<img src="https://upload.wikimedia.org/wikipedia/commons/e/ea/From_Continuous_To_Discrete_Fourier_Transform.gif">
<p>[<a href="https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif">Image1</a>]</p>
<h2>Introduction</h2>
<p>
Hey it's a me again <a href="https://peakd.com/@drifter1">@drifter1</a>!
</p>
<p>
Today we continue with my mathematics series about <strong>Signals and Systems</strong>.
There's much to talk about <strong>Signals</strong> and so let's first get into the <strong>Basics</strong>...
</p>
<p>
So, without further ado, let's get straight into it!
</p>
<hr>
<h2>Signal Categorization</h2>
<h3>Continuous and Discrete Time</h3>
<p>
Signals are functions used for specific purposes, that can be split into two types based on how "often" samples are taken of them:
<ul>
  <li><strong>Continuous Time</strong> - Independent Variable <em>t</em></li>
  <li><strong>Discrete Time</strong> - Independent Variable <em>n</em></li>
</ul>
</p>
<p>
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.
</p>
<h3>Deterministic and Non-Deterministic</h3>
<p>
Signals can also be categorized based on their deterministic or non-deterministic nature.
</p>
<h4>Deterministic</h4>
<p>
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.
</p>
<h4>Non-Deterministic</h4>
<p>
On the other hand, non-deterministic signals are of random and so uncertain nature.
Such signals can only be modelled in probabilistic terms.
</p>
<h3>Even and Odd</h3>
<p>
<ul>
  <li><strong>Even</strong>: Signals that satisfy the condition <em>x(t) =  x(-t)</em></li>
  <li><strong>Odd </strong>: Signals that satisfy the condition <em>x(t) = -x(-t)</em></li>
</ul>
Any signal can be written as a sum of an even and an odd function:<br><br>
<img src="https://quicklatex.com/cache3/d5/ql_3d2e347fd97fa417279197d9e3f4ecd5_l3.png">
<ul>
  <li><em>x(t)</em>: "original" signal</li>
  <li><em>x<sub>e</sub>(t)</em>: even part of <em>x(t)</em></li>
  <li><em>x<sub>o</sub>(t)</em>: odd part of <em>x(t)</em></li>
</ul>
</p>
<h3>Periodic and Aperiodic</h3>
<p>
Signals are periodic when they repeat a specific pattern every time period <em>T</em> or sampling <em>N</em>.
</p>
<p>
Mathematically speaking, any signal that satisfies the following condition(s) is periodic:<br><br>
<img src="https://quicklatex.com/cache3/60/ql_a5cbcbc5a2c761d4e11b9d3bb5697e60_l3.png">
</p>
<h3>Energy and Power</h3>
<p>
A signal is a <strong>energy signal</strong> if it has finite energy, whilst a signal is a <strong>power signal</strong> if it has finite power.
</p>
<p>
Its worth noting that a signal <strong>cannot</strong> be both, energy and power simultaneously, and that it may be neither of them.
</p>
<h4>Energy</h4>
<p>
The energy of a signal is calculated using:<br><br>
<img src="https://quicklatex.com/cache3/bc/ql_f8369b828d4b9777a4692c958a2ac0bc_l3.png">
</p>
<h4>Power</h4>
<p>
The power of a signal is calculated using:<br><br>
<img src="https://quicklatex.com/cache3/6d/ql_b68e72d68bb911d51d10707bd8a9216d_l3.png">
</p>
<h3>Real, Imaginary and Complex</h3>
<p>
Lastly, signals are also categorized as real and imaginary:
<ul>
  <li><strong>Real</strong>: A signal is real when it has no imaginary part, meaning that the imaginary part is zero.</li>
  <li><strong>Imaginary</strong>: A signal is imaginary when it has no real part, meaning that the real part is zero.</li>
</ul>
A signal where both parts (real and imaginary) are non-zero is considered complex.
</p>
<p>
An easy way to check if a signal is real or imaginary is using the complex conjugate of the signal:<br><br>
<img src="https://quicklatex.com/cache3/88/ql_4c6a669d8bc357014d2aa430a6120f88_l3.png">
</p>
<hr>
<h2>Basic Signal Types</h2>
<h3>Unit Step Function</h3>
<p>
The unit step function, <em>u(t)</em>, is defined as:<br><br>
<img src="https://quicklatex.com/cache3/e6/ql_a14ae787875a218df846abdc06f5e4e6_l3.png"><br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Step_function.svg/239px-Step_function.svg.png"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Step_function.svg">Image 2</a>]<br><br>
This function is the best test signal.
