/F3F4F.F4F6F/F<F/F-*.F)F/F0
F/F3F4F.F4F6F/FAF/F-F/FBF/F?F/#F-F5*(,&*.F)F/F0F/F3F4F.F4-F*6#,$F9#!#D
F5F/F<F/F/*.F)F/F?F/F3F4F.F4FTF/FAF/F/F/FBF/F0F/F4*(,&*.F)F/F?F/F3F4F.
F4FTF/F<F/F/*.F)F/F0F/F3F4F.F4FTF/FAF/F-F/FBF/F?F/F4" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 89 "The plot below shows that the function clearly \+
has a frequency localized around  5 and -5" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "plot(FTf,w=-10..10);" }}{PARA 13 "" 1 "" {INLPLOT "6%
-%'CURVESG6$7_s7$$!#5\"\"!$!1)4ZrTfyX%!#H7$$!1nmm;p0k&*!#:$!1$)f')>n**
\\'*!#F7$$!1LL$3<XZ=*F1$!1$*4Xs\"\\H*f!#D7$$!1mmmT%p\"e()F1$!1e&zvQz)
\\L!#B7$$!1nmm\"4m(G$)F1$!15*4bf<Xu)!#A7$$!1LL$3i.9!zF1$!1\"G>YF[(GD!#
@7$$!1nm;/R=0vF1$\"11<n$p!3`E!#>7$$!1ML$3i_+I(F1$\"1DKcxb.+9!#=7$$!1,+
]P8#\\4(F1$\"1ErV^d5`cFX7$$!1nm;/y%)))pF1$\"1,R$)fBZu5!#<7$$!1ML$3FuF)
oF1$\"1VCQEV;Z>F]o7$$!1,+]P2qwnF1$\"1v*)Q]8OuLF]o7$$!1mm;/siqmF1$\"1Ft
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!H1sFO\"!#;7$$!1mmmTrU`jF1$\"1_Ll`,S-?Fgp7$$!1****\\(y$pZiF1$\"1jYP%*R
7HGFgp7$$!1K$ek.M*QhF1$\"1;Z^&))z_(QFgp7$$!1mmT&Gu,.'F1$\"1AcX?eg\"4&F
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ajc;tbF1$\"1?g'fa@o'**Fgp7$$!1KLeRyEDbF1$\"1z(G)*40%>5F17$$!1lTgF*=8]&
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cMjX.\"F17$$!1mmm\">s%HaF1$\"1T\\YTe;M5F17$$!1***\\7)*G;K&F1$\"1s]s7PZ
k**Fgp7$$!1LL$3x&y8_F1$\"1%H*pl\\(f**)Fgp7$$!1+]ilT')f^F1$\"1;5F7c)[I)
Fgp7$$!1nmTgD%f5&F1$\"1)e!zJ,F$\\(Fgp7$$!1L$3_&4-_]F1$\"1'[l-WA%zlFgp7
$$!1******\\$*4)*\\F1$\"1BLNpw2&e&Fgp7$$!1++]Pj&R%\\F1$\"1m)=&yI;IXFgp
7$$!1+++DL\")*)[F1$\"1:vp')o&[W$Fgp7$$!1++]7.nN[F1$\"1L(e*[V&fN#Fgp7$$
!1++++t_\"y%F1$\"1')RIZ\"3(*G\"Fgp7$$!1++](G%QFZF1$\"1!*y(puUyq#F]o7$$
!1+++v7CtYF1$!1@KF$yhny'F]o7$$!1++]i#)4>YF1$!1e%e)H+%)R:Fgp7$$!1+++]_&
\\c%F1$!1t9Zw?x(H#Fgp7$$!1++++;ggWF1$!1*fi#)*[wJMFgp7$$!1+++]zCcVF1$!1
;))*f.Se6%Fgp7$$!1+++D62/VF1$!1wK:8;4&H%Fgp7$$!1++++V*=D%F1$!1Pd0No%\\
P%Fgp7$$!1++]()e!eA%F1$!16nKL7c!Q%Fgp7$$!1+++vur*>%F1$!1%p,fm*4lVFgp7$
$!1++]i!HO<%F1$!1)*egEr,IVFgp7$$!1+++]1aZTF1$!1Cyqa1$oF%Fgp7$$!1LL3FW,
eRF1$!1d@<xh/$\\$Fgp7$$!1mm;/#)[oPF1$!1V\"QS9>0T#Fgp7$$!1++v$>BJa$F1$!
