Difference between revisions of "HW5.2 Bavorndej Chanyasak ECE301Fall2008mboutin" - Rhea

Latest revision as of 11:31, 16 September 2013

Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform

Suppose we have a signal:

$ e^{-2(t-1)}u(t-1)\, $

The formula of Fourier Transform is:

$ X(w) = \int_{-\infty}^{ \infty} x(t)e^{-jwt}dt\, $

Substituting:

$ X(w) = \int_{-\infty}^{ \infty} e^{-2(t-1)}u(t-1)e^{-jwt}dt\, $

From the step function, the range becomes 1 to $ \infty $, so the equation becomes:

$ X(w) = \int_{1}^{ \infty} e^{-2(t-1)}e^{-jwt}dt\, $

$ X(w) = \int_{1}^{ \infty} e^{2-(2+jw)t}dt\, $

Integrating yields:

$ X(w) = {\left. -\frac{e^{2-(2+jw)t}}{2+jw} \right]_{1}^{\infty}}\, $

$ X(w) = 0 - -\frac{e^{2-(2+jw)}}{2+jw} \, $

$ X(w) = \frac{e^{2-2-jw}}{2+jw} \, $

$ X(w) = \frac{e^{-jw}}{2+jw} \, $

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