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# Lecture 4: Symmetry and Its Effects

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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

Examples of even functions are and . An odd function is a function such that

Examples of odd functions are and . There are several facts about even and odd functions that can help us simplify and interpret some computations.
1. The product rules:
• even even = even
• even odd = odd even = odd
• odd odd = even.
For example, is an even function. is an odd function. (The rules are the same as the rules for adding even and odd numbers.)

2. Integration. When integrating over a symmetric interval about the origin,

Let us use these facts in relation to Fourier series. Suppose we want to compute the F.S. of an even function (such as the square wave signal example). Then

To compute the F.S. of an odd signal,

Review the signals transformed so far in light of these symmetries.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 23). Lecture 4: Symmetry and Its Effects. 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_3node3.html. This work is licensed under a Creative Commons License