Digital Signal and Image Processing Using MATLAB

Chapter 7

Random Variables

7.1    Random phenomena in signal processing

In many practical circumstances, the phenomena observed show important variations, when in fact the relevant information itself has not changed. Thus, if you record the signals obtained when pronouncing several times the sound “A”, a simple observation will tell you that all the recordings are different even though they all sound basically the same. It is neither easy nor relevant to try to find a deterministic equation to describe the evolution in time of such a phenomenon. The model used to describe this variability is based on the concept of random variables (r.v.), defined by the Probability Theory. A common misuse of language consists of saying that this is a random phenomenon.

Another example of a random phenomenon is the background noise heard with radio reception. It seems difficult to describe this noise without using statistical characteristics. In the complete chain of communications, an example we will come back to later, every device, as well as the transmission medium, causes background noise. But with such a system, this is not the only “source of randomness”. From the receiver’s point of view, the message itself must also be considered random.

In fact, any device or physical phenomenon has a random part to it. Deciding how important that random part is to the system is the only factor in determining what type of model is used to describe the phenomena. If the amount of information that we cannot ...