The Fourier Transforms: Difference between revisions

Revision as of 10:55, 12 October 2007

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

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

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

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

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

@@ Line 12: / Line 12: @@
 ==== 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 ====