Think DSP

Chapter 5. Autocorrelation

In the previous chapter I characterized white noise as “uncorrelated”, which means that each value is independent of the others, and Brownian noise as “correlated”, because each value depends on the preceding value. In this chapter I define these terms more precisely and present the autocorrelation function, which is a useful tool for signal analysis.

The code for this chapter is in chap05.ipynb, which is in the repository for this book (see “Using the Code”). You can also view it at http://tinyurl.com/thinkdsp05.

Correlation

In general, correlation between variables means that if you know the value of one, you have some information about the other. There are several ways to quantify correlation, but the most common is the Pearson product-moment correlation coefficient, usually denoted ρ. For two variables, x and y, that each contain N values:

where μ_x and μ_y are the means of x and y, and σ_x and σ_y are their standard deviations.

Pearson’s correlation is always between –1 and +1 (including both). If ρ is positive, we say that the correlation is positive, which means that when one variable is high, the other tends to be high. If ρ is negative, the correlation is negative, so when one variable is high, the other tends to be low.

The magnitude of ρ indicates the strength of the correlation. If ρ is 1 or –1, the variables are perfectly ...