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

   ${ℱ}[a*x(t)+b*y(t)]=a*X(f)+b*Y(f)$

Shifting the function changes the phase of the spectrum

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

frac{x(t) - x(-t)}{2}</math>