What is a signal ?

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
Continuous Time Signal

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.

Discrete Signal

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:

Continuous Time Signal from Mathematical Expression

Image: Continuous Time Signal from Mathematical Expression

A discrete-time signal can be defined in several ways:

  • as a function:
\[x[n] = \left\{\begin{matrix} -1 &, \text{ for } n < 0\\ 1 &, \text{ for } n > 0\\ 0 &, \text{ for } n = 0 \end{matrix} \right. \]
  • as a table:
n-3-2-10123
x[n]0210120
  • as a sequence of values:
\[x[n]= \left \{…, 0, 2, 1, 0, 1, 2, 0, … \right \} \]
  • as a vector:
\[x[n]=[… \text{ } 0\text{ } 2\text{ } 1\text{ } 0\text{ } 1\text{ } 2\text{ } 0\text{ } …]\]
  • as graphical representation
Graphical Definition of a Discrete Signal

Image: Graphical Definition of a Discrete Signal

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One Response

  1. herve Bernad

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