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Colorado State University, Ft. Collins Spring 2016 ECE 312: Linear Systems Analysis II (Signal and Systems) Homework 4 Assigned on: 03/22/2016, Due by: 04/07/2016 4.1 Determine the z-transform for each of the following sequences. Indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. (a) n 5 (b) 0.5 un n 1 (c) 3 n2 un 2 n 1 (d) 2 un un 1 4 n 4.2 Given the following unilateral z-transforms, (i) (ii) (iii) (iv) 0.5z 2 z 1z 0.5 0.5 z X z z 1z 0.5 0.5 X z z 1z 0.5 z X z 2 z z 1 X z (a) Find the inverse unilateral z-transform of each function (b) Evaluate each xn in part (a) for the first three nonzero values. 1 (c) Use the final-value property to evaluate x for each function. Compare the results with those obtained from xn in part (a), explain why or why not they agree with each other. (d) Use the initial-value property to evaluate x0 for each function. Compare the results with those obtained from xn in part (a), explain why or why not they agree with each other. 4.3 For each of the following difference equations and associated input and initial conditions, determine the zero-input and zero-state responses by using the unilateral z-transform. (a) yn 3 yn 1 xn n 1 xn un 2 y 1 1 (b) yn 1 1 yn 1 xn xn 1 2 2 xn un y 1 0 (c) yn 1 1 yn 1 xn xn 1 2 2 xn un y 1 1 4.4 If X z denotes the unilateral z-tranform of xn , determine, in terms of X z , the unilateral z-transform of: 2 (a) xn 3 n (b) xn 3 (c) xk k 0 4.5 The following is known about a discrete-time LTI system with input xn and output yn : 1. If xn 2 for all n , then yn 0 for all n . n 2 un for all n , then yn for all n is of the form 2. If xn 1 n n 1 yn n a un 4 where a is a constant. (a) Determine the value of the constant a . (b) Determine the final value of the output if the input is xn un. 3