The Fourier Transforms: Difference between revisions
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=== Symmetries ==== |
=== Symmetries ==== |
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''' |
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* if x(t) is real, then<math> X(-f) = F(t)^*</math> |
* if x(t) is real, then<math> X(-f) = F(t)^*</math> |
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* if x(t) is even, then <math>X(-f) = X(f)$</math> |
* if x(t) is even, then <math>X(-f) = X(f)$</math> |
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* if x(t) is odd, then<math> X(-f) = -X(f)$.</math> |
* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>''' |
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Revision as of 10:03, 12 October 2007
The Fourier transform was named after Joseph Fourier, a French mathematician. A Fourier Transform takes a function to its frequency components.
Properties of a Fourier Transform:
Properties of a Fourier Transform:
Linearity
![{\displaystyle {\mathcal {F}}[a*x(t)+b*y(t)]=a*X(f)+b*Y(f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286cb04cdacd549bdd10c6a24be8c36b2bea67a1)
= Shifting the function changes the phase of the spectrum
![{\displaystyle {\mathcal {F}}[x(t-a)]=X(t)e^{j2\pi fa}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe03600d8fec85913994d0b0e192f040aac9fcf)
Frequency and amplitude are affected when changing spatial scale inversely
![{\displaystyle {\mathcal {F}}[x(a*t)]={\frac {1}{a}}X({\frac {f}{a}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9208b96cb19f4bbb63669d354bdbeaf3da689746)
Symmetries =
* if x(t) is real, then
* if x(t) is imaginary, then
* if x(t) is even, then
* if x(t) is odd, then
