THE UNIVERSITY OF BRITISH COLUMBIA
Department of Electrical and Computer Engineering
ELEC 221 – Signals and Systems
Instructor: Dr. Joseph Yan
Final Exam for April 17, 2018
Three Double-sided 8.5”x11” sheets of note
...
THE UNIVERSITY OF BRITISH COLUMBIA
Department of Electrical and Computer Engineering
ELEC 221 – Signals and Systems
Instructor: Dr. Joseph Yan
Final Exam for April 17, 2018
Three Double-sided 8.5”x11” sheets of notes are permitted. No calculators are permitted.
Time: 180 minutes.
This examination consists of 10 physical pages (last two pages intentionally blank for your
overflow work). Please check that you have a complete copy. You may use both sides of
each sheet as needed.
# MAX GRADE
1 51
2 21
3 28
4 22
5 23
6 35
TOTAL 180
IMPORTANT NOTE: The announcement “stop writing” will be made at the end of the examination.
Anyone writing after this announcement will receive a score of 0. No exceptions, no excuses.
All writings must be on this booklet. The blank sides on the reverse of each page may also be used.
Each candidate should be prepared to produce, upon request, his/her Library/AMS card.
Read and observe the following rules:*
No candidate shall be permitted to enter the examination room after the expiration of one-half hour, or
to leave during the first half-hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or
ambiguities in examination-questions.
Caution - Candidates guilty of any of the following, or similar, dishonest practices shall be immediately
dismissed from the examination and shall be liable to disciplinary action:
Making use of any books, papers or memoranda, calculators, audio or visual cassette players or
other memory aid devices, other than as 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.
Surname First name
Student Number
READ THIS2
1. “Short” Answer Questions:
a) (3pts) Determine the z-transform (including RoC) of 𝑥ሾ𝑛ሿ ൌ 2𝛿ሾ𝑛ሿ െ 0.5𝛿ሾ𝑛 െ 2ሿ 𝛿ሾ𝑛 െ 3ሿ.
b) (5pts) Consider two voltage signals 𝑣ଵሺ𝑡ሻ ൌ 𝐴 cosሺ2000𝜋 𝑡 𝜑ଵሻ and 𝑣ଶሺ𝑡ሻ ൌ 𝐴 cosሺ2000𝜋 𝑡 𝜑ଶሻ
and their sum 𝑣ଷ ൌ 𝑣ଵ 𝑣ଶ. Describe quantitatively how the frequency and the amplitude of 𝑣ଷ
changes with the difference 𝜑ଵ െ 𝜑ଶ (identifying specific notable values). Assume 𝐴 ∈ ℝ is constant.3
c) (8pts) The Z-transform of an LTI system’s impulse response is given by 𝐻ሺ𝑧ሻ ൌ ଵ
௭ି.ହ
ଵ
௭ାଶ
ଵ
௭మାଶ.
List every possible RoC, indicating whether the resulting system is stable, causal, anti-causal, and/or
two-sided.4
d) (7pts) A continuous-time signal 𝑥ሺ𝑡ሻ ൌ 4 cosሺ20𝜋𝑡ሻ 3 sinሺ30𝜋𝑡ሻ 5cos ሺ42𝜋𝑡 𝜋/4ሻ is sampled
at 30 Hz. Determine the reconstructed signal 𝑤 ൌ 𝐼𝑑𝑒𝑎𝑙𝐼𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑜𝑟்ሺ𝑆𝑎𝑚𝑝𝑙𝑒𝑟்ሺ𝑥ሻሻ for T=(1/30)s
(NB: The result is not as straightforward as in MT3 so consider carefully the effect of the phase shift.
Note that the sampling occurs at integer multiples of T).
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