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Aug 27th, 2009 00:37
techi gity, Knud van Eeden,
  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]  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.   Summarizing:  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   Note: The whole idea is thus similar to using polar coordinates, where you have an angle and a scaled radius length.   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, ...)   Internet: see also:   http://blogtact.com/ http://uddannelsepainternet.blogtact.com/ http://anxietygk.blogtact.com/ http://insuranceru.blogtact.com/ http://ajandekotlet.blogtact.com/ http://reisenenglandschottland.blogtact.com/ http://voyageangleterreecosse.blogtact.com/ http://travelengscoru.blogtact.com/ http://voyageafrique.blogtact.com/ http://automobilecommentaires.blogtact.com/ http://geschenkeidee.blogtact.com/ http://itavaltahotellit.blogtact.com/ http://petsru.blogtact.com/ http://insurancebg.blogtact.com/ http://haziallatellatas.blogtact.com/ http://landerderwelt.blogtact.com/ http://europetravelru.blogtact.com/ http://flyvninger.blogtact.com/ http://geschenkideeen.blogtact.com/