[Image1]

## Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about **Signals and Systems**.
There's much to talk about **Signals** and so let's first get into the **Basics**...

So, without further ado, let's get straight into it!

## Signal Categorization

### Continuous and Discrete Time

Signals are functions used for specific purposes, that can be split into two types based on how "often" samples are taken of them:

**Continuous Time**- Independent Variable*t***Discrete Time**- Independent Variable*n*

If samples are taken continuously, as time goes by (and so in respect to time itself), then the signal is of continuous-time, whilst when samples are taken after specific sampling intervals the signal is of discrete-time.

### Deterministic and Non-Deterministic

Signals can also be categorized based on their deterministic or non-deterministic nature.

#### Deterministic

A signal is deterministic if there is no uncertainty in respect to its value at any instance of time. This basically means that a deterministic signal can be perfectly defined by a mathematical formula.

#### Non-Deterministic

On the other hand, non-deterministic signals are of random and so uncertain nature. Such signals can only be modelled in probabilistic terms.

### Even and Odd

**Even**: Signals that satisfy the condition*x(t) = x(-t)***Odd**: Signals that satisfy the condition*x(t) = -x(-t)*

*x(t)*: "original" signal*x*: even part of_{e}(t)*x(t)**x*: odd part of_{o}(t)*x(t)*

### Periodic and Aperiodic

Signals are periodic when they repeat a specific pattern every time period *T* or sampling *N*.

Mathematically speaking, any signal that satisfies the following condition(s) is periodic:

### Energy and Power

A signal is a **energy signal** if it has finite energy, whilst a signal is a **power signal** if it has finite power.

Its worth noting that a signal **cannot** be both, energy and power simultaneously, and that it may be neither of them.

#### Energy

The energy of a signal is calculated using:

#### Power

The power of a signal is calculated using:

### Real, Imaginary and Complex

Lastly, signals are also categorized as real and imaginary:

**Real**: A signal is real when it has no imaginary part, meaning that the imaginary part is zero.**Imaginary**: A signal is imaginary when it has no real part, meaning that the real part is zero.

An easy way to check if a signal is real or imaginary is using the complex conjugate of the signal:

## Basic Signal Types

### Unit Step Function

The unit step function, *u(t)*, is defined as:

[Image 2]

This function is the best test signal.

### Unit Impulse Function

The unit impulse function, *δ(t)*, is defined as:

[Image 3]

### Ramp Signal

The ramp signal, *r(t)*, is defined as:

[Image 4]

### Parabolic Signal

A parabolic signal *t ^{2}/2*can be easily defined using

*r(t)*or

*u(t)*as:

### Signum Signal

Turning the unit step function into an odd function [*u(t) = -u(-t)*] creates the so called signum or sign function, *sgn(x)*:

[Image 5]

### Exponential Signal

Exponential signals are of the generic form:

The shape of the exponential depends on the value of the parameter *a*:

*a = 0*→ e^{0}= 1*a < 0*→ decaying exponential*a > 0*→ raising exponential

### Sinusoidal Signal

Any signal of the form:

or

[Image 6]

### Sinc and Sampling Functions

The sinc function, *sinc(t)*, is defined as:

The sampling function, *sa(t)*, is defined as:

[Image 7]

## RESOURCES:

### References

- Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.
- https://www.tutorialspoint.com/signals_and_systems/

### Images

- https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif
- https://commons.wikimedia.org/wiki/File:Step_function.svg
- https://commons.wikimedia.org/wiki/File:Unit_impulse.gif
- https://commons.wikimedia.org/wiki/File:Ramp_function.svg
- https://commons.wikimedia.org/wiki/File:Signum_function.svg
- https://commons.wikimedia.org/wiki/File:Sinusoidal.png
- https://commons.wikimedia.org/wiki/File:Si_sinc.svg

Mathematical equations used in this article, where made using quicklatex.

## Previous articles of the series

- Introduction → Signals, Systems

## Final words | Next up

And this is actually it for today's post!

Next time we will dive even more into Signals...

See Ya!

Keep on drifting!