The questions on the "half-sheets" of paper that I gave you on
Wednesday fell into three main categories: (1) linear, time-invariant
systems, (2) difference equations, and (3) MATLAB. I hope that class
today and Monday will tell you more about difference equations, and
that Lab 2 will help answer your MATLAB
questions.
Below is a practice problem for classifying systems
according to linearity and time-invariance.
You do not have to submit a solution, but please
work the problem for Monday, and I will look at your solutions
and answer your questions during class.
Consider the following three systems, where x(t) is the
input signal and y(t) is the output signal.
- y(t) = 2 x(t) + 1
- Differentiator: y(t) = dx / dt
- Half-wave rectifier: y(t) = x(t) if x(t) >= 0, y(t) = 0 if x(t)
< 0.
Sketch the output
y_{1}(t), y_{2}(t), y_{3}(t),
y_{4}(t), and
y_{5}(t)
from each system when the input is
x_{1}(t), x_{2}(t), x_{3}(t),
x_{4}(t), and x_{5}(t)
defined as follows:
- x_{1}(t) in the graph to the right
- x_{2}(t) = -2 x_{1}(t)
- x_{3}(t) = x_{1}(t-3)
- x_{4}(t) = 1 for all t
- x_{5}(t) = x_{1}(t) + x_{4}(t)
I would suggest that you sketch the input and output signal
for each case.
What can you conclude about linearity and time-invariance for
each system?
If you would like to practice with discrete-time convolution,
here is a problem that I will ask you to solve next week for homework.
Find and sketch the sequence y[n] = x[n] * h[n],
where x[n] and h[n] are defined below and * denotes convolution.
[GRAPHIC NOT AVAILABLE IN HTML FILE -- SEE PAPER VERSION]
You can
check your result using MATLAB and the conv command
as follows:
>> h = [1.5, 1.5, 1.5];
>> x = [2, 2, 2, 2, 2];
>> y = conv(x, h);
>> n = 0:length(y)-1;
>> stem(n, y)
A note about MATLAB help:
Remember that MATLAB has on-line help available.
For example, typing help conv will describe the
convolution command and how to use it.
Also, if you type hthelp on the Suns,
then a hypertext version of help will be opened in a new window.
This might be useful to view the available commands and follow
links to related commands.