Math 267 Final Exam
Section 202
April 18, 2011
Duration: 150 minutes
Name:
Student Number:
Do not open exam until so instructed.
• This exam has 22 pages, including this cover.
• One 8.5x11” double-sided page of notes is allowed. No textbooks, calculators, or other
aids are allowed.
• Cell phones and other electronic devices
must be switched off and left at the front
of the room.
• You must remain in the exam room until the
exam is over.
Problem
Out of
1&2
8
3&4
8
5
8
6
12
7&8
12
9
10
10
5
• Write answers in the indicated ’boxes’. Explain your work, using the back of the previous page if necessary.
11
6
12
5
• Relax.
13
5
14
5
15
16
Total
100
Score
Rules governing all UBC examinations:
(i) Each candidate must be prepared to produce, upon request, a Library/AMS card for identification.
(ii) Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors
or ambiguities in examination questions.
(iii) No candidate shall be permitted to enter the examination room after the expiration of one-half hour
from the scheduled starting time, or to leave during the first half hour of the examination.
(iv) Candidates suspected of any of the following, or similar, dishonest practices shall be immediately
dismissed from the examination and shall be liable to disciplinary action.
• Having at the place of writing any books, papers or memoranda, calculators, computers, audio
or video cassette players or other memory aid devices, other than those authorized by the
examiners.
• Speaking or communicating with other candidates.
• Purposely exposing written papers to the view of other candidates. The plea of accident or
forgetfulness shall not be received.
(v) Candidates must not destroy or mutilate any examination material; must hand in all examination
papers; and must not take any examination material from the examination room without permission
of the invigilator.
April 2011
Math 267
Name:
Page 2 out of 22
Problem 1
(4 Marks)
√
Let z = − 3 + i. Write z in polar form, z = Reiθ . Calculate and simplify z 3 − z 3 .
z=
z3 − z3 =
Problem 2
Calculate, I =
(4 Marks)
0
�
2π
e−i 15 k .
k=−15
I=
April 2011
Problem 3
Math 267
Name:
Page 3 out of 22
�
n
�
(4 Marks)
Sketch the graph of non-periodic function f [n] = 2 u[n] − u[n − 4] .
Problem 4
(4 Marks)
Find the complex Fourier coefficients of the function g(t) = cos(t/3) + sin(t/4).
with period T = 24 \pi.
g�[k] =





























April 2011
Math 267
Name:
Page 4 out of 22
Problem 5
(8 Marks)
Find all values σ > 0 so that the following BVP has a non-trivial solution :
 ��
�
�
�
2

 X (x) − 2X (x) + 1 + σ X(x) = 0,
X(0) = 0,


X(1) = 0.
σ=
April 2011
Math 267
Name:
Page 5 out of 22
Problem 6
Let f (t) be the 2π-periodic function that satisfies,
�
3
if 0 < t ≤ π
f (t) =
−1 if π < t ≤ 2π
Part A
(2 Marks)
Sketch the graph of f (t). Show at least three periods.
Part B
(6 Marks)
Calculate the complex fourier coefficients f�[k] of f (t).
(Continued Next Page)
April 2011
Math 267
Name:
Page 6 out of 22
(Problem 6, Part B Continued)
f�[k] =
Part C
Find the value of J =



















(4 Marks)
�∞
k=−∞
f�[k](−1)k .
J=
April 2011
Math 267
Name:
Page 7 out of 22
Problem 7
Part A
(4 Marks)
Find the Fourier transform of f (t) = sinc(t − 2).
f�(ω) =
Part B
Calculate K =
�
(4 Marks)
∞
−∞
|sinc(ω)|2 dω.
K=
April 2011
Math 267
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Page 8 out of 22
Problem 8
Find the inverse Fourier transform of g�(ω) = e−i3ω
g(t) =
�
�
(4 Marks)
�
i(ω − 1) + 2
��
� .
i(ω − 1) + 1 i(ω − 1) + 3
April 2011
Math 267
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Page 9 out of 22
Problem 9
Consider the following Wave Equation:
 2
∂t u(x, t) = ∂x2 u(x, t),






 u(x, 0) = 0


if −1 ≤ x ≤ 0
2



∂t u(x, 0) = h(x) = −2 if 0 ≤ x ≤ 1 ,






0
otherwise.
where t ≥ 0 and x ∈ R.
Part A
(6 Marks)
Use D’Alembert’s formula to find the solution u(x, t).
(Continued Next Page)
April 2011
Math 267
Name:
Page 10 out of 22
(Problem 9, Part A Continued)
u(x, t) =
where F (x) =
G(x) =









Part B
Sketch the solution u(x, t) at t = 2. Label the x-axis and y-axis clearly.
(4 Marks)
April 2011
Math 267
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Page 11 out of 22
Problem 10
(5 Marks)
Calculate the discrete Fourier transform of �x = [1, 1, 1, 0]. Give your answer as a vector,
fully simplified.
�x
�=
April 2011
Math 267
Name:
Page 12 out of 22
Problem 11
Consider periodic discrete-time signal, x[n] = cos( 2π
n) − sin( 2π
n).
6
6
Part A
with period N= 6
(4 Marks)
Find the discrete-time Fourier series (DTFS) coefficients x
�[k] of x[n].
x
�[k] =
Part B
Plot the phase of x
�[k].



















(2 Marks)
April 2011
Math 267
Name:
Page 13 out of 22
Problem 12
(5 Marks)
Consider non-periodic discrete-time signal
� �n
1
x[n] =
u[n − 1].
Discrete-time Fourier
3
transform
Find the DTFT, x
�DT (ω).
Calculate directly from the definition.
Do not use the time-shift property, or any examples you might know.
x
�DT (ω) =
April 2011
Math 267
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Page 14 out of 22
Problem 13
(5 Marks)
Find the z-transform of h[n] = 2n u[n]−3n u[−n−1], and determine its region of convergence.
H(z) =
ROC :
April 2011
Math 267
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Page 15 out of 22
Problem 14
(5 Marks)
Find the inverse z-transform of
H(z) =
z6
1
,
− 4z 4
h[n] =
with ROC |z| > 2.
April 2011
Math 267
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Page 16 out of 22
Problem 15
A causal discrete-time LTI system, with input x[n] and output y[n], is described by the
difference equation,
y[n] = y[n + 1] − 2y[n − 1] + x[n]
Part A
(6 Marks)
Find the system function H(z), and determine its region of convergence.
H(z) =
ROC :
(Continued Next Page)
April 2011
Math 267
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Page 17 out of 22
(Problem 15, Continued)
Part B
(6 Marks)
Find the impulse response h[n].
h[n] =
(Continued Next Page)
April 2011
Math 267
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Page 18 out of 22
(Problem 15, Continued)
Part C
(4 Marks)
Does the discrete-time Fourier transform of impulse response h[n] exist?
If yes, find �
hDT (ω). If no, give a brief explanation.
the discrete-time
Fourier transform.
�
hDT (ω) =
or
explanation
End of Exam
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