Difference between revisions of "CosFT"

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[[CosFT]], or Cosinus Transform, refers to the integral transform with kernel $K(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)$;
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[[CosFT]], or Cosinus Transform, refers to the integral transform with kernel \(K(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)\);
   
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for function $f$, the [[CosFT]]$f$ appears as $g$ defined with
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for function \(f\), the [[CosFT]]\(f\) appears as \(g\) defined with
   
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$\displaystyle g(x)=\,$[[CosFT]]$f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \cos(xy) \, f(y) \, \mathrm d y$
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\(\displaystyle g(x)=\,\)[[CosFT]]\(f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \cos(xy) \, f(y) \, \mathrm d y\)
   
 
==[[SinFT]] and [[CosFT]]==
  
==[[SinFT]] and [[CosFT]]==
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[[CosFT]] often appears together with [[SinFT]];
  
[[CosFT]] often appears together with [[SinFT]];
   
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the sine transform [[SinFT]] of function $f$ appears as $g=\,$[[SinFT]]$f$ with rofmula
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the sine transform [[SinFT]] of function \(f\) appears as \(g=\,\)[[SinFT]]\(f\) with rofmula
   
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$\displaystyle g(x)=\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y$
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\(\displaystyle g(x)=\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y\)
   
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It is assumed that function $f$ decays (or, at least, quickly oscillates) at infinity, in such a way that the integral converges.
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It is assumed that function \(f\) decays (or, at least, quickly oscillates) at infinity, in such a way that the integral converges.
   
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Then, [[SinFT]]$^2=\,$[[CosFT]]$^2=\hat 1$, id est, the identity transform.
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Then, [[SinFT]]\(^2=\,\)[[CosFT]]\(^2=\hat 1\), id est, the identity transform.
   
 
==Numerical implementation==
  
==Numerical implementation==
   
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[[CosFT]] can be implemented numerically through the [[CFT]] transform at the uniform grid at $N\!+\!1$ nodes; for array $f$, the SFT $g$ is defined with
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[[CosFT]] can be implemented numerically through the [[CFT]] transform at the uniform grid at \(N\!+\!1\) nodes; for array \(f\), the SFT \(g\) is defined with
   
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$g_m=\,$[[CFT]]$\displaystyle f_m=$ $\displaystyle
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\(g_m=\,\)[[CFT]]\(\displaystyle f_m=\) \(\displaystyle
 
\frac{1}{2}\, f_0+\frac{(-1)^m}{2}\, f_{N} +
  
\frac{1}{2}\, f_0+\frac{(-1)^m}{2}\, f_{N} +
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\sum_{n=1}^{N-1} \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n$
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\sum_{n=1}^{N-1} \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\)
   
 
The [[Numerical recipes in C]] (http://numerical.recipes)
  
The [[Numerical recipes in C]] (http://numerical.recipes)
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ftp://ftp.cs.stanford.edu/cs/robotics/scohen/nr/cosft.c</ref>
  
ftp://ftp.cs.stanford.edu/cs/robotics/scohen/nr/cosft.c</ref>
   
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At given number $N$ of nodes,
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At given number \(N\) of nodes,
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the set of the nodes can be denoted with $x_n$ for $n=0 .. N$,
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the set of the nodes can be denoted with \(x_n\) for \(n=0 .. N\),
   
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$\displaystyle
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\(\displaystyle
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x_n=\sqrt{\frac{\pi}{N}}~ n$
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x_n=\sqrt{\frac{\pi}{N}}~ n\)
   
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then, for $f_n=f(x_n)$, at large $N\gg 1$,
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then, for \(f_n=f(x_n)\), at large \(N\gg 1\),
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the transform $~g(x)=\frac{2}{\pi}\int_0^\infty f(y)\,\cos(x\,y)\,\mathrm d y~$ is approximated with
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the transform \(~g(x)=\frac{2}{\pi}\int_0^\infty f(y)\,\cos(x\,y)\,\mathrm d y~\) is approximated with
   
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$\displaystyle
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\(\displaystyle
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g(x_m) \approx g_m = \sqrt{\frac{2}{N}}\, \left( \frac{f_0+(-1)^m f_N}{2} + \sum_{n=1}^{N-1} \, \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\right)$
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g(x_m) \approx g_m = \sqrt{\frac{2}{N}}\, \left( \frac{f_0+(-1)^m f_N}{2} + \sum_{n=1}^{N-1} \, \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\right)\)
   
 
==Eigenfunctions==
  
==Eigenfunctions==
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[[File:Cosftes16t300.jpg|360px|right]]
  
[[File:Cosftes16t300.jpg|360px|right]]
   
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Eigenfunctions $F$ of the [[CosFT]] appear as even [[Oscillator function]]s.
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Eigenfunctions \(F\) of the [[CosFT]] appear as even [[Oscillator function]]s.
   
