What is the Fourier transform?

What is the Fourier transform?

What is the Fourier transform and why is everyone talking about it? In this video I present the Fourier transform and explain its relation with the Laplace transform and mention some of its applications. Moreover, I present some intuition behind the Fourier transform, which hopefully makes it easier to understand. Below you can find a link to a useful demo, which I also mention in this video. This is a must-see for anyone interested in partial differential equations or electrical engineering or signal processing.

[ianfowler]

As a high school math teacher, these kinds of "intuitive" explanations/analogies, while not rigorous, were the bread and butter of my existence. Sort of like looking "under the hood" to see what's really going on. I guess that's why I love them so much. Thanks for this. It helped me.

[alipourzand]

So glad you mentioned the use of FT in signal processing!
I remember that our professor explained FT in one lecture since we haven't seen it yet in the math class ☺

[ruud]

Thanks Dr Peyam. You're a really good educator.

[srikanthtupurani]

Heisenberg uncertainty principle, cramer rao inequality are connected to fourier transform. We see the same ideas. This is a very powerful subject in math. All the complicated combinatorics can be converted in to a problems in fourier analysis.

[celestindupilon]

Hello Dr. Peyam! Thanks for the great video. But for the record player example, I fear there aren't that many people in the age of streaming who know what device you are talking about. Cheers, Célestin

[wasimakram]

I want a video on Noether's theorem

[johnanderson]

Why is cos(x) being used for e^(ix)? Shouldn’t it be cos(x) + isin(x)?

Perhaps this is good enough for intuitional purposes (to illustrate the oscillating behavior).

[afseraph]

I'd say that in signal processing and control theory, the Laplace transform is often more powerful than the Fourier transform. In many applications you can treat Fourier as a special case of Laplace, by limiting the whole complex frequency domain to the imaginary axis.

[BerndSchnabl]

where were you when I had to study this ? .... oh wait ... not yet born 😂

[vxbx]

Excelent, Thank you for your informative video. I do appreciate it
Please, I am currently working on the equation:

exp(2πix) + exp(2πi * n / x) - 2 = 0

where a, b, and n are positive integers, and I am interested in determining how many solutions this equation has within the interval [a, b].

Additionally, I would like to know if it is possible to express the number of solutions as a function N(a, b, n), which would return the number of solutions for the given values of a, b, and n.

Could you please suggest a method or approach that would help in determining the number of solutions for this equation in the specified interval?

Thank you for your time and assistance. I look forward to your insights.

Best regards,

[MridulGupta]

I have learned about the fourier coefficients \hat{f}(k) as being \int_{-π}^π f(x)e^{-ikx}dx. I never learned this form. Can you point out the differences and why this form should work?

[ianthehunter]

Is this some sort of deja vu? I thought we already calculated the FT of the bell curve. Coincidentally, we're just taking this in school.

[indovash]

Your video's thumbnail just screams an innuendo.