New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Final Exam

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New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Name: Solve problems 1 3 and two from problems 4 7.

1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Name: Solve problems 1 3 and two from problems 4 7. Circle below which two of problems 4 7 you wish to be graded. Prob. 1 Prob. 2 Prob. 3 Prob. 4 Prob. 5 Prob. 6 Prob. 7 Total / 20 points / 20 points / 20 points / 20 points / 20 points / 20 points / 20 points / 100 points

2 Prob. 1 The linear, constant-coefficient differential equation (LCCDE) of a continuous-time (CT), linear, time-invariant (LTI) system is given by Assume at rest conditions. 1 d 2 y(t) dy(t) + y(t) = 1 dx(t) + x(t) d 2 t 1000 dt 100 dt (a) Draw a block diagram of the system using differentiators and not integrators. (b) Determine the frequency response, H(jω) of the system. 1

3 Prob. 1 (cont.) (c) Determine the impulse response, h(t) of the system. (d) Let the input to the system be, x(t) = e j100t. Use any method you wish to determine the output, y(t). Solve LCCDE for y(t), convolve h(t) x(t), transform-domain F 1 {H(jω)X(jω)}, or eigenfunction theory 2

4 Prob. 2 The linear, constant-coefficient difference equation (LCCDE) of a discrete-time (DT) LTI system is given by Assume at rest conditions. (a) Draw a block diagram of the system. y[n] 1 y[n 2] = x[n] 2x[n 1]. 4 (b) Determine the frequency response, H(e jω ) of the system. 3

5 Prob. 2 (cont.) (c) Determine the impulse response, h[n] of the system. (d) Let the input to the system be, x[n] = e j2n. Use any method you wish to determine the output, y[n]. Solve LCCDE for y[n], convolve h[n] x[n], transform-domain F 1 { H(e jω )X(e jω ) }, or eigenfunction theory 4

6 Prob. 3 A CT LTI system with frequency response, H(jω) is constructed by cascading two CT LTI systems with frequency responses, H 1 (jω) and H 2 (jω) as depicted below; obviously H(jω) = H 1 (jω)h 2 (jω). H(jω) x(t) H 1 (jω) H 2 (jω) y(t) The two figures below show the straight-line approximations of the Bode magnitude plots of H 1 (jω) and H(jω). Complete the following parts to determine H 2 (jω). 20log 10 H 1 (jω) 24 db 20log 10 H(jω) 20 db +20 db/decade 20 db/decade 40 db/decade 6 db ω (rads/s) ω (rads/s) (a) H 1 (jω) (b) H(jω) (a) Determine the three break frequencies of H 1 (jω). 5

7 Prob. 3 (cont.) (b) Determine the constant factor for H 1 (jω), A 1 such that H 1 (jω) = A 1 (jω + ω 1 ) (jω + ω 2 )(jω + ω 3 ) where ω 1, ω 2, and ω 3 are the break frequencies from (a). Hint: From the plot, 20 log 10 H 1 (j0) = 6 or H 1 (j0) = 2. (c) Determine the break frequency(ies) and constant factor for H(jω). (d) Determine H 2 (jω) = H(jω)/H 1 (jω) using your results in (b) and (c). 6

8 Prob. 4 Part I: Let t 2, 0 t < 1 x(t) = t 2, 1 t < 0 0, otherwise F X(jω) For this problem, the actual value of X(jω) does not matter. The first solution is given as an example. (a) Graph x 1 (t) = x(t 1.6) and write X 1 (jω) in terms of X(jω). Solution: X 1 (jω) = e j1.6ω X(jω) x 1 (t) t (s) (b) Graph x 1 (t) = x ( 12 ) t and write X 1 (jω) in terms of X(jω). (c) Graph x 1 (t) = x (3t + 1) and write X 1 (jω) in terms of X(jω). 7

9 Prob. 4 (cont.) Part II: Let x[n] = sin(πn/4) πn X(e jω ) is periodic with a period of 2π. { F 1, 0 ω π/4 X(e jω ) = 0, π/4 < ω π (d) x 1 [n] = x[n 3]. Graph the magnitude and phase response of X 1 (e jω ) for π ω π. (e) x 1 [n] = x[n] e jπn/2. Graph the magnitude and phase response of X 1 (e jω ) for π ω π. (f) X 1 (e jω ) = e j4ω X(e jω ). Write x 1 [n] in terms of x[n] and graph the magnitude and phase response of X 1 (e jω ) for π ω π. 8

10 Prob. 5 (a) Let x(t) = u(t) and h(t) = e 1000t u(t) + e 10t u(t). < t < using graphical convolution(s). Determine y(t) = h(t) x(t) for 9

11 Prob. 5 (cont.) ( ) n ( 1 (b) Let x[n] = u[n] and h[n] = u[n] + 1 n u[n]. Determine y[n] = h[n] x[n] for 2 2) < n < using graphical convolution(s). 10

12 Prob. 6 (a) Let x(t) = { 1 e t, 1 t < 0 e t 1, 0 t < 1 be a periodic signal with a fundamental period T = 2 s. Graph x(t) over the interval 3 t 3 and determine the Fourier Series (FS) coefficients, a k. x(t) t a k =? 11

13 Prob. 6 (cont.) (b) Let x[n] = δ[n] 2δ[n 1] + 4δ[n 2] 2δ[n 3] + δ[n 4] be a periodic signal with a fundamental period N = 5. Graph x(t) over the interval 5 n 9 and determine the Discrete-Time Fourier Series (DTFS) coefficients, a k. x[n] n a k =? 12

14 Prob. 7 Let the input signal be and the impulse response be x(t) = cos(20πt) + cos(200πt) + cos(2000πt) h(t) = 10000te 100t u(t). Use eigenfunction theory (not phasors or convolution) to determine the output signal, y(t). Express y(t) in terms of cosines. 13

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

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 /