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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Learning SDR And DSP Hack Chat

Join us on Wednesday, November 11th at noon Pacific for Learning SDR and DSP Hack Chat with Marc Lichtman!

“Revolution” is a term thrown about with a lot less care than it probably should be, especially in fields like electronics. It’s understandable, though — the changes to society that have resulted from the “Transistor Revolution” or the “PC Revolution” or more recently, the “AI Revolution” have been transformative, often for good and sometimes for ill. The common thread, though, is that once these revolutions came about, nothing was ever the same afterward.

Such is the case with software-defined radio (SDR) and digital signal processing (DSP). These two related fields may not seem as transformative as some of the other electronic revolutions, but when you think about it, they really have transformed the world of radio communications. SDR means that complex radio transmitters and receivers, no longer have to be implemented strictly in hardware as a collection of filters, mixers, detectors, and amplifiers; instead, they can be reduced to a series of algorithms running on a computer.

Teamed with DSP, SDR has resulted in massive shifts in the RF field, with powerful, high-bandwidth radio links being built into devices almost as an afterthought. But the concepts can be difficult to wrap one’s head around, at least when digging beyond the basics and really trying to learn how SDR and DSP work. Thankfully, Dr. Marc Lichtman, an Adjunct Professor at the University of Maryland, literally wrote the book on the subject. “PySDR: A Guide to SDR and DSP using Python” is a fantastic introduction to SDR and DSP that’s geared toward those looking to learn how to put SDR and DSP to work in practical systems. Dr. Lichtman will stop by the Hack Chat to talk about his textbook, to answer your questions on how best to learn about SDR and DSP, and to discuss what the next steps are once you conquer the basics.

join-hack-chatOur Hack Chats are live community events in the Hackaday.io Hack Chat group messaging. This week we’ll be sitting down on Wednesday, November 11 at 12:00 PM Pacific time. If time zones baffle you as much as us, we have a handy time zone converter.

Click that speech bubble to the right, and you’ll be taken directly to the Hack Chat group on Hackaday.io. You don’t have to wait until Wednesday; join whenever you want and you can see what the community is talking about.

[Banner image credit: Dsimic, CC BY-SA 4.0, via Wikimedia Commons]

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Intuition About Signals And Systems

Signals and systems theory is a tough topic. Terms like convolution and impulse response can be hard to understand on a visceral level and most books that talk about these things emphasize math over intuition. [Discretised] has a YouTube channel that already has several videos that promise to tackle these topics with “minimum maths, maximum intuition.” We particularly noticed the talks on convolution and impulse response.

We think that often math and intuition don’t always come together. It is one thing, for example, to know that E=I times R, and power is I times E, but it is another to realize that a half-watt transmitter delivers 5V into a 50Ω load and that one watt will take just over 7V into that same load.

The example used is computing how much smoke you can expect to create by setting off fireworks. We presume the math models are notional since we imagine a real model would be pretty complex and involve things like wind data. But it still makes a nice example.

If you don’t know anything about the topic, these might not be the right ones to try to learn the basics. But we do applaud people sharing their intuition on these complex subjects.

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DSP Spreadsheet: Talking To Yourself Using IQ

We’ve done quite a bit with Google Sheets and signal processing: we’ve generated signals, created filters, and computed quadrature signals. We can pull all that together into an educational model for two SDRs talking to each other, but it’s going to require two parts: modulation and demodulation. Guess what? We can do that with a spreadsheet.

The first step is to generate a reference clock for the carrier. You’ll need a cosine wave (I) and sine wave (Q). Of course, you also need the time base. That’s columns A-C in the spreadsheet and works like other signal generation we’ve seen.

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DSP Spreadsheet: IQ Diagrams

In previous installments of DSP Spreadsheet, we’ve looked at generating signals, mixing them, and filtering them. If you start trying to work with DSP, though, you’ll find a topic that always rears its head: IQ signals. It turns out, these aren’t as hard as they appear at first and, as usual, we’ll tackle them in a spreadsheet.

What does IQ stand for? The I stands for “in phase” and the Q stands for quadrature. By convention, the I signal is a cosine wave and the Q signal is a sine wave. Another way to say that is that the I and Q signals are 90 degrees out of phase. By manipulating the amplitude of I and Q, you can create complex modulation or, conversely, demodulate signals. We’ll see a spreadsheet that shows that completely next time.

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DSP Spreadsheet: FIR Filtering

There’s an old saying: Tell me and I forget, teach me and I may remember, involve me and I learn. I’m guilty of this in a big way — I was never much on classroom learning. But if I build something or write some code, I’m more likely to understand how it works and why.

Circuit simulation and software workbooks like Matlab and Jupyter are great for being able to build things without a lot of overhead. But these all have some learning curve and often use clever tricks, abstractions, or library calls to obscure what’s really happening. Sometimes it is easier to build something in a spreadsheet. In fact, I often do little circuit design spreadsheets or even digital design because it forces me to create a mathematical model which, in turn, helps me understand what’s really going on.

In this article I’m going to use Google Sheets — although you could do the same tricks in just about any spreadsheet — to generate some data and apply a finite impulse response (FIR) filter to it. Of course, if you had a spreadsheet of data from an instrument, this same technique would work, too.

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Tutorial On Signal Processing In Linux With Octave

If you’ve taken any digital signal processing classes at a college or university, you’ve probably been exposed to MATLAB. However, if you want to do your own work, you might think about Linux and one of the many scientific computing applications available for it.

[David Duarte] recently published a three-part tutorial on using Octave to do scientific audio processing. The first part covers basic reading, writing, and playing of audio files. Part two covers synthesis of signals, plotting, and some basic transformations.  Modulation is the topic of the third part. If you prefer your tutorials on video, you can check out the video below.

We’ve talked about MATLAB before in the context of message cracking. Then again, some of the best signal processing is done by humans. If you don’t like Octave, you might try Scilab, another Linux package that is similar. There’s also Freemat, Sage, and Spyder. Of course, you can also run MATLAB under Linux.