Problem 2. Consider the LTI system defined as y[n] = 0.5y[n-1]+2x[n]+ x[n–1] Q1. Is the system stable? Causal?
Problem 2. Consider the LTI system defined as y[n] = 0.5y[n-1]+2x[n]+ x[n–1] Q1. Is the system stable? Causal?
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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![### Problem 2: Understanding an LTI System
Consider the Linear Time-Invariant (LTI) system defined by the equation:
\[ y[n] = 0.5y[n-1] + 2x[n] + x[n-1] \]
#### Q1. Is the System Stable? Causal?
- **Stability:** A system is stable if, for any bounded input, the output is also bounded. To determine this, check the system's impulse response or the bound on the coefficients.
- **Causality:** A system is causal if the output at any time depends only on past and present inputs and not on future inputs. This means \( y[n] \) should only depend on \( x[n], x[n-1], \dots \) and not on \( x[n+k] \) for \( k > 0 \).
#### Q2. Determine the Output Signal \( y[n] \)
Given the input \( x[n] = u[-n] \) where \( u[n] \) is the unit step sequence. The unit step sequence \( u[n] \) is defined as:
\[ u[n] = \begin{cases}
1 & \text{if } n \geq 0 \\
0 & \text{if } n < 0
\end{cases} \]
To find \( y[n] \), recall that the input sequence can be written as the sum of a causal sequence and an anticausal sequence. You will need to:
1. Establish the expression for \( x[n] \).
2. Derive the output \( y[n] \) using the given system equation.
### Detailed Explanation for Q2:
1. **Write the Input Sequence:**
\[ x[n] = u[-n] = \begin{cases}
1 & \text{if } n \leq 0 \\
0 & \text{if } n > 0
\end{cases} \]
2. **Determine the Output Sequence:** Using the recursion formula given by the system,
\[ y[n] = 0.5y[n-1] + 2x[n] + x[n-1] \]
substitute \( x[n] \) and solve iteratively for \( y[n] \).
#### Notes:
- Calculations or further steps to derive \( y[n] \) will](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1345a87f-3d4a-45ee-a13b-cb772771cc65%2F32cc8ae9-d39f-4009-92bd-afd556db996e%2Fj10mum_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2: Understanding an LTI System
Consider the Linear Time-Invariant (LTI) system defined by the equation:
\[ y[n] = 0.5y[n-1] + 2x[n] + x[n-1] \]
#### Q1. Is the System Stable? Causal?
- **Stability:** A system is stable if, for any bounded input, the output is also bounded. To determine this, check the system's impulse response or the bound on the coefficients.
- **Causality:** A system is causal if the output at any time depends only on past and present inputs and not on future inputs. This means \( y[n] \) should only depend on \( x[n], x[n-1], \dots \) and not on \( x[n+k] \) for \( k > 0 \).
#### Q2. Determine the Output Signal \( y[n] \)
Given the input \( x[n] = u[-n] \) where \( u[n] \) is the unit step sequence. The unit step sequence \( u[n] \) is defined as:
\[ u[n] = \begin{cases}
1 & \text{if } n \geq 0 \\
0 & \text{if } n < 0
\end{cases} \]
To find \( y[n] \), recall that the input sequence can be written as the sum of a causal sequence and an anticausal sequence. You will need to:
1. Establish the expression for \( x[n] \).
2. Derive the output \( y[n] \) using the given system equation.
### Detailed Explanation for Q2:
1. **Write the Input Sequence:**
\[ x[n] = u[-n] = \begin{cases}
1 & \text{if } n \leq 0 \\
0 & \text{if } n > 0
\end{cases} \]
2. **Determine the Output Sequence:** Using the recursion formula given by the system,
\[ y[n] = 0.5y[n-1] + 2x[n] + x[n-1] \]
substitute \( x[n] \) and solve iteratively for \( y[n] \).
#### Notes:
- Calculations or further steps to derive \( y[n] \) will
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