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The state-space representation of a causal LTI system is given by, õ[n + 1] = A · õ[n] + B • x[n] y[n] = C · õ[n] + D· x[n] where x[n] and y[n] are the input and output to the system, respectively. The state-vector is given by matrix, [v[n – 2]] [u]4 lv[n – 1]] The matrices in the equation above are, го 1 1 B = C A = |2 %3D 6] 3] 35 21 C = D = [1] 24 а) Sketch the Direct-Form II block-diagram representation of the given causal LTI System b) Determine the difference equation relating x[n] and y[n]. c) ) Is this system BIBO stable?

The state-space representation of a causal LTI system is given by, õ[n + 1] = A · õ[n] + B • x[n] y[n] = C · õ[n] + D· x[n] where x[n] and y[n] are the input and output to the system, respectively. The state-vector is given by matrix, [v[n – 2]] [u]4 lv[n – 1]] The matrices in the equation above are, го 1 1 B = C A = |2 %3D 6] 3] 35 21 C = D = [1] 24 а) Sketch the Direct-Form II block-diagram representation of the given causal LTI System b) Determine the difference equation relating x[n] and y[n]. c) ) Is this system BIBO stable?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### State-Space Representation of a Causal LTI System The state-space representation of a causal Linear Time-Invariant (LTI) system is given by: \[ \mathbf{v}[n + 1] = A \cdot \mathbf{v}[n] + B \cdot x[n] \] \[ y[n] = C \cdot \mathbf{v}[n] + D \cdot x[n] \] where \( x[n] \) and \( y[n] \) are the input and output to the system, respectively. The state-vector is given by the matrix, \[ \mathbf{v}[n] = \begin{bmatrix} v[n-2] \\ v[n-1] \end{bmatrix} \] #### The matrices in the equation above are: \[ A = \begin{bmatrix} 0 & 1 \\ \frac{2}{9} & -\frac{1}{3} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} -\frac{35}{24} & \frac{2}{9} \end{bmatrix}, \quad D = [1] \] #### Exercises: a) **Sketch the Direct-Form II block-diagram representation of the given causal LTI system** - This part asks to sketch the Direct-Form II block diagram. You would need to visualize or draw a specific layout of how the system in state-space form is implemented using delay elements, adders, and multipliers. b) **Determine the difference equation relating \( x[n] \) and \( y[n] \)** - Convert the given state-space representation to a difference equation, showing a direct relationship between the input \( x[n] \) and the output \( y[n] \). c) **Is this system BIBO stable?** - Verify whether the system is Bounded Input, Bounded Output (BIBO) stable. This can be determined by analyzing whether the system's response remains bounded for any given bounded input. ### Explanation of Graphs or Diagrams - Since no diagrams or graphs are provided in the text, ensure any block diagrams or illustrations reflect the mathematical relationships and transformations specified in the equations and state-space matrices.
Transcribed Image Text:### State-Space Representation of a Causal LTI System The state-space representation of a causal Linear Time-Invariant (LTI) system is given by: \[ \mathbf{v}[n + 1] = A \cdot \mathbf{v}[n] + B \cdot x[n] \] \[ y[n] = C \cdot \mathbf{v}[n] + D \cdot x[n] \] where \( x[n] \) and \( y[n] \) are the input and output to the system, respectively. The state-vector is given by the matrix, \[ \mathbf{v}[n] = \begin{bmatrix} v[n-2] \\ v[n-1] \end{bmatrix} \] #### The matrices in the equation above are: \[ A = \begin{bmatrix} 0 & 1 \\ \frac{2}{9} & -\frac{1}{3} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} -\frac{35}{24} & \frac{2}{9} \end{bmatrix}, \quad D = [1] \] #### Exercises: a) **Sketch the Direct-Form II block-diagram representation of the given causal LTI system** - This part asks to sketch the Direct-Form II block diagram. You would need to visualize or draw a specific layout of how the system in state-space form is implemented using delay elements, adders, and multipliers. b) **Determine the difference equation relating \( x[n] \) and \( y[n] \)** - Convert the given state-space representation to a difference equation, showing a direct relationship between the input \( x[n] \) and the output \( y[n] \). c) **Is this system BIBO stable?** - Verify whether the system is Bounded Input, Bounded Output (BIBO) stable. This can be determined by analyzing whether the system's response remains bounded for any given bounded input. ### Explanation of Graphs or Diagrams - Since no diagrams or graphs are provided in the text, ensure any block diagrams or illustrations reflect the mathematical relationships and transformations specified in the equations and state-space matrices.
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