CD Player

A CD is a digitally recorded sound wave. Since the CD is digital it must have definite numerical values for the magnitude of that wave at any given time, and since there is only a finite amount of storage on a CD those definite values only change every so often. So if you were to look at a what is stored on the CD you would see a stair stepping function similar to the one below.

Insert Image

The basic idea of turning this stair step into the smooth sound wave that it used to be is to run it through a digital to analog converter then through a low pass filter and then out to a speaker. While this would work your signal would be substantially distorted due to its conversion to and then from a digital signal. We will look at a few different ways to compensate for this.

Signal in Time and Frequency

Insert Image

This shows the progression of your signal both in the time and frequency domains. You will notice that the frequency representation of your stair step wave, which is stored on the CD, does not look like the original signal. It is too tall in the middle and drops off too fast of the edges, it also has too many high frequency (both positive and negative) components that aren't wanted. The basic idea to fix this is to pre-distort the signal by multiplying by a function that would increase the edges of the signal and shrink the middle so that when it gets distorted it ends up being what you want. But there is also the problem of filtering out the unwanted high frequencies. To do this we would need a low pass filter with a very very sharp edge, this is an expensive piece of electronics and we would like to eliminate it if we could, and as I'm sure you guessed, we can.

Over Sampling

The following is only included for my sake, so I can look at the syntax.

The Fourier Series

A Fourier series is an expansion of a periodic function $f$ in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.

$f(t)=\sum _{k=-\infty }^{\infty }\alpha _{k}e^{\frac {j2\pi kt}{T}}$.

see also:Orthogonal Functions

Principle author of this page: Aric Goe