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The april 2007 exam for mathematics 267: mathematical methods for electrical and computer engineering at the university of british columbia. The exam covers topics such as elastic strings, fourier series, fourier transforms, and lti systems.
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[15] 1. (a) An elastic string of length 4 with fixed ends has an initial shape u(x, 0) = f (x), where
f (x) =
{ (^0) if 0 ≤ x < 1 1 if 1 ≤ x ≤ 3 0 if 3 < x ≤ 4
It is released from rest at time t = 0. Assume that the displacement u(x, t) satisfies
uxx = utt, 0 ≤ x ≤ 4 , t > 0.
Find u(x, t).
(b) Sketch u(x, 0) and u(x, 1).
(a)
(a)
(b)
(c)
(a)
(b)
(c)
[15] 4. In this problem you will analyze this circuit:
x(t)
y(t)
The input signal is a time-varying voltage x(t) and the output signal is the voltage y(t) mea- sured across the inductor. Low-frequency signals face little opposition to flow through the inductor, so they get dissipated mostly by the resistor. High-frequency signals flow easily through the capacitor, so they also get dissipated by the resistor. But signals of some inter- mediate frequency are opposed by both reactive components, and produce large-amplitude outputs. The signals described above are related by the constant coefficient differential equa- tion RLCy′′(t) + Ly′(t) + Ry(t) = Lx′(t).
(a) Let x̂(ω) and ̂y(ω) be the Fourier transforms of x(t) and y(t). Define
H(ω) =
y(ω) ̂ x(ω)
, A(ω) = |H(ω)|, H(ω) = A(ω)eiφ(ω).
Find simple algebraic expressions for H(ω), A(ω) and tan(φ(ω)). (b) Use calculus to find the value of ω > 0 at which A(ω) is maximized. This is the circuit’s resonant frequency. Express your answer in terms of L, R, and C. [Hint: Maximize |A(ω)|^2. ]
[15] 5. Consider the discrete time signal
x[n] = sin πn 2 cos(πn)
(a) Is x[n] periodic? If so, find a period N. (b) Is the discrete Fourier transform ̂x[k] of this signal periodic? If so, find a period for ̂x[k].
(c) Find the discrete Fourier transform x̂[k] of this signal.
(a)
(b)
(c)
(a)
(b)
(c)
(d)
The End
Be sure that this examination has 14 pages including this cover
The University of British Columbia Final Examinations - April, 2007
Mathematics 267 Mathematical Methods for Electrical and Computer Engineering
Closed book examination Time: 2 12 hours
Name Signature
Student Number Instructor’s Name
Section Number
To receive full credit, all answers must be supported by clear and correct derivations. No calculators, notes, or other aids are allowed. A formula sheet is provided with the exam. Use the backs of the sheets, if necessary, for additional work. But please write your final answers in the boxes provided.
Rules Governing Formal Examinations
Total 100
