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| author | drifter1 |
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| permlink | mathematics-signals-and-systems-exercises-on-system-representation-using-differential-equations |
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| title | "Mathematics - Signals And Systems - Exercises on System Representation using Differential Equations" |
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| body | "<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 System Representation using Differential Equations</strong>.
</p>
<p>
So, without further ado, let's dive straight into it!
</p>
<hr>
<h2>Differential Equation Representation</h2>
<p>
Let's quickly recap how continuous-time LTI systems are represented.
Such systems can be represented using linear constant-coefficient differential equations (LCCDE), which are of the form:<br><br>
<img src="https://quicklatex.com/cache3/45/ql_868f6a543a241771ded49053029b8445_l3.png">
</p>
<p>
This representation can lead us to very useful Block Diagram Representations, such as the Direct Form I and II implementations.
</p>
<p>
In the examples that follow we will convert from one to the other, to understand those representations better.
</p>
<hr>
<h2>Equation to Block Diagram [Based on P6.6 from <a href="https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/">Ref1</a>]</h2>
<p>
Let's first consider a LTI system that is represented by the following differential equation:<br><br>
<img src="https://quicklatex.com/cache3/ae/ql_a5cb7f6e2fc60580a7f1c1dea2313aae_l3.png">
</p>
<p>
Let's draw the Direct Form I and II realizations of this system.
</p>
<h3>Solution</h3>
<p>
The diagram for continous-time systems contains integrator blocks, and so the first thing that one should do is integrate both sides of the given differential equation.
This leads us to the following:<br><br>
<img src="https://quicklatex.com/cache3/8d/ql_ec898471b92df32e8cd6aa38cd70ba8d_l3.png">
</p>
<p>
Solving for <em>y(t)</em> leads to:<br><br>
<img src="https://quicklatex.com/cache3/de/ql_a6a74da0d0f1cda72ebefbf67a5618de_l3.png"><br><br>
which can be easily turned into the Direct Form I implementation.
</p>
<p>
<img src="https://i.ibb.co/pdMQ0SM/1-a.jpg">
</p>
<p>
Now for the Direct Form II implementation.
First of all, the system is linear and time-invariant and so the two "sub-systems" can be easily interchanged as follows:<br><br>
<img src="https://i.ibb.co/bJX3xNs/1-b.jpg">
</p>
<p>
Combining the two integrators gives us the Direct Form II implementation:<br><br>
<img src="https://i.ibb.co/zXbxF4w/1-c.jpg">
</p>
<hr>
<h2>Block Diagram to Equation [Based on P6.8 from <a href="https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/">Ref1</a>]</h2>
<p>
Let's now do the opposite.
Consider the following Direct Form I Block Diagram that represents an LTI system:<br><br>
<img src="https://i.ibb.co/C9fxwct/2-a.jpg">
</p>
<p>
Let's find the differential equation representation for this system.
</p>
<h3>Solution</h3>
<p>
Let's first find the equation that describes the intermediate signal <em>r(t)</em>.
Two integrations of <em>x(t)</em> are made and so:<br><br>
<img src="https://quicklatex.com/cache3/71/ql_64587b5a645b533a6b0177abc47a0471_l3.png">
</p>
<p>
Differentatiating twice yields the following result:<br><br>
<img src="https://quicklatex.com/cache3/bc/ql_4cbe148f67ff0611880ce760e4754dbc_l3.png"><br><br>
which shows us how <em>x(t)</em> and <em>r(t)</em> are related.
</p>
<p>
Now for the second "block". Relating <em>y(t)</em> and <em>r(t)</em> is as simple as:<br><br>
<img src="https://quicklatex.com/cache3/75/ql_c1107a01e4ffe23cd2c24b87a450eb75_l3.png">
</p>
<p>
Differentiating twice in order to get the second derivative of <em>r(t)</em> (and <em>y(t)</em> at the same time), and also solving for this derivative, gives us:<br><br>
<img src="https://quicklatex.com/cache3/44/ql_885310d12a3fc510a564a8876def9f44_l3.png">
</p>
<p>
Equating these two relations, gives us the final answer that equates <em>x(t)</em> and <em>y(t)</em>, which is:<br><br>
<img src="https://quicklatex.com/cache3/ae/ql_3eb360f56d7894f1826b980ef76b1aae_l3.png">
</p>
<p>
This final representation can also be taken by using the "theory" block diagram as an example, but its easier to make mistakes that way.
That's why I chose to insert an intermediate signal.
Furthermore, this whole process is more convenient, easier to remember, and more applicable to any situation (and so even on non-standard forms of block diagrams).
</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> → Signals, Systems</li>
<li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-signal-basics">Signal Basics</a> → 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> → 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> → 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> → 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> → 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> → 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> → 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> → 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> → 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> → 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> → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE 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> → 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> → 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> → 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> → 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> → 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 start getting into exercises on the Fourier Series and/or 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>" |
|---|
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