Difference between revisions of "CT DT Fourier transform ECE438F10" - Rhea

Revision as of 05:46, 23 October 2010

A work in progress.

The Continuous Time Fourier Transform (CTFT)

CTFT:

$X(\omega) = \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt$

Inverse CTFT:

$x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \! X(\omega)e^{j \omega t} dw$

Example:

Let $x(t) = \delta (t)$

$\begin{align} X(\omega) &= \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt \\ &= \int_{-\infty}^{\infty} \! \delta (t)e^{-j \omega t} dt \\ &= 1\end{align}$

Therefore, CTFT of $\delta (t) = 1$

@@ Line 12: / Line 12: @@
 '''Example:'''
-Let <math>x(t) = <math>\delta (t)</math>
+Let <math>x(t) = \delta (t)</math>
-<math>\begin{align}X(\omega) = \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt \\
+<math>
-&= \int_{-\infty}^{\infty} \! rect(t)e^{-j \omega t} dt \\
+\begin{align}
-&= \int_{-\frac{1}{2}}^{\frac{1}{2}} \! rect(t)e^{-j \omega t} dt \\
+X(\omega) &= \int_{-\infty}^{\infty} \! x(t)e^{-j \omega t} dt \\ &= \int_{-\infty}^{\infty} \! \delta (t)e^{-j \omega t} dt \\ &= 1\end{align}</math>
-&= \int_{-\frac{1}{2}}^{\frac{1}{2}} \! e^{-j \omega t} dt\end{align}</math>
-...because rect(t) has an area of 1 over the limits <math>[-\frac{1}{2}, \frac{1}{2}]</math>. So,
+Therefore, CTFT of <math>\delta (t) = 1</math>
-<math>\int_{-\frac{1}{2}}^{\frac{1}{2}} \! e^{-j \omega t} dt = \frac{1}{-j\omega}(e^{-j\frac{\omega}{2}} - e^{j\frac{\omega}{2}}) = \frac{1}{j\omega}(e^{j\frac{\omega}{2}} - e^{-j\frac{\omega}{2}}) = \frac{2}{\omega}(\frac{e^{j\frac{\omega}{2}} - e^{-j\frac{\omega}{2}}}{2j}) = \frac{2}{\omega}sin(\frac{\omega}{2})</math>
-Therefore,
-<math>X(\omega) = \frac{2}{\omega}sin(\frac{\omega}{2}) \ \ or \ \ \frac{1}{\omega/2}sin(\frac{\omega}{2}) = sinc(\frac{\omega}{2\pi})</math>
 == Properties ==

Difference between revisions of "CT DT Fourier transform ECE438F10" - Rhea

Revision as of 05:46, 23 October 2010

The Continuous Time Fourier Transform (CTFT)

Properties

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