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

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!

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

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

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

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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Reverse Engineering The Nintendo Wavebird

Readers who were firmly on Team Nintendo in the early 2000’s or so can tell you that there was no accessory cooler for the Nintendo GameCube than the WaveBird. Previous attempts at wireless game controllers had generally either been sketchy third-party accessories or based around IR, and in both cases the end result was that the thing barely worked. The WaveBird on the other hand was not only an official product by Nintendo, but used 2.4 GHz to communicate with the system. Some concessions had to be made with the WaveBird; it lacked rumble, was a bit heavier than the stock controllers, and required a receiver “dongle”, but on the whole the WaveBird represented the shape of things to come for game controllers.

Finding the center frequency for the WaveBird

Given the immense popularity of the WaveBird, [Sam Edwards] was somewhat surprised to find very little information on how the controller actually worked. Looking for a project he could use his HackRF on, [Sam] decided to see if he could figure out how his beloved WaveBird communicated with the GameCube. This moment of curiosity on his part spawned an awesome 8 part series of guides that show the step by step process he used to unlock the wireless protocol of this venerable controller.

Even if you’ve never seen a GameCube or its somewhat pudgy wireless controller, you’re going to want to read though the incredible amount of information [Sam] has compiled in his GitHub repository for this project.

Starting with defining what a signal is to begin with, [Sam] walks the reader though Fourier transforms, the different types of modulations, decoding packets, and making sense of error correction. In the end, [Sam] presents a final summation of the wireless protocol, as well as a simple Python tool that let’s the HackRF impersonate a WaveBird and send button presses and stick inputs to an unmodified GameCube.

This amount of work is usually reserved for those looking to create their own controllers from the ground up, so we appreciate the effort [Sam] has gone through to come up with something that can be used on stock hardware. His research could have very interesting applications in the world of “tool-assisted speedruns” or even automating mindless stat-grinding.

Control System Fundamentals By Video

If you’ve had the classic engineering education, you probably have a hazy recollection of someone talking about control theory. If you haven’t, you’ve probably at least heard of PID controllers and open loop vs closed loop control. If you don’t know about control theory or even if you just want a refresher, [Brian Douglas] has an excellent set of nearly 50 video lectures that will give you a great introduction to the topic. You can watch the first lecture, below.

You might think that control systems are only useful in electronics when you are trying to control a process like a chemical plant or a temperature. However, control theory shows up in a surprising number of places from filters to oscillators, to the automatic gain control in a receiver. You’ll find the background behind many familiar results inside control theory. Sort of like when you take calculus and you discover how they came up with all the formulas you memorized in geometry.

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