Download Math 267 Final Exam December 2011 and more Exams Mathematical Methods in PDF only on Docsity!
Math 267 Final Exam
Section 101
December 10, 2011
Duration: 150 minutes
Name: Student Number:
Do not open exam until so instructed.
- This exam has twenty-one pages, including
this cover, and four blank pages at the end.
- One 8.5x11” double-sided page of notes is al-
lowed. 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.
- Write answers in the indicated ’boxes’. Ex-
plain your work, using the back of the previ-
ous page if necessary.
- Problems are not organized by difficulty.
- Relax.
Pages Marks Score
2-3 14
4-6 16
7-9 16
10-12 19
13-15 19
16-17 16
Total 100
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.
Problem 1 (4 Marks)
Plot complex numbers α = 2e
−i
π (^6) and β = e
+i
3 π (^4) , in the space provided. Label axes appro-
priately.
Problem 2 (3 Marks)
Rewrite cos(x) cos(3x) as a simple sum of cosines.
cos(x) cos(3x) =
Problem 4
For both parts of this question, consider g(x), defined only for x ∈ [0, 2 π],
g(x) =
0 for 0 ≤ x ≤
π
1 for
π
< x < π
0 for π ≤ x ≤ 2 π
Part A (2 Marks)
For x ∈ [− 2 π, 2 π], sketch the odd extension g odd (x) in the space provided,
Part B (6 Marks)
Compute the coefficients c k,odd for the Fourier series of g odd (x).
(Continued Next Page)
(Problem 4, Part B Continued)
c k,odd
Problem 6
Show your work. You may use any formula or shortcut discussed in lecture.
Part A (5 Marks)
Compute the Fourier transform of,
f (ω) = 3 sinc(ω + 1)
F f (ω) =
(Continued Next Page)
Part B (6 Marks)
Compute the Fourier transform of,
g(t) = e
it e
−t u(t)
F g(t) =
(Continued Next Page)
Problem 7 (7 Marks)
Using the definition of convolution, compute,
(q ∗ q)(x), where, q(x) = x u(x)
(q ∗ q)(x) =
Problem 8 (6 Marks)
Compute (f ~ g)[n], where the discrete-time signals f [n] and g[n] are given by,
Hint: You may leave your answer in terms of g[n].
(f ~ g)[n] =
Problem 10
Consider the discrete-time LTI described by the difference equation,
y[n] − y[n − 1] = x[n] + x[n − 1]
Part A (7 Marks)
Use the difference equation to find the impulse response, h[n], with h[−1] = 0.
h[n] =
(Continued Next Page)
Part B (1 Marks)
Is your answer to Part A right-sided, left-sided, neither, or both?
Part C (7 Marks)
Use the difference equation and your answer to Part B to find the system function, H(z).
H(z) =
ROC =
Problem 12
Part A (6 Marks)
Find the DFT of, a[n] = [0, 1 , 0 , 1]. Give your answer as a vector, fully-simplied.
̂ a[k] = [ , , , ]
Part B (4 Marks)
Use your answer to Part A to find the DFT of b[n] = [1, 0 , 1 , 0].
b[k] = [ , , , ]
Problem 13 (6 Marks)
Compute the DFT of x[n] = sin[
π
5
n].
Hint: Rewrite x[n] as complex exponentials.
̂ x[k] =
End of Exam
Blank Page
Blank Page
