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In Search of The Fourth Wave

Allen DowneySeptember 25, 20212 comments

Last year I participated in the first DSP Related online conference, where I presented a short talk called "In Search of The Fourth Wave". It's based on a small mystery I encountered when I was working on Think DSP.  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 the talk, I suggest four ways to solve this mystery, and in the Q&A, one of the attendees suggests a fifth.

You can watch the talk here:


And you can run the Jupyter notebook on Colab by clicking here.




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Comment by kevin_keeganOctober 1, 2021

Which was the suggested 5th method?

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Comment by AllenDowneyOctober 1, 2021

Convolving a square wave with itself.

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