Entry

Math: Transform: Fourier: Can you tell more about f(x) . e^( x . i )? [fast Fourier transform]

Apr 16th, 2005 03:48
Knud van Eeden,

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--- Knud van Eeden --- 16 April 2005 - 00:41 am ----------------------

Math: Transform: Fourier: Can you tell more about f(x) . e^( x . i )? 
[fast Fourier transform]

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1. The definition of a complex number (in a rectangular coordinate
   system) is:

  z = (real number) + (imaginary number)

  or thus

  (horizontal component) + (vertical component) . i

  is a representation of a vector

                          x . i
 2. It can be shown that e      is just a special case of such a vector
    (that is it is a unit vector (thus with length 1), rotated over an
    angle x, in a unit circle.

        x . i
 3. So e      is a unit vector (with length 1), rotated over an angle 
x, in a unit circle.

         x . i
 4. 1 . e      is a scaled unit vector (with length 1), rotated over 
an angle x, in a unit circle.

         x . i
 5. 8 . e      is a scaled vector (with length equal to 8), rotated 
over an angle x, in a circle.

                x . i
 6. constant . e      is a scaled vector (with length equal to this 
constant), rotated over an angle x, in a circle.

         x . i
 7. y . e      is a scaled vector (with length equal to this y), 
rotated over an angle x, in a circle.

            x . i
 8. f(x) . e      is a scaled vector (with length equal to this f
(x) ), rotated over an angle x, in a circle.


             1. x . i
 9. f(x) . e          is a scaled vector (with length equal to this f
(x) ), rotated over an angle 1 . x, in a circle.


             8 . x . i
10. f(x) . e          is a scaled vector (with length equal to this f
(x) ), rotated over an angle 8 . x, in a circle.


             8 . x . i + 1
11. f(x) . e               is a scaled vector (with length equal to 
this f(x) ), rotated over an angle 8 . x and starting at an angle 1, 
in a circle.


             8 . x . i + 4
12. f(x) . e               is a scaled vector (with length equal to 
this f(x) ), rotated over an angle 8 . x and starting at an angle 4, 
in a circle.


             8 . x . i + constant1
13. f(x) . e                      is a scaled vector (with length 
equal to this f(x) ), rotated over an angle 8 . x and starting at an 
angle constant1, in a circle.


             constant2 . x . i + constant1
14. f(x) . e                              is a scaled vector (with 
length equal to this f(x) ), rotated over an angle constant2 . x and 
starting at an angle constant1, in a circle.


                          scale.(independent variable).i+translation
15. (dependent variable).e

    is a scaled vector (with length equal to this dependent variable),
    rotated over an angle ('scalefactor' times 'independent variable')
    plus that translationconstant, in a circle.

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Summarizing:

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Steps: Overview:

 1. -Choose the value of your independent variable
     (e.g. 1, 2, 3, ...)

 2. -Calculate the value of the corresponding dependent
     variable (e.g if y = x^2, you get 1^2, 2^2, 3^2, ...
     or thus 1, 4, 9, ...)

 3. -Choose a scalefactor for the rotation

 4. -Choose a translationconstant for the rotation

 5. -Then you rotate that angle of that vector first over an angle
     equal to that scalefactor times the given independent variable.

 6. -You add the translation constant to this result

           /
          /
         / angle equal to this scalefactor times independent variable 
plus a translationconstant, thus an angle equal to e.g.  constant2 . 1 
+ constant1, constant2 . 2 + constant1, constant 2 . 3 + constant1, ...
         -----

 7. -Then you finally scale the length of the vector to be equal to
     that calculated value of the dependent variable


             /
            /
           /
          /
         / length equal to this independent variable, thus an length 
equal to e.g.  1, 4, 9, ...
         -----

 8. -If you need to create a sum of this vectors, as e.g. in a Fourier
     Transform, you just vector add this vectors

     1. -Add the horizontal component of each vector to a sum

     2. -Add the vertical component of each vector to a sum

 9. -This will give thus a final resulting vector representing this
     sums

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Note:

The whole idea is thus similar to using polar coordinates,
where you have an angle and a scaled radius length.

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Note:

You see thus clearly that this is a special case of a transformation
(as a transformation is per definition just some combination of
 operations like e.g. scaling, translation, rotation, ...)

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Internet: see also:

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