The Fourier Transforms: Difference between revisions

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=== Symmetries ====
=== Symmetries ====


     * if f(x) is real, then $F(-\omega) = F(\omega)^*$
     * if x(t) is real, then<math> X(-f) = F(t)^*</math>
     * if f(x) is imaginary, then $F(-\omega) = -F(\omega)^*$
     * if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>
     * if f(x) is even, then $F(-\omega) = F(\omega)$
     * if x(t) is even, then <math>X(-f) = X(f)$</math>
     * if f(x) is odd, then $F(-\omega) = -F(\omega)$.
     * if x(t) is odd, then<math> X(-f) = -X(f)$.</math>

Revision as of 11:01, 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

   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)ej2πfa

Frequency and amplitude are affected when changing spatial scale inversely

   ℱ[x(a*t)]=1aX(fa)

Symmetries =

   * if x(t) is real, thenX(−f)=F(t)*
   * if x(t) is imaginary, then X(−f)=−X(f)*
   * if x(t) is even, then X(−f)=X(f)$
   * if x(t) is odd, thenX(−f)=−X(f)$.
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