### Allen Downey (@AllenDowney)

## In Search of The Fourth Wave

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:

A sawtooth wave contains harmonics at integer multiples of the fundamental frequency, and their amplitudes drop off in proportion to 1/f. A square wave contains only odd multiples of the fundamental, but they also drop off...## Autocorrelation and the case of the missing fundamental

[UPDATED January 25, 2016: One of the examples was broken, also the IPython notebook links now point to nbviewer, where you can hear the examples.]

For sounds with simple harmonic structure, the pitch we perceive is usually the fundamental frequency, even if it is not dominant. For example, here's the spectrum of a half-second recording of a saxophone.

The first three peaks are at 464, 928, and 1392 Hz. The pitch we perceive is the fundamental, 464 Hz, which is close to...

## Generating pink noise

In one of his most famous columns for Scientific American, Martin Gardner wrote about pink noise and its relation to fractal music. The article was based on a 1978 paper by Voss and Clarke, which presents, among other things, a simple algorithm for generating pink noise, also known as 1/f noise.

The fundamental idea of the algorithm is to add up several sequences of uniform random numbers that get updated at different rates. The first source gets updated at...

## Amplitude modulation and the sampling theorem

I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.

Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key figures.

Convolution...

## Differentiating and integrating discrete signals

I am back at work on Think DSP, adding a new chapter on differentiation and integration. In the previous chapter (which you can read here) I present Gaussian smoothing, show how smoothing in the time domain corresponds to a low-pass filter in the frequency domain, and present the Convolution Theorem.

In the current chapter, I start with the first difference operation (diff in Numpy) and show that it corresponds to a high-pass filter in the frequency domain. I use historical stock...

## Bayes meets Fourier

Joseph Fourier never met Thomas Bayes—Fourier was born in 1768, seven years after Bayes died. But recently I have been exploring connections between the Bayes filter and the Fourier transform.

By "Bayes filter", I don't mean spam filtering using a Bayesian classifier, but rather recursive Bayesian estimation, which is used in robotics and other domains to estimate the state of a system that evolves over time, for example, the position of a moving robot. My interest in...

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