Fourier Series and Transform: Difference between revisions
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(Created page with "A Fourier series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain is much simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as <math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{nx}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{nx}{L}})}</math> ==Intuition==...") |
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A Fourier series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain is much simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as <math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{nx}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{nx}{L}})}</math> | A Fourier series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain is much simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as: <br><math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{nx}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{nx}{L}})}</math> | ||
==Intuition== | ==Intuition== | ||
Many physical systems can be modeled by square waves. Consider systems with on-off behavior, similar to an on-and-off switch. A square wave looks like this: | Many physical systems can be modeled by square waves. Consider systems with on-off behavior, similar to an on-and-off switch. A square wave looks like this:<br> | ||
[[File:squarewave.png|500px|]] | |||
Revision as of 23:03, 5 December 2022
A Fourier series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain is much simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as:
[math]\displaystyle{ f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{nx}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{nx}{L}})} }[/math]
Intuition
Many physical systems can be modeled by square waves. Consider systems with on-off behavior, similar to an on-and-off switch. A square wave looks like this:
