**Problem Statement:** *The response of a certain Linear Time-Invariant (LTI) system to the input \( x[n] = a^n u[n] \) is given by the sequence \( g[\cdot] \). Express the impulse response \( h[\cdot] \) of the system in terms of \( g[\cdot] \).* --- In this problem, we are dealing with a Linear Time-Invariant (LTI) system. The input to the system is \( x[n] = a^n u[n] \), where \( u[n] \) is the unit step function. The response of the system to this input is given by \( g[\cdot] \). We are tasked with expressing the impulse response \( h[\cdot] \) of the system in terms of the sequence \( g[\cdot] \). ### Explanation: - \( x[n] = a^n u[n] \): - This represents an exponential sequence starting at \( n = 0 \). - \( u[n] \) is the unit step function, which is 0 for \( n < 0 \) and 1 for \( n \geq 0 \). - \( g[\cdot] \): - This is the response of the LTI system to the input \( x[n] \). ### Approach: To find the impulse response \( h[n] \), we use the properties of LTI systems, primarily the principle of superposition and convolution. Since the input \( x[n] = a^n u[n] \) is not an impulse directly, but a more complex input, the response \( g[n] \) is related to the convolution of the input \( x[n] \) with the impulse response \( h[n] \): \[ g[n] = x[n] * h[n] \] This can be written as: \[ g[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \] Given \( x[k] = a^k u[k] \), the convolution sum becomes: \[ g[n] = \sum_{k=0}^{n} a^k h[n-k] \] Our goal is to find \( h[n] \) by working with this relation. ### Conclusion: Expressing \( h[\cdot] \) in terms of \( g[\cdot
**Problem Statement:** *The response of a certain Linear Time-Invariant (LTI) system to the input \( x[n] = a^n u[n] \) is given by the sequence \( g[\cdot] \). Express the impulse response \( h[\cdot] \) of the system in terms of \( g[\cdot] \).* --- In this problem, we are dealing with a Linear Time-Invariant (LTI) system. The input to the system is \( x[n] = a^n u[n] \), where \( u[n] \) is the unit step function. The response of the system to this input is given by \( g[\cdot] \). We are tasked with expressing the impulse response \( h[\cdot] \) of the system in terms of the sequence \( g[\cdot] \). ### Explanation: - \( x[n] = a^n u[n] \): - This represents an exponential sequence starting at \( n = 0 \). - \( u[n] \) is the unit step function, which is 0 for \( n < 0 \) and 1 for \( n \geq 0 \). - \( g[\cdot] \): - This is the response of the LTI system to the input \( x[n] \). ### Approach: To find the impulse response \( h[n] \), we use the properties of LTI systems, primarily the principle of superposition and convolution. Since the input \( x[n] = a^n u[n] \) is not an impulse directly, but a more complex input, the response \( g[n] \) is related to the convolution of the input \( x[n] \) with the impulse response \( h[n] \): \[ g[n] = x[n] * h[n] \] This can be written as: \[ g[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \] Given \( x[k] = a^k u[k] \), the convolution sum becomes: \[ g[n] = \sum_{k=0}^{n} a^k h[n-k] \] Our goal is to find \( h[n] \) by working with this relation. ### Conclusion: Expressing \( h[\cdot] \) in terms of \( g[\cdot
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...
Related questions
Question
![**Problem Statement:**
*The response of a certain Linear Time-Invariant (LTI) system to the input \( x[n] = a^n u[n] \) is given by the sequence \( g[\cdot] \). Express the impulse response \( h[\cdot] \) of the system in terms of \( g[\cdot] \).*
---
In this problem, we are dealing with a Linear Time-Invariant (LTI) system. The input to the system is \( x[n] = a^n u[n] \), where \( u[n] \) is the unit step function. The response of the system to this input is given by \( g[\cdot] \).
We are tasked with expressing the impulse response \( h[\cdot] \) of the system in terms of the sequence \( g[\cdot] \).
### Explanation:
- \( x[n] = a^n u[n] \):
- This represents an exponential sequence starting at \( n = 0 \).
- \( u[n] \) is the unit step function, which is 0 for \( n < 0 \) and 1 for \( n \geq 0 \).
- \( g[\cdot] \):
- This is the response of the LTI system to the input \( x[n] \).
### Approach:
To find the impulse response \( h[n] \), we use the properties of LTI systems, primarily the principle of superposition and convolution.
Since the input \( x[n] = a^n u[n] \) is not an impulse directly, but a more complex input, the response \( g[n] \) is related to the convolution of the input \( x[n] \) with the impulse response \( h[n] \):
\[ g[n] = x[n] * h[n] \]
This can be written as:
\[ g[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]
Given \( x[k] = a^k u[k] \), the convolution sum becomes:
\[ g[n] = \sum_{k=0}^{n} a^k h[n-k] \]
Our goal is to find \( h[n] \) by working with this relation.
### Conclusion:
Expressing \( h[\cdot] \) in terms of \( g[\cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd07e8233-789e-4760-ba71-e70869af7b54%2F87038433-47f8-4642-aca3-e88464e1b853%2F7ijz1tm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
*The response of a certain Linear Time-Invariant (LTI) system to the input \( x[n] = a^n u[n] \) is given by the sequence \( g[\cdot] \). Express the impulse response \( h[\cdot] \) of the system in terms of \( g[\cdot] \).*
---
In this problem, we are dealing with a Linear Time-Invariant (LTI) system. The input to the system is \( x[n] = a^n u[n] \), where \( u[n] \) is the unit step function. The response of the system to this input is given by \( g[\cdot] \).
We are tasked with expressing the impulse response \( h[\cdot] \) of the system in terms of the sequence \( g[\cdot] \).
### Explanation:
- \( x[n] = a^n u[n] \):
- This represents an exponential sequence starting at \( n = 0 \).
- \( u[n] \) is the unit step function, which is 0 for \( n < 0 \) and 1 for \( n \geq 0 \).
- \( g[\cdot] \):
- This is the response of the LTI system to the input \( x[n] \).
### Approach:
To find the impulse response \( h[n] \), we use the properties of LTI systems, primarily the principle of superposition and convolution.
Since the input \( x[n] = a^n u[n] \) is not an impulse directly, but a more complex input, the response \( g[n] \) is related to the convolution of the input \( x[n] \) with the impulse response \( h[n] \):
\[ g[n] = x[n] * h[n] \]
This can be written as:
\[ g[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]
Given \( x[k] = a^k u[k] \), the convolution sum becomes:
\[ g[n] = \sum_{k=0}^{n} a^k h[n-k] \]
Our goal is to find \( h[n] \) by working with this relation.
### Conclusion:
Expressing \( h[\cdot] \) in terms of \( g[\cdot
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,