EECE 340 Homework, Spring 2008

American University of Beirut Department of Electrical and Computer Engineering EECE 440 Signals and Systems Homework 4 Part I Problem 1 Consider a discrete-time system with input x[n] and output y[n]. The input- output relationship for this system is: y[n] = x[n]x[n-2] a. Is the system memoryless? b. Determine the output of the system when the input is A , where A is any real or complex number. c. Is this system invertible? Problem 2 Consider a discrete-time system with input x[n] and output y[n] related by: where is a finite positive integer. a. Is this system linear? b. Is this system time-invariant? c. If x[n] is known to be bounded by a finite integer B, it can be shown that y[n] is bounded by a finite number C. We conclude that the given system is stable. Express C in terms of B and . Problem 3 Consider a system S with input x[n] and output y[n] related by: y[n] = x[n]{g[n] + g[n-1]} a. If g[n] = 1 for all n, show that S is time invariant. b. If g[n]=n, show that S is not time invariant c. If g[n] = (1)+(-1) , show that S is time invariant. 1 Problem 4 a. Is the following statement true or False “The series interconnection of two linear time-invariant systems is itself a linear time-invariant system”. Justify your answer. b. Is the following statement true of false: “The series interconnection of two nonlinear systems is itself nonlinear” Justify your answer. c. Consider three systems with the following input-output relationships: System 1: System 2: System 3: y[n]=x[2n] Suppose that these systems are connected in series. Find the input- output relationship for the overall interconnected system. In this system stable? Is it time invariant? Problem 5. Evaluate 1. y[n] =u[n] * u[n-3] 2. 3. 4. Problem 6 The input-output relationship for the first-order recursive system is given by: . Let the input be given by : . Use convolution to find the output of this system, assuming that b and that the system is causal. 2 Part II 1. Stability using MATLAB. You are requested to determine a procedure to check on the stability of a LTI discrete-time using MATLAB. Give procedure and a minimum of two examples. 2. State Variable Analysis using MATLAB. Determine a MATLAB procedure for representing a discrete-time system defined in a State-variable representation. Give a minimum of two examples. 3