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.

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

    ${\mathcal {F}}[x(t-a)]=X(t)e^{j2\pi fa}$

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

   - 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 18: / 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>
-frac{x(t) - x(-t)}{2}</math>
+    - if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>
+    - if x(t) is even, then <math>X(-f) = X(f)$</math>
+    - if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''