MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.003,Signals and Systems—Fall 2003
Problem Set 3
Issued,September 18,2003 Due,September 26,2003
Reading Assignments,
Lectures #5-6 & PS#2,Chapter 3 of Oppenheim and Willsky (O&W)
Lectures #3-4 & PS#3,Chapters 3&4 of Oppenheim and Willsky (O&W)
Exercise for home study (not to be turned in,although we will provide solutions),
O&W 3.46 (a),(c)
Problems to be turned in:
Problem 1 O&W 3.22 (a) - only the signal in Figure p3.22 (c)
Problem 2 O&W 3.23 (a)
Problem 3 Determine the Fourier series coe?cients for the periodic signal x[n] depicted
below,Plot the magnitude and phase of these coe?cients.
x[n]
2
12?6 0 6 12
· · · · · ·
n
1
1
Problem 4 O&W 3.29 (a)
1
Problem 5 Consider the following CT periodic signals,x(t),y(t),and z(t).
x(t)
1
t
6?5?4?3?2?1 0 1 2 3 4 5 6
y(t)
6?4?2 2 4 6
1
1
t
z(t)
6?5
4?3
2?1
1 2 3 4 5 6
1
1
t
(a) Determine the fundamental frequency,period,and Fourier series coe?cients, a
k
,for
x(t),
(b) Determine the fundamental frequency,period,and Fourier series coe?cients,b
k
,for
y(t),
(c) Determine the fundamental frequency and period for z(t),Also,using the results of
parts (a) and (b),determine the Fourier series coe?cients,c
k
for z(t),
Problem 6 Let x(t) be a periodic signal with fundamental period T and Fourier series
coe?cients a
k
,Derive the Fourier series coe?cients of each of the following signals in terms
of a
k
,
(a) Od{x(t? T/2)}
(b) x(T/4? t)
2
Problem 7 O&W 3.31 (also determine a
0
)
Problem 8 O&W 3.51
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