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Lecture 4: Energy of Singals and Parseval's Relationships

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Schedule :: Periodic Signals :: Fourier Spectrum :: Symmetry :: Fundamental Frequency :: Smoothness of the Function :: Fourier Series :: Exponential Series :: Spectra :: Bandwidth :: Energy of Signals :: Geometric Viewpoint

It is possible and often theoretically useful to examine the energy of signals in both the time domain and the frequency (Fourier series) domain. We will develop an important relationship. Suppose $f(t)$ is a periodic function with F.S. representation
\begin{displaymath}f(t) = \sum_n D_n e^{jn\omega_0 t} \end{displaymath}

and is a periodic function with the same period and a F.S. representation
$g(t)$ \begin{displaymath}g(t) = \sum_n E_n e^{jn\omega_0 t} \end{displaymath}

Now consider an average energy kind of term
\begin{displaymath}\frac{1}{T_0}\int_{T_0} f(t)g^*(t) dt \end{displaymath}

Substituting in for each of the F.S. gives (taking advantage of the orthogonality of the exponential function)
\begin{displaymath}\frac{1}{T_0}\int_{T_0} f(t)g^*(t) dt = \sum_{n} D_n E_n^* \end{displaymath}

We can write this in a convenient inner product notation. We can define the inner product between two series $\{D_n\}$ and $\{E_n\}$ as
\begin{displaymath}\la D_n,E_n\ra = \sum_{n} D_n E_n^* \end{displaymath}

Then we can write (using our complex inner product for functions)
\begin{displaymath}\frac{1}{T_0} \la f(t),g(t)\ra = \la D_n,E_n \ra \end{displaymath}

A relationship such as this is known as a Parseval's relationship, named after some guy.

As a special case, take $g(t) = f(t)$ . Then $\frac{1}{T_0} \la f(t),f(t) \ra$ is the average energy of $f(t)$ . By the Parseval's relationship,

\begin{displaymath}\frac{1}{T_0} \la f(t),f(t) \ra = \la D_n,D_n\ra \end{displaymath}


\begin{example}Find the sum of the series
\begin{displaymath}\sum_n \left(\frac{...
...2} \end{displaymath}This would have been hard to do any other way!
\end{example}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 23). Lecture 4: Energy of Singals and Parseval\'s Relationships. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/4_11node11.html. This work is licensed under a Creative Commons License Creative Commons License