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BC:5.1 In the flow diagram shown in Fig. 1, the blocks with the "D" represent a one-step time delay. Find the difference equation in the form N M Σaky[n - k] = Σban - p] k=0 p=0 where ao = = 1. q[n] D 0.24 ↓ x[n]- -5- Σ 2 w[n] -0.5 D Σ y[n]

BC:5.1 In the flow diagram shown in Fig. 1, the blocks with the "D" represent a one-step time delay. Find the difference equation in the form N M Σaky[n - k] = Σban - p] k=0 p=0 where ao = = 1. q[n] D 0.24 ↓ x[n]- -5- Σ 2 w[n] -0.5 D Σ 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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**BC:5.1** In the flow diagram shown in Fig. 1, the blocks with the “D” represent a one-step time delay. Find the difference equation in the form \[ \sum_{k=0}^{N} a_k y[n-k] = \sum_{p=0}^{M} b_p x[n-p] \] where \( a_0 = 1 \). **Flow Diagram Explanation:** - The diagram describes a digital filter system with input \( x[n] \) and output \( y[n] \). - **"D" Blocks:** Represent a one-step time delay. - **Paths:** - The input \( x[n] \) branches into two paths. - In the upper path, \( x[n] \) is multiplied by 2 and added to the signal before the summation that leads to \( y[n] \). - In the lower path, \( x[n] \) is multiplied by -5 and summed. - Before reaching the final summation to output \( y[n] \): - A delayed version of a signal, \( w[n] \), is present. - There is another path into this summation from a previously delayed signal \( q[n] \), multiplied by 0.24. - **Intermediate Nodes:** - Initial summation node adds \( -5x[n] \) and the delayed \( q[n] \). - The result is then multiplied by -0.5 and delayed once more before proceeding to the final summation node. This diagram represents a recursive filter characterized by the feedback and feedforward paths, whose system equation needs to be derived from the given structure. The goal is to identify the coefficients \( a_k \) and \( b_p \) that relate the output \( y[n] \) to the input \( x[n] \) and its delayed versions.
Transcribed Image Text:**BC:5.1** In the flow diagram shown in Fig. 1, the blocks with the “D” represent a one-step time delay. Find the difference equation in the form \[ \sum_{k=0}^{N} a_k y[n-k] = \sum_{p=0}^{M} b_p x[n-p] \] where \( a_0 = 1 \). **Flow Diagram Explanation:** - The diagram describes a digital filter system with input \( x[n] \) and output \( y[n] \). - **"D" Blocks:** Represent a one-step time delay. - **Paths:** - The input \( x[n] \) branches into two paths. - In the upper path, \( x[n] \) is multiplied by 2 and added to the signal before the summation that leads to \( y[n] \). - In the lower path, \( x[n] \) is multiplied by -5 and summed. - Before reaching the final summation to output \( y[n] \): - A delayed version of a signal, \( w[n] \), is present. - There is another path into this summation from a previously delayed signal \( q[n] \), multiplied by 0.24. - **Intermediate Nodes:** - Initial summation node adds \( -5x[n] \) and the delayed \( q[n] \). - The result is then multiplied by -0.5 and delayed once more before proceeding to the final summation node. This diagram represents a recursive filter characterized by the feedback and feedforward paths, whose system equation needs to be derived from the given structure. The goal is to identify the coefficients \( a_k \) and \( b_p \) that relate the output \( y[n] \) to the input \( x[n] \) and its delayed versions.
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