# 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

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:

$\begin{split} \bbox[#FFFF9D]{x_n=x[n]=x(t_n)} \end{split}$

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:
$x[n] = \left\{\begin{matrix} -1 &, \text{ for } n < 0\\ 1 &, \text{ for } n > 0\\ 0 &, \text{ for } n = 0 \end{matrix} \right.$
• as a table:
 n … -3 -2 -1 0 1 2 3 … x[n] … 0 2 1 0 1 2 0 …
• 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

Image: Graphical Definition of a Discrete Signal

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