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**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)
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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**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
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
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