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Be sure this exam has 11 pages including the cover The University of British Columbia Mathematics 267, Mathematical Methods for EE and CS StudentsFinal Exam – December 2009
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This exam consists of 5 questions worth (^) permitted. 100 marks in total. No notes, calculators aids are
Problem max score score
- 20
- 20
- 20
- 20
- 20 total 100
- Each candidate should be prepared to produce his library/AMS card upon request 2. 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 leaveduring the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguitiesin examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from theexamination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those authorized by the examiners.(b) Speaking or communicating with other candidates. shall not be received.(c) Purposely exposing written papers to the view of other candidates.^ The plea of accident or forgetfulness
- Smoking is not permitted during examinations.
(20 points) 1. Find the solution u(x, t) of the wave equation:
uutt(x, t) = (^) π^12 uxx(x, t), 0 ≤ x ≤ 1 , t > 0 , ut((x,x, 0) =^ 0) = 0 f (,x^ ), 00 ≤≤^ xx^ ≤≤^11 ,, u(0, t) = u(1, t) = 0, t ≥ 0. Here f (x) = { (^103) x, if 0 ≤ x ≤ 13 , 3(1 20 − x), if 13 ≤ x ≤ 1.
(20 points) 2. Let g(x) be the 2π-periodic triangle wave and g(x) = { (^) π + x, if − π ≤ x ≤ 0 , π − x, if 0 ≤ x ≤ π. (a) Plot the graph of g(x) in at least three period. (b) Find the real Fourier series of g(x). (c) Find 1 + 312 + 512 + 712 + · · ·.
(d) Find the z-transform of x[n] = δ[n] + 2nu[−n].
Answer =
(e) Find the inverse Fourier transform of fˆ (ω) = (^) 1+^1 ω 2.
Answer =
(20 points) 4. Consider the system with input x(t) and output y(t) which is characterized by the ODE y′′(t) + 5y′(t) + 4y(t) = 3x(t). (a) Find the transfer function Hˆ(ω) and the impulse response H(t). (b) Find y(t) if x(t) = δ(t − 4). (c) Find y(t) if x(t) = e−^4 tu(t − 2).
(20 points) 5. Assume that H(z) is the z-transform of the discrete signal h[n] and H(z) = (^) z 2 z (^) −^2 − (^32) z^3 + 1z. (a) Find the regions of convergence so that H(z) is causal. Find h[n] is this case. (b) Find the regions of convergence so that H(z) is stable. Find h[n] is this case.
