Suppose a discrete-time LTI system is characterized by the following system func- 2-¹2-1 4 1-212-¹2-2 + 4 (a) What are the possible ROCs for H(z)? for each possible ROC, sketch the pole-zero plot. (b) For each possible ROC, determine if the system is causal, non-causal, or anti-causal. (c) For each possible ROC, determine if the system is BIBO stable or not. (d) For each possible ROC, determine the impulse response h(n). tion: H(z) = 5
Suppose a discrete-time LTI system is characterized by the following system func- 2-¹2-1 4 1-212-¹2-2 + 4 (a) What are the possible ROCs for H(z)? for each possible ROC, sketch the pole-zero plot. (b) For each possible ROC, determine if the system is causal, non-causal, or anti-causal. (c) For each possible ROC, determine if the system is BIBO stable or not. (d) For each possible ROC, determine the impulse response h(n). tion: H(z) = 5
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**Discrete-Time LTI System Analysis**
Suppose a discrete-time Linear Time-Invariant (LTI) system is characterized by the following system function:
\[
H(z) = \frac{2 - \frac{21}{4} z^{-1}}{1 - \frac{21}{4} z^{-1} + \frac{5}{4} z^{-2}}
\]
### Tasks:
(a) **Region of Convergence (ROC):**
- Determine the possible ROCs for \( H(z) \).
- For each possible ROC, sketch the pole-zero plot.
(b) **Causality:**
- For each possible ROC, determine if the system is causal, non-causal, or anti-causal.
(c) **Stability:**
- For each possible ROC, determine if the system is BIBO (Bounded Input Bounded Output) stable or not.
(d) **Impulse Response:**
- For each possible ROC, determine the impulse response \( h(n) \).
---
**Explanation of System Function:**
The given LTI system function is expressed in terms of \( z \) and consists of a ratio of polynomials in \( z^{-1} \).
- **Numerator:** \( 2 - \frac{21}{4} z^{-1} \)
- **Denominator:** \( 1 - \frac{21}{4} z^{-1} + \frac{5}{4} z^{-2} \)
This form is typical for rational functions used in the analysis of discrete-time systems. The poles and zeros of the system function are key to understanding its behavior, including causality, stability, and the nature of the impulse response.
**Note:**
- Poles are values of \( z \) that make the denominator zero.
- Zeros are values of \( z \) that make the numerator zero.
- The ROC determines the range of \( z \) values for which the \( H(z) \) converges.
For solving the tasks:
1. **ROC and Pole-Zero Plot:** Identify poles and zeros, and mark them on the complex plane.
2. **Causality:** Determine the ROC in relation to the outermost pole.
3. **Stability:** Check if all poles are within the unit circle in the z-plane for BIBO stability.
4. **Impulse Response:** Analyze the inverse Z-transform of \( H(z)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb23cde1-d29d-434b-98b1-3e624e29f5ef%2Fded80d59-f3c1-4ca7-813a-aca8d3812ec9%2F6plvdhy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Discrete-Time LTI System Analysis**
Suppose a discrete-time Linear Time-Invariant (LTI) system is characterized by the following system function:
\[
H(z) = \frac{2 - \frac{21}{4} z^{-1}}{1 - \frac{21}{4} z^{-1} + \frac{5}{4} z^{-2}}
\]
### Tasks:
(a) **Region of Convergence (ROC):**
- Determine the possible ROCs for \( H(z) \).
- For each possible ROC, sketch the pole-zero plot.
(b) **Causality:**
- For each possible ROC, determine if the system is causal, non-causal, or anti-causal.
(c) **Stability:**
- For each possible ROC, determine if the system is BIBO (Bounded Input Bounded Output) stable or not.
(d) **Impulse Response:**
- For each possible ROC, determine the impulse response \( h(n) \).
---
**Explanation of System Function:**
The given LTI system function is expressed in terms of \( z \) and consists of a ratio of polynomials in \( z^{-1} \).
- **Numerator:** \( 2 - \frac{21}{4} z^{-1} \)
- **Denominator:** \( 1 - \frac{21}{4} z^{-1} + \frac{5}{4} z^{-2} \)
This form is typical for rational functions used in the analysis of discrete-time systems. The poles and zeros of the system function are key to understanding its behavior, including causality, stability, and the nature of the impulse response.
**Note:**
- Poles are values of \( z \) that make the denominator zero.
- Zeros are values of \( z \) that make the numerator zero.
- The ROC determines the range of \( z \) values for which the \( H(z) \) converges.
For solving the tasks:
1. **ROC and Pole-Zero Plot:** Identify poles and zeros, and mark them on the complex plane.
2. **Causality:** Determine the ROC in relation to the outermost pole.
3. **Stability:** Check if all poles are within the unit circle in the z-plane for BIBO stability.
4. **Impulse Response:** Analyze the inverse Z-transform of \( H(z)
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