Fourier integral and Fourier Transform

In summary: There is no one "good" way to make an FFT. Some common approaches include using libraries or modules that come with a programming language, or using programming languages that have built-in support for FFTs.
  • #1
Jhenrique
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Which is the difference between the Fourier integral and Fourier transform? Or they are the same thing!?

Fourier integral:
image.png

 
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  • #3
The Fourier integral is one way to calculate the Fourier transform of a function:
$$\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$$
This definition only makes sense for certain kinds of functions. Since ##|f(x) e^{-i \omega x}| = |f(x)|##, the Fourier integral is defined if and only if ##|f|## is integrable. (In the case of Lebesgue integration, we say that ##f \in L^1##.)

But the Fourier transform can be defined for a larger class of functions, and even some objects that are not functions.

For example, we can define the Fourier transform of a function which is square-integrable but not integrable (i.e. ##f \in L^2 \setminus L^1##). To do this, we approximate ##f## by a sequence of functions ##f_n \in L^1 \cap L^2## and define ##\hat{f} = \lim \hat{f_n}##. Of course there are lot of details to check, such as the existence of such a sequence, and the fact that the limit does not depend on a particular choice for the sequence.

It is also possible to define the Fourier transform of certain types of distributions, which can be thought of as generalized functions. See here for example:

http://en.wikipedia.org/wiki/Fourier_transform#Tempered_distributions

We can also define a Fourier transform on other types of objects, such as groups (a special case of this is the discrete Fourier transform):

http://en.wikipedia.org/wiki/Fourier_transform_on_finite_groups

So, the Fourier transform is the more general concept, and the Fourier integral is how it is defined/computed in the case of integrable functions.
 
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  • #4
^I agree. The Fourier integral is a method of calculating the Fourier transform. In many cases it is not useful to distinguish between the two. Be aware that there are different Fourier transforms and using a slightly different one can cause confusion.
 
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  • #5
Is there a good way to make FFTs without premade modules in programming? Just an aside...
 

What is Fourier integral?

The Fourier integral is a mathematical tool that is used to represent a periodic function as a combination of sine and cosine waves. It is used in signal processing and other fields to analyze and manipulate signals.

What is the difference between Fourier integral and Fourier transform?

The Fourier transform is a special case of the Fourier integral, where the function being transformed is not necessarily periodic. The Fourier transform allows us to analyze non-periodic signals, while the Fourier integral is specifically for periodic signals.

What is the purpose of using Fourier transforms in signal processing?

In signal processing, Fourier transforms are used to convert a signal from its original domain (such as time or space) to a representation in the frequency domain. This allows us to analyze and manipulate signals by their frequency components.

How is the Fourier transform calculated?

The Fourier transform is calculated using an integral that involves complex numbers and the function being transformed. This integral is known as the Fourier integral and can be solved using mathematical techniques such as integration by parts or the use of tables.

What are some applications of Fourier transforms?

Fourier transforms have many applications in fields such as engineering, physics, and mathematics. Some common applications include signal processing, image processing, data compression, and solving differential equations. They are also used in everyday devices such as cell phones and Wi-Fi routers.

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