<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>Exercises on Continuous-Time Fourier Series</strong>.
</p>
<p>
So, without further ado, let's dive straight into it!
</p>
<hr>
<h2>Continuous-Time Fourier Series Recap</h2>
<p>
Let's quickly refresh our knowledge on the Fourier Series.
The Fourier Series is a way of representing periodic signals as a sum of weighted, harmonically related, complex exponentials.
This procedure can be split into two parts:
<ul>
  <li>Fourier Synthesis</li>
  <li>Fourier Analysis</li>
</ul>
</p>
<p>
The Fourier Synthesis equation, which is the sum that forms the signal, is:<br><br>
<img src="https://quicklatex.com/cache3/de/ql_6a68fe1d2374d150103d00e1d908bbde_l3.png">
</p>
<p>
whilst the Fourier Analysis equation, which gives us the corresponding coefficients, is:<br><br>
<img src="https://quicklatex.com/cache3/20/ql_c1e984b7d240ab573495af690bdf9e20_l3.png">
</p>
<hr>
<h2>Fourier Series Coefficients Calculation [Based on 7.3 from <a href="https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/">Ref1</a>]</h2>
<p>
For a fundamental frequency <em>ω<sub>o</sub> = 2π</em>, find the Fourier series coefficients for each of the following signals:<br><br>
<img src="https://quicklatex.com/cache3/44/ql_125114eccf95777d59ff8ebb302ba644_l3.png">
</p>
<h3>Solution</h3>
<h4>(a)</h4>
<p>
From Euler's Formula we know that a complex exponential can be turned into a sum of a sine and cosine trigonometric function, as follows:<br><br>
<img src="https://quicklatex.com/cache3/40/ql_b561c47b27e115985dea58a370302240_l3.png">
</p>
<p>
As such, the cosine represents the real part of the complex exponential, whilst the sine represents the imaginary part.
Due to conjugate symmetry, relating the sine to the complex exponential is as simple as:<br><br>
<img src="https://quicklatex.com/cache3/97/ql_5753cf819b111be805a84fe8dc173397_l3.png">
</p>
<p>
Thus, the signal <em>x(t)</em> can be re-written as:<br><br>
<img src="https://quicklatex.com/cache3/63/ql_561b79f93c16e62fd83ff612517c1663_l3.png">
</p>
<p>
Because the fundamental frequency is chosen to be <em>2π</em>, the Fourier Series is of the form:<br><br>
<img src="https://quicklatex.com/cache3/2e/ql_8824d7ab3b4a427f8eb378b85414df2e_l3.png">
</p>
<p>
From the Euler expansion that we did for the signal, its easy to notice that only <em>k = 4</em> and <em>k = - 4</em> give non-zero results.
So, the final coefficients are:<br><br>
<img src="https://quicklatex.com/cache3/e3/ql_ca1be3144615ac691b31624c33cb36e3_l3.png">
</p>
<h4>(b)</h4>
<p>
The cosine signal can be represented in a similar sense using the following formula:<br><br>
<img src="https://quicklatex.com/cache3/2b/ql_20213c9d0e33e9e6eca37f550ceeb32b_l3.png">
</p>
<p>
Using this representation the signal <em>x(t)</em> can be defined as:<br><br>
<img src="https://quicklatex.com/cache3/56/ql_cee0e99e2c5add56d991e4185cd49956_l3.png">
</p>
<p>
which leads us to the following coefficients:<br><br>
<img src="https://quicklatex.com/cache3/c6/ql_d16d979d4d295cbb4f415db403f5a6c6_l3.png">
</p>
<hr>
<h2>Fourier Series from Signal Graph [Based on 7.4 from <a href="https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/">Ref1</a>]</h2>
<p>
Consider the following periodic signal:<br><br>
<img src="https://i.ibb.co/MB5QzL2/graph-1.jpg"><br><br>
Let's evaluate the Fourier series analysis equation.
</p>
<h3>Solution</h3>
<p>
From the graph its easy to notice that the period of the signal is <em>T<sub>o</sub> = 4</em> and so the fundamental frequency is:<br><br>
<img src="https://quicklatex.com/cache3/9f/ql_604491b5b8936b640583578433ec559f_l3.png">
</p>
<p>
Chosing a period of integration from <em>- 2</em> to <em>2</em>, can give us the Fourier coefficients as follows:<br><br>
<img src="https://quicklatex.com/cache3/ad/ql_68173374c778794b391af7640751c7ad_l3.png">
</p>
<p>
But, <em>x(t)</em> is inverse symmetrical when taken at <em>- 2</em> to <em>-1</em> and <em>1</em> to <em>2</em> correspondingly.
Let's also note that <em>x(0) = 0</em>.
Thus, its possible to re-write the formula as:<br><br>
<img src="https://quicklatex.com/cache3/8c/ql_85747641b2b37d8f2b9559bfd9535c8c_l3.png"><br><br>
where <em>x(t)</em> is completely gone.
</p>
<p>
Let's continue with the calculations to find the final equation for the coefficients...<br><br>
<img src="https://quicklatex.com/cache3/62/ql_0b15cfd8707fed5344c90c89621c6e62_l3.png">
</p>
<p>
In order to calculate the value of <em>a<sub>o</sub></em>, because it leads to <em>0 / 0</em>, L'Hopital's rule can be applied.
</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>
<h3>Basics</h3>
<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>
  </ul>
<h3>LTI Systems and Convolution</h3>
<ul>
  <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>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-exercises-on-lti-system-properties">Exercises on LTI System Properties</a> &rarr; Superposition, Impulse Response and System Classification Examples</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-exercise-on-convolution">Exercise on Convolution</a> &rarr; Discrete-Time Convolution Example with the help of visualizations</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-exercises-on-system-representation-using-difference-equations">Exercises on System Representation using Difference Equations</a> &rarr; Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-exercises-on-system-representation-using-differential-equations">Exercises on System Representation using Differential Equations</a> &rarr; Equation to Block Diagram Example, Direct Form I to Equation Example</li>
</ul>
<h3>Fourier Series and Transform</h3>
<ul>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-continuous-time-periodic-signals-and-fourier-series">Continuous-Time Periodic Signals & Fourier Series</a> &rarr; Input Decomposition, Fourier Series, Analysis and Synthesis</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-continuous-time-aperiodic-signals-and-fourier-transform">Continuous-Time Aperiodic Signals & Fourier Transform</a> &rarr; Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-continuous-time-fourier-transform-properties">Continuous-Time Fourier Transform Properties</a> &rarr; Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-discrete-time-fourier-series-and-transform">Discrete-Time Fourier Series & Transform</a> &rarr; Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-discrete-time-fourier-transform-properties">Discrete-Time Fourier Transform Properties</a> &rarr; Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities</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 get into exercises on the continuous-time 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>