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Lecture 5: System Analysis Using Fourier Transforms

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Schedule :: Fourier Series Formulas :: Nonperiodic Functions :: Useful Functions :: Simple Functions :: Fourier Transforms :: Properties :: System Analysis :: Signal Distortion :: Filters :: Parseval's Formulas

We have seen that we can use Laplace transforms for analysis of LTIC systems. We can do similar things with Fourier transforms, except that we will be able to obtain only steady-state responses, not the transient response. For the differential equation
\begin{displaymath}Q(D)y(t) = P(D)f(t) \end{displaymath}

we can use the time-differentiation property of the F.T.:
\begin{displaymath}D^k y(t) \Leftrightarrow (j\omega)^k Y(\omega) \end{displaymath}

so that we can write
\begin{displaymath}Q(j\omega)Y(\omega) = P(j\omega)F(\omega) \end{displaymath}

or
\begin{displaymath}Y(\omega) = H(\omega)F(\omega), \end{displaymath}

where
\begin{displaymath}H(\omega) = \frac{P(j\omega)}{Q(j\omega)}. \end{displaymath}

The function $H(\omega)$ is the transfer function of the system, and is related to the transfer function we saw for Laplace transforms.

Analysis with the F.T. is not generally preferable to analysis with the L.T., because the system must always be asymptotically stable and we have to lug around $j\omega$ , instead of the more compact $s$ . However, for systems or signals that are noncausal and Fourier transformable, the analysis may be easier using F.T.
\begin{example}Let
\begin{displaymath}H(s) = \frac{1}{s+2} \end{displaymath}and...
...gin{displaymath}y(t) = \frac{1}{2}(1-e^{-2t})u(t) \end{displaymath}\end{example}

\begin{example}For the same system, find the response when
\begin{displaymath}f...
...aymath}y(t) = \frac{1}{3}[e^tu(-t) + e^{-2t}u(t)] \end{displaymath}\end{example}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 26). Lecture 5: System Analysis Using Fourier Transforms. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/5_7node7.html. This work is licensed under a Creative Commons License Creative Commons License