{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 9 "Filtering" }}{PARA 3 "" 0 "" {TEXT -1 31 "Filter out high frequency noise" }}{PARA 0 "" 0 "" {TEXT -1 57 "The goal here is to define a signal, compute its fourier " }}{PARA 0 "" 0 "" {TEXT -1 54 "series and then either filter the ser ies (to eliminate" }}{PARA 0 "" 0 "" {TEXT -1 53 "high frequency noise ) or compress it (to send it more" }}{PARA 0 "" 0 "" {TEXT -1 64 "effi ciently). An inititial signal is defined as the function f. " }}{PARA 0 "" 0 "" {TEXT -1 39 "Other sample signals appear at the end." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "First def ine the signal." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f:=x->exp(-x^2/10)*(sin(2*x)+2*cos( 4*x)+0.4*sin(x)*sin(50*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR 6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#,$*$)9$\"\"#\"\"\"#!\"\"\" #5\"\"\",(-%$sinG6#,$F3F4F9-%$cosG6#,$F3\"\"%F4*&-F<6#F3F9-F<6#,$F3\"# ]F9$FCF7F9F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Bascially, th e signal f involves a smoothly varying term (given first) together wi th" }}{PARA 0 "" 0 "" {TEXT -1 86 "some high frequency noise (the term involving sin(50x)). Let's see how Fourier series" }}{PARA 0 "" 0 " " {TEXT -1 53 "filters out the high frequency noise. 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$$\"1at]6FG6DF*$!1$yg(3JY&Q\"F*7$$\"129Z13kEDF*$!1bW[RXPU7F*7$$\"1gaV, *)*>a#F*$!1Zy;z>wU6F*7$$\"1IT`dDF*$! 1h%p/E(326F*7$$\"1r9t\\HzkDF*$!1-;gJRx36F*7$$\"1T,$e'4RsDF*$!1(p_'G6X< 6F*7$$\"1&[@SFh0e#F*$!1JK/^T)y7\"F*7$$\"1GG@#eJ()e#F*$!1z8T\"[KB8\"F*7 $$\"1sTS!*=!pf#F*$!1\"=:([#HY7\"F*7$$\"1:bf)>s]g#F*$!1njH7$>25\"F*7$$ \"1I)Rh\"*y9i#F*$!1]>qb,A,5F*7$$\"1YToLc)yj#F*$!15DJ7ux]&)F-7$$\"16--, 5unEF*$!1@6;3;RGiF-7$$\"1W.'\\2Y%*p#F*$!1$)='466H`&F-7$$\"1azSGrQIFF*$ !14xJ*e],$\\F-7$$\"1*onnbFRw#F*$!1;G\"=sQml#F-7$$\"1R)QDN\\Hz#F*$!1S7* )>GEGF*$\"1,d\"45aP?\"Fcu7$$\"1m7rCjhdGF*$\"1'p$)z_l I_)Fcu7$$\"1&QjL0['))GF*$\"1T\"eoRvfY#F-7$$\"1*[()zY/(>HF*$\"1NlLK^BER F-7$$\"1Ak0X+Q^HF*$\"1)y`.,>_`%F-7$$\"1S85Ec&\\)HF*$\"1A4TqtDE^F-7$$\" 1r_3Y6u:IF*$\"1K!py.)e]gF-7$$\"1I=swd$4.$F*$\"1JM0jb " 0 "" {MPLTEXT 1 0 69 "(1/(2*Pi))*Int(f(x),x=-Pi..Pi); a0:=evalf(%); #the coefficient of a_0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$Int G6$*&-%$expG6#,$*$)%\"xG\"\"#\"\"\"#!\"\"\"#5\"\"\",(-%$sinG6#,$F/F0F5 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-1 53 "to the dominant Fourier coefficients. Then, the guess" }}{PARA 0 "" 0 "" {TEXT -1 53 "can be checked against the comp utation of a_n and b_n" }}{PARA 0 "" 0 "" {TEXT -1 28 "given in the fo llowing code." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#Evaluate \+ some sample a(n), b(n)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f or n from 10 to 12 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "n; a(n); b (n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+r-*[#\\!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$!+\\T\\+l!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@A8 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 56 "Now set up the a partial Fourier series \+ that filters out" }}{PARA 0 "" 0 "" {TEXT -1 32 "the high frequency co efficients." