View source for File:FFTexample16T.png

== Summary ==
{{Image_Details|user
|description  = Application of the [[FFT]] operator to the array that approximates the [[self-Fourier]] gaussian
|author       = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
|date-created =  9 October 2011
|pub-country  = Japan
|notes        = Comparison of the discrete Fourier transform, shown with red, 
of a [[self–Fourier function]] <math>A(x)=\exp(-x^2/2)</math>, shown with black dots, to the result of the numerical evaluation of the the [[Fourier operator]] of array <math>A</math>, shown with blue. The discrete representation is performed with number of nodes <math>n\!=\!16</math>.

The function <math>A</math> is represented at the mesh with nodes <math>x_n=\sqrt{\frac{\pi}{8}} (n\!-\!8)~,~ n=0 .. 15</math> in such a way that <math>A_n=A(x_n)</math>

The [[FFT]] of array <math>A</math> is performed with the routine 

 void fft(z_type *a, int N, int o) // Arry is FIRST!
 {z_type u,w,t;  int i,j,k,l,e=1,L,p,q,m;
 q=N/2;  p=2; for(m=1;p<N;m++) p*=2; 
 p=N-1;  z_type y=z_type(0.,M_PI*o); j=0;
 for(i=0;i<p;i++){ if(i<j) { t=a[j]; a[j]=a[i]; a[i]=t;}
                   k=q; while(k<=j){ j-=k; k/=2;} 
                   j+=k; }
 for(l=0;l<m;l++)
 { L=e; e*=2; u=1.; w=exp(y/((double)L));
  for(j=0;j<L;j++)
  {  for(i=j;i<N;i+=e){k=i+L; t=a[k]*u; a[k]=a[i]-t; a[i]+=t;}
     u*=w;
 } }
 }

However, the transformation with the routine [[fafo.cin]] would return practically the same array, while the routine fafo is the discrete analogy of the [[Fourier operator]] and function <math>A</math> is [[self Fourier]]. 

The figure shows difference between [[FFT]] and the discrete implementation of the [[Fourier operator]]. For the self-Fourier Gaussian, the Fourier operator returns the same array, while the [[FFT]] returns the saw-like structure. For the physical applications, [[fafo.cin]] seems to be more suitable.

==[[C++]] generator of curves==

// The picture is generated with the 
//File [[fafo.cin]] should be loaded to the working directory for the compilation of the code below.

 #include<math.h>
 #include<stdio.h>
 #include <stdlib.h>
 #include <complex>
 using namespace std;
 #define z_type complex<double>
 #define Re(x)  x.real()
 #define Im(x)  x.imag()
 #define RI(x)  x.real(),x.imag()
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)

 void ado(FILE *O, int X, int Y)
 {       fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
        fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
        fprintf(O,"/M {moveto} bind def\n");
        fprintf(O,"/L {lineto} bind def\n");
        fprintf(O,"/S {stroke} bind def\n");
        fprintf(O,"/s {show newpath} bind def\n");
        fprintf(O,"/C {closepath} bind def\n");
        fprintf(O,"/F {fill} bind def\n");
        fprintf(O,"/o {.025 0 360 arc C F} bind def\n");
        fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
        fprintf(O,"/W {setlinewidth} bind def\n");
        fprintf(O,"/RGB {setrgbcolor} bind def\n");}
 // #include "ado.cin"

 #include"fafo.cin"

 // DB F(DB x){DB u=x*x; return u*(-3.+u)*exp(-x*x/2.);}
 DB F(DB x){ return exp(-x*x/2.);}

 main(){z_type * a, *b, c; int j,m,n, N=16; FILE *o;
        double step=sqrt(2*M_PI/N),x,y,u;
        a=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
        b=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
 //for(j=0;j<N;j++) { x=step*(j-N/2); u=x*x; a[j]=b[j]=(3.+u*(-6.+u))*exp(-x*x/2); }
 for(j=0;j<N;j++) { x=step*(j-N/2); u=x*x; a[j]=b[j]=F(x); }
 fft(b,N,1);
 for(j=0;j<N;j++) printf("%2d %18.15f %18.15f  %18.15f %18.15f\n", j, RI(a[j]), RI(b[j])  );
 o=fopen("FFTexample16.eps","w"); ado(o,1024,780); 
 #define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 #define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
 fprintf(o,"522 340 translate 100 100 scale\n");
 // M(-5,0) L(5,0) M(0,0) L(0,1) fprintf(o,".01 W S\n");
 // M(-5,1) L(5,1) M(-5,-1) L(5,-1)
 for(m=-5;m<6;m++) {M(m,-3) L(m,3)} fprintf(o,".004 W S\n");
 for(m=-3;m<4;m++) {M(-5,m) L(5,m)} fprintf(o,".004 W S\n");
 fprintf(o,"1 setlinejoin 1 setlinecap\n");
 DB *X; X=(DB *) malloc((size_t)((N+1)*sizeof(DB))); DO(j,N){ x=step*(j-N/2); X[j]=x; }
 DO(j,N){x=X[j]; M(x,0)L(x,.15)} fprintf(o,".01 W S\n");
 DO(j,N){x=X[j];y=Re(a[j]); if(j==0)M(x,y)else L(x,y);} fprintf(o,".04 W 0 .4 1 RGB S\n");
 DO(j,N){x=X[j];y=Re(b[j]); if(j==0)M(x,y)else L(x,y);} fprintf(o,".04 W 1 0 0 RGB S\n");
 // DO(j,N){x=X[j];y=Im(b[j]); if(j==0)M(x,y)else L(x,y);} fprintf(o,".04 W 1 0 0 RGB S\n");
 // DO(j,N){x=X[j];y=100.*(Re(b[j])-F(x)); if(j==0)M(x,y)else L(x,y);} fprintf(o,"0.007 W 0 0 .3 RGB S\n");
 printf("X[0]=%9.5f step=%9.6f\n",X[0],step);
 // DO(m,101){x=-5.+.1*m; y=F(x); if(m/2*2==m)M(x,y)else L(x,y);} fprintf(o,".01 W 0 0 0 RGB S\n");
 fprintf(o,".01 W 0 0 0 RGB S\n");
 DO(m,101){x=-5.+.1*m; y=F(x); o(x,y)}
 fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  system("epstopdf FFTexample16.eps");
  system(    "open FFTexample16.pdf"); //these 2 commands may be specific for macintosh
  getchar(); system("killall Preview");// if run at another operational system, may need to modify
  free(a);
  free(b);
  free(X);
 }
The image is generated in the following way.