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)H_J?-QF]o7$$!1LLLeW%o7$F1$!1(p?kygsY#F]o7$$!1LL$ef(QJIF1$!1U&*e-Ut]:F
]o7$$!1LLLL2$f$HF1$!1-**G)o;'R%*FX7$$!1mmT&o_Qr#F1$!15oyWSmAEFX7$$!1++
]PYx\"\\#F1$!11@$>A6S1'FR7$$!1MLLL7i)4#F1$!1\"\\,80.so#!#?7$$!1****\\P
'psm\"F1$!1\"eLJt'p#y\"FF7$$!1****\\74_c7F1$\"1k(z,k3A^\"F@7$$!1:LL$3x
%z#)Fgp$\"1beSTE*[w%F:7$$!1ILL3s$QM%Fgp$\"1#QIq.^Oo)F47$$!1^omm;zr)*FX
$\"1g')f-r[q5!#G7$$\"1WLLezw5VFgp$\"15'>ayuYP)F47$$\"1.++v$Q#\\\")Fgp$
\"1#4cCY3u@%F:7$$\"1NLLe\"*[H7F1$\"1['*y@.Dq7F@7$$\"1++++dxd;F1$!1?pFZ
mG&[\"FF7$$\"1,++D0xw?F1$!17a-Pd_0AFc_l7$$\"1,+]i&p@[#F1$!1U9m$*4(pm&F
R7$$\"1,+v=GB2FF1$!1Le'z+vu^#FX7$$\"1+++vgHKHF1$!1Ui=^P)pD*FX7$$\"1mm
\"zu5M.$F1$!1Z7t79fm:F]o7$$\"1LL$3UDX8$F1$!1TwDZpkdDF]o7$$\"1***\\P4Sc
B$F1$!1&zo6/3)GSF]o7$$\"1lmmmZvOLF1$!1g[K$>MC7'F]o7$$\"1LLLexn_NF1$!1Z
*p4Y%fC8Fgp7$$\"1,++]2goPF1$!1VJ$R!z:6CFgp7$$\"1nm\"H2fU'RF1$!1,TrH3]E
NFgp7$$\"1KL$eR<*fTF1$!1*e,q0AUI%Fgp7$$\"1\\(=Unbm=%F1$!15auMP*)\\VFgp
7$$\"1mTg_RR8UF1$!11Hu()euvVFgp7$$\"1$e*)4BK,C%F1$!1:b'35x,Q%Fgp7$$\"1
+]P40(oE%F1$!10]X'4l;O%Fgp7$$\"1Me9mqM?VF1$!1Y*=?NW-D%Fgp7$$\"1nm\"HiB
QP%F1$!1!>392'oJSFgp7$$\"1N$ektw2[%F1$!1c],yK\\ZKFgp7$$\"1-++])Hxe%F1$
!1HY*QD?B*>Fgp7$$\"1NeRAr/QYF1$!1#30.'p^\\7Fgp7$$\"1o;z%Rk$)o%F1$!1)H;
<U7i@%F]o7$$\"1,v=n;oQZF1$\"1PZ=/#zMy%F]o7$$\"1MLeR*)**)y%F1$\"1\\&Ha`
nXV\"Fgp7$$\"1n\"z>@;$R[F1$\"1Z)\\$QZ&)GCFgp7$$\"1+]P%[L'*)[F1$\"1,ZEa
6BTMFgp7$$\"1L3xc2&*R\\F1$\"1eqR<Ne]WFgp7$$\"1mm;H!o-*\\F1$\"1FC(zL!RN
aFgp7$$\"1LL3xS'G/&F1$\"1iP#Rg(f:kFgp7$$\"1+++D,Y&4&F1$\"1t\\s=\"))HK(
Fgp7$$\"1mm\"H<c![^F1$\"12rb4`)o8)Fgp7$$\"1ML$3A_1?&F1$\"1$)er4(*eR))F
gp7$$\"1nmm;V%eI&F1$\"1$G<2wW\"f)*Fgp7$$\"1,+]7k.6aF1$\"1F/v*H?>.\"F17
$$\"1=/wPvuOaF1$\"18fcI`eM5F17$$\"1M3-j'eCY&F1$\"1XbS7e)R.\"F17$$\"1^7
G)yp\")[&F1$\"1wp_BH@I5F17$$\"1o;a84)Q^&F1$\"18a-0xPB5F17$$\"1,D1kJIlb
F1$\"1z1[.R0,5F17$$\"1MLe9as;cF1$\"16b8Y5N#o*Fgp7$$\"1+]i:*p&>dF1$\"1V
9o@NQp()Fgp7$$\"1mmm;WTAeF1$\"1h2^<K9@wFgp7$$\"1+]7yI3IfF1$\"1B+IZT81j
Fgp7$$\"1KLeR<vPgF1$\"1w='><&[-]Fgp7$$\"1l;/,/UXhF1$\"1u)*>))*ow!QFgp7
$$\"1****\\i!*3`iF1$\"1%fh)*)=I#y#Fgp7$$\"1nm\"z\\%[gkF1$\"18kgHD)eN\"
