EECE 340 Midterm, Summer 2004

AMERICAN UNIVERSITY OF BEIRUT ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT ___________________________________________________________ EECE 440 SIGNALS AND SYSTEMS Summer 2004 Prof. Kabalan Midterm Exam July 26, 2004 __________________________________________________________ __________________________________________________________ NAME: ID NUMBER: ___________________________________________________________ INSTRUCTIONS: 1. Open-Book Exam 2. Duration: Two Hours 3. Provide your answers on the question sheet. Scratch sheet will not be considered in grading 4. 17 pages, 50 pts 5. On this sheet, write your name and ID # in the provided space. ___________________________________________________________ Problems on Chapter 1 Problem 1 (2 pts) What is the fundamental period of the signal . Write if the signal is not periodic. EECE 440 – Midterm Exam 1/17 Problem 2 (3 pts) Consider a continuous-time system with real input x(t) and real output y(t). The input-output relationship is given by: a. Is this system linear? Show your work. b. Is this system time invariant? Show your work. c. Is this system causal? Show your work. Problem 3 (3 pts) EECE 440 – Midterm Exam 2/17 Consider the following Continuous time system The boxes represent LTI Systems, and the enclosed expressions are their impulse responses. Determine g(t). Problem 4 (3 pts) EECE 440 – Midterm Exam e tu )t(  g(t)  + + 3/17 Consider the following two signals and The signal is defined as the convolution 1. For what value (or values) of t is non-zero? 2. For what value (or values) of t is maximum? 3. What is the maximum of ? EECE 440 – Midterm Exam 4/17 Problem 5 (2 pts) Determine the transfer function of the linear control system whose block diagram is shown below Problem 6 (2 pts) EECE 440 – Midterm Exam )s( G1 )s( H2 )s( G2 )s( H1 R(s) C(s) + + + - 5/17 The dynamic equations of a linear time-invariant system whose output is y(t) and input r(t) are given by: Determine the stability of the above system. Problem 7 (2 pts) The open loop transfer function of a unity feedback control system is given by: EECE 440 – Midterm Exam 6/17 Use the RH criterion to determine the exact number of roots of the CE in each half of the complex plane. Problems on Chapter 2 Problem 8 (2 pts) What is the fundamental period of the signal EECE 440 – Midterm Exam 7/17 . Write if the signal is not periodic. Problem 9 (3 pts) Consider the following Discrete-time system with real input x[n] and real output y[n]. The input-output relationship is given by: EECE 440 – Midterm Exam 8/17 a. Is this system linear? Show your work. b. Is this system time-invariant? Show your work c. Is this system causal? Show your work Problem 10 (3 pts) Consider two discrete LTI systems characterized by their impulse responses EECE 440 – Midterm Exam 9/17 Determine the impulse response of the system formed by connecting the two above systems. a. In parallel. (1 pt) b. In series. (2 pts) Problem 11 (4 pts) Consider the discrete-time linear causal system described by the difference equation y[n]-3/4 y[n-1] + 1/8 y[n-2] = x[n] + 1/3 x[n-1] EECE 440 – Midterm Exam 10/17 a. Represent this system in Direct Form II. b. Is this system stable? Why? Problem 12 (3 pts) For the system described by the difference equation y[n+1] – 0.5 y[n] = x[n] a. Find the zero-input response of the system if y[-1]=2 EECE 440 – Midterm Exam 11/17 b. Find the impulse response. c. Is the system stable? Why? Problems on Chapter 3 Problem 13 (4 pts) Given the z-transform of x[n] below, use the properties of the z-transform to derive the transform of the following signals EECE 440 – Midterm Exam 12/17 a. y[n] = x[n] – x[n-1] b. y[n] = x[n] Problem 14 (4 pts) For each z-transform specified below, determine the corresponding discrete-time signal. a. EECE 440 – Midterm Exam 13/17 b. Problem 15 (4 pts) Consider a system S which is known to be LTI, causal, and stable and which has a real-valued impulse response h[n]. The corresponding transfer function H(z) has 3 poles and 3 zeros in the finite z-plane but no poles or zeros at the origin. Moreover, a. Could the system function S have one of its poles z at (z=0.9- j0.9)? Justify your answer. EECE 440 – Midterm Exam 14/17 b. Does H(z) have a zero on the real axis? Justify your answer. Problem 16 (3 pts) EECE 440 – Midterm Exam 15/17 Consider a causal LTI system whose input x[n] and output y[n] are related through the block diagram shown above. a. Determine a difference equation relating y[n] to x[n]. (2 pts) b. Is this system stable? (1 pt) Problem 17 (3 pts) EECE 440 – Midterm Exam 16/17 A pole-zero plot for a filter is shown above. a. Write down the transfer function of this filter (2 pts) b. What is the ROC for this filter to be causal? (1 pt) EECE 440 – Midterm Exam 17/17