Digitization is the process of converting an analog signal to a digital one. Digitization is actually a combination of two distinct processes: sampling and quantization. A signal is a simply a function from some domain to some range (I'm using 'domain' and 'range' in the mathematical sense here). Sampling converts the signal from a continuous domain to a discrete domain. Quantization converts the signal from a continuous range to a discrete range.

Here is an example to make this definition more concrete. Suppose a live musical performance is recorded using a microphone and laptop directly onto a CD-RW. The sound waves generated by the musical instruments are converted into an electrical signal by the microphone. The electrical signal is then digitized by the laptop's sound card and then written onto the CD-RW.

Now, as you all know, sound is a compression wave. Pressure variations propagate outward from the source of the disturbance (e.g. a guitar) through the surrounding medium (e.g. air). At any stationary point -- for example, where the microphone is located -- the pressure variation is a function of time. In other words, the input signal at the microphone is a function P(t) ('P' for 'pressure'). This pressure variation is an analog signal. The microphone converts this into another analog signal, V(t) ('V' for 'voltage').

Now the sound card must convert this analog signal V(t) into a sequence of numbers that constitutes the output digital signal. First, it looks at analog signal at even intervals and records their values. Those values are called samples. For example, the sound card may record the values every millisecond:

```time (ms)   0       1       2       3       4       5     ...
voltage   1.2742  3.1845 -2.2834 -1.1934 -0.6256  0.2998  ...
```
This is the sampling part of digitization. The domain of the signal, time, is discretized.

The rate that the sound card takes these samples is called the sampling frequency. For example, many sound cards take 48,000 samples per second. In other words, they have a sampling frequency of 48kHz.

Next the sound card looks at each voltage value and rounds it to, say, the nearest tenths:

```time (ms)  0    1    2    3    4    5   ...
voltage   1.3  3.2 -2.3 -1.2 -0.6  0.3  ...
```
This is the quantization part of digitization. The range of the signal, voltage, is discretized.

The number of bits required to represent each voltage value is called the sample size. Many sound cards support sample sizes of 8, 16, or 24 bits.

The resulting sequence of quantized voltage values constitute the digital signal. It is represented in binary and the on/off sequence is written onto the CD-RW.

Even in the ideal case (as outlined above), the digitization process is far from perfect. The discretization of the signal results in information loss. For the example above this information loss will manifest itself as unpleasant (screeching) noise when the music is played back. The sampling process may cause aliasing, and the quantization process may cause quantization error or quantization noise. The obvious 'solution' to the problem of aliases is to take samples at finer intervals (i.e. use a higher sampling frequency). Similarly, quantization noise may be alleviated somewhat by using more digits to represent each sample (i.e. use a larger sample size). Another way to deal with these errors is to filter them out. These problems, and ways to deal with them, are studied in depth in the subjects of signal processing and information theory, respectively.

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