EECS
700: Discrete-Time Processing
Objective
The goal of this exercise is to gain a thorough understanding of
the discrete-time differentiator and the discrete-time integrator.
Part of this lab exercise is MATLAB based, and part is pencil-and-paper
based. You may submit the pencil-and-paper portion through the
usual methods (i.e. the EECS front desk, in class, or in the
lab). You may submit the MATLAB part electronically via the
Blackboard Digital Drop Box (preferred) or also via e-mail to the class
TA. Please complete the MATLAB portion in a single .m file that
can generate all the necessary plots when it is executed. The
resulting plots should be cut-and-pasted into a Word document for
submission along with your comments and observations. Your .m
file can be submitted as an appendix to the Word document. The
Word document should have a rough outline that matches the flow of the
lab below (i.e. Part I, numbers 1-7, Part II, numbers 1-4).
Part I: Discrete-Time
Differencer
- Complete Exercise 3.36
in the class textbook.
- Reproduce Figure
3.3.17 in the
class textbook. Note that you must select Wc*T = pi, which yields
Equation (3.59). For the purposes of computing Equation (3.59),
you
may assume T=1.
- Complete Exercise 3.37
in the class textbook.
- Reproduce Figure
3.3.18 in the
class textbook. I found the following MATLAB functions to be
helpful: blackman(), freqz().
- Using whichever method
you prefer
(pencil and paper, or MATLAB), determine the frequency response of a
discrete-time "differencer,"
i.e. the system with y(n) = x(n) - x(n-1). You should find that
the
differencer does an adequate job as long as the sample rate is high
enough.
- Sample a known signal
[such as
cos(wt)] using a sample rate that is at
least 5 times greater than the Nyquist
rate. Pass this signal through the first central difference
approximation and plot it along with its analytical derivative.
What
observations do you have? I found the MATLAB filter() function to be helpful.
- Sample a known signal
[such as
cos(wt)] using a sample rate that is close
to the Nyquist rate.
Pass this signal through the first central difference approximation and
plot it along with its analytical derivative. What observations
do you
have?
Part II: Discrete-Time
Integrator
Here are the steps for the final
part of the exercise:
- Complete Exercise 3.39
in the class textbook.
- Reproduce Figure
3.3.22 in the
class textbook.
- Sample a known signal
[such as
cos(wt)] using a sample rate that is at
least 5 times greater than
the Nyquist
rate. Pass this signal through the discrete-time integrator using
the
trapezoidal rule and plot it along with its analytical integral.
What
observations do you have?
- Sample a known signal
[such as
cos(wt)] using a sample rate that is close
to the Nyquist rate. Pass this signal through the
discrete-time integrator using the trapezoidal rule and plot it along
with its analytical integral. What
observations do you have?
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