 
The simplest of them is just Gaussian;
 
The simplest of them is just Gaussian;
   
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$F(x)=\exp(-x^2/2)$
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\(F(x)=\exp(-x^2/2)\)
   
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This property is used in the [[C++]] test. The thick segmented line in the explicit plots at right show the discrete representation of the Gaussian at the discrete grid with $N\!=\!4$ and $N\!=\!16$.
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This property is used in the [[C++]] test. The thick segmented line in the explicit plots at right show the discrete representation of the Gaussian at the discrete grid with \(N\!=\!4\) and \(N\!=\!16\).
   
 
The thin segmented line shows its [[SinFT]] transform, as it is approximated at this grid. The discrete representation and its CFT practically coincide; the deviation is smaller than the thicknesses of the lines.
  
The thin segmented line shows its [[SinFT]] transform, as it is approximated at this grid. The discrete representation and its CFT practically coincide; the deviation is smaller than the thicknesses of the lines.

Latest revision as of 18:47, 30 July 2019

CosFT, or Cosinus Transform, refers to the integral transform with kernel \(K(x,y)=\sqrt{\frac{2}{\pi}} \cos(xy)\);

for function \(f\), the CosFT\(f\) appears as \(g\) defined with

\(\displaystyle g(x)=\,\)CosFT\(f\,(x) \displaystyle =\sqrt{\frac{2}{\pi}} \int_0^\infty \cos(xy) \, f(y) \, \mathrm d y\)

SinFT and CosFT

CosFT often appears together with SinFT;

the sine transform SinFT of function \(f\) appears as \(g=\,\)SinFT\(f\) with rofmula

\(\displaystyle g(x)=\sqrt{\frac{2}{\pi}} \int_0^\infty \sin(xy) \, f(y) \, \mathrm d y\)

It is assumed that function \(f\) decays (or, at least, quickly oscillates) at infinity, in such a way that the integral converges.

Then, SinFT\(^2=\,\)CosFT\(^2=\hat 1\), id est, the identity transform.

Numerical implementation

CosFT can be implemented numerically through the CFT transform at the uniform grid at \(N\!+\!1\) nodes; for array \(f\), the SFT \(g\) is defined with

\(g_m=\,\)CFT\(\displaystyle f_m=\) \(\displaystyle \frac{1}{2}\, f_0+\frac{(-1)^m}{2}\, f_{N} + \sum_{n=1}^{N-1} \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\)

The Numerical recipes in C (http://numerical.recipes) suggest the implementation through routines four1 and realft; however, for the serious applications, specification "float" should be replaced to something appropriate, for example, double, or complex double. [1][2][3][4]

At given number \(N\) of nodes, the set of the nodes can be denoted with \(x_n\) for \(n=0 .. N\),

\(\displaystyle x_n=\sqrt{\frac{\pi}{N}}~ n\)

then, for \(f_n=f(x_n)\), at large \(N\gg 1\), the transform \(~g(x)=\frac{2}{\pi}\int_0^\infty f(y)\,\cos(x\,y)\,\mathrm d y~\) is approximated with

\(\displaystyle g(x_m) \approx g_m = \sqrt{\frac{2}{N}}\, \left( \frac{f_0+(-1)^m f_N}{2} + \sum_{n=1}^{N-1} \, \cos\left( \frac{\pi}{N} \,m\,n \right) \, f_n\right)\)

Eigenfunctions

Cosftes04t300.jpg
Cosftes16t300.jpg

Eigenfunctions \(F\) of the CosFT appear as even Oscillator functions.

The simplest of them is just Gaussian;

\(F(x)=\exp(-x^2/2)\)

This property is used in the C++ test. The thick segmented line in the explicit plots at right show the discrete representation of the Gaussian at the discrete grid with \(N\!=\!4\) and \(N\!=\!16\).

The thin segmented line shows its SinFT transform, as it is approximated at this grid. The discrete representation and its CFT practically coincide; the deviation is smaller than the thicknesses of the lines.

References

  1. ↑ ftp://ftp.cpc.ncep.noaa.gov/wd51we/random_phase/four1.c
  2. ↑ ftp://ftp.cpc.ncep.noaa.gov/wd51we/random_phase/realft.c
  3. ↑ ftp://ftp.cs.stanford.edu/cs/robotics/scohen/nr/sinft.c
  4. ↑ ftp://ftp.cs.stanford.edu/cs/robotics/scohen/nr/cosft.c

Keywords

CosFT, FFT, Integral transform, SinFT,,,

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