The Fourier Transforms

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

   ${\mathcal {F}}[a*x(t)+b*y(t)]=a*X(f)+b*Y(f)$

= Shifting the function changes the phase of the spectrum

   ${\mathcal {F}}[x(t-a)]=X(t)e^{j2\pi fa}$

Frequency and amplitude are affected when changing spatial scale inversely

   ${\mathcal {F}}[x(a*t)]={\frac {1}{a}}X({\frac {f}{a}})$

Symmetries =

   * if f(x) is real, then $F(-\omega) = F(\omega)^*$
   * if f(x) is imaginary, then $F(-\omega) = -F(\omega)^*$
   * if f(x) is even, then $F(-\omega) = F(\omega)$
   * if f(x) is odd, then $F(-\omega) = -F(\omega)$.