New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Spring 2018 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points / 25 points / 100 points Use pencil for all math, not pen. Color may be used for plots if desired, but not red. Show all work on exam, additional paper not accepted.
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 n. 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 [u(t + 1) u(t 1)]. Is x(t) an energy signal or power signal? Justify your answer mathematically. (b) Suppose x[n] = ejπn. Is x[n] an energy signal or power signal? Justify your answer mathematically. N Hint: n 2 = 1 N(N + 1)(2N + 1) 6 n=0 1
Prob. 1 (cont.) Signal Properties (Periodicity/Symmetries) (c) Suppose x[n] = j n + e jπn. Is x[n] periodic? If so, what is the fundamental period? (d) Suppose x(t) = ( 1 4 α)t, where α = e j9. Is x(t) periodic? If so, what is the fundamental period? 2
Prob. 1 (cont.) Signal Properties (Periodicity/Symmetries) (e) Suppose x(t) = 2u(t + 3) + 2δ(t 1). 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
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) = tx(t 2 )x(t) MEMORYLESS CAUSAL BIBO STABLE TIME INVARIANT LINEAR 4
(Additional Paper for Math Calculations) 5
Prob. 2 (cont.) System Properties (b) y[n] = x[n] + (n 1)δ[n 1]x 2 [n + 2] Hint: You may want to simplify the expression. MEMORYLESS CAUSAL BIBO STABLE TIME INVARIANT LINEAR 6
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 4] u[n + 1] + ( ) k= 2 n 1 is given by x[n] = 4 (u[n + 1] u[n 1]). 2 Hint: you may want to draw h[n] and simplify before computing the convolution. 2 δ[n k] and the input signal y[n] n Is the system memoryless? Is the system causal? Is the system BIBO stable? 7
(Additional Paper for Math Calculations) 8
Prob. 3 (cont.) Convolution/LTI System Properties (b) Let h[n] = u[n + 1] u[n 4] and the input signal is given by x[n] = 2δ[n 9] + 3δ[n 10] + 2δ[n 11]. 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
(Additional Paper for Math Calculations) 10
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. 1, t 1 (a) Let h(t) = 1, 1 < t 2 and the input signal is given by x(t) = 2δ(t + 4) + 2δ(t) + 2δ(t 4). 0, else 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
(Additional Paper for Math Calculations) 12
Prob. 4 (cont.) Convolution/LTI System Properties t, 0 < t 1 1, 1 < t 4 (b) Let h(t) = and the input signal is given by x(t) = u(t + 3) u(t + 2.5). What t + 5, 4 < t 5 0, else is the output of the system y(t)? 13
(Additional Paper for Math Calculations) 14
y(t) t Is the system memoryless? Is the system causal? Is the system BIBO stable? 15