EECE 340 Midterm, Summer 2003

AMERICAN UNIVERSITY OF BEIRUT ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT _____________________________________________________________________ EECE 440 SIGNALS AND SYSTEMS SUMMER 03-04 Prof. Kabalan Midterm Exam- Solution July 21, 2003 _____________________________________________________________________ Problem 1 (2 pts) What is the fundamental period of ? Write if the signal is not periodic. T=24 s Problem (2 pts) Answer by True or False If the signal is applied to an LTI system, the output will be proportional to the input signal regardless of the value of s. True Problem 3 (2 pts) Answer by True or False The integral of an even periodic signal over one fundamental period is equal to zero. False Problem 4 (2 pts) Answer by True or False If a signal x(t) has odd symmetry, we know that it must be zero when t=0. True Problem 5 (6 pts) Fill each cell of the table with a Y if the system has the corresponding property and N if the system does not have the property. These systems have an input x(t) and output y(t). Memoryless Causal Stable Time- Invariant Linear y(t)=-7+x(t) Y Y Y Y N N N N Y Y N Y Y Y N Problem 6 (6 pts) EECE 440-Midterm Exam Solution 1/8 The transfer function of a control system is given by: a. Use the RH table to determine the number of poles of the transfer function in each half of the complex plane. (4 pts) Q(p)= P6 3 7 5 1 P5 6 5 1 P4 9 9 2 P3 -3 -1 P2 3 1 P 0 6 A(p)=3p2 + 1 P0 1 Number of roots in each half of the complex plane: 2 in the rhp (from RH table), 2 on the jw axis( from A(s), and 2 in the lhp. (2,2,2) b. Is the system stable? Justify your answer. (2 pts) N, system has poles in the rhp and on the imaginary axis. Problem 7 (6 pts) EECE 440-Midterm Exam Solution 2/8 a. Draw the block diagram for a circuit that implements the following transfer function You may use amplifiers, adders, and integrators. b. Determine the state equation for this system. c. Determine the output equation for this system. d. Is the state representation for an LTI system unique? Explain. No EECE 440-Midterm Exam Solution -1/6 -1/2 -5/12 1/2 1/3 3/8 Problem 8 (2 pts) What is the fundamental period of ? Write if the signal is not periodic. Problem 9 (2 pts) Answer by True or False The signals and are equal for all values of n if . False Problem 10 (2 pts) Answer by True or False Any finite-valued periodic discrete-time signal can be exactly represented as a sum of n harmonically related complex exponentials. True Problem 11 (2 pts) Use the following discrete-time signal x[n] to answer the questions below. You may assume that x[n] = 0 for the remaining values of n. 1. What is the signal energy of x[n]? 18 Joules 2. Draw 0.5x[n+3] EECE 440-Midterm Exam Solution 4/8 Problem 12 (4 pts) Fill each cell of the table with a Y if the system has the corresponding property and N if the system does not have the property. These systems have an input x[n] and output y[n]. Memoryless Causal Stable Time- Invariant Linear Y Y N N N y[n]=x[2n-7] N N Y N Y Problem 13 (4 pts) Given , and . Determine n 0 1 2 3 4 5 k h[k] x[n] 4 3 3 1 0 0 x[n-1] 4 3 3 1 1 2 x[n-2] 4 3 3 1 2 4 x[n]h[0] 0 0 0 0 0 0 x[n-1]h[1] 0 8 6 6 2 0 x[n-2]h[2] 0 0 16 12 12 4 y[n] 0 8 22 18 14 4 EECE 440-Midterm Exam Solution 5/8 Problem 14 (4 pts) a. Write a difference equation for the following system y[n]-y[n-1]-0.25y[n-2]=x[n]-0.25x[n-1] b. Determine system impulse response. Knowing that the system is physically realizable, and using the difference equation the following is obtained h[0] = 1, h[1]=0.75, h[2] = 0.5, …… Problem 15 (6 pts) An LTI system is represented by the following difference equation y[n]-2y[n-2]=x[n]-x[n-1]+6x[n-2] a. Represent this system in Direct Form I EECE 440-Midterm Exam Solution 6/8 + D D + + x[n] y[n] 1 - 0.25 - 0.25 b. Represent this system in Direct Form II. c. Write the state equation and output equation for this system. EECE 440-Midterm Exam Solution D D D D -1 6 2 x[n] y[n] D x[n] y[n] D -1 6 2 7/8 EECE 440-Midterm Exam Solution 8/8