<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> in order to cover <strong>Continuous-Time Periodic Signals & Fourier Series</strong>.
</p>
<p>
I highly suggest reading the previous article about Discrete-Time, as the representation is quite similar, in some sense.
</p>
<p>
So, without further ado, let's get straight into it!
</p>
<hr>
<h2>Input Decomposition</h2>
<p>
Let's start simple, by basically refreshing our knowledge around linear systems, and why they are so useful...
</p>
<p>
First of all, due to the properties of linear systems, and mainly due to the principle of superposition, its possible to decompose any signal into a set of basic signals.
The expected outcome of this procedure is a linear combination of basic signals for which the response can be easily computed.
</p>
<p>
Consider a signal <em>x(t)</em> in continuous-time.
For linear systems, this signal can be decomposed to as a linear combination of <em>φ<sub>k</sub></em> 's in the following way:<br><br>
<img src="https://quicklatex.com/cache3/50/ql_668c097986726fbb5bc7529ca8aed650_l3.png">
</p>
<p>
In the end, what matters is that the response of the system will be some combination of easy to compute <em>ψ<sub>k</sub></em> 's:<br><br>
<img src="https://quicklatex.com/cache3/7c/ql_18c557dbb52e473902ab1d76f458957c_l3.png"><br><br>
Those <em>ψ<sub>k</sub></em> 's are the result of convolution for each of the basic <em>φ<sub>k</sub></em> 's.
In continuous-time, the convolution integral can be represented as:<br><br>
<img src="https://quicklatex.com/cache3/00/ql_c682809a6a9fe383bc66d2db74bbd200_l3.png">
</p>
<hr>
<h2>Basic Signals and the Concept of Fourier Analysis</h2>
<p>
As we already mentioned in some previous article, the main concept of Fourier analysis is using complex exponentials as the set of basic signals.
Thus, the basic signals will be of the form:<br><br>
<img src="https://quicklatex.com/cache3/50/ql_27a9f5360e8a3515062047a187f2e350_l3.png"><br><br>
Substituting <em>s<sub>k</sub></em> with <em>jω<sub>k</sub></em>, its easier to see the "complex" part:<br><br>
<img src="https://quicklatex.com/cache3/7f/ql_4ff6a98657b9c0b8fa79d29fcb47017f_l3.png"><br><br>
When <em>s<sub>k</sub></em> is complex the transform is also known as a Laplace transform.
</p>
<p>
The system response for such a signal is:<br><br>
<img src="https://quicklatex.com/cache3/2e/ql_81cd1745d89c9728c387d0b9af2db02e_l3.png"><br><br>
For periodic signals the result will be represented by a Fourier Series, whilst for aperiodic signals (which will be covered next time) the outcome is a Fourier Transform. 
</p>
<hr>
<h2>Fourier Series - Sum of Complex Exponentials</h2>
<p>
A Fourier Series is basically a weighted sum of complex exponentials, which are harmonically related.
Its known that a signal is periodic only when the following relation is true:<br><br>
<img src="https://quicklatex.com/cache3/57/ql_68bcd1fb1ac37668db4511715d1c5257_l3.png"><br><br>
where <em>T<sub>o</sub></em> is the period.
</p>
<p>
The period <em>T<sub>o</sub></em> can be substituted using the following equation:<br><br>
<img src="https://quicklatex.com/cache3/7e/ql_d7dfc98464c4d3501ea2490276ec937e_l3.png"><br><br>
which allows us to replace the <em>ω<sub>k</sub></em>'s by the product <em>k ω<sub>o</sub></em> :<br><br>
<img src="https://quicklatex.com/cache3/8f/ql_8dd8d25a58e06a635f42e938f185788f_l3.png">
</p>
<p>
In this notion, a periodic signal <em>x(t)</em> can be decompsed into a sum of complex exponentials as such:<br><br>
<img src="https://quicklatex.com/cache3/de/ql_6a68fe1d2374d150103d00e1d908bbde_l3.png">
</p>
<hr>
<h2>Fourier Synthesis</h2>
<p>
The coefficients, <em>a<sub>k</sub></em>, can be calculated using:<br><br>
<img src="https://quicklatex.com/cache3/20/ql_c1e984b7d240ab573495af690bdf9e20_l3.png"><br><br>
for which I will skip the proof.
</p>
<h3>Trigonometric form</h3>
<p>
In the article about complex exponentials, it was shown how Euler's Relation can be used to get a trigonometric form.
In this case, the trigonometric form looks as follows:<br><br>
<img src="https://quicklatex.com/cache3/35/ql_e8ccb314415e41309cdea48595dd6635_l3.png"><br><br>
which results in the following complete decomposition of <em>x(t)</em> :<br><br>
<img src="https://quicklatex.com/cache3/dc/ql_79bc8260ec99d266a7e7b75aa8e60ddc_l3.png"><br><br>
where <em>a<sub>k</sub> = B<sub>k</sub> + jC<sub>k</sub></em> and the harmonic relation between the complex exponentials is also taken into consideration.
</p>
<p>
The coefficients, <em>a<sub>k</sub></em>, can be still calculated using the same equation as before, with <em>a<sub>o</sub></em> having the special property of being the average value of <em>x(t)</em> during a period.
</p>
<p>
This is basically an alternative way of representing the complex exponentials that form the Fourier Series, where each exponential is a linear combination of sines and cosines, or sinusoids in general, of different phase angles.
</p>
<hr>
<h2>Signal Approximation</h2>
<p>
Incorporating only <em>2N + 1</em> terms of the Fourier Series, its possible to end up with an approximation of <em>x(t)</em> in the form of signal <em>x<sub>N</sub>(t)</em>.
This signal is represented by the sum:<br><br>
<img src="https://quicklatex.com/cache3/75/ql_0624bf6e191994153b18859482f0a175_l3.png">
</p>
<p>
Doing that we introduced an error <em>e<sub>N</sub>(t)</em>, which can be calculated by:<br><br>
<img src="https://quicklatex.com/cache3/5e/ql_1942796824332e843fdd10a2edafc35e_l3.png">
</p>
<p>
As <em>N</em> increases, the error <em>e<sub>N</sub>(t)</em> is generally decreasing, as we end up with a better approximation of the signal <em>x(t)</em>.
</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>
</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>
</ol>
<p>Mathematical equations used in this article were made using <a href="http://quicklatex.com/">quicklatex</a>.</p>
<p>Block diagrams and other visualizations were made using <a href="https://app.diagrams.net/">draw.io</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>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-signal-basics">Signal Basics</a> &rarr; Signal Categorization, Basic Signal Types</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-signal-operations-with-examples">Signal Operations with Examples</a> &rarr; Amplitude and Time Operations, Examples</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-system-classification-with-examples">System Classification with Examples</a> &rarr; System Classifications and Properties, Examples</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-sinusoidal-and-complex-exponential-signals">Sinusoidal and Complex Exponential Signals</a> &rarr; Sinusoidal and Exponential Signals in Continuous and Discrete Time</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-lti-system-response-and-convolution">LTI System Response and Convolution</a> &rarr; Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral </li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-lti-convolution-properties">LTI Convolution Properties</a> &rarr; Commutative, Associative and Distributive Properties of LTI Convolution</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-system-representation-in-discrete-time-using-difference-equations">System Representation in Discrete-Time using Difference Equations</a> &rarr; Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-system-representation-in-continuous-time-using-differential-equations">System Representation in Continuous-Time using Differential Equations</a> &rarr; Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)</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 cover Aperiodic Signal in Continuous-Time, which are represented using something known as the Fourier Transform.</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>