# Difference between revisions of "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)

### 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>Textbook, 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 $\infty$

### 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 $\omega_0 = 2\pi f_0 = \frac{2\pi}{T_0}$.

Andrew Roth

Brandon Vazquez