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.

[ - ]
Comment by kevin_keeganOctober 1, 2021

Which was the suggested 5th method?

[ - ]
Comment by AllenDowneyOctober 1, 2021

Convolving a square wave with itself.

To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.

Registering will allow you to participate to the forums on ALL the related sites and give you access to all pdf downloads.

Sign up

I agree with the terms of use and privacy policy.

Try our occasional but popular newsletter. VERY easy to unsubscribe.
or Sign in