A signal can be defined as an abstraction of any quantity that can be measured, which is function of at least one independent variable (time or space). A signal is the functional representation of a physical quantity or variable, and it contains information about the behavior of the physical quantity.
Example of signals:
- electrical: voltage and current
- mechanical: position and speed
- hydraulic: pressure and flow
The sound coming to our ears is also a signal. A picture within a book, a movie on the TV screen, they are also signals.
Mathematically, a signal is represented as a function of an independent variable t. Most of the time t stand for time. The mathematical notation of a signal is x(t).
Depending on the continuity of the contained information signals can be:
- continuous-time
- discrete-time
Image: Continuous Time Signal
A continuous-time signal has a value for all instants in time or space. Example of a continuous-time signal is the voltage of a battery or the position of a pendulum.
A discrete-time signal has a value only at discrete moments in time. Example of a discrete signal is the weight of a human measured early or the daily average temperature measure in a specific area.
Image: Discrete Signal
Very often, discrete-time signals are sampled versions of continuous time signal. For example, if we measure the voltage of a battery at a specified time interval (e.g. every second) we’ll end up with a discrete-time signal.
If x(t) is a continuous-time signal, and we sample it at t0, t1, … tn, we get samples:
\[x(t_0), x(t_1), … , x(t_n)\]In a shorter form we can write the samples as:
\[x[0], x[1], … , x[n]\]or even shorter:
\[x_0, x_1, … , x_n\]Generally speaking:
\[ \begin{equation} \begin{split}\bbox[#FFFF9D]{x_n=x[n]=x(t_n)}
\end{split} \end{equation} \]
The time interval between two adjacent samples is called sampling interval. When the sampling interval is constant for the whole measurement, we’ll have an uniform sampling, which means:
\[x_n=x[n]=x(n \cdot T_s)\]where Ts is the sampling interval.
A continuous-time signal can be defined with the help of mathematical function. For example:
\[ \begin{equation*} \begin{split}t &= – 2 \pi … 2 \pi\\
x(t) &= sin^2(t)+cos(t)\\
\end{split} \end{equation*} \]
The graphical representation of this signal is:
Image: Continuous Time Signal from Mathematical Expression
A discrete-time signal can be defined in several ways:
- as a function:
- as a table:
| n | … | -3 | -2 | -1 | 0 | 1 | 2 | 3 | … |
| x[n] | … | 0 | 2 | 1 | 0 | 1 | 2 | 0 | … |
- as a sequence of values:
- as a vector:
- as graphical representation
Image: Graphical Definition of a Discrete Signal
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herve Bernad
Merci pour votre def de ce qu’est un signal !