</p>
<h3>Unit Impulse Function</h3>
<p>
The unit impulse function, <em>δ(t)</em>, is defined as:<br><br>
<img src="https://quicklatex.com/cache3/50/ql_d8af4c3291add78f80416101a6497350_l3.png"><br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/8/88/Unit_impulse.gif"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Unit_impulse.gif">Image 3</a>]
</p>
<h3>Ramp Signal</h3>
<p>
The ramp signal, <em>r(t)</em>, is defined as:<br><br>
<img src="https://quicklatex.com/cache3/8d/ql_310afce6c6f65285ac8af6b30f14538d_l3.png"><br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Ramp_function.svg/320px-Ramp_function.svg.png"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Ramp_function.svg">Image 4</a>]
</p>
<h3>Parabolic Signal</h3>
<p>
A parabolic signal <em>t<sup>2</sup>/2</em>can be easily defined using <em>r(t)</em> or <em>u(t)</em> as:<br><br>
<img src="https://quicklatex.com/cache3/5f/ql_6eb437b5da8468d5a642813841e47d5f_l3.png">
</p>
<h3>Signum Signal</h3>
<p>
Turning the unit step function into an odd function [<em>u(t) = -u(-t)</em>] creates the so called signum or sign function, <em>sgn(x)</em>:<br><br>
<img src="https://quicklatex.com/cache3/4a/ql_644bbe9e8f15934b3075f160c8d7644a_l3.png"><br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Signum_function.svg/500px-Signum_function.svg.png"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Signum_function.svg">Image 5</a>]
</p>
<h3>Exponential Signal</h3>
<p>
Exponential signals are of the generic form:<br><br>
<img src="https://quicklatex.com/cache3/d6/ql_9b1374077727ef302d4bcc336327b9d6_l3.png"><br><br>
The shape of the exponential depends on the value of the parameter <em>a</em>:
<ul>
  <li><em>a = 0</em> &rarr; e<sup>0</sup> = 1</li>
  <li><em>a &lt; 0</em> &rarr; decaying exponential</li>
  <li><em>a &gt; 0</em> &rarr; raising exponential</li>
</ul>
</p>
<h3>Sinusoidal Signal</h3>
<p>
Any signal of the form:<br><br>
<img src="https://quicklatex.com/cache3/88/ql_b169033dea27c35051701ccafef0b688_l3.png"><br><br>
or<br><br>
<img src="https://quicklatex.com/cache3/fa/ql_d446cad85a90a23b0038b92c13b4abfa_l3.png"><br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/3/3a/Sinusoidal.png"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Sinusoidal.png">Image 6</a>]
</p>
<h3>Sinc and Sampling Functions</h3>
<p>
The sinc function, <em>sinc(t)</em>, is defined as:<br><br>
<img src="https://quicklatex.com/cache3/24/ql_4ef637d708f9404ed59ea8704ed2b624_l3.png"><br><br>
The sampling function, <em>sa(t)</em>, is defined as:<br><br>
<img src="https://quicklatex.com/cache3/c0/ql_6f2c00298a578f3355765339039019c0_l3.png"><br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/59/Si_sinc.svg/500px-Si_sinc.svg.png"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Si_sinc.svg">Image 7</a>]
</p>
<hr>
<h2>RESOURCES:</h2>
<h3>References</h3>
<ol>
  <li><a href="https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011">Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.</a></li>
  <li><a href="https://www.tutorialspoint.com/signals_and_systems/">https://www.tutorialspoint.com/signals_and_systems/</a></li>
</ol>
<h3>Images</h3>
<ol>
  <li><a href="https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif">https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Step_function.svg">https://commons.wikimedia.org/wiki/File:Step_function.svg</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Unit_impulse.gif">https://commons.wikimedia.org/wiki/File:Unit_impulse.gif</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Ramp_function.svg">https://commons.wikimedia.org/wiki/File:Ramp_function.svg</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Signum_function.svg">https://commons.wikimedia.org/wiki/File:Signum_function.svg</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Sinusoidal.png">https://commons.wikimedia.org/wiki/File:Sinusoidal.png</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Si_sinc.svg">https://commons.wikimedia.org/wiki/File:Si_sinc.svg</a></li>
</ol>
<p>Mathematical equations used in this article, where made using <a href="http://quicklatex.com/">quicklatex</a>.</p>
<hr>
<h2>Previous articles of the series</h2>
<ul>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-introduction">Introduction</a> &rarr; Signals, Systems</li>
</ul>
<hr>
<h2>Final words | Next up</h2>
<p>And this is actually it for today's post!</p>
<p>Next time we will dive even more into Signals...</p>
<p>See Ya!</p>
<p><img src="https://steemitimages.com/0x0/https://media.giphy.com/media/ybITzMzIyabIs/giphy.gif" width="500" height="333"/></p>
<p>Keep on drifting!</p>