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Lecture 5: Properties of 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 saw earlier a variety of properties associated with the Laplace transform: linearity, time shift, convolution, differentiation, and integration. These opened up a variety of applications. There are similar the properties of F.T. which are also very useful.
Linearity
The F.T. is linear:
\begin{displaymath}\Fc[af(t) + bg(t)] = a \Fc[f(t)] + b\Fc[g(t)] \end{displaymath}

This follows from the linearity of the integral.
Time/Frequency Duality
The duality property is one that is not shared by the Laplace transform. While slightly confusing perhaps at first, it essentially doubles the size of our F.T. table. The duality property follows from the similarity of the forward and inverse F.T. It states that if
\begin{displaymath}f(t) \Leftrightarrow F(\omega) \end{displaymath}

then
\begin{displaymath}F(t) \Leftrightarrow 2\pi f(-\omega) \end{displaymath}

where the function on the left is the function of time and the function on the right is the function of frequency.


\begin{proof}
\begin{displaymath}f(t) = \frac{1}{2\pi}\int_{-\infty}^\infty F(x)...
...T. of $F(x)$. Now
replace $t \rightarrow \omega$\ to see the result. \end{proof}


\begin{example}We have seen that
\begin{displaymath}\rect(t/T) \Leftrightarrow T...
...row 2\pi \rect(-\omega/T) = 2\pi
\rect(\omega/T).
\end{displaymath}\end{example}

\begin{example}We have already seen another example of the duality property
in t...
...displaymath}1 \Leftrightarrow 2\pi \delta(\omega) \end{displaymath}\end{example}
In terms of the alternative notation $F(f)$ (in Hz), we have that if

\begin{displaymath}g(t) \Leftrightarrow G(f)
\end{displaymath}

then
\begin{displaymath}G(t) \Leftrightarrow g(-f)
\end{displaymath}

Scaling Property
We have seen this for Laplace transforms: If
\begin{displaymath}f(t) \Leftrightarrow F(\omega) \end{displaymath}

then
\begin{displaymath}f(at) \Leftrightarrow \frac{1}{\vert a\vert} F(\omega/a) \end{displaymath}


\begin{proof}
By student...
\end{proof}
Discuss in terms of time and bandwidth: when a signal is expanded in time, it is compressed in frequency, and vice versa. We cannot be simultaneously short in time and short in frequency. This is the basis for the famed Heisenberg uncertainty principle .

Time-Shift Property
If
\begin{displaymath}f(t) \Leftrightarrow F(\omega) \end{displaymath}

then
\begin{displaymath}f(t-t_0) \Leftrightarrow F(\omega)e^{-j\omega t_0} \end{displaymath}

In other words, a shift in time corresponds to a change in phase in the F.T.


\begin{proof}
By student...
\end{proof}
What is interesting is that higher frequencies experience a greater phase shift than lower frequencies. A plot of the phase is linear in frequency.


\begin{example}Find the F.T. of $f(t) = e^{-\vert t-t_0\vert}$. From the table w...
...ert}] = \frac{2a}{a^2 + \omega^2}e^{-j\omega t_0} \end{displaymath}\end{example}

Frequency-shift Property
This innocuous-looking property forms a basis for every radio and TV transmitter in the world! It simply states that if
\begin{displaymath}f(t) \Leftrightarrow F(\omega) \end{displaymath}

then
\begin{displaymath}f(t)e^{j\omega_0 t} \Leftrightarrow F(\omega - \omega_0) \end{displaymath}

Note that this is a dual of the time-shift property.


\begin{proof}
By student...
\end{proof}
Show a block diagram of what happens. Since $e^{j\omega_0 t}$ is a complex signal, this is really only a mathematical description. But we can multiply by $\cos(\omega_0 t)$ . Using linearity we get

\begin{displaymath}f(t)\cos(\omega_0 t) \Leftrightarrow \frac{1}{2}[F(\omega-\omega_0)
+ F(\omega + \omega_0)] \end{displaymath}

What we get out is two images in the frequency domain, at positive and negative frequencies.

As an application, suppose that $f(t)$ is some information-bearing signal, say, the signal from a microphone. Plot. Now multiply by $\cos(\omega_0 t)$ and plot the result, and show the effect in frequency. This is what KSL does! This kind of modulation is called amplitude modulation .

Observe that we can modulate signals onto a variety of different frequencies. This makes it possible to have several radio (or TV) stations. The higher frequencies also propagate further through the air than the baseband frequencies.

Convolution Property
If
\begin{displaymath}f_1(t) \Leftrightarrow F_1(\omega) \quad\mbox{and}\quad f_2(t)
\Leftrightarrow F_2(\omega)
\end{displaymath}

then
\begin{displaymath}f_1(t)*f_2(t) \Leftrightarrow F_1(\omega)F_2(\omega) \end{displaymath}

(where $*$ is convolution) and
\begin{displaymath}f_1(t)f_2(t) \Leftrightarrow \frac{1}{2\pi}F_1(\omega)*F_2(\omega)
\end{displaymath}


\begin{proof}
By student...
\end{proof}


\begin{example}Show that
\begin{displaymath}\int_{-\infty}^t f(x)dx \Leftrightar...
... $\Fc[u(t)] = 1/j\omega + \pi \delta(\omega)$\ the result
follows.
\end{example}

Time Differentiation
If
\begin{displaymath}f(t) \Leftrightarrow F(\omega) \end{displaymath}

then
\begin{displaymath}\frac{df}{dt} \Leftrightarrow j\omega F(\omega) \end{displaymath}


\begin{proof}
By student...
\end{proof}

Time Integration
We saw above that
\begin{displaymath}\int_{-\infty}^t f(x)dx \Leftrightarrow \frac{F(\omega)}{j\omega} +
\pi F(0) \delta(\omega) \end{displaymath}

If $f$ is zero mean (i.e. $F(0) = 0$ ) then
\begin{displaymath}\int_{-\infty}^t f(x)dx \Leftrightarrow \frac{F(\omega)}{j\omega}
\end{displaymath}


\begin{example}Use the time differentiation property to find the F.T. of
$f(t) =...
...2 T} \sin^2(\omega T/4) = T/2
\sinc^2(\omega T/4) \end{displaymath}\end{example}


\begin{example}Find the F.T. of $f(t) = \vert t\vert$. This is hard to do by oth...
...begin{displaymath}F(\omega) = \frac{-2}{\omega^2} \end{displaymath}\end{example}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 26). Lecture 5: Properties of 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_6node6.html. This work is licensed under a Creative Commons License Creative Commons License