Fourier

17 Articles

Treating Functions As Vectors In Hilbert Space

December 25, 2025 by 29 Comments

Perhaps the most beautiful aspect of mathematics is that it applies to literally everything, even things that do not exist in this Universe. In addition to this there are a number of alternative ways to represent reality, with Fourier space and its related transforms being one of the most well-known examples. An alternative to Euclidian vector space is called Hilbert space, as a real or complex inner product space, which is used in e.g. mathematical proofs. In relation to this, [Eli Bendersky] came up with the idea of treating programming language functions as vectors of a sort, so that linear algebra methods can be applied to them.

Of course, to get really nitpicky, by the time you take a function with its arguments and produce an output, it is no longer a vector, but a scalar of some description. Using real numbers as indices also somewhat defeats the whole point and claim of working in a vector space, never mind Hilbert space.

As with anything that touches upon mathematics there are sure to be many highly divisive views, so we’ll leave it at this and allow our esteemed readers to flex their intellectual muscles on this topic. Do you think that the claims made hold water? Does applying linear algebra to every day functions make sense in this manner, perhaps even hold some kind of benefit?

Tetris Goes Full Circle

February 7, 2024 by 10 Comments

As a game concept, Tetris gave humanity nearly four solid decades of engagement, but with the possibility for only seven possible puzzle pieces it might seem a little bit limiting. Especially now that someone has finally beaten the game, it could be argued that as a society it might be time to look for something new. Sinusoidal Tetris flips these limits on their head with a theoretically infinite set of puzzle pieces for an unmistakable challenge.

Like Tetris, players control a game piece as it slowly falls down the screen. Instead of blocks, however, the game piece is a sinusoid that stretches the entire width of the screen. Players control the phase angle, amplitude, and angular frequency in order to get it to cancel out the randomly-generated wave in the middle of the screen. When the two waves overlap, a quick bit of math is done to add the two waves together. If your Fourier transformation skills aren’t up to the task, the sinusoid will eventually escape the playing field resulting in a game over. The goal then is to continually overlap sinusoids to play indefinitely, much like the original game.

While we’re giving Tetris a bit of a hard time, we appreciate the simplicity of a game that’s managed to have a cultural impact long after the gaming systems it was originally programmed for have become obsolete, and this new version is similar in that regard as well. The game can be quite addictive with a lot to take in at any given moment. If you’re more interested in the programming for these types of games than the gameplay, though, take a look at this deep-dive into Tetris for the NES.

DSP Spreadsheet: The Goertzel Algorithm Is Fourier’s Simpler Cousin

November 13, 2020 by 30 Comments

You probably have at least a nodding familiarity with the Fourier transform, a mathematical process for transforming a time-domain signal into a frequency domain signal. In particular, for computers, we don’t really have a nice equation so we use the discrete version of the transform which takes a series of measurements at regular intervals. If you need to understand the entire frequency spectrum of a signal or you want to filter portions of the signal, this is definitely the tool for the job. However, sometimes it is more than you need.

For example, consider tuning a guitar string. You only need to know if one frequency is present or if it isn’t. If you are decoding TouchTones, you only need to know if two of eight frequencies are present. You don’t care about anything else.

A Fourier transform can do either of those jobs. But if you go that route you are going to do a lot of math to compute things you don’t care about just so you can pick out the one or two pieces you do care about. That’s the idea behind the Goertzel. It is essentially a fast Fourier transform algorithm stripped down to compute just one frequency band of interest.  The math is much easier and you can usually implement it faster and smaller than a full transform, even on small CPUs.

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Fourier Explained: [3Blue1Brown] Style!

July 13, 2019 by 12 Comments

If you ask most people to explain the Fourier series they will tell you how you can decompose any particular wave into a sum of sine waves. We’ve used that explanation before ourselves, and it is not incorrect. In fact, it is how Fourier first worked out his famous series. However, it is only part of the story and master video maker [3Blue1Brown] explains the story in his usual entertaining and informative way. You can see the video below.

Paradoxically, [3Blue1Brown] asserts that it is easier to understand the series by thinking of functions with complex number outputs producing rotating vectors in a two-dimensional space. If you watch the video, you’ll see it is an easier way to work it out and it also lets you draw very cool pictures.

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Interactive Demo Shows The Power Of Fourier Transforms

April 10, 2019 by 14 Comments

When it comes to mathematics, the average person can probably get through most of life well enough with just basic algebra. Some simple statistical concepts would be helpful, and a little calculus couldn’t hurt. But that leaves out a lot of interesting mathematical concepts that really do have applications in everyday life and are just plain fascinating in their own right.

Chief among these concepts is the Fourier transform, which is the key to understanding everything from how JPEGs work to how we can stream audio and video over the Internet. To help get your mind around the concept, [Jez Swanson] has this interactive Fourier transform visualizer that really drives home the important points. This is high-level stuff; it just covers the basic concepts of a Fourier transform, how they work, and what they’re good for in everyday life. There are no equations, just engaging animations that show how any function can be decomposed into a set of sine waves. One shows the approximation of a square wave with a slider to control to vary the number of component sine waves; a button lets you hear the resulting sound getting harsher as it approaches a true square wave. There’s also a great bit on epicycles and SVGs, and one of the best introductions to encoding images as JPEGs that we’ve seen. The best part: all the code behind the demos is available on GitHub.

In terms of making Fourier transform concepts accessible, we’d put [Jez]’s work right up there with such devices as the original Michelson harmonic analyzer, or even its more recent plywood reproduction. Plus the interactive demos were a lot of fun to play with.

[via the Adafruit blog]

Explaining Fourier Again

December 21, 2018 by 28 Comments

One of the nice things about living in the Internet age is that creating amazing simulations and animations is relatively simple today. [SmarterEveryDay] recently did a video that shows this off, discussing a blog post (which was in Turkish) to show how sine waves can add together to create arbitrary waveforms. You can see the English video, below.

We’ve seen similar things before, but if you haven’t you can really see how a point on a moving circle describes a sine wave. Through adding those waves, anything can then be done.

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All The Stuff You Wished You Knew About Fourier Transforms But Were Afraid To Ask

February 10, 2018 by 51 Comments

The Fourier transform underpins so much of our technological lives, in most cases probably without our realising it. The ability to mathematically split a waveform into its frequency components and vice versa underpins much of the field of digital signal processing, and DSP has become an essential part of many electronic devices we take for granted.

But while most of us will know what a Fourier transform is, fewer of us will know anything of how one works. They are a function called from a library rather than performed in themselves. Even when they are taught in schools or university courses they remain something that not all students “get”, and woe betide you if (as your scribe did) you have a sub-par maths lecturer.

The video below the break then is very much worth a look if Fourier transforms are a bit of a mystery to you. In it [Grant Sanderson] explains them through a series of simple graphical examples in a style that perhaps may chalk-and-talk mathematics teachers should emulate. You may still only use Foruier transforms through a library, but after watching this video perhaps some of their mysteries will be revealed.

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