The lines are drawn in the [[EPS]] format by the [[C++]] code below. The result is concerted to [[PDF]] format.

The labels are added in the [[latex]] document below.

The result is concerted to the [[Portable Network Graphics|PNG]] format with default resolution.

|versions     =  The image is loaded from http://tori.ils.uec.ac.jp/TORI/index.php/File:FFTexample16T.png
}}

==Latex generator of labels==
The labels are drawn with the [[Latex]] document below. File FFTexample16.pdf is supposed to be already generated with the [[C++]] code above and stored in the working directory.

<nowiki>

\documentclass[12pt]{article} %<br>
\usepackage{geometry} %<br>
\paperwidth 1012pt %<br>
\paperheight 740pt %<br>
\textheight  800pt
\topmargin -104pt %<br>
\oddsidemargin -92pt %<br>
\parindent 0pt %<br>
\pagestyle{empty} %<br>
\usepackage{graphicx} %<br>
\newcommand \sx \scalebox %<br>
\begin{document} %<br>
\begin{picture}(1024,742) %<br>
\put(4,0){\includegraphics{FFTexample16}} %<br>
%\put(510,732){\sx{2.4}{$y$}} %<br>
\put(510,634){\sx{2.5}{$y$}} %<br>
\put(510,532){\sx{2.4}{$2$}} %<br>
\put(510,432){\sx{2.4}{$1$}} %<br>
%\put(510,320){\sx{2.4}{$0$}} %<br>
\put(496,232){\sx{2.4}{$-\!1$}} %<br>
\put(496,132){\sx{2.4}{$-\!2$}} %<br>
\put(496, 32){\sx{2.4}{$-\!3$}} %<br>
%<br>
\put(104,310){\sx{2.4}{$-4$}} %<br>
\put(204,310){\sx{2.4}{$-3$}} %<br>
\put(304,310){\sx{2.4}{$-2$}} %<br>
\put(404,310){\sx{2.4}{$-1$}} %<br>
\put(522,310){\sx{2.4}{$0$}} %<br>
\put(622,310){\sx{2.4}{$1$}} %<br>
\put(722,310){\sx{2.4}{$2$}} %<br>
\put(822,310){\sx{2.4}{$3$}} %<br>
\put(923,310){\sx{2.4}{$4$}} %<br>
\put(1016,311){\sx{2.5}{$x$}} %<br>
\put(19,360){\sx{2.6}{$x_0$}} %<br>
\put(79,360){\sx{2.6}{$x_1$}} %<br>
\put(140,360){\sx{2.6}{$x_2$}} %<br>
\put(199,360){\sx{2.6}{$x_3$}} %<br>
\put(261,360){\sx{2.6}{$x_4$}} %<br>
\put(327,360){\sx{2.6}{$x_5$}} %<br>
\put(388,360){\sx{2.6}{$x_6$}} %<br>
\put(450,360){\sx{2.6}{$x_7$}} %<br>
\put(518,360){\sx{2.6}{$x_8$}} %<br>
\put(578,360){\sx{2.6}{$x_9$}} %<br>
\put(640,360){\sx{2.6}{$x_{10}$}} %<br>
\put(704,360){\sx{2.6}{$x_{11}$}} %<br>
\put(767,360){\sx{2.6}{$x_{12}$}} %<br>
\put(829,360){\sx{2.6}{$x_{13}$}} %<br>
\put(890,360){\sx{2.6}{$x_{14}$}} %<br>
\put(950,360){\sx{2.6}{$x_{15}$}} %<br>
\end{picture} %<br>
\end{document}   
%</nowiki>

==Keywords==

[[Fourier operator]],
[[Explicit plots]],
[[FFT]]

== Licensing ==
{{CC|by|3.0}}