EECE 340 Homework 2, Spring 2016

1 | P a g e AMERICAN UNIVERSITY OF BEIRUT ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT EECE 340 Homework II - Topic: System Properties Problem 1 The input-output relationship of a system is given by:    d e x t y t ( ) )1 ( ( )       a. Is this system stable? Justify your answer. b. Is this system linear? Justify your answer. c. Is this system Time invariant? Justify your answer. Problem 2 The input-output relationship of a system is given by: ( ) ( ) ( 1) 2 ( 2) y t x t x t x t     a. Is this system linear? Justify your answer. b. Is this system time-invariant? Justify your answer. Problem 3 Consider a system whose input-output relationship is given by: 2x )t( 4ty )t( dt dy )t(   a. Is this system linear? Justify your answer. b. Is the system causal? Justify your answer. c. Is the system memory-less? Justify your answer. 2 | P a g e Problem 4 Determine whether each of the statements is true or False. You must Justify your answer to get a grade a. If y(t) is the output of a linear time-invariant system for an input x(t), then y(-t) is the output for the input x(-t). b. For an unstable system, every bounded input x(t) yields an output that is not bounded. c. If x(t) is a periodic signal, then x(t)+x(at) is periodic for any real number a. Problem 5 The response of an LTI system to a unit step input x(t)= 4u(t) is y(t)= )u )t( e 4 1(  2t . What is the response to an input of )1 4u t( 4u )t( x )t(    ? Problem 6 a. Prove whether or not the system defined by: 𝑦(𝑡) = 𝑥(𝑡)cos⁡[𝑥(𝑡)] ∙ sin⁡[𝑥(𝑡)] Is time invariant or not b. Prove if the system defined by 𝑦(𝑡) = [𝑥(𝑡 − 1)] + 2 is linear or not c. Is the system defined by 𝑦(𝑡) = 𝑥 ( 𝑡 3) causal? Why or why not? Problem 7 A system is defined by the input-output relationship given by: y(t) = x(t) sin(2t) + 1 a. Is this system linear? Justify your answer. b. Is this system time-invariant? Justify your answer? c. Is this system stable? Justify your answer. d. Is this system causal? Justify your answer. 3 | P a g e Problem 8 A system takes an input x(t) and produces the output y(t) given by     t 0 x( )d y )t( a. Is the system linear? Justify your answer. b. Is the system time-invariant? Justify your answer. c. Is the system stable? Justify your answer. d. Is the system causal? Justify your work. Problem 9 Let H denote a continuous system such that the relationship between the input f(t) and the output y(t) is given by the equation ( ) )1 2 ( 1 ( ) tf t y t y t    Is the system linear? Justify your answer Problem 10 The output y(t) of a continuous-time system is related to its input x(t) by   x )t( )1 cos 2x t( y )t(    a. Is the system linear? Justify your answer. b. Is the system time-invariant? Justify your answer. c. Is the system causal? Justify your answer. d. Is the system memoryless? Justify your answer. e. Is the system stable? Justify your answer. Problem 11 Consider a continuous-time system which has input of signal x(t) and output of y(t) = x(t)u(t). Is this system time invariant? Justify your answer.