# Difference between revisions of "Chapter 22--Fourier Series: Fundamental Period, Frequency, and Angular Frequency"

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Please refrain from error checking until the 2 reviewers do their job. Thank you! |
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I can't figure out how to reference something. If someone knows how please let me know. Thanks! |
I can't figure out how to reference something. If someone knows how please let me know. Thanks! |
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+ | ==Period, Frequency, and Angular Frequency== |
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[[Image:Sinewave.png|500px|thumb|right|Picture of a Sine Wave where f(x)=sin(x)]] |
[[Image:Sinewave.png|500px|thumb|right|Picture of a Sine Wave where f(x)=sin(x)]] |
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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. |
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. |
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− | A signal <math>f(t)</math> is periodic if, for some <math>T > 0</math> and all ''t'', |
+ | <center>A signal <math>f(t)</math> is periodic if, for some <math>T > 0</math> and all ''t'',</center> |

− | <math>f(t+T) = f(t)</math><ref>Textbook, 22.1</ref> |
+ | <center><math>f(t+T) = f(t)</math><ref>Textbook, 22.1</ref></center> |

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+ | Where T is the period |
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The picture to the right shows the plot of the standard sine function whose period is <math>2\pi</math>. What the plot does not show is that the line keeps extending and repeating the bumps and valleys over the whole x axis, or <math>(-\infty,\infty)</math>. But wait! Can't the period also be <math>4\pi</math> or <math>6\pi</math>? In fact it can. Because the graph of sin(x) repeats itself every <math>2\pi</math> units, the period of the function is actually <math>2\pi n</math> where n is any whole number from zero to <math>\infty</math> |
The picture to the right shows the plot of the standard sine function whose period is <math>2\pi</math>. What the plot does not show is that the line keeps extending and repeating the bumps and valleys over the whole x axis, or <math>(-\infty,\infty)</math>. But wait! Can't the period also be <math>4\pi</math> or <math>6\pi</math>? In fact it can. Because the graph of sin(x) repeats itself every <math>2\pi</math> units, the period of the function is actually <math>2\pi n</math> where n is any whole number from zero to <math>\infty</math> |
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+ | ===Frequency and Angular Frequency=== |
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+ | The '''Frequency''' is the number of periods per second and is defined mathematically as |
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+ | <math>f = \frac{1}{T}</math> |
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+ | The standard unit of measurement for frequency is Hz (Hertz). 1 Hz = 1 cycle/second |
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+ | The '''Angular Frequency''' is defined as |
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+ | <math>\omega = 2\pi f = \frac{2\pi}{T}</math> |
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+ | The standard unit of measurement for angular frequency is in radians/second. |
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+ | ===Fundamental Period, Frequency, and Angular Frequency=== |
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+ | The '''fundamental period''' is the smallest positive real number <math>T_0</math> for which the periodic equation <math>f(t+T) = f(t)</math> holds true. |
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+ | The '''fundamental frequency''' is defined as <math>f_0 = \frac{1}{T_0}</math>. |
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+ | The '''fundamental angular frequency''' is defined as <math>\omega_0 = 2\pi f_0 = \frac{2\pi}{T_0}</math>. |
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==References== |
==References== |
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{{reflist}} |
{{reflist}} |
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+ | ==Author== |
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+ | Andrew Roth |
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+ | ==Reviewers== |
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+ | ==Readers== |

## Revision as of 22:57, 5 January 2010

22 lines (currently)

1 reference

1 figure

123 points

Please refrain from error checking until the 2 reviewers do their job. Thank you!

I can't figure out how to reference something. If someone knows how please let me know. Thanks!

## Contents

## Period, Frequency, and Angular Frequency

### 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.

*t*,

Where T is the period

The picture to the right shows the plot of the standard sine function whose period is . What the plot does not show is that the line keeps extending and repeating the bumps and valleys over the whole x axis, or . But wait! Can't the period also be or ? In fact it can. Because the graph of sin(x) repeats itself every units, the period of the function is actually where n is any whole number from zero to

### Frequency and Angular Frequency

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

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

The **Angular Frequency** is defined as

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 for which the periodic equation holds true.

The **fundamental frequency** is defined as .

The **fundamental angular frequency** is defined as .

## References

## Author

Andrew Roth