Analog to digital transformation is the first necessary step to load multimedia signals into digital devices. It contains two operations called sampling and quantization. The theoretical background of sampling is given by the famous sampling theorem. The first attempts to formulate and prove the sampling theorem date back to the beginning of the twentieth century. In this chapter we present Shannon's elegant proof of the sampling theorem. Consequences of sampling “too slowly” in the time and frequency domains are discussed. Quantization is the main operation which determines the quality–compression ratio tradeoff in all lossy compression systems. We consider different types of quantizer commonly used in modern multimedia compression systems.

Analog and digital signals

First, we introduce some definitions.

• A function f(x) is continuous at a point x = a if limx→af(x) = f(a). We say a function is continuous if it is continuous at every point in its domain (the set of its input values).

• We call a set of elements a discrete set if it contains a finite or countable number of elements (elements of a countable set can be enumerated).

In the real world analog signals are continuous functions of continuous arguments such as time, space, or any other continuous physical variables, although we often use mathematical models with not continuous analog signals such as the saw-tooth signal. We consider mainly time signals which can take on a continuum of values over a defined interval of time.