## 2.1. Theoretical background

A signal is a physical support for information, depending on one or several independent variables, such as: time, range, temperature, pressure, etc. The signal concept generally refers to its mathematical or physical model, chosen in the most appropriate manner for describing the complexity of real signals.

According to whether a signal depends on only one variable or two variables, it is called one-dimensional (1D) or two-dimensional (2D). As a general rule, a multidimensional signal is defined as a function of several variables.

A discrete-time signal is represented by a digital series, uniformly or non-uniformly sampled. The first case is considered in the following and corresponds to a constant time delay between each two successive samples.

Usually, *s(k)* stands for the *k*^{th} sample of the discrete-time signal {*s(k)*}_{k = 1, 2, ...}. However, the two notations usually overlap in order to simplify the theoretical presentation.

The signal theory is mainly related to the signal mathematical representation in the original or a transformed space, and to its algorithmic processing in order to extract the useful information.

### 2.1.1. *Mathematical model of 1D and 2D discrete-time signals*

The mathematical model of a discrete-time signal can be defined in terms of the function indicated below:

so that:

while, for 2D discrete-time signals *n* = (*n*_{1}, *n*_{2}) → *x*(*n*_{1}, *n*_{2}

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