# 1.3. Systems – Digital Filters Design for Signal and Image Processing

## 1.3. Systems

A system carries out an operation chain, which consists of processing applied to one or several input signals. It also provides one or several output signals. A system is therefore characterized by several types of variables, described below:

– inputs: depending on the situation, we differentiate between the commands (which are inputs that the user can change or manipulate) and the driving processes or excitations which usually are not accessible;

– outputs;

– state variables that provide information on the “state” of the system. By the term “state” we mean the minimal number of parameters, stored usually in a vector, that can characterize the development of the system, where the inputs are supposed to be known;

– mathematical equations that link input and output variables.

In much the same way as we classify signals, we speak of digital systems (respectively analog) if the inputs and outputs are digital (respectively analog).

When we consider continuous physical systems, if we have two inputs and two outputs, the system is then a quadrupole. We wish to impose a given variation law on the output according to the input. If the relation between input and output is given in the form of a differential linear equation with constant coefficients, we then have a linear system that is time-invariant and continuous. Depending on the situation, we use physical laws to develop equations; in electronics, for example, we employ Kirchhoff's laws and Thévenin's and Norton's theorems or others to establish our equations.

Later in this text, we will discuss discrete-time systems in more detail. These are systems that transform a discrete-time input signal x(k) into a discrete-time output signal y(k) in the following manner: By way of example, we see that y(k) = x(k), y(k) = x(k − 1) and y(k) = x(k + 1) respectively express the identity, the elementary delay and the elementary lead.