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

. 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

. There are several facts about even and odd functions that can help us simplify and interpret some computations.

The product rules:
- even $\times$ even = even
- even $\times$ odd = odd $\times$ even = odd
- odd $\times$ 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.)
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 November 22, 2009, 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.

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	Info Lecture 4: Symmetry and Its Effects Document Actions 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 . 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 . There are several facts about even and odd functions that can help us simplify and interpret some computations. 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.) 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 November 22, 2009, 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.	Reuse Course Download this course

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