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 2: / Line 2: @@
+== Properties of a Fourier Transform: ==
-== Headline text ==
 Properties of a Fourier Transform:
+==== Linearity ====
-;Convolution
-::::{|
-|<math>f(t)* g(t) \,</math>
+    <math>\mathcal{F}[a*x(t) + b*y(t)] = a*X(f) + b*Y(f)</math>
-|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
-\sqrt{2\pi}\cdot F(\omega)\cdot G(\omega) \,</math>
+==== Shifting the function changes the phase of the spectrum ====
-| &nbsp; &nbsp; (unitary convention)
-|-
+    <math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>
-|
-|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
+==== Frequency and amplitude are affected when changing spatial scale inversely ====
-F(\omega)\cdot G(\omega) \,</math>
-| &nbsp; &nbsp; (non-unitary convention)
+    <math>\mathcal{F}[x(a*t)] = \frac{1}{a}X(\frac{f}{a})</math>
-|-
-|
+=== Symmetries ====
-|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
+'''
-F(f)\cdot G(f) \,</math>
+    - if x(t) is real, then <math> X(-f) = F(t)^*</math>
-| &nbsp; &nbsp; (ordinary frequency)
-|}
+    - 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>'''