The Fourier Transforms: Difference between revisions

Latest revision as of 12:44, 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 7: / Line 7: @@
 ==== Linearity ====
-<math>\mathcal{F}[a*X(t)]x_{\mathrm{t}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ((2k-1)2\pi ft \right )}\over(2k-1)}</math>
+    <math>\mathcal{F}[a*x(t) + b*y(t)] = a*X(f) + b*Y(f)</math>
+==== Shifting the function changes the phase of the spectrum ====
+    <math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>
+==== Frequency and amplitude are affected when changing spatial scale inversely ====
+    <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)$</math>
+    - if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''