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The causal system described by the difference equation y[n] + y[n – 2] = x[n] – 3x[n – 1] is such that x[n] = y[n] = 0 at all times n < 0. If the input x[n] = 8[n] + 28[n – 1] – 38[n – 2] | is applied to this system, determine the output y[n] for all n > 0. (There are at most six distinct cases for n.)

The causal system described by the difference equation y[n] + y[n – 2] = x[n] – 3x[n – 1] is such that x[n] = y[n] = 0 at all times n < 0. If the input x[n] = 8[n] + 28[n – 1] – 38[n – 2] | is applied to this system, determine the output y[n] for all n > 0. (There are at most six distinct cases for 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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### Difference Equations and Causal Systems This lesson focuses on understanding the behavior of a causal system described by a difference equation and how to determine its output given a specific input. #### The Difference Equation The system is defined by the following difference equation: \[ y[n] + y[n-2] = x[n] - 3x[n-1] \] For this configuration: - \( y[n] \) represents the output at time \( n \). - \( x[n] \) represents the input at time \( n \). The system is described as causal, meaning that \( x[n] = y[n] = 0 \) for all times \( n < 0 \). #### The Input The input signal \( x[n] \) is specified as: \[ x[n] = \delta[n] + 2\delta[n-1] - 3\delta[n-2] \] Here, \( \delta[n] \) represents the discrete-time unit impulse function, which is defined as: \[ \delta[n] = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{otherwise} \end{cases} \] #### Determining the Output The primary task is to determine the output \( y[n] \) for all \( n \geq 0 \). Given that “There are at most six distinct cases for \( n \),” the solution will involve sequentially solving the difference equation for different values of \( n \). Understanding these steps will enable you to solve for \( y[n] \) based on the given inputs and initial conditions of the system. Continue to the next sections for detailed calculations and interpretations for solving these kinds of equations.
Transcribed Image Text:### Difference Equations and Causal Systems This lesson focuses on understanding the behavior of a causal system described by a difference equation and how to determine its output given a specific input. #### The Difference Equation The system is defined by the following difference equation: \[ y[n] + y[n-2] = x[n] - 3x[n-1] \] For this configuration: - \( y[n] \) represents the output at time \( n \). - \( x[n] \) represents the input at time \( n \). The system is described as causal, meaning that \( x[n] = y[n] = 0 \) for all times \( n < 0 \). #### The Input The input signal \( x[n] \) is specified as: \[ x[n] = \delta[n] + 2\delta[n-1] - 3\delta[n-2] \] Here, \( \delta[n] \) represents the discrete-time unit impulse function, which is defined as: \[ \delta[n] = \begin{cases} 1 & \text{if } n = 0 \\ 0 & \text{otherwise} \end{cases} \] #### Determining the Output The primary task is to determine the output \( y[n] \) for all \( n \geq 0 \). Given that “There are at most six distinct cases for \( n \),” the solution will involve sequentially solving the difference equation for different values of \( n \). Understanding these steps will enable you to solve for \( y[n] \) based on the given inputs and initial conditions of the system. Continue to the next sections for detailed calculations and interpretations for solving these kinds of equations.
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