Lecture 4: Determining the Fundamental Frequency

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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

The trigonometric Fourier series can be used to represent any periodic function. In periodic functions, every frequency in the Fourier series representation is an integral multiple of some fundamental frequency. Such frequencies are said to be harmonically related. The ratio of any two harmonically related frequencies is a rational number (i.e., a number which can be represented as the ratio of two integers). (Interesting mathematical fact: there are more irrational numbers than there are rational numbers. ) Any number which involves a transcendental number such as $\pi$ or , or which involves square roots which cannot be simplified down to ratios of integers (such as $\sqrt{2}$ ) is an irrational number. For functions which are harmonically related, the fundamental frequency is the greatest common divisor of the frequencies.

$\begin{example}Is the function $f_1(t) = 2 + 7 \cos(\frac{1}{2} t + \theta_1) + ... ...D}(1/2, 2/3, 7/6) = 1/6. \end{displaymath}(The GCD(num)/LCM(den)). \end{example}$

$\begin{example}Is the function $f(t) = 3 \sin(3\sqrt{2} t + \theta) + 7 \cos(6 \... ...= \frac{\sqrt{2}}{2\sqrt{3}} \end{displaymath}which is irrational. \end{example}$

$\begin{example}Is $f(t) = 3 \sin(3\sqrt{2} t + \theta) + 7 \cos(6 \sqrt{2} t )$\... ...math}\mbox{GCD}(3\sqrt{2},6\sqrt{2}) = 3\sqrt{2}. \end{displaymath}\end{example}$

Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 23). Lecture 4: Determining the Fundamental Frequency. 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_4node4.html. This work is licensed under a Creative Commons License.

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	Info Lecture 4: Determining the Fundamental Frequency 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 The trigonometric Fourier series can be used to represent any periodic function. In periodic functions, every frequency in the Fourier series representation is an integral multiple of some fundamental frequency. Such frequencies are said to be harmonically related. The ratio of any two harmonically related frequencies is a rational number (i.e., a number which can be represented as the ratio of two integers). (Interesting mathematical fact: there are more irrational numbers than there are rational numbers. ) Any number which involves a transcendental number such as $\pi$ or , or which involves square roots which cannot be simplified down to ratios of integers (such as $\sqrt{2}$ ) is an irrational number. For functions which are harmonically related, the fundamental frequency is the greatest common divisor of the frequencies. $\begin{example}Is the function $f_1(t) = 2 + 7 \cos(\frac{1}{2} t + \theta_1) + ... ...D}(1/2, 2/3, 7/6) = 1/6. \end{displaymath}(The GCD(num)/LCM(den)). \end{example}$ $\begin{example}Is the function $f(t) = 3 \sin(3\sqrt{2} t + \theta) + 7 \cos(6 \... ...= \frac{\sqrt{2}}{2\sqrt{3}} \end{displaymath}which is irrational. \end{example}$ $\begin{example}Is $f(t) = 3 \sin(3\sqrt{2} t + \theta) + 7 \cos(6 \sqrt{2} t )$\... ...math}\mbox{GCD}(3\sqrt{2},6\sqrt{2}) = 3\sqrt{2}. \end{displaymath}\end{example}$ Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 23). Lecture 4: Determining the Fundamental Frequency. 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_4node4.html. This work is licensed under a Creative Commons License.	Reuse Course Download this course

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Lecture 4: Determining the Fundamental Frequency

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