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

From Class Wiki
Jump to: navigation, search
Line 1: Line 1:
Under Construction
 
  +
22 lines (currently)
  +
  +
1 reference
  +
  +
1 figure
  +
  +
123 points
   
 
Please refrain from error checking until the 2 reviewers do their job. Thank you!
 
Please refrain from error checking until the 2 reviewers do their job. Thank you!
Line 5: Line 11:
 
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!
   
==Period==
 
  +
==Period, Frequency, and Angular Frequency==
 
[[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)]]
  +
 
===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.
 
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 <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>
  +
  +
Where T is the period
   
 
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>
  +
  +
===Frequency and Angular Frequency===
  +
  +
The '''Frequency''' is the number of periods per second and is defined mathematically as
  +
  +
<math>f = \frac{1}{T}</math>
  +
  +
The standard unit of measurement for frequency is Hz (Hertz). 1 Hz = 1 cycle/second
  +
  +
The '''Angular Frequency''' is defined as
  +
  +
<math>\omega = 2\pi f = \frac{2\pi}{T}</math>
  +
  +
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 <math>T_0</math> for which the periodic equation <math>f(t+T) = f(t)</math> holds true.
  +
  +
The '''fundamental frequency''' is defined as <math>f_0 = \frac{1}{T_0}</math>.
  +
  +
The '''fundamental angular frequency''' is defined as <math>\omega_0 = 2\pi f_0 = \frac{2\pi}{T_0}</math>.
  +
   
 
==References==
 
==References==
 
{{reflist}}
 
{{reflist}}
  +
  +
==Author==
  +
Andrew Roth
  +
  +
==Reviewers==
  +
  +
==Readers==

Revision as of 21:57, 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!

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}.


References

Template:Reflist

Author

Andrew Roth

Reviewers

Readers