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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

\begin{displaymath}f(t) = f(-t) \end{displaymath}

Examples of even functions are $f(t) = \cos(t)$ and $f(t) = t^2$ . An odd function is a function such that

\begin{displaymath}f(t) = -f(-t) \end{displaymath}

Examples of odd functions are $f(t) = \sin(t)$ and $f(t) = t^3$ . There are several facts about even and odd functions that can help us simplify and interpret some computations.
  1. The product rules:
    • even $\times$ even = even
    • even $\times$ odd = odd $\times$ even = odd
    • odd $\times$ odd = even.
    For example, $t^2 \cos(t)$ is an even function. $t \cos(t)$ 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,

    \begin{displaymath}\int_{-T_0/2}^{T_0/2} \mbox{even}(t) dt = 2\int_{0}^{T_0/2}
\mbox{even}(t) dt \end{displaymath}
    \begin{displaymath}\int_{-T_0/2}^{T_0/2} \mbox{odd}(t) dt = 0 \end{displaymath}

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

\begin{displaymath}a_0 = \frac{1}{T_0} \int_{-T_0/2}^{T_0/2} f(t) dt = \frac{2}{T_0}
\int_{0}^{T_0/2} f(t) dt.\end{displaymath}
\begin{displaymath}a_n = \frac{2}{T_0} \int_{-T_0/2}^{T_0/2} f(t)\cos(n\omega_0 t) dt
= \frac{4}{T_0} \int_{0}^{T_0/2} f(t)\cos(n\omega_0 t) dt.\end{displaymath}
\begin{displaymath}b_n = \frac{2}{T_0} \int_{-T_0/2}^{T_0/2} f(t)\sin(n\omega_0 t) dt
= 0
\end{displaymath}

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

\begin{displaymath}a_0 = \frac{1}{T_0} \int_{-T_0/2}^{T_0/2} f(t) dt = 0 \end{displaymath}
\begin{displaymath}a_n = \frac{2}{T_0} \int_{-T_0/2}^{T_0/2} f(t)\cos(n\omega_0 t) dt
= 0 \end{displaymath}
\begin{displaymath}b_n = \frac{2}{T_0} \int_{-T_0/2}^{T_0/2} f(t)\sin(n\omega_0 t) dt
= \frac{4}{T_0} \int_{0}^{T_0/2} f(t)\sin(n\omega_0 t) dt.\end{displaymath}

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 Creative Commons License