Signals are function that map one or more input to an output. One-dimensional signals are those that describe the evolution of a system over time — in fact, the input variable is often the time, $t$. Two-dimensional signals are those that describe the value of a function in a certain spot — they are often used to represent images and they take two values, $x$ and $y$, which are the pixel coordinates.

Of course, there are many more signals and they can describe different things. However, in the context of Image and Signal Processing, these are the two most important kinds of signal.

## Mathematical functions

Mathematical functions are usually described as $f \longrightarrow \R \to \R$, which basically means that function $f$ takes a real value (value in $\R$) and outputs another real value.

Functions have so many properties that covering all of them here would be an understatement. They have, nonetheless, two important properties that we should not forget:

- they can be
. A function is said to be even if $f(t) = f(-t)$;*even* - they can be
. A function is said to be odd if $f(t) = -f(-t)$.*odd*

## Signals

Some signals are periodic, which means that they repeat over time. The two basic periodic signals are:

- $f(t) = \cos 2 \pi \mu_0 t$
- $f(t) = \sin 2 \pi \mu_0 t$

Variable $\mu_0$ is a frequency — indeed it is measured in Hertz (Hz, which is $[1/s] \text{ or } [s^{-1}]$). $\mu_0 = 1/T_0$, where $T_0$ is the time (measured in seconds) taken to make a full cycle.

We call
$2 \pi \mu_0 = \frac{2 \pi}{T_0} = \omega_0$ the *pulsation* or, better, the ** angular velocity**.

Signals can also be translated. It is possible to **delay** them by **subtracting** some quantity from the input varialbe.
$f(t) \rightarrow f(t - \tau)$ is shifted **to the right**: it *starts* later than
$f(t)$;
$f(t) \rightarrow f(t + \tau)$ is shifted **to the left**: it starts *before* earlier than
$f(t)$.

### Common signals

Some common signals are presented afterwards.

#### Box Signal

#### Generalized Box Signal

#### Dirac’s Delta Signal (impulse)

Dirac’s Delta is thus:

#### Triangle Signal

#### Sign Signal

#### Step Signal

#### Sinc Signal

##### Sampling property

The impulse has a great property, known as the “sampling property”. It states that:

This means that if we place the
$\delta$ signal at some point in time (
$x_0$) and multiply it by *any* other signal
$f$, what we get is the so-called ** sampling** of signal
$f$ at time
$x_0$. This property — which may look pretty complex and hard to understand — is just the formal way of representing the behaviour of two things we use everyday: microphones and cameras.

Microphones and cameras are indeed based on sampling: they sample signals from the surroundings (sounds and photons, respectively).