imported>Meg Taylor |
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| == Summary == | | == Summary == |
| {{Image_Details|user
| | Importing file |
| |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}}
| |
