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Suppose we know the system function H(z) of a discrete-time LTI system has a pair of complex-conjugate 0.4e¹ and p2 = 0.4e-, double poles at p3 = P4 = -2, poles at p₁ = two zeros at 21 = 22=1, and double zeros at the origin z3 = z4 = 0. (a) What is the expression for H(z)? (b) Suppose h(n) = Z-¹ [H(z)] is a causal system. What is ROC of H(z)? Determine h(n). Is the system BIBO stable? (c) Suppose h(n) = Z-¹[H(z)] is an anti-causal system. What is ROC of H(z)? Determine h(n). Is the system BIBO stable?

Suppose we know the system function H(z) of a discrete-time LTI system has a pair of complex-conjugate 0.4e¹ and p2 = 0.4e-, double poles at p3 = P4 = -2, poles at p₁ = two zeros at 21 = 22=1, and double zeros at the origin z3 = z4 = 0. (a) What is the expression for H(z)? (b) Suppose h(n) = Z-¹ [H(z)] is a causal system. What is ROC of H(z)? Determine h(n). Is the system BIBO stable? (c) Suppose h(n) = Z-¹[H(z)] is an anti-causal system. What is ROC of H(z)? Determine h(n). Is the system BIBO stable?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Suppose we know the system function \( H(z) \) of a discrete-time LTI system has a pair of complex-conjugate poles at \( p_1 = 0.4e^{j\frac{\pi}{3}} \) and \( p_2 = 0.4e^{-j\frac{\pi}{3}} \), double poles at \( p_3 = p_4 = -2 \), two zeros at \( z_1 = \frac{1}{2} \), \( z_2 = -\frac{1}{2} \), and double zeros at the origin \( z_3 = z_4 = 0 \). (a) What is the expression for \( H(z) \)? (b) Suppose \( h(n) = \mathcal{Z}^{-1}[H(z)] \) is a causal system. What is ROC of \( H(z) \)? Determine \( h(n) \). Is the system BIBO stable? (c) Suppose \( h(n) = \mathcal{Z}^{-1}[H(z)] \) is an anti-causal system. What is ROC of \( H(z) \)? Determine \( h(n) \). Is the system BIBO stable? (d) Suppose \( h(n) = \mathcal{Z}^{-1}[H(z)] \) is a BIBO stable system. What is ROC of \( H(z) \)? Determine \( h(n) \).
Transcribed Image Text:Suppose we know the system function \( H(z) \) of a discrete-time LTI system has a pair of complex-conjugate poles at \( p_1 = 0.4e^{j\frac{\pi}{3}} \) and \( p_2 = 0.4e^{-j\frac{\pi}{3}} \), double poles at \( p_3 = p_4 = -2 \), two zeros at \( z_1 = \frac{1}{2} \), \( z_2 = -\frac{1}{2} \), and double zeros at the origin \( z_3 = z_4 = 0 \). (a) What is the expression for \( H(z) \)? (b) Suppose \( h(n) = \mathcal{Z}^{-1}[H(z)] \) is a causal system. What is ROC of \( H(z) \)? Determine \( h(n) \). Is the system BIBO stable? (c) Suppose \( h(n) = \mathcal{Z}^{-1}[H(z)] \) is an anti-causal system. What is ROC of \( H(z) \)? Determine \( h(n) \). Is the system BIBO stable? (d) Suppose \( h(n) = \mathcal{Z}^{-1}[H(z)] \) is a BIBO stable system. What is ROC of \( H(z) \)? Determine \( h(n) \).
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