An interactive guide to the Fourier transform (2012)

I don’t like the casting of the frequency domain view as the ‘recipe’ and the time domain view as the ‘product’. The point of Fourier is that you can switch between these perspectives losslessly - they contain equivalent information. The ‘smoothie’ metaphor of ‘unmixing’ the smoothie to get the ingredients, and then blending it to get the smoothie back conjures the impression that Fourier transformation is some sort of entropy reversing process, which is misleading.

A time domain function has a frequency domain interpretation - that doesn’t mean the frequency domain function is what ‘made’ the time domain function. It’s a chicken and an egg - both make each other.

Just like a function has a derivative, and you can recover the original function (modulo it’s y-offset) by taking its antiderivative - that doesn’t make the derivative the ‘recipe’ for the function.

I don't think there's a satisfying concise answer to this question. You're asking to boil down multiple semesters of engineering and mathematics courses to explain something that isn't simple.

The "why" we need it is because it's a convenient mathematical tool to simplify complex problems dealing with sequences/series. It decomposes sequences of numbers (*) into another sequence of numbers that represents a summation of repeating patterns in the original sequence.

The "where" we use it is anywhere that knowing about which patterns show up in sequence of numbers might be more convenient than looking at the original sequence.

There are other nice properties, too. There's a mathematical operation called convolution that's very useful in domains operating with sequences of numbers (machine learning, control systems, audio and image processing, etc). It happens that convolution in the original signal's domain is equivalent to multiplication in the fourier domain. Why that's useful is that convolution is an O(N^2) algorithm but we can compute the Fourier transform in O(nlogn) complexity (**) and multiplication is constant. We can even do things like deconvolution (taking an output signal that we know was convolved with something else) and extract one of the inputs.

Another nice property of the Fourier transform is that most sequences of numbers do not have lots of energy (meaning magnitude) in "high frequency" (meaning fastly repeating patterns in the original domain). The FT of a sequence of numbers will naturally compact most of the information of the sequence into few places. We can exploit this for lossy compression (***).

Ultimately, it's a way of taking information that is hard to grok and transforming it to a domain that's more meaningful and operations/analysis are easier. Both for humans and machines.

* And it works on continuous functions too, but let's not get into that

** Technically this is a variant called "circular" convolution

*** The FT is not the most convenient tool for this job, so very closely related transforms like the DCT can be used, which have the same computational advantages of the FT.

Approximate square.

It is a least squares fit of sine waves. (A^T)A = I because sines are orthogonal.

Use Euler's eqn to convert e^jw into sin + cos and just work through the algebra.

Some of the more well-known examples are:

Low-bitrate audio compression (MP3) use Fourier transforms.

Image and video compression use Fourier transforms.

Less-well-known:

AC electricity has to run at (almost) exactly 60hz / 50hz. The power plants use Fourier transforms to measure the exact frequency. (And then use the measurement to adjust accordingly.)

Some telecommunications techniques may use Fourier transforms: IE, the dial-up modems used in the 1990s used Fourier transforms to interpret the analog signal and determine what the bits were.

This article explains using a Fourier transform to remove the "dots" from an image that was printed in a book: https://matzjb.se/2015/08/08/smoothing-a-halftone-photo-usin...