Advanced Engineering Mathematics, Lecture 3.7: Fourier transforms

Advanced Engineering Mathematics, Lecture 3.7: Fourier transforms.

The complex Fourier series of a periodic function defined on [-L,L] expresses it as an infinite sum of exponentials (harmonics), corresponding to a discrete set of frequencies. Given a function or signal f(x) that vanishes outside of a finite interval [-L,L], we can extend it to be periodic and compute its Fourier series. The original function is clearly the limit as L goes to infinity, and taking the limit of the Fourier series essentially turns a Riemann sum into an integral, which is called the Fourier transform of f(x). It can be thought of as a way to express it as a function of frequency instead of time or position. After a brief introduction, we compute the Fourier transform of a rectangular pulse, which is a function called sinc(x). Unfortunately, there are four different definitions of the Fourier transform which are commonly used, and we compare and contrast them. We conclude by revealing the connection between the Fourier and Laplace transforms, and we summarize a number of basic properties that they both share.

[dennydaydreamer]

I'm enjoying this great series of lectures. Certainly wish I had access to such wonderful resources back when I was an undergrad.

One minor note: I think starting from 6:51, when the 2πL-period extension of f(x) was first written down, an extra factor of π was included in the numerator of the index of the exponential. I think the way it was written down in the video corresponds to a 2L-period function.

[thomasjefferson]

Bruh, im super glad ive had analysis. I saw the Riemann sum miles away!