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Write the difference equation of the following system. x[n] D DD -2 →y[n]

Write the difference equation of the following system. x[n] D DD -2 →y[n]

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
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**Task: Write the Difference Equation for the System** **Diagram Explanation:** The diagram is a block diagram representing a discrete-time system. The components and their connections are as follows: 1. **Input:** \( x[n] \) 2. **Adder Nodes:** There are several summing nodes in the system, which combine signals. 3. **Delay Elements (D):** These blocks introduce a delay of one time unit to the signals. There are three delay elements in the system. 4. **Multiplier (\(\times -2\)):** This block multiplies the incoming signal by -2. 5. **Output:** \( y[n] \) **Signal Flow:** - The input \( x[n] \) splits into two paths: - The first path goes directly to the output summing node. - The second path goes through a node where it is summed with the output of a delayed signal from another branch. - The second path enters a delay block \( D \) and then proceeds through the multiplier \(\times -2\). - The result of the multiplication is fed into another delay block \( D \), whose output is connected to an adder node where it is combined with \( x[n] \). - The summed output from the second adder node is passed through another delay block \( D \) before forming part of \( y[n] \). **Writing the Equation:** The system can be decomposed to form the relationship and derive the difference equation. Use the signals at various points to express relationships in mathematical terms. **Solution Approach:** 1. Use intermediate variables to represent signals at different points in the system. 2. Write equations based on the signal flows and transformations (additions, delays, multiplications). 3. Combine these to achieve the difference equation that relates \( x[n] \) and \( y[n] \). By analyzing the block diagram in this way, students will understand how to derive difference equations from discrete-time system diagrams.
Transcribed Image Text:**Task: Write the Difference Equation for the System** **Diagram Explanation:** The diagram is a block diagram representing a discrete-time system. The components and their connections are as follows: 1. **Input:** \( x[n] \) 2. **Adder Nodes:** There are several summing nodes in the system, which combine signals. 3. **Delay Elements (D):** These blocks introduce a delay of one time unit to the signals. There are three delay elements in the system. 4. **Multiplier (\(\times -2\)):** This block multiplies the incoming signal by -2. 5. **Output:** \( y[n] \) **Signal Flow:** - The input \( x[n] \) splits into two paths: - The first path goes directly to the output summing node. - The second path goes through a node where it is summed with the output of a delayed signal from another branch. - The second path enters a delay block \( D \) and then proceeds through the multiplier \(\times -2\). - The result of the multiplication is fed into another delay block \( D \), whose output is connected to an adder node where it is combined with \( x[n] \). - The summed output from the second adder node is passed through another delay block \( D \) before forming part of \( y[n] \). **Writing the Equation:** The system can be decomposed to form the relationship and derive the difference equation. Use the signals at various points to express relationships in mathematical terms. **Solution Approach:** 1. Use intermediate variables to represent signals at different points in the system. 2. Write equations based on the signal flows and transformations (additions, delays, multiplications). 3. Combine these to achieve the difference equation that relates \( x[n] \) and \( y[n] \). By analyzing the block diagram in this way, students will understand how to derive difference equations from discrete-time system diagrams.
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