What is the general form of the response of H(z) = excited by a step sequence? z² + 2z+1 (z − 1)(z-0.5 + j0.6) (z - 0.5 - j0.6)
What is the general form of the response of H(z) = excited by a step sequence? z² + 2z+1 (z − 1)(z-0.5 + j0.6) (z - 0.5 - j0.6)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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![### Transcription for Educational Website:
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#### Problem Statement:
**Question:** What is the general form of the response of
\[ H(z) = \frac{z^2 + 2z + 1}{(z - 1)(z - 0.5 + j0.6)(z - 0.5 - j0.6)} \]
when excited by a step sequence?
---
### Detailed Explanation:
In this problem, we are asked to determine the general form of the response of a given system, \( H(z) \), when it is excited by a step sequence.
The transfer function \( H(z) \) is given by:
\[ H(z) = \frac{z^2 + 2z + 1}{(z - 1)(z - 0.5 + j0.6)(z - 0.5 - j0.6)} \]
This function can be broken down as follows:
- The numerator \( z^2 + 2z + 1 \) is a polynomial in \( z \).
- The denominator consists of three factors:
- \( (z - 1) \)
- \( (z - 0.5 + j0.6) \)
- \( (z - 0.5 - j0.6) \)
To find the response when excited by a step sequence, we need to perform the following steps:
1. **Partial Fraction Decomposition:**
Break down the rational function into partial fractions.
2. **Inverse Z-Transform:**
Apply the inverse Z-transform to find the time-domain response.
3. **Step Sequence Effect:**
Consider the effect of the step sequence on the identified response in the time domain.
### Graph/Diagram Explanation:
There are no graphs or diagrams provided with this problem. However, if a diagram were to be included, it would typically involve the pole-zero plot of the transfer function, which provides a visual representation of the location of the poles and zeros of \( H(z) \) in the complex plane.
---
This explanation breaks down the problem into manageable steps, providing a clear pathway to solve for the system's response to a step sequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F256d0281-9872-4731-9411-f33987b406a3%2F63354737-d6d9-46e7-ad22-595993f5b25a%2Fcj7qcop_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Website:
---
#### Problem Statement:
**Question:** What is the general form of the response of
\[ H(z) = \frac{z^2 + 2z + 1}{(z - 1)(z - 0.5 + j0.6)(z - 0.5 - j0.6)} \]
when excited by a step sequence?
---
### Detailed Explanation:
In this problem, we are asked to determine the general form of the response of a given system, \( H(z) \), when it is excited by a step sequence.
The transfer function \( H(z) \) is given by:
\[ H(z) = \frac{z^2 + 2z + 1}{(z - 1)(z - 0.5 + j0.6)(z - 0.5 - j0.6)} \]
This function can be broken down as follows:
- The numerator \( z^2 + 2z + 1 \) is a polynomial in \( z \).
- The denominator consists of three factors:
- \( (z - 1) \)
- \( (z - 0.5 + j0.6) \)
- \( (z - 0.5 - j0.6) \)
To find the response when excited by a step sequence, we need to perform the following steps:
1. **Partial Fraction Decomposition:**
Break down the rational function into partial fractions.
2. **Inverse Z-Transform:**
Apply the inverse Z-transform to find the time-domain response.
3. **Step Sequence Effect:**
Consider the effect of the step sequence on the identified response in the time domain.
### Graph/Diagram Explanation:
There are no graphs or diagrams provided with this problem. However, if a diagram were to be included, it would typically involve the pole-zero plot of the transfer function, which provides a visual representation of the location of the poles and zeros of \( H(z) \) in the complex plane.
---
This explanation breaks down the problem into manageable steps, providing a clear pathway to solve for the system's response to a step sequence.
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