<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>System Representation in Continuous-Time using Differential Equations</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>Linear Constant-Coefficient Differential Equation</h2>
<p>
In Continuous-Time, Linear Time-Invariant (LTI) Systems are quite easily representable using linear differential equations with constant-coefficients (LCCDE).
Such equations are of the form:<br><br>
<img src="https://quicklatex.com/cache3/45/ql_868f6a543a241771ded49053029b8445_l3.png"><br><br>
where each derivative of the system response and input is in Leibniz notation.
</p>
<h3>Homogeneous Equation</h3>
<p>
In order to find the solution, let's first set the sum that contains the system response equal to zero.
That way its easy to end up with a homogeneous equation with solution <em>y<sub>h</sub></em>:<br><br>
<img src="https://quicklatex.com/cache3/b0/ql_eeff00ab25dc3c6c0a42ca8d51db9bb0_l3.png">
</p>
<p>
Due to the nature of those equations, whenever some particular <em>y<sub>p</sub>(t)</em> satisfies the general equations, there has to exist some <em>y<sub>h</sub>(t)</em>, for which their sum <em>y<sub>p</sub>(t) + y<sub>h</sub>(t)</em> satisfies the homogeneous equation.
</p>
<p>
Notation explained:
<ul>
  <li><em>y<sub>p</sub>(t)</em>: Particular solution</li>
  <li><em>y<sub>h</sub>(t)</em>: Homogeneous solution</li>
</ul>
</p>
<h3>Homogeneous Solution</h3>
<p>
The homogeneous solution has to be of the following exponential form:<br><br>
<img src="https://quicklatex.com/cache3/05/ql_d97b541e68fab7fc8e9c69c4a2626a05_l3.png">
</p>
<p>
Substituting this form in the homogeneous equation results in:<br><br>
<img src="https://quicklatex.com/cache3/de/ql_70f7515cc421351628a0c53adc41bbde_l3.png">
</p>
<p>
The equation:<br><br>
<img src="https://quicklatex.com/cache3/21/ql_20d360f1c3a19fd4a2b83334f27ca321_l3.png"><br><br>
has N roots, <em>s<sub>i</sub></em>, which give us the homogeneous solution:<br><br>
<img src="https://quicklatex.com/cache3/17/ql_b7747e7aae4ada215518ac87f02b9517_l3.png">
</p>
<h3>Auxiliary Conditions</h3>
<p>
Combining such a homogeneous solution with a particular solution, its possible to calculate the solution of the differential equation.
In other words, the solution will be of the form:<br><br>
<img src="https://quicklatex.com/cache3/8b/ql_0efdf78638924c158cd1daea4a551b8b_l3.png">
</p>
<p>
For this to be true, N auxiliary conditions are needed, which are basically the values of the derivatives of the system response <em>y</em> for some initial time <em>t = t<sub>o</sub></em>:<br><br>
<img src="https://quicklatex.com/cache3/9f/ql_cc4559682190f06df8274c2b31875b9f_l3.png">
</p>
<p>
For linear systems the initial conditions are usually 0, whilst for causal LTI systems, at initial rest:<br><br>
<img src="https://quicklatex.com/cache3/63/ql_f7e2e3c2d527d15f62b528e717edc063_l3.png">
</p>
<h3>Explicit Recursive Solution</h3>
<p>
Assuming causality, when given:<br><br>
<img src="https://quicklatex.com/cache3/c3/ql_526f00028721e73312f1b2f1c2300ac3_l3.png"><br><br>
its possible to calculate <em>y(t<sub>o</sub>)</em>.
</p>
<p>
Similarly, this newly calculated value can be used in order to calculate <em>y(t<sub>o</sub> + 1)</em>.
Thus, this procedure can be continued on indefinitely, and so recursively.
</p>
<p>
As such, at initial rest, where the rest solution is known from auxiliary solutions, any other solution will be calculated from:<br><br>
<img src="https://quicklatex.com/cache3/3b/ql_c45f7ea9ac311812c4af9a36550d1b3b_l3.png">
</p>
<hr>
<h2>Block Diagram Representation</h2>
<p>
Replacing the delay with an integrator, the block diagram for continuous-time is basically the same as the block diagram for discrete-time.
Integration is visualized using an integration block, whilst the various coefficients are put on top of the connection lines.
</p>
<h3>Direct Form I Implementation</h3>
<p>
In Direct Form I, a system in continuous-time can be represented as follows:<br><br>
<img src="https://i.ibb.co/bQkvTGG/direct-form-I.jpg"><br><br>
</p>
<p>
The two subsystems, or chains, can also be connected in reverse, resulting into the following representation:<br><br>
<img src="https://i.ibb.co/CMnQ2yx/direct-form-I-reversed.jpg">
</p>
<h3>Direct Form II Implementation</h3>
<p>
Combining the two chains of integrators into one chain, its possible to compact the design even more.
This form is known as the Direct Form II Implementation, and is visualized as follows:<br><br>
<img src="https://i.ibb.co/FzwwVwB/direct-form-II.jpg">
</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>
</ul>
<hr>
<h2>Final words | Next up</h2>
<p>And this is actually it for today's post!</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>