$\%$ FAST FOURIER TRANSFORM ALGORITHM $8\mathrm{.}3\%\%$ To compute the coefficients in the discrete approximation $\%$ for the data $($x$($J$),$y$($J$)),0<=$J$<=2$m$-1$ where m$=2^$p and $\%$ x$($J$)=-$pi$+$J$*$pi$/$m for $0<=$J$<=2$m$-1.\%\%$ INPUT: m; y$(0),$y$(1),...$y$(2$m$-1).\%\%$ OUTPUT: complex numbers c$(0),...,$c$(2$m$-1)$; real numbers $\%$ a$(0),...,$a$($m$)$; b$(1),...,$b$($m$-1).$ syms$(\text{'}$OK$\text{'},$ 'FLAG', $\text{'}$M$\text{'},\text{'}$N$\text{'},\text{'}$JJ$\text{'},\text{'}$J$\text{'},\text{'}$Y$\text{'},\text{'}$A$\text{'})$; syms$(\text{'}$NAME$\text{'},$ 'INP', $\text{'}$F$\text{'},\text{'}$Z$\text{'},$ 'OUP', $\text{'}$TW$\text{'},\text{'}$N$2\text{'},\text{'}$C$\text{'})$; syms$(\text{'}$NG$\text{'},\text{'}$NU$1\text{'},\text{'}$YY$\text{'},\text{'}$WW$\text{'},\text{'}$X$\text{'},\text{'}$W$\text{'},\text{'}$K$\text{'},\text{'}$L$\text{'},\text{'}$M$1\text{'})$; syms$(\text{'}$NP$\text{'},\text{'}$T$1\text{'},\text{'}$T$3\text{'},\text{'}$x$\text{'},\text{'}$s$\text{'})$; TRUE $=1$; FALSE $=0$; fprintf$(1,$'This is the Fast Fourier Transform.

$\text{'})$; fprintf$(1,$'The file IBR.m is used by this program.

$\text{'})$; fprintf$(1,$'The user must make provisions if the

$\text{'})$; fprintf$(1,$'interval is not $(-$pi$,$pi$).$

$\text{'})$; OK $=$ FALSE; while OK $==$ FALSE fprintf$(1,$'Choice of input method:

$\text{'})$; fprintf$(1,\text{'}1.$ Input entry by entry from keyboard

$\text{'})$; fprintf$(1,\text{'}2.$ Input data from a text file

$\text{'})$; fprintf$(1,\text{'}3.$ Generate data using a function F

$\text{'})$; fprintf$(1,$'Choose $1,2,$ or $3$ please

$\text{'})$; FLAG $=$ input$(\text{'}\text{'})$; if FLAG $==1|$ FLAG $==2|$ FLAG $==3$ OK $=$ TRUE; end; end; if FLAG $==1$ OK $=$ FALSE; while OK $==$ FALSE fprintf$(1,$'Input m

$\text{'})$; M $=$ input$(\text{'}\text{'})$; if M $>0$ OK $=$ TRUE; N $=2*$M; Y $=$ zeros$(1,$N$)$; for JJ $=1$ : N J $=$ JJ$-1$; fprintf$(1,$'Input y$(\%$d$).$

$\text{'},$ J$)$; Y$($JJ$)=$ input$(\text{'}\text{'})$; end; else fprintf$(1,$'Number must be a positive integer.

$\text{'})$; end; end; end; if FLAG $==2$ fprintf$(1,$'Has a text file been created with the $\text{'})$; fprintf$(1,$'entries y$(0),...,$y$(2$m$-1)$

$\text{'})$; fprintf$(1,$'separated by a blank?

$\text{'})$; fprintf$(1,$'Enter Y or N

$\text{'})$; A $=$ input$(\text{'}\text{'},\text{'}$s$\text{'})$; if A $==\text{'}$Y$\text{'}|$ A $==\text{'}$y$\text{'}$ fprintf$(1,$'Input the file name in the form $-\text{'})$; fprintf$(1,$'drive:$\backslash$

ame.ext

$\text{'})$; fprintf$(1,$'for example: A:$\backslash \backslash$DATA$.$DTA

$\text{'})$; NAME $=$ input$(\text{'}\text{'},\text{'}$s$\text{'})$; INP $=$ fopen$($NAME$,\text{'}$rt$\text{'})$; OK $=$ FALSE; while OK $==$ FALSE fprintf$(1,$'Input number m$.$

$\text{'})$; M $=$ input$(\text{'}\text{'})$; N $=2*$M; Y $=$ zeros$(1,$N$)$; if N $>0$ for JJ $=1$ : N Y$($JJ$)=$ fscanf$($INP$,\text{'}\%$f$\text{'},1)$; end; fclose$($INP$)$; OK $=$ TRUE; else fprintf$(1,$'Number must be a positive integer.

$\text{'})$; end; end; else fprintf$(1,$'The program will end so the input file can $\text{'})$; fprintf$(1,\text{'}$be created.

