van Lint: The mathematics of the CD-player

The main point of this lecture is to show what kind of mathematical ideas played a central role in achieving the superb quality of music on the Compact Disc. To show how error-correcting codes work, we take as example printed English instead of recorded music and use simple codes that the audience can check. The principles are exactly the same as on the CD. Practically no mathematical background is required.

If we see a printed (long) word with a few misprints in it, then we are able to correct these misprints. The reason we can do this is the fact that there is only one way to make a few changes in the printed word such that the result is a word in the language that we use. It is this idea that is used in the mathematical systems with which music is encoded for the CD.

We first show a simple example of an error-correcting code. By adjoining three redundant symbols to strings of four bits (a bit is a 0 or a 1) by an explicit rule, we obtain so-called codewords consisting of seven symbols 0 or 1. The rule is chosen in such a way that if we alter one of the seven symbols, we can calculate which one has been altered.

Next, we show how music is digitalized into strings of symbols called bytes (a byte is an eighttuple of 0's and 1's). For the CD, these bytes are encoded (again by adjoining extra symbols that have nothing to do with the music) into a so-called Reed-Solomon code. To explain this important step, we oversimplify and use a completely analogous system to encode a printed book in such a way that misprints can be corrected by mathematical calculations (simple arithmetic !).

One of the main reasons for the success of the CD is a method in which two simple (and easily decodable) codes join forces and thus, together, manage to handle situations with vast numbers of errors. An example of such a cooperation demonstrates the idea.

The bytes mentioned above are not recorded on the compact disc. Instead, a so-called recording code is used. It is this code that makes it possible for the laserbeam to follow the track on the disc without jumping to an adjacent track. It is also used to maintain focus and makes it possible to handle smudges.

It is perhaps useful to point out in this abstract that a high quality CD-disc, carefully handled by the buyer, can easily contain 500000 errors! After processing of the signal in the CD-player, these errors do not lead to a single click or other disturbing noise. Not all the praise goes to mathematics, but without error-correcting codes there would be no CD.

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Last modified: July 21, 1998