Think DSP

Chapter 6. Discrete Cosine Transform

The topic of this chapter is the Discrete Cosine Transform (DCT), which is used in MP3 and related formats for compressing music; JPEG and similar formats for images; and the MPEG family of formats for video.

The DCT is similar in many ways to the Discrete Fourier Transform (DFT), which we have been using for spectral analysis. Once we learn how the DCT works, it will be easier to explain the DFT.

Here are the steps to get there:

1. We’ll start with the synthesis problem: given a set of frequency components and their amplitudes, how can we construct a wave?

2. Next we’ll rewrite the synthesis problem using NumPy arrays. This move is good for performance, and also provides insight for the next step.

3. We’ll look at the analysis problem: given a signal and a set of frequencies, how can we find the amplitude of each frequency component? We’ll start with a solution that is conceptually simple but slow.

4. Finally, we’ll use some principles from linear algebra to find a more efficient algorithm. If you already know linear algebra, that’s great, but I’ll explain what you need as we go.

The code for this chapter is in chap06.ipynb, which is in the repository for this book (see “Using the Code”). You can also view it at http://tinyurl.com/thinkdsp06.

Synthesis

Suppose I give you a list of amplitudes and a list of frequencies, and ask you to construct a signal that is the sum of these frequency components. Using objects in the thinkdsp module, there ...