# Solving a Problem With Fourier Transforms: Heaviside Unit Step Function

• Telemachus
In summary, jbunni was having trouble with a Fourier transform exercise, and he asked for help. He defined the odd extension for the function and found its Fourier integral representation. He then calculated the value for \int_0^{\infty}\frac{1-cos (\omega l)}{\omega}\sin (\omega l) d\omega.
Telemachus
Hi there. I'm starting with the Fourier transforms, and I'm having some trouble with my first exercise on this topic.
The problem says: Given $$f(x)=H(x)-H(x-l)$$ (H(x) is the Heaviside unit step function).
a) Consider the odd extension for f and find its Fourier integral representation.
b) Using the previous incise calculate the value for $$\int_0^{\infty}\frac{1-cos (\omega l)}{\omega}\sin (\omega l) d\omega$$

Well, for a) I think I should get the Fourier transfor for the sign function, I don't know if this is right, but anyway I've defined the function like this:

Then the Fourier integral representation:
$$g(x)=\displaystyle\frac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty}g(x)e^{-i \omega x}dx e^{i \omega x}d\omega$$

But g(x) is odd, then:

And the thing is that the integral for the cosine diverges as I see it, but I'm probably doing something wrong.

PD, I don't know why latex isn't working in some cases, so I've attached some images.

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Can you tell us how your g(x) is related to the f(x) in the problem statement?

Also, is there any assumption regarding whether $l$ is positive, negative, or zero?

If $l$ is positive, then f(x) should be 1 for $0 < x < l$ and 0 elsewhere. (Change $<$ to $\leq$ as appropriate, depending on how you defined the Heaviside function.)

P.S. Your Latex looks fine, but be aware that it doesn't seem to render correctly in your own browser after you first post it. If you refresh the page, it should fix itself.

Hi jbunni. Thanks for posting.

Can you tell us how your g(x) is related to the f(x) in the problem statement?
Yes, g(x) is the odd extension for f(x). Here you have a graph of both, I take l=2, but l is arbitrary (I think that when I find the Fourier Integral I must be doing something like l->infinity, so I think I should get the sign function):

You're right about that it must be 1 at x=0. But it doesn't make any difference I think.

About latex, I've realized that sometimes I have to refresh to see the "pictures" instead of the code, but today it just didn't work, and for some commands (the brackets for example, to define a trunked function) it just doesn't work for me, I've tried to refresh, but it didn't work, so then I've uploaded the pictures.

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• graph.PNG
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The odd function can be also defined as $$g(x)=H(x)-H(x-l)-H(-x)+H(-x-l)$$

## 1. What is a Fourier transform?

A Fourier transform is a mathematical tool used to break down a function into its frequency components. It can be thought of as a way to represent a function as a sum of sine and cosine waves of different frequencies.

## 2. How can Fourier transforms be used to solve problems?

Fourier transforms are commonly used in a variety of fields such as signal processing, image processing, and engineering to analyze and solve problems involving functions. They can help identify patterns, extract specific components of a signal, and transform a function into a more manageable form for further analysis.

## 3. What is the Heaviside unit step function?

The Heaviside unit step function, also known as the unit step function or the Heaviside function, is a mathematical function that is used to represent a sudden change or "jump" in a function. It is defined as 0 for negative values and 1 for positive values.

## 4. How can the Heaviside unit step function be solved using Fourier transforms?

The Heaviside unit step function can be solved using Fourier transforms by taking the inverse Fourier transform of the function's Fourier transform. This will result in a representation of the function as a sum of sine and cosine waves, making it easier to analyze and solve problems involving the function.

## 5. What are some real-world applications of solving problems with Fourier transforms and the Heaviside unit step function?

Fourier transforms and the Heaviside unit step function have many practical applications in various fields. They are used in signal processing to remove noise from signals, in image processing to enhance images, in control systems to analyze and adjust signals, and in solving differential equations in physics and engineering. They are also commonly used in modeling and analyzing communication systems, electrical circuits, and mechanical systems.

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