The Fourier Transforms
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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/08202a4a4747e62cfd1be15be3d5ce0b64752fed)
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)
frac{x(t) - x(-t)}{2}</math>