2020 >

## In Search of the Fourth Wave

Allen Downey - Watch Now - Duration: 28:40

When I was working on Think DSP, I encountered a small mystery.  As you might know:

1.  A sawtooth wave contains harmonics at integer multiples of the fundamental frequency, and their amplitudes drop off in proportion to 1/f.
2. A square wave contains only odd multiples of the fundamental, but they also drop off like 1/f.
3. A triangle wave also contains only odd multiples, but they drop off like 1/f².

This pattern suggests that there is a fourth simple waveform that contains all integer multiples (like a sawtooth) and drops off like 1/f² (like a triangle wave).  Do you know what it is?

In this talk, I'll suggest three ways we can solve this mystery and show how to compute each of them using Python, NumPy, and SciPy.  This talk is appropriate for beginners in both DSP and Python.

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TheBOOM
Score: 2 | 2 years ago | 1 reply

How about all even harmonics? :)

AllenDowneySpeaker
Score: 1 | 2 years ago | 1 reply

Interesting question. I guess that would be the fundamental at 1000 Hz plus harmonics at 2000, 4000, 6000, etc. Not sure what that would look/sound like. I guess that'll be next year's talk.

greghunter
Score: 0 | 2 years ago | 1 reply

1 2 4 6 is not the even series, because of the 1
Even would be 2 4 6 which is the same integer ratio as a sawtooth.

AllenDowneySpeaker
Score: 0 | 2 years ago | 1 reply

If you start at 2, then 2 is the fundamental, and 4, 6, 8 are the even and odd harmonics.

greghunter
Score: 0 | 2 years ago | no reply

Yes true, unless you said that DC is always the even series fundamental...?
Perhaps you could somehow harness the DC Fourier analysis partial and freq shift it up to hear the even wave or something ? ?
Thanks for the great presentation btw, very interesting.

Puru
Score: 0 | 2 years ago | no reply

Nice explanation covering most of the DSP topics

Leandro
Score: 0 | 2 years ago | 1 reply

Hello,
Is it possible to have the slides of this presentation? Thanks.

AllenDowneySpeaker
Score: 0 | 2 years ago | no reply

The Jupyter notebook is the slides. I used RISE to display the notebook in slide format. You can run the notebook here: https://tinyurl.com/mysterywave

robertdown
Score: 0 | 2 years ago | no reply

this is awesome one

MrFox
Score: 0 | 2 years ago | 1 reply

Very cool talk! I can't help but point out though, the complex exponential at 23:30 is missing the imaginary constant i or j!

AllenDowneySpeaker
Score: 0 | 2 years ago | no reply

Yes, I noticed that during the playback of the talk. It's fixed now in the slides. Thanks for letting me know!

doc_cls
Score: 2 | 2 years ago | no reply

The sawtooth & square wave spectra fall off as 1/f since the wave has discontinuities in the series itself (in the continuous representation); the spectrum of the triangle wave drops as 1/f^2 since the discontinuities are in the first derivative.

CoryClark
Score: 2 | 2 years ago | no reply

Very enjoyable talk Allen, much appreciated, will check out your blog and your ThinkDSP book.

TheBOOM
Score: 1 | 2 years ago | 1 reply

Why did you pick the parabola-looking wave, instead of the sine-ish looking wave as the representative shape?

AllenDowneySpeaker
Score: 0 | 2 years ago | 1 reply

No special reason other than the shape was recognizable as a parabola, which led me to Method 3. And then the integration property tells us that the parabola is the integral of the sawtooth.

TheBOOM
Score: 0 | 2 years ago | no reply

personally, i'm "partial" to the leaning-sine :)

TheBOOM
Score: 1 | 2 years ago | 1 reply

13:07 sounds distorted

AllenDowneySpeaker
Score: 0 | 2 years ago | no reply

I think we picked up a delayed copy of the wave from my speakers to my mike, so there's some interference.