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# Is this FFT Really Necessary?

December 13, 2009

I wasted a lot of time reading Mad magazine as a kid. I remember reading one issue that explained why Tom Jones was more successful than Engelbert Humperdinck. According to Mad, a few people would notice Jones leaving a concert, ask him for an autograph, and he could provide them and dash off before he was seriously mobbed. Humperdinck, on the other hand, would take so long to sign his name that he'd be mobbed and injured. Like I said, I wasted a lot of time.

I wonder sometimes if there aren't similar forces at work in the software world where cool names and acronyms displace ideas where the names are not quite as melodious. For example, if I asked you to decode some DTMF (Dual Tone Multi Frequency – touch tones from a phone) it is a good bet you'd want to use a Fourier transform.

Why not? A Fourier transform takes a time domain signal and converts it into a frequency domain signal. So a transform will show us how much energy is at each frequency and we can use that to do lots of things from speech analysis to tomography, to decoding DTMF. And the most common way to do that even has a cool name: The Fast Fourier Transform (FFT) and lots of libraries.

But would you think of using the Goertzel algorithm? It isn't new. It has been around since 1958 (according to Wikipedia, anyway). The Goertzel doesn't require nearly as much math as an FFT and you can precompute a lot of it based on your sampling rate and frequencies of interest. The Goertzel is basically a filter for the frequency (or frequencies) of interest. For DTMF you only care about 8 tones (there is an extra column of keys you don't normally see on modern phones). Each row and column produce a distinct tone and by finding which two of the 8 tones are “on” you can figure out which key is active.

Maybe the poor Goertzel just needs a better brand name.

Want to read more? The Wikipedia article is a good start and has some references. You might also enjoy a page from Microstar Laboratories. If you want the hairy math, try MIT.

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