---**Discrete-Time LTI System Analysis**Suppose a discrete-time Linear Time-Invariant (LTI) system is characterized by the following system function:\[ H(z) = \frac{2 - \frac{21}{4} z^{-1}}{1 - \frac{21}{4} z^{-1} + \frac{5}{4} z^{-2}} \]### Tasks:(a) **Region of Convergence (ROC):** - Determine the possible ROCs for \( H(z) \). - For each possible ROC, sketch the pole-zero plot.(b) **Causality:** - For each possible ROC, determine if the system is causal, non-causal, or anti-causal.(c) **Stability:** - For each possible ROC, determine if the system is BIBO (Bounded Input Bounded Output) stable or not.(d) **Impulse Response:** - For each possible ROC, determine the impulse response \( h(n) \).---**Explanation of System Function:**The given LTI system function is expressed in terms of \( z \) and consists of a ratio of polynomials in \( z^{-1} \). - **Numerator:** \( 2 - \frac{21}{4} z^{-1} \)- **Denominator:** \( 1 - \frac{21}{4} z^{-1} + \frac{5}{4} z^{-2} \)This form is typical for rational functions used in the analysis of discrete-time systems. The poles and zeros of the system function are key to understanding its behavior, including causality, stability, and the nature of the impulse response.**Note:**- Poles are values of \( z \) that make the denominator zero.- Zeros are values of \( z \) that make the numerator zero.- The ROC determines the range of \( z \) values for which the \( H(z) \) converges.For solving the tasks:1. **ROC and Pole-Zero Plot:** Identify poles and zeros, and mark them on the complex plane.2. **Causality:** Determine the ROC in relation to the outermost pole.3. **Stability:** Check if all poles are within the unit circle in the z-plane for BIBO stability.4. **Impulse Response:** Analyze the inverse Z-transform of \( H(z)

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Suppose a discrete-time LTI system is characterized by the following system func- 2-¹2-1 4 1-212-¹2-2 + 4 (a) What are the possible ROCs for H(z)? for each possible ROC, sketch the pole-zero plot. (b) For each possible ROC, determine if the system is causal, non-causal, or anti-causal. (c) For each possible ROC, determine if the system is BIBO stable or not. (d) For each possible ROC, determine the impulse response h(n). tion: H(z) = 5

Suppose a discrete-time LTI system is characterized by the following system func- 2-¹2-1 4 1-212-¹2-2 + 4 (a) What are the possible ROCs for H(z)? for each possible ROC, sketch the pole-zero plot. (b) For each possible ROC, determine if the system is causal, non-causal, or anti-causal. (c) For each possible ROC, determine if the system is BIBO stable or not. (d) For each possible ROC, determine the impulse response h(n). tion: H(z) = 5

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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