# Lecture 4: Energy of Singals and Parseval's Relationships

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

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,

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by the Contributing Authors.
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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.
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