### Problem 4: LTI System Analysis#### Problem StatementAn LTI (Linear Time-Invariant) system accepts the input:\[ x[n] = 2^n u[n] \]and produces the output:\[ y[n] = c\delta[n] + (-1)^n u[n] \]where \( c \neq 0 \) is a real constant.##### Tasks:1. **Determine the system function \( H(z) \) and its region of convergence, expressing your answer in terms of \( c \).**2. **If the input**\[ x_1[n] = (-4)^n, \quad n \in \mathbb{Z} \]results in zero output at all times, determine the value of \( c \). Also, determine the output \( y_2[n] \) produced by the input:\[ x_2[n] = 3^n, \quad n \in \mathbb{Z} \]##### Notes:- It is not necessary to invert \( z \)-transforms in this problem.#### Step-by-Step Solution**(i) Determining the System Function \( H(z) \)**1. Using the input-output relationship of the LTI system, the output in the z-domain \( Y(z) \) can be expressed as:\[ Y(z) = H(z)X(z) \]2. Given the input \( x[n] = 2^n u[n] \), taking the z-transform:\[ X(z) = \frac{1}{1 - 2z^{-1}} \quad \text{for } |z| > 2 \]3. Given the output \( y[n] = c\delta[n] + (-1)^n u[n] \), the z-transform of \( y[n] \) is:\[ Y(z) = c + \frac{1}{1 + z^{-1}} \quad \text{for } |z| > 1 \]4. Therefore, the system function \( H(z) \) is:\[ H(z) = \frac{Y(z)}{X(z)} = \left( c + \frac{1}{1 + z^{-1}} \right) \left( 1 - 2z^{-1} \right) \]**Region of Convergence:**\[ |z| > 2 \text

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LTI system accepts the input x[n] = 2"u[n] nd produces the output y[n] = c8[n] + (-1)"u[n] here c # 0 is a real constant. O Determine the system function H(z) and its region of convergence, expressing your answer in terms of c. i) If the input x1[n] = (-4)", ne Z results in zero output at all times, determine the value of c. Also, determine the output Y2[n] produced by the input x2[n] = 3", n E Z (It is not necessary to invert z-transforms in this problem).

LTI system accepts the input x[n] = 2"u[n] nd produces the output y[n] = c8[n] + (-1)"u[n] here c # 0 is a real constant. O Determine the system function H(z) and its region of convergence, expressing your answer in terms of c. i) If the input x1[n] = (-4)", ne Z results in zero output at all times, determine the value of c. Also, determine the output Y2[n] produced by the input x2[n] = 3", n E Z (It is not necessary to invert z-transforms in this problem).

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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