Signals, the basics

Published May 19, 2022 - By Marco Garosi

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

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

\Pi(x) = \begin{cases} 1 &\text{if } -\frac{1}{2} \le x \le \frac{1}{2} \\ 0 &\text{otherwise} \end{cases}

Box Signal

Generalized Box Signal

A \Pi(\frac{x}{b}) = \begin{cases} A &\text{if } -\frac{b}{2} \le x \le \frac{b}{2} \\ 0 &\text{otherwise} \end{cases}

Generalized Box Signal

Dirac’s Delta Signal (impulse)

\delta(x) = \lim_{b \to \infty} \frac{1}{b} \Pi \frac{x}{b}

with the constraint that

\int_{-\infty}^{\infty} \delta(x) dx = 1

Dirac’s Delta is thus:

\delta(x) = \begin{cases} \infty &\text{if } x = 0 \\ 0 &\text{otherwise} \end{cases}

Dirac's Delta Signal (impulse)

Triangle Signal

tri(x) = \begin{cases} 1 - |x| &\text{if } |x| \lt 1 \iff -1 \lt x \lt 1 \\ 0 &\text{otherwise} \end{cases}

Triangle Signal

Sign Signal

sgn(x) = \begin{cases} 1 &\text{if } x \gt 0 \\ 0 &\text{if } x = 0 \\ -1 &\text{if } x \lt 0 \end{cases}

Triangle Signal

Step Signal

tri(x) = \begin{cases} 0 &\text{if } x \lt 0 \\ 1 &\text{if } x \ge 0 \end{cases}

Step Signal

Sinc Signal

\text{sinc}(t) = \frac{\sin \pi t}{\pi t}

Sinc Signal

It intersects the x-axis in

t = \pm 1, 2, 3, \dots

and

\lim_{t \to \infty} \text{sinc}(t) = 0

Sampling property

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

\int_{-\infty}^{\infty} f(x) \delta(x - x_0) dx = f(x_0)

Sampling property

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).