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Problem 1. A Linear Time Invariant (LTI) system is defined by the Linear Difference Equation y[n]=1.2728 y[n-1]-0.81y[n– 2]+ x[n]– x[n– 1] with x[n], y[n] input and output sequences respectively. Q1. Determine its transfer function H(z), poles, zeros and Region of Convergence (ROC); Q2. Determine the output signal when the input is x[n]=1+2cos(0.5An-0.75); Q3. Approximately, around which frequency 0

Problem 1. A Linear Time Invariant (LTI) system is defined by the Linear Difference Equation y[n]=1.2728 y[n-1]-0.81y[n– 2]+ x[n]– x[n– 1] with x[n], y[n] input and output sequences respectively. Q1. Determine its transfer function H(z), poles, zeros and Region of Convergence (ROC); Q2. Determine the output signal when the input is x[n]=1+2cos(0.5An-0.75); Q3. Approximately, around which frequency 0

Power System Analysis and Design (MindTap Course List)
Power System Analysis and Design (MindTap Course List)
6th Edition
ISBN:9781305632134
Author:J. Duncan Glover, Thomas Overbye, Mulukutla S. Sarma
Publisher:J. Duncan Glover, Thomas Overbye, Mulukutla S. Sarma
Chapter6: Power Flows
Section: Chapter Questions
Problem 6.22P
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Please help with the solution to this LTI problem

### Problem 1 A Linear Time Invariant (LTI) system is defined by the Linear Difference Equation: \[ y[n] = 1.2728y[n-1] - 0.81y[n-2] + x[n] - x[n-1] \] where \( x[n] \) and \( y[n] \) are the input and output sequences, respectively. **Questions:** 1. **Determine its transfer function \( H(z) \), poles, zeros, and Region of Convergence (ROC):** - For this, you'll need to apply the Z-transform to the difference equation and solve for \( H(z) = \frac{Y(z)}{X(z)} \). Identify the values of \( z \) for which the denominator is zero (poles) and the zeros. Determine the ROC for stability. 2. **Determine the output signal when the input is \( x[n] = 1 + 2 \cos(0.5\pi n - 0.75) \):** - Substitute the given \( x[n] \) into the difference equation and solve for \( y[n] \). 3. **Approximately, around which frequency \( 0 \leq \omega \leq \pi \) is the magnitude of the frequency response you think is the largest?** - Analyze the frequency response \( H(e^{j\omega}) \) to find the frequency at which the magnitude is maximized. Please refer to relevant course materials or textbooks for detailed methodologies and mathematical steps to solve these questions.
Transcribed Image Text:### Problem 1 A Linear Time Invariant (LTI) system is defined by the Linear Difference Equation: \[ y[n] = 1.2728y[n-1] - 0.81y[n-2] + x[n] - x[n-1] \] where \( x[n] \) and \( y[n] \) are the input and output sequences, respectively. **Questions:** 1. **Determine its transfer function \( H(z) \), poles, zeros, and Region of Convergence (ROC):** - For this, you'll need to apply the Z-transform to the difference equation and solve for \( H(z) = \frac{Y(z)}{X(z)} \). Identify the values of \( z \) for which the denominator is zero (poles) and the zeros. Determine the ROC for stability. 2. **Determine the output signal when the input is \( x[n] = 1 + 2 \cos(0.5\pi n - 0.75) \):** - Substitute the given \( x[n] \) into the difference equation and solve for \( y[n] \). 3. **Approximately, around which frequency \( 0 \leq \omega \leq \pi \) is the magnitude of the frequency response you think is the largest?** - Analyze the frequency response \( H(e^{j\omega}) \) to find the frequency at which the magnitude is maximized. Please refer to relevant course materials or textbooks for detailed methodologies and mathematical steps to solve these questions.
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