The Fourier Transforms: Difference between revisions

Revision as of 11:04, 12 October 2007

The Fourier transform was named after Joseph Fourier, a French mathematician. A Fourier Transform takes a function to its frequency components.

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

    $ℱ [x (t - a)] = X (t) e^{j 2 π f a}$

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

   * 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) $$

   * if x(t) is odd, then  $X (- f) = - X (f) $ .$

@@ Line 20: / Line 20: @@
 === Symmetries ====
 '''
-     * if x(t) is real, then<math> X(-f) = F(t)^*</math>
+     * if x(t) is real, then <math> X(-f) = F(t)^*</math>
      * if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>
@@ Line 26: / Line 26: @@
      * if x(t) is even, then <math>X(-f) = X(f)$</math>
-     * if x(t) is odd, then<math> X(-f) = -X(f)$.</math>'''
+     * if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''