SIGNALS AND SYSTEMS II LABORATORY 1

INTRODUCTION

The Laplace transform offers significant advantages for modeling continuous-time physical systems versus
linear differential equations with constant coefficients. The transformed differential equations are algebraic,
and hence are easier to solve. We have learned many ways to solve differential equations using Laplace
transforms last semester in lab 4 (The lab is located on the ECE 311 website under the Computer Labs
link). In this lab we will review how to use the Laplace transform to solve differential equations, change the
result from frequency-domain to time-domain using the inverse Laplace transform, and simulate the result
using MATLAB’s SIMULINK toolbox. The other goal of this lab is to build analog filters using
SIMULINK and study their function by filtering various (noisy) signals.

REVIEW OF MATLAB TOOLS

In ECE 311, we learned:

1. Finding the roots of a polynomial using ‘roots’

2. Multiplying Polynomials using ‘conv’

3. Adding Polynomials . Only polynomials with the same length can be added together.
m=length(x); n=length(y);
if m>=n
z=x+[zeros(1,m-n),y];
else
z=y+[zeros(1,n-m),x];
end

4. Evaluating Polynomials using ‘polyval’

5. For typical systems the transfer function can be ex pressed as a rational function, such as

6. Partial Fraction Expansion using ‘residue’
»b=[1 0 1]; % B(s)
»a=[1 6 11 6]; % A(s)
»[gamma,alpha,k]=residue(b,a)
gamma =
5.0000
-5.0000
1.0000
alpha =
-3.0000
-2.0000
-1.0000
k = []

7. Inverse Laplace Transform using ‘ilaplace’ (this was not introduced in ECE 311)
» syms F s
» F=(s^2+1)/(s^3+6*s^2+11*s+6);
»ilaplace(F)
ans =
5*exp(-3*t)-5*exp(-2*t)+exp(-t)

8. Plotting Complex Frequency Response of H(s) using ‘freqs’
» b=[1 0 1];
» a=[1 6 11 6];
» freqs(b,a)
% The output of this is seen in Figure One below.

Figure One Output of ‘freqs’

9. Create Bode Diagrams using ‘bode’ (frequency response with dB magnitude plot)
» sys=tf([1 0 1],[1 6 11 6]);
» bode(sys)
% This is shown in Figure Two below.

Figure Two Bode plot of the same system from Figure One

SIMULINK

Now we wish to use SIMULINK to simulate the system, e.g., see the output when the input is a step
function (see the Figure Three below). SIMULINK is a powerful simulation tool provided by MATLAB.
It al lows for analysis /simulation of interconnections of dynamic systems (both continuous-time and
discrete-time).

You can easily build models from scratch, or take an existing model and edit it. Simulations are interactive,
so you can change parameters “on the fly” and immediately see what happens. In this lab you will be
learning to use SIMULINK.

Figure Three SIMULINK diagram for the step response
of the system used in Figures One and Two

Running, Plotting, Printing

In Math -software/second-order-ode-solver.html">order to see a demonstration of a ( fairly complex ) SIMULINK diagram type »thermo at the MATLAB
prompt (see Figure Four below). Open the scope block, labeled “Thermo Plots” by double clicking, and
then run the simulation using the buttons or pull down menus provided. You can print the plot of the
simulation output (scope block) and also print the simulation model itself.

Model Building

Figure Three above shows a SIMULINK model for an ODE. You can launch the SIMULINK library
browser from within MATLAB by using the button or typing: »simulink at the command prompt. Then
open a new model (using button or pull down menus), and build a copy of the above model. This is
achieved by dragging components from the library to the model and connecting them using the mouse.
Double clicking a box then allows you to edit the contents, such as entering values for the transfer function
(as shown above).

Look around at the (many) available blocks in SIMULINK. You will certainly need to look in Sources,
Sinks, Continuous, Math. When you have built a copy of the model save it with the name “myode” (it will
actually be saved as myode.mdl). You can then launch this model later from MATLAB simply by typing
»myode at the command line. Go under Simulation to Parameters and you can change the simulation
time. You can then run the simulation and print the results from the scope block (note the autoscale button
on the scope).

Having completed this exercise you should have a model plot and a simulation run. You can now try
varying the ODE and input parameters, and easily see how they affect the solution (note also that you can
enter variable names in SIMULINK blocks if you like , and it will read them from the MATLAB
workspace).

Figure Four Thermo demo in SIMULINK

ANALOG FILTERS

We have learned about analog filters such as Butterworth and Chebychev filters in lab 4 for ECE 311. In
this lab we’ll use SIMULINK to filter noisy signals using these filters.

Figure Five SIMULINK diagram in ‘square.mdl’