### Problem Statement on the Transfer Function of a Causal LTI System#### 1. The transfer function of a causal LTI system is:\[ H(z) = \frac{\left(1 - 0.5e^{j\frac{\pi}{3}}z^{-1}\right)\left(1 - 0.5e^{-j\frac{\pi}{3}}z^{-1}\right)}{1 - 0.25z^{-2}} \]_(a) Write down the difference equation representation of the system._**Note:** \(\cos\left(\frac{\pi}{3}\right) = 0.5\)._(b) Find the natural response of the system \(y[n]\). Your answer may contain unknown coefficients, \(A_1\) and \(A_2\).__(c) Find the forced response of the system when \(x[n] = u[n]\). You can assume that the forced response is of the form \(y_p[n] = K u[n]\), where \(K\) is a constant you have to determine.__(d) Using your answers in parts (b) and (c), find the complete response of the system, given that the system is initially relaxed.__(e) Using your answer in part (d), find the steady state response of the system._---#### Explanation of Content**1. The Transfer Function:**The transfer function provided defines the behavior of a Linear Time-Invariant (LTI) system in terms of the complex frequency variable \(z\). It comprises a numerator and a denominator, each representing parts of the system's behavior.**Difference Equation Representation:**The difference equation corresponds to the time-domain representation of the transfer function. This is a crucial step for implementing the LTI system using digital filters.**Natural Response:** The natural response of the system refers to its behavior with no external input, determined by its characteristic equation. The coefficients \(A_1\) and \(A_2\) will be generalized constants representing initial conditions.**Forced Response:**The forced response depicts the system’s reaction to an external input, in this case, the unit step function \(u[n]\). The form \(y_p[n] = K u[n]\) simplifies calculations, with \(K\) being a proportional constant identifying the steady-state gain from input to output.**Complete Response:**Combining the natural and forced responses

Note: We are authorized to answer three subparts at a time since you have not mentioned which part…

1. The transfer function of a causal LTI system is: (1– 0.5 e# :-") (1– 0.5e# H(2) = z-1) (1– 0.5 e 1 – 0.25 z-2 (a) Write down the difference equation representation of the system. Note: cos (품) %3D 0.5. (b) Find the natural response of the system y n]. Your answer may contain unknown coefficients, A1 and A2. (c) Find the forced response of the system when x[n] the forced response is of the form ypn] = Kun], where K is a constant you have to u n]. You can assume that determine.

1. The transfer function of a causal LTI system is: (1– 0.5 e# :-") (1– 0.5e# H(2) = z-1) (1– 0.5 e 1 – 0.25 z-2 (a) Write down the difference equation representation of the system. Note: cos (품) %3D 0.5. (b) Find the natural response of the system y n]. Your answer may contain unknown coefficients, A1 and A2. (c) Find the forced response of the system when x[n] the forced response is of the form ypn] = Kun], where K is a constant you have to u n]. You can assume that determine.

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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Question

### Problem Statement on the Transfer Function of a Causal LTI System

#### 1. The transfer function of a causal LTI system is:

\[ H(z) = \frac{\left(1 - 0.5e^{j\frac{\pi}{3}}z^{-1}\right)\left(1 - 0.5e^{-j\frac{\pi}{3}}z^{-1}\right)}{1 - 0.25z^{-2}} \]

_(a) Write down the difference equation representation of the system._

**Note:** \(\cos\left(\frac{\pi}{3}\right) = 0.5\).

_(b) Find the natural response of the system \(y[n]\). Your answer may contain unknown coefficients, \(A_1\) and \(A_2\)._

_(c) Find the forced response of the system when \(x[n] = u[n]\). You can assume that the forced response is of the form \(y_p[n] = K u[n]\), where \(K\) is a constant you have to determine._

_(d) Using your answers in parts (b) and (c), find the complete response of the system, given that the system is initially relaxed._

_(e) Using your answer in part (d), find the steady state response of the system._

---

#### Explanation of Content

**1. The Transfer Function:**
The transfer function provided defines the behavior of a Linear Time-Invariant (LTI) system in terms of the complex frequency variable \(z\). It comprises a numerator and a denominator, each representing parts of the system's behavior.

**Difference Equation Representation:**
The difference equation corresponds to the time-domain representation of the transfer function. This is a crucial step for implementing the LTI system using digital filters.

**Natural Response:**
The natural response of the system refers to its behavior with no external input, determined by its characteristic equation. The coefficients \(A_1\) and \(A_2\) will be generalized constants representing initial conditions.

**Forced Response:**
The forced response depicts the system’s reaction to an external input, in this case, the unit step function \(u[n]\). The form \(y_p[n] = K u[n]\) simplifies calculations, with \(K\) being a proportional constant identifying the steady-state gain from input to output.

**Complete Response:**
Combining the natural and forced responses

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