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "n:='n'; temp :=a0; #set up the fourier partial sum" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG$!+D4!\\V*!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for n from 1 to 5 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "temp:=temp+a(n)*cos(n*x)+b(n)*sin(n *x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG,($!+D4!\\ V*!#7\"\"\"-%$cosG6#%\"xG$\"+KK'RH#!#6-%$sinGF,$\"+M'3sI\"!#5" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%tempG,,$!+D4!\\V*!#7\"\"\"-%$cosG6# %\"xG$\"+KK'RH#!#6-%$sinGF,$\"+M'3sI\"!#5-F+6#,$F-\"\"#$!+G$[?H%F0-F2F 7$\"+q+@SvF5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%tempG,0$!+D4!\\V*!# 7\"\"\"-%$cosG6#%\"xG$\"+KK'RH#!#6-%$sinGF,$\"+M'3sI\"!#5-F+6#,$F-\"\" #$!+G$[?H%F0-F2F7$\"+q+@SvF5-F+6#,$F-\"\"$$\"+soC9GF5-F2F@$\"+&)Hzh8F5 " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%tempG,4$!+D4!\\V*!#7\"\"\"-%$co sG6#%\"xG$\"+KK'RH#!#6-%$sinGF,$\"+M'3sI\"!#5-F+6#,$F-\"\"#$!+G$[?H%F0 -F2F7$\"+q+@SvF5-F+6#,$F-\"\"$$\"+soC9GF5-F2F@$\"+&)Hzh8F5-F+6#,$F-\" \"%$\"+c2F'\\\"!\"*-F2FI$!+-[gH%%tempG,8$!+D4!\\V*!#7\"\"\"-%$cosG6#%\"xG$\"+KK'RH#!#6-%$sinGF,$\"+M '3sI\"!#5-F+6#,$F-\"\"#$!+G$[?H%F0-F2F7$\"+q+@SvF5-F+6#,$F-\"\"$$\"+so C9GF5-F2F@$\"+&)Hzh8F5-F+6#,$F-\"\"%$\"+c2F'\\\"!\"*-F2FI$!+-[gH " 0 "" {MPLTEXT 1 0 7 "n:='n';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "S:=temp; #the partial four ier series of through 5th order frequency" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG,8$!+D4!\\V*!#7\"\"\"-%$cosG6#%\"xG$\"+KK'RH#!#6- %$sinGF,$\"+M'3sI\"!#5-F+6#,$F-\"\"#$!+G$[?H%F0-F2F7$\"+q+@SvF5-F+6#,$ F-\"\"$$\"+soC9GF5-F2F@$\"+&)Hzh8F5-F+6#,$F-\"\"%$\"+c2F'\\\"!\"*-F2FI $!+-[gH " 0 "" {MPLTEXT 1 0 79 "plot(S,x=-Pi..Pi,numpoints=200,thickness=0); #plot of the 5th order fourier sum" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6 $7dy7$$!1++JZEfTJ!#:$\"1')[1'HkLf)!#;7$$!1^B,%Rv]7$F*$\"1Z;;**y%or)F-7 $$!1-ZrS\"e&3JF*$\"1)Qxg=R;!))F-7$$!1!pPpc'=%4$F*$\"1f9l#)F-7$$!19\\y&pUD&HF*$\" 1**H$>V#*Hy(F-7$$!1\"H.X=a9#HF*$\"1;$)ydHKEsF-7$$!1hKm,CI*)GF*$\"1hM`x 3!*[lF-7$$!1h&)**HPDdGF*$\"1#>J)3!RLy&F-7$$!1F*3>t(GCGF*$\"1$z)*=shl\" \\F-7$$!15y%GE^_z#F*$\"18MCH9Z*4%F-7$$!1')R&f2jDw#F*$\"1#[pj>)4NJF-7$$ !1fTAo1uHFF*$\"1g8'>s4n8#F-7$$!19A`'>5\")p#F*$\"1>bh$R/O;\"F-7$$!1(e#) zp'QpEF*$\"1%H8v,a>&G!#<7$$!1>3,!oJ_j#F*$!1-F4ML6VtFgq7$$!1\\wm**yH1EF *$!1%\\Gc8F:c\"F-7$$!1XzsM5ksDF*$!1()eA\"o/OY#F-7$$!19+K1!\\Ga#F*$!1_! 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Then we " }} {PARA 0 "" 0 "" {TEXT -1 44 "eliminate all the small Fourier coefficie nts" }}{PARA 0 "" 0 "" {TEXT -1 47 "(set them equal to zero) to obtain a compressed" }}{PARA 0 "" 0 "" {TEXT -1 42 "signal. By changing the \+ tolerance, we can " }}{PARA 0 "" 0 "" {TEXT -1 41 "change the degree t o which the signal has" }}{PARA 0 "" 0 "" {TEXT -1 37 "been compressed . Then we can plot the" }}{PARA 0 "" 0 "" {TEXT -1 39 "compressed sign al and compare it to the" }}{PARA 0 "" 0 "" {TEXT -1 16 "original sign al." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Firs t define the signal h" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "h: =x->exp(-x^2/10)*(sin(2*x)+2*cos(4*x)+0.4*sin(x)*sin(10*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(*&-% $expG6#,$*$)9$\"\"#\"\"\"#!