Fgp7$$\"1NLLL*zym'F1$\"1rq#R99Un&F]o7$$\"1NL3Fe#Rx'F1$\"1Hg<HCE@MF]o7$
$\"1OL$3sr*zoF1$\"1<,`(ysn(>F]o7$$\"1OLe9w,')pF1$\"17r=?;J#4\"F]o7$$\"
1OLL3N1#4(F1$\"1k87,%ebv&FX7$$\"1-+vo!*R-tF1$\"1<I#oK>gP\"FX7$$\"1pm;H
Yt7vF1$\"1>T\\-![xZ#FR7$$\"1-+++xG**yF1$!1p[,g(\\LX#FL7$$\"1qmmT6KU$)F
1$!1,QkKd9:!)FF7$$\"1OLLLbdQ()F1$!1g`g]RBiRF@7$$\"1++]i`1h\"*F1$!1%)Q<
#>'pDwF:7$$\"1-+]P?Wl&*F1$!1y[6Hm!o\\*F47$$\"#5F*F+-%'COLOURG6&%$RGBG$
F[em!\"\"F*F*-%+AXESLABELSG6$%\"wG%!G-%%VIEWG6$;F(Fjdm%(DEFAULTG" 2 
533 283 283 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 16322 -26406 0 0 0 0 0 0 }}}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 26 "Windowed Fourier Transform" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 18 "General definition" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "unassign('g'
,'p','q','h'):\nWFTh := Int(h(t)*g(t-p)*exp(-I*q*t),t=-infinity..infin
ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WFThG-%$IntG6$*(-%\"hG6#%
\"tG\"\"\"-%\"gG6#,&F,F-%\"pG!\"\"F--%$expG6#,$*(%\"IGF-%\"qGF-F,F-F3F
-/F,;,$%)infinityGF3F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The tra
nsform may also be calculated with the command below once the specific
 function are known." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fourier(h(t
)*g(t-p),t,q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(fourierG6%*&-%\"h
G6#%\"tG\"\"\"-%\"gG6#,&F*F+%\"pG!\"\"F+F*%\"qG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Windowing functi
on" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g := unapply(exp(-t^2)
,t);\nplot(g(t),t=-5..5);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:
6#%\"tG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$9$\"\"#!\"\"F(F(" }}
{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7ho7$$!\"&\"\"!$\"1-k\\'Q%z)Q
\"!#E7$$!1LLLe%G?y%!#:$\"1HW1A.Br6!#D7$$!1mmT&esBf%F1$\"1#pP<vG2$pF47$
$!1LL$3s%3zVF1$\"1LW\\km!op%!#C7$$!1ML$e/$QkTF1$\"1=<C!)RaSH!#B7$$!1nm
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Far$\"1D%*y$yh?0*Far7$$!1lm;/'=><#Far$\"1$)zc%=I#R&*Far7$$!1K$3Fpy7k\"
