The Fourier Transforms: Difference between revisions

Latest revision as of 13:44, 28 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 10: / Line 10: @@
-==== Shifting the function changes the phase of the spectrum ===
+==== Shifting the function changes the phase of the spectrum ====
-    <math>\mathcal{F}[x(t-a)] = X(t)e^{-j2\pi f a}\</math>
+    <math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>
 ==== Frequency and amplitude are affected when changing spatial scale inversely ====
@@ Line 19: / Line 18: @@
      <math>\mathcal{F}[x(a*t)] = \frac{1}{a}X(\frac{f}{a})</math>
+=== Symmetries ====
+'''
+    - if x(t) is real, then <math> X(-f) = F(t)^*</math>
+    - if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>
+    - if x(t) is even, then <math>X(-f) = X(f)$</math>
-frac{x(t) - x(-t)}{2}</math>
+    - if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''