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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 is a periodic function with F.S. representation

and is a periodic function with the same period and a F.S. representation

Now consider an average energy kind of term

Substituting in for each of the F.S. gives (taking advantage of the orthogonality of the exponential function)

We can write this in a convenient inner product notation. We can define the inner product between two series and as

Then we can write (using our complex inner product for functions)

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

As a special case, take . Then is the average energy of . By the Parseval's relationship,

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