y(t)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems—Fall 2003
Problem Set 10
Issued,November 25,2003 Due,December 5,2003
REMINDER,Computer Lab 3 is also due on December 5,
Reading Assignments,
Lectures #21-22 & PS#10,Chapters 10 & 11 (through Subsection 11.3.4) of O&W
Lectures #23-24 & PS#11,Chapters 10 & 11 (through Subsection 11.3.4) of O&W
Exercise for home study (not to be turned in,although we will provide solutions),
(E1) O&W 11.32 (a) through (d)
Problems to be turned in:
Problem 1 Consider the following feedback con?guration.
x(t)
+
e(t)
K
G(s)
Sketch the root loci for K > 0 and K < 0 for each of the following:
1
(a) G(s) =,
s + 1
1
(b) G(s) =
(s? 5)(s + 3)
s + 1
(c) G(s) =
2
,For this part,clearly indicate the point at which the closed-loop system
s
has a double-pole,
1
y(t)
Problem 2 Consider the system shown below:
x(t)
+
e(t)
K(s)
1
s(s + 10)
(a) Compute the steady state tracking error,e(?),due to a unit step input x(t) = u(t)
when K(s) = K,Does the steady state tracking error change as K changes?
(b) Compute the steady state tracking error,e(?),due to a ramp input x(t) = tu(t) when
K(s) = K,Does the steady state tracking error change as K changes?
(c) Assume,for this part,that K(s) = 1,Find systems K
f
(s) and K
s
(s) in the modi?ed
system shown below such that the steady state tracking error due to the ramp input
x(t) = tu(t) becomes zero,Hint,One of the two systems,K
f
(s) and K
s
(s),is a
constant gain,Note that the tracking error is de?ned to be e(t) = x(t)? y(t),
x(t) K
s
(s) +
+
K
f
(s)
K(s)
1
s(s
y(t)
+ 10)
Problem 3 O&W 11.27
Problem 4 Determine the z-transform for each of the following sequences,Sketch pole-zero
plot and indicate the region of convergence,Indicate whether or not the Fourier transform
of the sequence exists,
(a) x[n] = 2?[n + 3][n? 2]
(b) x[n] = 2
n
u[n? 1] + 4
n
u[?n]
2
Problem 5 For each of the following z-transforms,determine the inverse z-transform
5
(a) X(z) = 12z
4
z
1
+ 6 + 9z
2
8z
5 1 1
(b) X(z) =
1 +
1
z
1
1
z
2
,< |z| <
3 2
6 6
Problem 6 Consider a signal y[n] which is related to two signals x
1
[n] and x
2
[n] by
y[n] = x
1
[?n? 2]? x
2
[n + 4]
where

1
n
1
n
x
1
[n] =? u[n] and x
2
[n] = u[n],
2 4
Determine the z-transform Y (z) of y[n],together with its ROC,
Reminder,The?rst 20 problems in each chapter of O&W have answers included at the
end of the text,Consider using these for additional practice,either now or as you study for
tests,
3