Chapter 22--Fourier Series: Fundamental Period, Frequency, and Angular Frequency

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Period, Frequency, and Angular Frequency

Picture of a Sine Wave where f(x)=sin(x)<ref>http://en.wikipedia.org/wiki/File:Sine.svg</ref>

Period

Long long ago, in a high school class called trigonometry, we leaned about periodic functions. A periodic function is a function that repeats itself over and over for infinity. The period of the function is the distance of one iteration that is infinitely repeating.

A signal $f(t)$ is periodic if, for some $T>0$ and all t, $f(t+T)=f(t)$<ref>DeCarlo/Lin, Linear Circuit Analysis--Time Domain, Phasor, and Laplace Transform Approaches, Second Edition. Figure 22.1</ref>

Where T is the period

The picture to the right shows the plot of the standard sine function whose period is $2\pi$. What the plot does not show is that the line keeps extending and repeating the bumps and valleys over the whole x axis, or $(-\infty ,\infty )$. But wait! Can't the period also be $4\pi$ or $6\pi$? In fact it can. Because the graph of sin(x) repeats itself every $2\pi$ units, the period of the function is actually $2\pi n$ where n is any whole number from zero to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}

Frequency and Angular Frequency

The Frequency is the number of periods per second and is defined mathematically as

$f={\frac {1}{T}}$

The standard unit of measurement for frequency is Hz (Hertz). 1 Hz = 1 cycle/second

The Angular Frequency is defined as

$\omega =2\pi f={\frac {2\pi }{T}}$

The standard unit of measurement for angular frequency is in radians/second.

Fundamental Period, Frequency, and Angular Frequency

The fundamental period is the smallest positive real number $T_{0}$ for which the periodic equation $f(t+T)=f(t)$ holds true.

The fundamental frequency is defined as $f_{0}={\frac {1}{T_{0}}}$.

The fundamental angular frequency is defined as $\omega _{0}=2\pi f_{0}={\frac {2\pi }{T_{0}}}$.

References

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Author

Andrew Roth

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