$\text{'})$; OK $=$ FALSE; end; end; if FLAG $==3$ fprintf$(1,$'Input the function F$($x$)$ in terms of x

$\text{'})$; fprintf$(1,$'for example: cos$($x$)$

$\text{'})$; s $=$ input$(\text{'}\text{'})$; F $=$ inline$($s$,\text{'}$x$\text{'})$; OK $=$ FALSE; while OK $==$ FALSE fprintf$(1,$'Input the number m$.$

$\text{'})$; M $=$ input$(\text{'}\text{'})$; N $=2*$M; Y $=$ zeros$(1,$N$)$; if N $>0$ for JJ $=1$ : N Z $=-$pi$+($JJ$-1)*$pi$/$M; Y$($JJ$)=$ F$($Z$)$; end; OK $=$ TRUE; else fprintf$(1,$'Number must be a postive integer.

$\text{'})$; end; end; end; if OK $==$ TRUE fprintf$(1,$'Choice of output method:

$\text{'})$; fprintf$(1,\text{'}1.$ Output to screen

$\text{'})$; fprintf$(1,\text{'}2.$ Output to text file

$\text{'})$; fprintf$(1,$'Please enter $1$ or $2.$

$\text{'})$; FLAG $=$ input$(\text{'}\text{'})$; if FLAG $==2$ fprintf$(1,$'Input the file name in the form $-$ drive:$\backslash$

ame.ext

$\text{'})$; fprintf$(1,$'for example: A:$\backslash \backslash$OUTPUT$.$DTA

$\text{'})$; NAME $=$ input$(\text{'}\text{'},\text{'}$s$\text{'})$; OUP $=$ fopen$($NAME$,\text{'}$wt$\text{'})$; else OUP $=1$; end; fprintf$($OUP$,$ 'FAST FOURIER TRANSFORM

$\text{'})$; TW $=$ log$(2)$; $\%$ STEP $1\%$ use N$2$ for m$,$ NG for p$,$ NU$1$ for q$,$ WW for zeta N$2=$ floor$($N$/2)$; $\%$ STEP $2$ C $=$ zeros$(1,$N$)$; for JJ $=1$ : N C$($JJ$)=$ Y$($JJ$)$; end; Z $=$ N; NG $=$ round$($log$($Z$)/$TW$)$; NU$1=$ NG$-1$; YY $=2*$pi$*$sqrt$(-1)/$N; WW $=$ exp$($YY$)$; $\%$ STEP $3$ W $=$ zeros$(1,$N$)$; for JJ $=1$ : N$2$ X $=1$; YY $=1$; for J $=1$ : JJ YY $=$ X$*$WW; X $=$ YY; end; W$($JJ$)=$ X; W$($N$2+$JJ$)=-$X; end; $\%$ STEP $4$ K $=0$; $\%$ STEP $5$ for L $=1$ : NG $\%$ STEP $6$ while K $<$ N$-1\%$ STEP $7$ for JJ $=1$ : N$2\%$ STEP $8$ Z $=$ exp$($NU$1*$TW$)$; M$1=$ round$($Z$)$; M$1=$ floor$($K$/$M$1)$; $\%$ IBR does the bit reversal NP $=$ IBR$($M$1,$NG$)$; $\%$ T$1$ is eta T$1=$ C$($K$+$N$2+1)$; $\%$ STEP $9$ if NP ~$=0$ X $=$ T$1$; T$1=$ X$*$W$($NP$)$; end; C$($K$+$N$2+1)=$ C$($K$+1)-$T$1$; C$($K$+1)=$ C$($K$+1)+$T$1$; $\%$ STEP $10$ K $=$ K$+1$; end; $\%$ STEP $11$ K $=$ K$+$N$2$; end; $\%$ STEP $12$ K $=0$; N$2=$ floor$($N$2/2)$; NU$1=$ NU$1-1$; end; $\%$ STEP $13$ while K $<$ N$-1\%$ STEP $14$ JJ $=$ IBR$($K$,$NG$)$; $\%$ STEP $15$ if JJ $>$ K T$3=$ C$($K$+1)$; C$($K$+1)=$ C$($JJ$+1)$; C$($JJ$+1)=$ T$3$; end; $\%$ STEP $16$ K $=$ K$+1$; end; $\%$ STEPS $17$ and $18$ fprintf$($OUP$,$ 'Coefficients c$(0),...,$ c$(2$m$-1)$

$\text{'})$; for JJ $=1$ : N YY $=-($JJ$-1)*$pi$*$sqrt$(-1)$; X $=$ exp$($YY$)$; YY $=$ X$*$C$($JJ$)$; C$($JJ$)=$ YY$/(0\mathrm{.}5*$N$)$; K $=$ JJ$-1$; fprintf$($OUP$,\text{'}\%3$d $\%\mathrm{.}8$f $\%\mathrm{.}8$f

$\text{'},$ K$,$ real$($C$($JJ$)),$imag$($C$($JJ$)))$; end; fprintf$($OUP$,\text{'}$

Coefficients a$(0),...,$ a$($m$)$

$\text{'})$; for JJ $=1$ : M$+1$ fprintf$($OUP$,\text{'}\%\mathrm{.}8$f

$\text{'},$ real$($C$($JJ$)))$; end; fprintf$($OUP$,\text{'}$

Coefficients b$(1),...,$ b$($m$-1)$ <

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