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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Department of Electrical and Computer Engineering
ECE 401 Signal and Image Analyais
Spring 2022
EXAM 1
Wednesday, September 28, 2022
• This is a CLOSED BOOK exam.
- You are permitted one sheet of handwritten notes, 8.5x11.
- Calculators and computers are not permitted.
- If you’re taking the exam online, you will need to have your webcam turned on. Your exam will appear
on Gradescope at exactly 1:00pm; you will need to photograph and upload your answers by exactly
2:00pm.
- There will be a total of 100 points in the exam. Each problem specifies its point total. Plan your work
accordingly.
• You must SHOW YOUR WORK to get full credit.
Name:
NetID:
Phasors
A cos(2πf t + θ) = <
Ae
jθ e
j 2 πf t
e
−jθ e
−j 2 πf t
e
jθ e
j 2 πf t
Spectrum
Scaling: y(t) = Gx(t) =
N ∑
k=−N
(Gak) e
j 2 πfk t
Add a Constant: y(t) = x(t) + C = (a 0
+ C) +
k 6 =
a k
e
j 2 πfk t
Add Signals: If fk = f
′
n
= f
′′
m
then ak = a
′
n
′′
m
Time Shift: y(t) = x(t − τ ) =
N ∑
k=−N
ake
−j 2 πf k
τ
e
j 2 πf k
t
Frequency Shift: y(t) = x(t)e
j 2 πF t
=
N ∑
k=−N
ake
j 2 π(fk +F )t
Differentiation: y(t) =
dx
dt
N ∑
k=−N
(j 2 πfkak) e
j 2 πfk t
Fourier Series
Analysis: X k
T 0
T 0
0
x(t)e
−j 2 πkt/T 0 dt
Synthesis: x(t) =
∞ ∑
k=−∞
Xke
j 2 πkt/T 0
Sampling and Interpolation:
x[n] = x
t =
n
F
s
f a
= min (f mod F s
, −f mod F s
z a
z f mod Fs < −f mod Fs
z
∗ f mod F s
−f mod F s
y(t) =
∞ ∑
n=−∞
y[n]p(t − nT s
- (25 points) x(t) is a signal with a period of 0.01 seconds, and with the following shape:
x(t) =
1 0 < t < 0. 001
0 0. 001 < t < 0. 005
− 1 0. 005 < t < 0. 006
0 0. 006 < t < 0. 01
(a) What are the Fourier series coefficients X k
for k 6 = 0? Your answer should contain no variables
other than k, but you don’t need to simplify.
(b) Suppose that y(t) is a signal such that x(t) =
dy
dt
. Express the Fourier series coefficients Yk in terms
of the Fourier series coefficients Xk. Note that you don’t need to solve part (a) in order to solve
this part of the problem.
(b)
x(t) = 3 cos
2 π 12 , 000 t +
π
What is z(t)?
- (25 points) Suppose that
x[n] = sin
πn
1 n is odd, and
n− 1
2
is even
− 1 n is odd, and
n− 1
2
is odd
0 otherwise
You would like to generate a continuous-time audio signal, y(t), using the interpolation formula
y(t) =
∞ ∑
n=−∞
x[n]g (t − n)
In each of the following cases, specify the value of y(t) over the range 0 ≤ t ≤ 4. You may specify y(t)
by drawing a plot of the function (if your plot clearly shows the value at each point in time in the range
0 ≤ t ≤ 4), or by using an equation or a set of cases.
(a) What is y(t) if
g(t) =
1
2
≤ t ≤
1
2
0 otherwise