\"\"\"#5\"\"\",(-%$sinG6#,$F3F4F9-%$cosG6#, $F3\"\"%F4*&-F<6#F3F9-F<6#,$F3F8F9$FCF7F9F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "plot(h,-Pi..Pi); #plot the signal" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7bt7$$!1++JZEfTJ!#:$\"1k7H`o:au!#;7$$! 1ue[-KZCJF*$\"1)3AA,e9k(F-7$$!1[i!>&zF-7$$!1q$*=Ba*f0$F*$ \"1fA'fPj?(zF-7$$!1W_Oyf()QIF*$\"1O-s2sS_zF-7$$!1=6aLlv@IF*$\"1A$4I3\" *[*yF-7$$!1#*pr)3PY+$F*$\"1!4!z$47>!yF-7$$!1+)*4R\\0XHF*$\"1b%>PN-BC(F -7$$!13E[*ysa)GF*$\"1/TGwk>4kF-7$$!1Y&))>2g9v#F*$\"13,h#)fK()QF-7$$!1B .g^&4So#F*$\"1\\gA)ej7@#F-7$$!1,@@J!flh#F*$\"1GUOmcE)\\\"!#<7$$!1TKP\" 4H%\\DF*$!1(4[dZUmA#F-7$$!1#QM::*H#[#F*$!1q;_xw4HYF-7$$!1;L2/61?CF*$!1 ,#=@HVBV'F-7$$!1]AhcI#yN#F*$!1)[!R8\\]atF-7$$!1nyJ$37usOsF-7$$!1=Z Vj\"zLH#F*$!1qQGA'Q+-(F-7$$!1_f%o@d6E#F*$!1&orWW%GCjF-7$$!1'=d-FN*GAF* $!1F*$\"1E%ojgY!3mF -7$$!1@I3>^V%*=F*$\"1:;L>ZVs#)F-7$$!1<(4AN*4E=F*$\"1FT!o%RDY%*F-7$$!1Q Q*3**=dq\"F*$\"1*)\\xz-$G>\"F*7$$!1:&yM5fzj\"F*$\"1))[e$Q:qQ\"F*7$$!1 \">jg@*>q:F*$\"1?H@x$4Rc\"F*7$$!1pt0(o*=`:F*$\"1;4O3%)y%f\"F*7$$!1Z:0e ,=O:F*$\"1m*=*)oaph\"F*7$$!1Dd/H1<>:F*$\"1'G&y@O!)G;F*7$$!1/*R+5h@]\"F *$\"1Jb9!*[$)G;F*7$$!1g#G?/U\"o9F*$\"1x]TL9D)e\"F*7$$!1%G8F*7$$!1d=2_cbo8F*$\"1`K^*>H 26\"F*7$$!1x%*4')>xN8F*$\"1s\\2V)3DR)F-7$$!1)4F,K))HI\"F*$\"1T2iwO@\"F*$!1!3Y'yS@_XF-7$$!1*4!R#[0R=\"F*$!1/wGzZT6yF-7$$!16\")QH_][6 F*$!1,TjBr^W6F*7$$!1BhQw\\586F*$!1DWN()=ro9F*7$$!1NTQBZqx5F*$!1Ba(*4Ou W:)=_m*o>F*7$$!1o6z[:FB)*F-$!1_,.IR'oB#F*7$$!1p3 w$R)\\B#*F-$!1iW6X\\w$R#F*7$$!1]#[4j>e_)F-$!1&Qs\"fZK&\\#F*7$$!1Jc8o39 GyF-$!1$3]eW8jb#F*7$$!18Te())\\Pn(F-$!1tp:_JNlDF*7$$!1%fKq!*e$>vF-$!1! 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We store the new coef ficients in the " }}{PARA 0 "" 0 "" {TEXT -1 22 "arrays aa(n) and bb(n )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "n:='n'; tol:=.05; # fr ee up n and the value of tolerance set to .05" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tolG$\" \"&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for n from 1 to \+ 25 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if abs(aa(n)) < tol then a a(n):=0 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "if abs(bb(n)) < tol \+ then bb(n):=0 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "#Print the Fourier Coefficients" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for n from 1 to 25 do" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n; aa(n); bb(n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M'3sI\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +q+@Sv!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+xM:7G!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+& )Hzh8!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Jue'\\\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;rq'z#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! 