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Lecture 5: Signal Distortion During Transmission

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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

Fourier transforms form a natural way of looking at the frequency response of filters, since they examine the response due to a a trigonometric exponential. We have already seen how to characterize a filter by its frequency response, using the Laplace transform. We can do the same thing (somewhat more naturally) using Fourier transforms.

The output of a signal with transfer function $H(\omega)$ is

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

Let us express this in polar form:
\begin{displaymath}\vert Y(\omega)\vert e^{j\angle Y(\omega)} = \vert F(\omega)\...
...vert H(\omega)\vert
e^{j(\angle F(\omega) + \angle H(\omega))} \end{displaymath}

The magnitude response of the output is therefore
\begin{displaymath}\vert Y(\omega)\vert = \vert F(\omega)\vert \vert H(\omega) \vert,
\end{displaymath}

so the magnitude response is the input magnitude response ``shaped'' by the system magnitude response. In a word, the output is filtered. We can also consider the phase response:
\begin{displaymath}\angle Y(\omega) = \angle F(\omega) + \angle H(\omega) \end{displaymath}

The phase angle of the output is the phase angle of the input plus whatever phase the system contributes.

There are applications in which we would like the system to not change the input signal as it passes through the system, other than to delay the signal and possibly uniformly attenuating the whole signal. For example, the signals that pass through the air to our radios: we would hope that they would arrive at the radio in the same form as they left the transmitter, except delayed by the propagation through the air and attenuated. A system which does not distort a signal (other than delaying it and possibly changing the overall gain) is said to be a distortionless system . The output of a distortionless system may be written as

\begin{displaymath}y(t) = k f(t-t_d) \end{displaymath}

where $k$ is the overall gain and $t_d$ is the delay introduced by the system. In terms of Fourier transforms, we have
\begin{displaymath}Y(\omega) = k F(\omega)e^{-j\omega t_d} \end{displaymath}

So the transfer function of a distortionless system is
\begin{displaymath}H(\omega) = ke^{-j\omega t_d} \end{displaymath}

Observe that the magnitude does not change with frequency, and the phase is a linear function of $\omega$ :
\begin{displaymath}\vert H(\omega)\vert = k \end{displaymath}


\begin{displaymath}\angle H(\omega) = -\omega t_d \end{displaymath}

In practice, distortionless transmission is never exactly obtained over all frequencies. Over a localized region of frequency, however, the magnitude response may be flat enough, and the phase response may be linear enough, so the distortion is acceptable. For any system, including those with nonlinear phase, we can compute the time delay of a signal as it passes through a system by

\begin{displaymath}t_d(\omega) = -\frac{d}{d\omega} \angle H(\omega) \end{displaymath}


\begin{example}Examine the distortion of the system with frequency response
\beg...
...displaymath}Plot. Note that the delay is a function of frequency.
\end{example}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 26). Lecture 5: Signal Distortion During Transmission. 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_8node8.html. This work is licensed under a Creative Commons License Creative Commons License