# Chapter 8. Filtering and Convolution

In this chapter I present one of the most important and useful ideas related to signal processing: the Convolution Theorem. But before we can understand the Convolution Theorem, we have to understand convolution. I’ll start with a simple example, smoothing, and we’ll go from there.

The code for this chapter is in `chap08.ipynb`, which is in the repository for this book (see “Using the Code”). You can also view it at http://tinyurl.com/thinkdsp08.

# Smoothing

Smoothing is an operation that tries to remove short-term variations from a signal in order to reveal long-term trends. For example, if you were to plot daily changes in the price of a stock, it would look noisy; a smoothing operator might make it easier to see whether the price was generally going up or down over time.

A common smoothing algorithm is a moving average, which computes the mean of the previous n values, for some value of n.

For example, Figure 8-1 shows the daily closing price of Facebook stock from May 17, 2012 to December 8, 2015. The gray line is the raw data, and the darker line shows the 30-day moving average. Smoothing removes the most extreme changes and makes it easier to see long-term trends.

Smoothing operations also apply to sound signals. As an example, I’ll start with a square wave at 440 Hz. As ...

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