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

   ${ℱ}[x(t-a)]=X(t)e^{j2\pi fa}$

Frequency and amplitude are affected when changing spatial scale inversely

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

Symmetries =

   - if x(t) is real, then $X(-f)=F(t{)}^{*}$

   - if x(t) is imaginary, then $X(-f)=-X(f{)}^{*}$

   - if x(t) is even, then $X(-f)=X(f)\mathrm{\$}$

   - if x(t) is odd, then $X(-f)=-X(f)\mathrm{\$}.$