New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Fall 2017 Exam #1

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1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2017 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points / 25 points / 100 points (Use pencil, not pen. Show all work on exam, additional paper not accepted.)

2 Prob. 1 Signal Properties (Energy and Power) Let E denote the total energy over the finite time interval t 1 t t 2 or n 1 n n 2, and E denote the total energy over the infinite time interval t or n. Similarly, let P denote the average power over the finite time interval t 1 t t 2 or n 1 n n 2, and P denote the average power over the infinite time interval t or t. Let the even part of x(t) be denoted as x e (t) = Ev{x(t)} and the odd part of x(t) be denoted as x o (t) = Od {x(t)}. (a) Suppose x(t) = e 2t+j(5t+13). Is x(t) an energy signal or power signal? Justify your answer mathematically. (b) Suppose x[n] = j n. Is x[n] an energy signal or power signal? Justify your answer mathematically. 1

3 Prob. 1 (cont.) Signal Properties (Periodicity/Symmetries) (c) Suppose x[n] = α n, where α = e j 2π. Is x[n] periodic? If so, what is the fundamental period? (d) Suppose x(t) = 4α t 1, where α = e j6. Is x(t) periodic? If so, what is the fundamental period? 2

4 Prob. 1 (cont.) Signal Properties (Periodicity/Symmetries) (e) Suppose x(t) = u(t + 3) u(t 2) + 4δ(t 5). Carefully sketch the graphs of x(t), the even part, x e (t), and odd part, x o (t) (be sure to label critical x- and y-axis values). 3

5 Prob. 2 System Properties For the following systems, check which properties are true ( checked) and provide a short and convincing reason or proof. For properties which are not true ( checked), provide a short and convincing reason, proof, or counter example. No credit for just check. (a) y(t) = t+t 0 x(τ)dτ, t 0 = 2 MEMORYLESS CAUSAL BIBO STABLE TIME INVARIANT LINEAR 4

6 (Additional Paper for Math Calculations) 5

7 Prob. 2 (cont.) System Properties (b) y[n] = x[n] + e jπ u[n + 1] MEMORYLESS CAUSAL BIBO STABLE TIME INVARIANT LINEAR 6

8 Prob. 3 Convolution/LTI System Properties An LTI system is described by the impulse response h[n] determine the output signal y[n] for the given input signal x[n]. Carefully graph the output (be sure to label critical x- and y-axis values) and answer the system property questions. (a) What is the output of the system y[n], if h[n] = u[n 1] u[n + 1] + is given by x[n] = 16 ( ) n 1 (u[n 2] u[n 5]). 2 2 δ[n k] and the input signal k= 2 y[n] n Is the system memoryless? Is the system causal? Is the system BIBO stable? 7

9 (Additional Paper for Math Calculations) 8

10 Prob. 3 (cont.) Convolution/LTI System Properties n + 2, 3 n 1 (b) Let h[n] = u[n] u[n 3] + δ[n 2] and the input signal is given by x[n] = n + 2, 0 n 2. 0, otherwise What is the output of the system y[n]? y[n] n Is the system memoryless? Is the system causal? Is the system BIBO stable? 9

11 (Additional Paper for Math Calculations) 10

12 Prob. 4 Convolution/LTI System Properties An LTI system is described by the impulse response h(t) determine the output signal y(t) for the given input signal x(t). Carefully graph the output (be sure to label critical x- and y-axis values) and answer the system property questions. (a) Let h(t) = sin(t)u(t+π) sin(t)u(t π). The input signal is given by x(t) = δ(t+2π)+δ(t) δ(t 2π). What is the output of the system y(t)? y(t) t Is the system memoryless? Is the system causal? Is the system BIBO stable? 11

13 (Additional Paper for Math Calculations) 12

14 Prob. 4 (cont.) Convolution/LTI System Properties (b) Let h(t) ( = u(t) u(t 1) + e 2(t 1) [u(t 1) u(t 2)]. The input signal is given by x(t) = u(t + 4) u t + 9 ). What is the output of the system y(t)? 2 y(t) t Is the system memoryless? Is the system causal? Is the system BIBO stable? 13

15 (Additional Paper for Math Calculations) 14

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1

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