EE202 Signals and Systems
KAIST, Spring 2023
Friday, March 10, 2023
Handout #
Due: Wednesday, March 22, 11:59pm
Problem Set 1
Announcement:
The grade of this problem set will be one of {0, 1, 2, 3, 4} based on rough estimate of the
performance (the amount of the problems you solved).
4: above 75%
3: 50% – 75%
2: 25% – 50%
1: 0% – 25%
Office hours (Zoom) by appointments.
Problem 1.1
Problem 1.5, 1.6, 1.8, 1.14 from Oppenheim and Willsky.
Problem 1.2
Problem 1.18, 1.19, 1.27(d),(e) from Oppenheim and Willsky.
There is a typo with O&W 1.19 (d):
it should be y(t) = Od{x(t)}, i.e. y[n] ⇒ y(t).
Problem 1.3 (Complex Exponential)
An exponential function f (z) = ez can be defined for a complex number z by
z
e =
∞
X
zk
k=0
k!
.
(a) Show that
ejθ = cos θ + j sin θ.
(b) Using the result of part (a), show that
ejθ1 ejθ2 = ej(θ1 +θ2 ) .
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Problem 1.4 (Linearity)
(a) Show that the following two conditions are equivalent for a continuous time system
T [·] :
(i) T [ax1 (t) + bx2 (t)] = aT [x1 (t)] + bT [x2 (t)] for all a, b, x1 (t), x2 (t).
(ii) T [x1 (t)+x2 (t)] = T [x1 (t)]+T [x2 (t)] for all x1 (t), x2 (t) and T [ax1 (t)] = aT [x1 (t)]
for all a, x1 (t).
(b) Show that if a continuous time system is linear, then the following superposition
property is satisfied:
m
m
X
X
m
T[
ak xk (t)] =
ak T [xk (t)] for all m, {ak }m
k=1 , {xk (t)}k=1 .
k=1
k=1
(c) Problem 1.40 (a) from Oppenheim and Willsky.
(d) Consider a linear operator T : Rn → Rm , which maps an n-dimensional vector
to an m-dimensional vector. As this operator is linear, it satisfies the following
superposition property:
X
X
ak T [xk ] for all scalars {ak } and vectors {xk ∈ Rn }.
ak x k ] =
T[
k
k
Show that for this linear operator T [·], there exists a matrix A ∈ Rm×n such that
T [x] = Ax.
( Hint: Any vector x ∈ Rn can be represented as a linear combination of simple
 
0
 0 
 . 
 . 
 . 
vectors, {e1 , e2 , . . . , en }, where ek =   (an elementary vector whose k th element
 1 
 . 
 .. 
0
is 1 and all other elements are zeros) as follows:


x1
n
 x2  X


x =  ..  =
xk e k .
 .  k=1
xn
Let ak = T [ek ], which is an m-dimensional vector, and find the matrix A. )
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