**Problem Statement:**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 the 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 the ROC of \( H(z) \)? Determine \( h(n) \). Is the system BIBO stable?**Explanation of Concepts:**- **Complex-Conjugate Poles:** These are poles located at complex numbers in conjugate pairs. For this problem, they are given by \( p_1 \) and \( p_2 \).- **Zeros and Poles:** These are specific points in the z-plane where the system function becomes zero (zeros) or undefined (poles).- **Region of Convergence (ROC):** For a given system, ROC is the range of \( z \)-values for which the z-transform is defined.- **BIBO Stability:** A system is BIBO (Bounded Input, Bounded Output) stable if for every bounded input, the output is also bounded. This is generally determined by checking if the ROC includes the unit circle.

For causal system, ROC must be outermost side of the outer most pole. For, anti causal system, ROC must be inner side of the innermost pole.

Engineering

Electrical Engineering

Suppose we know the system function H(z) of a discrete-time LTI system has a pair , double poles at p3 = P4 = -2, of complex-conjugate poles at pi 0.4e¹ and p2 0.4e = two zeros at z1 = 1,22 = − 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(2)] 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 , double poles at p3 = P4 = -2, of complex-conjugate poles at pi 0.4e¹ and p2 0.4e = two zeros at z1 = 1,22 = − 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(2)] 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)

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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