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## Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about **Signals and Systems** in order to cover **Exercises on System Representation using Differential Equations**.

So, without further ado, let's dive straight into it!

## Differential Equation Representation

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:

This representation can lead us to very useful Block Diagram Representations, such as the Direct Form I and II implementations.

In the examples that follow we will convert from one to the other, to understand those representations better.

## Equation to Block Diagram [Based on P6.6 from Ref1]

Let's first consider a LTI system that is represented by the following differential equation:

Let's draw the Direct Form I and II realizations of this system.

### Solution

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:

Solving for *y(t)* leads to:

which can be easily turned into the Direct Form I implementation.

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:

Combining the two integrators gives us the Direct Form II implementation:

## Block Diagram to Equation [Based on P6.8 from Ref1]

Let's now do the opposite.
Consider the following Direct Form I Block Diagram that represents an LTI system:

Let's find the differential equation representation for this system.

### Solution

Let's first find the equation that describes the intermediate signal *r(t)*.
Two integrations of *x(t)* are made and so:

Differentatiating twice yields the following result:

which shows us how *x(t)* and *r(t)* are related.

Now for the second "block". Relating *y(t)* and *r(t)* is as simple as:

Differentiating twice in order to get the second derivative of *r(t)* (and *y(t)* at the same time), and also solving for this derivative, gives us:

Equating these two relations, gives us the final answer that equates *x(t)* and *y(t)*, which is:

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).

## RESOURCES:

### References

### Images

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io

## Previous articles of the series

### Basics

- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples
- Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time

### LTI Systems and Convolution

- LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
- LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
- System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
- System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
- Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples
- Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
- Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example

### Fourier Series and Transform

- Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
- Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
- Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
- Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
- Discrete-Time Fourier Transform Properties → 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

## Final words | Next up

And this is actually it for today's post!

Next time we will start getting into exercises on the Fourier Series and/or Transform!

See Ya!

Keep on drifting!