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ELEC 221 Signals & Systems FINAL EXAM

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ELEC 221 18W1, Signals & Systems Final Exam, December 19th, 2018 1 of 9 Instructions Failure to follow any of the instructions below will result in a zero-mark for your exam. No appeals or excuses will be entertained. • There are 100 marks that can be earned on the exam • No calculators • Figures and formulas are found at the back of the test paper and may be removed • Personal formula sheet is allowed (8.5x11, double sided) • 150 min to complete the exam • Marks associated with each question are given in the [square brackets] • Continuous time 𝑡𝑡 is measured in seconds • Continuous frequency 𝜔𝜔 is measured in rad/sec • Discrete frequency Ω is measured in rad/sample Exam Booklets 1. Fill in your name and student number on the exam booklet. 2. All solutions must be written in exam booklet. You may ask for an additional booklet if you need more space. 3. At the end of the exam pass the tests to the closest aisle for pick-up by a TA Answer Format 1. All sketches and diagrams must be fully labelled for full marks. 2. Answers must be simplified in order to obtain full marks. 3. All answers must be clearly labelled in terms of the question number and part of the question.ELEC221 18W1, Final Exam 2 of 9 Question 1 [2] Consider the convolution 𝑦𝑦(𝑡𝑡) = 𝑥𝑥1(𝑡𝑡) ∗ 𝑥𝑥2(𝑡𝑡) where 𝑥𝑥1(𝑡𝑡) = �𝑡𝑡, 0 ≤ 𝑡𝑡 ≤ 1 0, otherwise 𝑥𝑥2(𝑡𝑡) = �1, 0 ≤ 𝑡𝑡 ≤ 2 0, otherwise 𝑦𝑦(𝑡𝑡) is a piecewise linear function. Find the number of non-zero piecewise regions in 𝑦𝑦(𝑡𝑡) (do not solve for 𝑦𝑦(𝑡𝑡)). Question 2 [2] For the signal 𝑥𝑥(𝑡𝑡) = sin(2𝜋𝜋𝜋𝜋) 𝑢𝑢(𝑡𝑡) find the even and odd components 𝑥𝑥𝑒𝑒(𝑡𝑡) and 𝑥𝑥𝑜𝑜(𝑡𝑡), respectively. Question 3 [3] For the system shown below find the following (‘D’ corresponds to a delay of one): a) [2] the z-transform 𝐻𝐻(𝑧𝑧) and the region of convergence b) [1] the impulse response ℎ[𝑛𝑛] Question 4 [5] Given the transfer function 1 1 1 3 2 2 1 1 ( ) 1 1 H z z z − − = − − + a) [2] Sketch the region of convergence corresponding to a stable system. b) [2] Find ℎ[𝑛𝑛] for the region of convergence found in part (a). c) [1] Classify ℎ[𝑛𝑛] as causal, anti-causal or non-causal. Question 5 [5] Determine if the following signals are periodic or aperiodic. If periodic, find the period. a) 𝑥𝑥(𝑡𝑡) = 6 sin2(𝑡𝑡) b) 𝑥𝑥(𝑡𝑡) = 𝑒𝑒−𝑡𝑡cos(2𝜋𝜋𝜋𝜋) c) 𝑥𝑥[𝑛𝑛] = cos �𝜋𝜋3 ⋅ 𝑛𝑛� + sin(𝜋𝜋4 ⋅ 𝑛𝑛) d) 𝑥𝑥[𝑛𝑛] = cos(1 4 ⋅ 𝑛𝑛) Question 6 [6] A system samples input signals at a rate of 𝑇𝑇𝑠𝑠 = 2 seconds to produce a discrete time output signal. Show using a sketch how this sampler maps the s-plane to the z-plane. a) [3] Identify the region of the s-plane that is uniquely mapped to the z-plane. b) [2] For 𝑠𝑠 = 𝜎𝜎 + 𝑗𝑗𝑗𝑗 identify the region 𝜎𝜎 > 0 and 𝑠𝑠 = 𝑗𝑗𝑗𝑗 in the z-plane. c) [1] Describe in words what would change in the s-plane and / or z-plane if 𝑇𝑇𝑠𝑠 → 0

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