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===Euler's Formula=== | ===Euler's Formula=== | ||
To understand the formula for the Fourier transform, it is helpful to have knowledge of Euler’s formula:<br /> | To understand the formula for the Fourier transform, it is helpful to have knowledge of Euler’s formula:<br /> | ||
| + | <math> e^{it} = \cos t + i\sin t </math> | ||
| − | + | One way of representing a unit circle is parametrically with sines and cosines:<br /> | |
| − | + | <math>x = \cos t</math><br /> | |
| − | One way of representing a unit circle is parametrically with sines and cosines: | + | <math>y = \sin t</math><br /> |
| − | + | Euler’s formula allows for an elegant representation of a unit circle in the complex plane. Instead of using sines and cosines like in our parameterization, these can be substituted out with a simpler expression of <math>e^{it}</math>. Another nice thing about Euler’s formula is that since it represents a unit circle where t is the angle in radians, t also represents the arclength traveled around the circle. | |
[[Category:MA271Fall2020Walther]] | [[Category:MA271Fall2020Walther]] | ||
Revision as of 12:28, 5 December 2020
Contents
Fourier Transforms
Author: Luke Oxley
Table of Contents:
- Introduction
- Euler’s Formula
- Formula Visualization
- Example
- Inverse Transform
- Applications
- References and Further Reading
Introduction
The Fourier transform is a method used to break a function down into a representation of its frequencies. This method can be thought of as transforming from the time domain (a function with time as the input variable) to the frequency domain (a function with frequency as the input). This function in the frequency domain is a way of representing how much of each frequency is prevalent in the function. The Fourier transform allows for signals to be broken into their individual frequency components. The advantage of extracting the individual frequencies is that this allows for the individual manipulation of each frequency and for a unique representation of the function. Once you have extracted the component frequencies, many times it is possible to express the original function with sine and cosine waves at these frequencies. I like to picture this representation as a Taylor series, but instead of being based on derivatives, it is based on frequencies. This transform is useful in many fields, especially for analyzing electrical and sound signals. The purpose of this page is for you to gain an understanding of the basics of this transform, including a way to mentally visualize what this formula is doing and its applications.
One area of confusion is the difference between the Fourier series and the Fourier transform. The Fourier transform can be thought of as a limited case of the Fourier series. The series is mainly concerned with periodic functions while the transform is concerned with nonperiodic functions.
Euler's Formula
To understand the formula for the Fourier transform, it is helpful to have knowledge of Euler’s formula:
$ e^{it} = \cos t + i\sin t $
One way of representing a unit circle is parametrically with sines and cosines:
$ x = \cos t $
$ y = \sin t $
Euler’s formula allows for an elegant representation of a unit circle in the complex plane. Instead of using sines and cosines like in our parameterization, these can be substituted out with a simpler expression of $ e^{it} $. Another nice thing about Euler’s formula is that since it represents a unit circle where t is the angle in radians, t also represents the arclength traveled around the circle.
