Five problems sets. To be announced towards the end of each chapter covered
by the class. To be turned in one week later.
Grading:
Each main question will typically carry an equal number of points. Each
problem set will carry 100 points.
Up to 10% extra points for particularly clever / simple solutions.
Be orderly and tidy! If the graders do not understand what you did
you may lose points.
SUPPLEMENT TO GRADING POLICY (March
07 2001, by popular demand)
HW 1 thru 3 will be graded on a scale of 0:100 points
HW 4 will be graded on a scale of 0:50 points
the project will be graded on a scale of 0:200 points
You may choose to shift up to 50 points from HW4 towards the project.
indicate the n. of points on the heading of both HW4 and the project.
By default: if you do not turn HW4 in then it will be worth 0
and the project will be worth 250 points.
Collaboration policy:
- Collaboration allowed in groups of 3 (max) students in figuring out how
to solve each problem. No collaboration allowed in writing the solutions
and in writing the Matlab code. Do not use the notes you wrote while collaborating
when you write your solution.
- You are allowed to compare the printouts of the results of your experiments.
You are not allowed to do so next to an open session of Matlab where you
may be tempted to write code in collaboration.
- You are encouraged to teach Matlab to each other: so it is OK to
work together on Matlab when you are not working on the HW.
Late homework:
Your homework is due in class before the beginning of the lecture
(i.e. before 1:05pm). Late homework is penalized by 1/3 if it is turned
in before the following lecture.
PROBLEM SET 1
PROBLEM SET 2
PROBLEM SET 3 (Due on Ms. Catharine Stebbins' desk, Moore 104, on Fri
March 2 at 12-noon)
-
Prove properties 10.5.1 thru 10.5.8 of the Z-transform
-
Prove the 12 common Z-transform pairs of table 10.2
-
Examine the conditions for stability of a discrete-time LTI system:
-
Define formally stability for a discrete-time system. Be careful with the
quantifiers (for any, for all ...).
-
Prove that if the impulse response of an LTI system is absolutely summable
then the system is stable.
-
Prove that this condition is not only sufficient but also necessary (be
careful with the quantifiers).
-
Prove that if the system function H(z) is rational then a causal (*)
system
is stable iff the poles are within the unit circle
(*) Correction made March 1, 2pm. Thanks to Philip Fung for spotting this
one!
PROBLEM SET 4 (Due on Ms Catharine Stebbins' desk, Moore 104, on Monday
March 12 at 1pm)
Read the text of problem 11.60, ignore the last question.
Instead of solving problem 11.60, solve a modified version:
The discrete-time part of the system (G(Z)) has sampling period T2.
The ``continuous'' part of the system (H(s)) is now discrete [ H(z1)=1/(1-z)
] with a sampling period T1. Assume T1<<T2.
-
Work out the conditions for stability of the new system (K as a function
of T1 and T2)
-
What happens for T2->0 ?
-
Simulate the system in Matlab and verify that your calculations are correct.
Show the impulse response of your system as a function of T1 and T2. Pick
some values of K within the interval of stability and some outside.
Matlab sample code