Far$\"1XU!>L;Ut*Far7$$!1++D\"yQ16\"Far$\"1?@>:eSx)*Far7$$!1L$3_D)=`%)F
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yYJFgp$\"1**eC>E5!***Far7$$!1EMLLe*e$\\F\\p$\"1(*4Hsjv****Far7$$\"1l;a
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\"HKk>uxFgp$\"1cA#39W(R**Far7$$\"1pmT5D,`5Far$\"1W7Fd*G(*))*Far7$$\"1q
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e,]6$Far$\"1(eSS[m_2*Far7$$\"1-+](=>Y2%Far$\"1%[=*f([-Z)Far7$$\"1QLe*[
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ibOO$*Far$\"1m%*=YY\\#=%Far7$$\"1++]i_QQ5F1$\"1:\"G8&R#>S$Far7$$\"1+](
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$\"1RG$*>#e]d)Fgp7$$\"1LLL$Qx$o;F1$\"1)p<gtzA='Fgp7$$\"1nm;z)Qjx\"F1$
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FV7$$\"1NLe9tOcPF1$\"1Kgf]z\"pW(FK7$$\"1,++]Qk\\RF1$\"1#\\lY&QNz;FK7$$
\"1NL$3dg6<%F1$\"1[tw[_-zFFE7$$\"1ommmxGpVF1$\"1bcnh@6<^F?7$$\"1++D\"o
K0e%F1$\"1mK<\\c%es(F47$$\"1,+v=5s#y%F1$\"19#eo`(\\j6F47$$\"\"&F*F+-%'
COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F(Ffcl
%(DEFAULTG" 2 573 289 289 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 15785 31591 0 0 
0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 54 "Check energy of windowing function (for normalization)" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Ng:=  sqrt(Int(g(t)^2,t=-in
finity..infinity));\nNg := value(Ng);\nevalf(Ng);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#NgG*$-%$IntG6$*$-%$expG6#,$*$%\"tG\"\"#!\"\"F0/F/;,$
%)infinityGF1F5#\"\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NgG,$*&
\"\"##\"\"\"F'*&F'F(%#PiGF(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+N^^>6!\"*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 26 "Windowing Kernel functions" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "We think of the functions " }{XPPEDIT 18 0 "g[p,q](t)" "-
&%\"gG6$%\"pG%\"qG6#%\"tG" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "g(t-p)*e
xp(-I*q*t)" "*&-%\"gG6#,&%\"tG\"\"\"%\"pG!\"\"F(-%$expG6#,$*(%\"IGF(%
\"qGF(F'F(F*F(" }{TEXT -1 113 " as our pure time-localized wave forms.
