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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Making A Spectrum Analyzer The Wrong Way On An ATtiny85

Everyone’s a critic, but it’s hard to argue with success. And that’s exactly what [agp.cooper] has with his ATtiny85-based spectrum analyzer devices.

The “normal” way to build a spectrum analyzer is to collect a bunch of samples and run a Fast Fourier Transform (FFT) on them all in one shot. As the name implies, the FFT is fast, and the result is the frequency components of the sampled data. [agp.cooper]’s “wrong” way to do it takes the Goertzel algorithm, which is used for detecting the intensity of a particular frequency, and scanning across the frequency range of interest. It’s a lot slower than a single FFT but, importantly for the ATtiny85 that he implements this on, it’s less demanding of the RAM.

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Detecting DTMF Tones From Scratch

If you’ve ever wondered about the best way to detect dial and DTMF tones from a phone line, [Debraj] is your man.

[Debraj] built a DTMF detector using the Goertzel algorithm. Normally, when we think about detecting tones, we pull FFT out of our bag of tricks. The Goertzel algorithm isn’t as computationally complex as FFT and can be implemented on even the smallest microcontrollers.

For the build, the first thing to solder is a nice audio transformer and some protection diodes. The ring tone from a phone line goes from +35 V to -35 V – a bit more than a microcontroller could handle. A PIC18F4520 dev board was used as the brain of the system with all the code is available on [Debraj]’s site.

Although implementations of the Goertzel algorithm are a little uncommon, [Debraj] has seen a few interesting projects using this technique. [Debraj]’s build could easily be modified into a guitar tuner with a few changes in the code, for example.

This project was built as the command and control for a home automation system and from the video after the break, we can’t wait for [Debraj] to get annoyed at the phrase, “To turn on the kitchen lights, please press 1…”

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