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The arr ay ifunc is also defined" }}{PARA 0 "" 0 "" {TEXT -1 55 "which contain s the imaginary parts of the input signal." }}{PARA 0 "" 0 "" {TEXT -1 52 "Of course, most signals are real valued and in these" }}{PARA 0 "" 0 "" {TEXT -1 47 "cases, the values of ifunc are all set to zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "func:=array([seq(evalf(f((j*step))),j=1..2^N)]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ifunc:=array([seq(0,j=1..2^N )]): #imaginary values=0" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "FFT(N,func,ifunc): #compute \+ FFT with 2^N data points" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The F FT computes the Discrete Fourier Transform and stores" }}{PARA 0 "" 0 "" {TEXT -1 50 "the real and imaginary parts of these coefficients" }} {PARA 0 "" 0 "" {TEXT -1 60 "in the arrays func and ifunc (the origina l values are lost)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n ';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "for n from 250 to 256 do #Now examine some of \+ them" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "n; func[n]; ifunc[n];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$]#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*:%oa)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JD*=v\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$^#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")21H)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+pwTb!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$`#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7Iz_5!\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+iT=Ga!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$a#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+*f!3*p\"!\" )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G2#o0%!\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"$b#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4!z'Rf! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,:7'>%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$c#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PpSlJ !\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+u=xc=!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The high frequency coefficients are the ones ce ntered about" }}{PARA 0 "" 0 "" {TEXT -1 55 "n=2^N/2 (i.e. the center \+ of the range of coefficients)." }}{PARA 0 "" 0 "" {TEXT -1 74 "To filt er out high frequency, set all Fourier coefficients above say, n=7" } }{PARA 0 "" 0 "" {TEXT -1 68 "and below 2^N - 7 equal to zero. Of cour se, the value 7 is arbitrary" }}{PARA 0 "" 0 "" {TEXT -1 66 "and can b e changed depending on what is considered high frequency." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for n from 7 to 249 do" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "func[n]:=0; ifunc[n]:=0;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Now compute the inverse FFT using \+ the package iFFT." }}{PARA 0 "" 0 "" {TEXT -1 36 "The real and imagina ry parts of the " }}{PARA 0 "" 0 "" {TEXT -1 44 "signal values will be computed and stored in" }}{PARA 0 "" 0 "" {TEXT -1 52 "the arrays fun c and ifunc (the Fourier coefficients " }}{PARA 0 "" 0 "" {TEXT -1 18 "will now be lost)." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "iFFT(N,func,ifunc):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "To plot this filtered signal we arrange a sequence" }} {PARA 0 "" 0 "" 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