 The p parameter is a location parameter and q is a frequency paramete
r.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Plot some windowed expon
entials. The location and frequency parameters are easily interpreted \+
in terms of the graph. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "g15 := \+
subs(p=1,q=5,g(t-p)*exp(-I*q*t)); \nRg15 := evalc(Re(g15));\nIg15 := e
valc(Im(g15));\nplot(\{Rg15,Ig15\},t=-5..5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$g15G*&-%$expG6#,$*$,&%\"tG\"\"\"!\"\"F-\"\"#F.F--F'6
#,$*&%\"IGF-F,F-!\"&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Rg15G*&-%
$expG6#,$*$,&%\"tG\"\"\"!\"\"F-\"\"#F.F--%$cosG6#,$F,\"\"&F-" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%Ig15G,$*&-%$expG6#,$*$,&%\"tG\"\"\"!\"\"F
.\"\"#F/F.-%$sinG6#,$F-\"\"&F.F/" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CU
RVESG6$7fs7$$!\"&\"\"!$!1hZuf!H*pI!#K7$$!1LLLe%G?y%!#:$!1C?#\\M-N%G!#I
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$!1_W90S?i6Fap7$Fefm$!1g())fMnsl\"Fcfm7$Fjfm$\"1saKO'GH5#Fao7$F_gm$\"1
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m%(DEFAULTG" 2 626 301 301 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -16419 -11112 0 
0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 46 "3D plots of imaginary part of windowing kernel" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 116 "Lets concentrate on the imaginary part a
nd look at some of the graphs. First let's look at the \"integration k
ernel\" " }{XPPEDIT 18 0 "Im(g(t-p)*exp(-I*q*t))= g(t-p)*sin((q*t))  \+
" "/-%#ImG6#*&-%\"gG6#,&%\"tG\"\"\"%\"pG!\"\"F,-%$expG6#,$*(%\"IGF,%\"
qGF,F+F,F.F,*&-F(6#,&F+F,F-F.F,-%$sinG6#*&F5F,F+F,F," }{TEXT -1 90 ". \+
Lets pick a couple of values of  q and vary p. Then we will reverse th
e roles of p and q" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 12 "Fix q vary p" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 12 "Fix q vary p" }}{PARA 0 "" 0 "" {TEXT -1 169 "We have pic
ked q=1 and q=-5 for the two different picutures.Note that as p moves \+
from -5 to 5 the ocillatory part of the kernel moves from right to rig
ht. The graph of  " }{XPPEDIT 18 0 "g(t-p)*sin((q*t))" "*&-%\"gG6#,&%
\"tG\"\"\"%\"pG!\"\"F(-%$sinG6#*&%\"qGF(F'F(F(" }{TEXT -1 19 "  as a f
unction of " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 145 " can be seen
 by tracing the undulating gridlines from left to right for a given va
lue of p.   Observe the effect of q on the rate of oscillation." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "gsp
 := g(t-p)*sin(q*t);\ngsp1 := subs(q=1,g(t-p)*sin(q*t));\ngsp5 := subs
(q=-5,g(t-p)*sin(q*t));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gspG*&
-%$expG6#,$*$,&%\"tG\"\"\"%\"pG!\"\"\"\"#F/F--%$sinG6#*&%\"qGF-F,F-F-
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gsp1G*&-%$expG6#,$*$,&%\"tG\"\"
\"%\"pG!\"\"\"\"#F/F--%$sinG6#F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%gsp5G*&-%$expG6#,$*$,&%\"tG\"\"\"%\"pG!\"\"\"\"#F/F--%$sinG6#,$F,!
\"&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "plot3d(gsp1,p= -5
..5, t=-5..5,\n       style = patch,\n       grid = [20,50],\n       o
rientation = [5,50],\n       axes=boxed,\n       colour =gsp1,\n      \+
 labelfont = [TIMES,BOLD,12]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 182 
"plot3d(gsp5,p= -5..5, t=-5..5,\n       style = patch,\n       grid = \+
[20,50],\n       orientation = [5,50],\n       axes=boxed,\n       col
our =gsp5,\n       labelfont = [TIMES,BOLD,12]); " }}{PARA 13 "" 1 "" 
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}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Fix p vary q" }}{PARA 5 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 243 "We have picked p=
-1 and p=2 for the two different pictures. Note that as p moves from -
10 to 10 the oscillations become more rapid. The selection of  p tells
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tion of " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 95 " can be seen by \+
tracing the undulating gridlines from left to right for a given value \+
of  q.   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "gsq := g(t-p)*sin(q*t);\ngsq1 := subs(p=-1,g(t-p)*sin
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on = [5,25],\n       axes=boxed,\n       colour =gsq1,\n       labelfo
nt = [TIMES,BOLD,12]); \nplot3d(gsq2,q= -5..5, t=-5..5,\n       style \+
= patch,\n       grid = [20,50],\n       orientation = [5,25],\n      \+
 axes=boxed,\n       colour =gsq2,\n       labelfont = [TIMES,BOLD,12]
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