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Comment by SilverSlash | Hacker News Reader

It's amazing how almost every one of these articles about the Fourier Transform actually only talk about Fourier Series. Perhaps because most of the people who write these articles themselves lack an intuition for Fourier transforms (hint: there isn't much)?

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There's this article https://jackschaedler.github.io/circles-sines-signals/ which is accompanied by some really neat interactive visuals.

I was just going to post that. I don't think any other explanation or tutorial does a more complete-yet-understandable job of explaining how the DFT actually works. I feel like there are a million articles out there that do basically what this article does, which is to explain kinda-sorta what the Fourier Transform is (i.e. "make a wave using other waves!") but never go into any detail about how you select which constituent waves contribute to the overall wave etc., and so I was always left unsatisfied.

That's true, but it's kind of the same idea, so it's okay. The best way to understand this, for people who are familiar with linear algebra, is to think of the Fourier transformations as change-of-basis operations that convert signals from a time basis to a frequency basis. Depending for the bases for the time domain and the frequency domain, you get the Fourier series, Fourier transform, or Discrete Fourier transform[1].

I find the Fourier series is the hardest to understand intuitively (compared to FT and DFT), since the time basis (periodic functions of time) and the frequency basis (infinite sequence of Fourier coefficients) look very different.

[1] https://minireference.com/static/excerpts/noBSguide2LA_previ...

I never really intuited how the time domain and frequency domain related until I looked at the animation mentioned in the article[1] - I really wish I had seen that gif earlier since it shows the frequency and time domains as orthogonal (literally).

Edit: I've realized that our comment was on series, not transforms.

1. https://upload.wikimedia.org/wikipedia/commons/5/50/Fourier_...

What do you mean, not much intuition for Fourier transforms? Sure, sometimes you get weird results and need to work with fairly generalized versions of functions, and a bit of background in functional analysis is helpful, but a strong intuition about the Fourier transform is both extremely useful, and not that hard to develop.

For starters, I'd recommend the course notes to Stanford's EE261 (https://see.stanford.edu/materials/lsoftaee261/book-fall-07....) - well written, very funny, a nice level of rigour, etc...

Thank you for the link! I used Stanford handouts for every subject I could find, they are usually excellent, like a better and more rigorous Wikipedia article.

I'm still trying to understand the convolution of two functions. Every God damn text ist only about how useful the Fourier series are - that's it. (Math student)

Since you mentioned that you are a math student, I highly recommend Stein and Shakarchi's "Fourier Analysis: An Introduction".

It is a rigorous treatment that covers a lot of the essentials of Fourier theory, while also covers some very interesting "math applications" (e.g Dirichlet's theorem on primes in arithmetic progressions at the end), as well as "mixed applications" (e.g Radon transforms that are used in imaging but are also of interest from a math perspective).

Note that this book requires an introductory analysis class as a prereq; basically you need to be familiar with the standard Riemann integral theory and rigorous treatment of limits, continuity, and derivatives.

It's much easier to get an intuitive understanding of convolution directly in the time domain, without worrying about Fourier Transforms.

Convolution with a top-hat kernel is just a moving average. Each point of the output is the average of the input signal over a given radius about that point.

Convolution with other kernels is a weighted moving average. Each point of the output is the average of the input signal over a region with weights depending on the displacement from that point.

The Fourier Transform allows implementing convolution as multiplication in frequency space (because the Fourier transform turns translations into multiplications), which is sometimes formally useful, and sometimes more efficient to evaluate.

I similarly struggled to get intuition on the convolution operation, until the following simple idea clicked. Think of convolution as similar to elementary school long multiplication, but without carries. (Also, if you are working with vectors of of some length N, the multiplication "wraps" modular-arithmetic style around the end of the vector back to the beginning.)

I think it's because they're meant to explain Fourier transforms to people who aren't familiar with transforms, and therefore are much more comfortable with Fourier coefficients rather than an actual transform in the frequency domain.

What is wrong with that?

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