Please convert the below content to rendered equations without any errors.In the pole–zero plots:
* **× = poles**
* **○ = zeros**
* **Unit circle** determines stability.
Key rules used:
1. **Causal system** → ROC outside outermost pole.
2. **Stable system** → all poles inside unit circle.
3. **FIR system** → no poles except possibly at origin.
4. **All-pass system** → poles inside unit circle with mirrored zeros outside.
5. **Minimum-phase** → all zeros inside unit circle.
6. **Linear phase** → symmetric zeros on unit circle.
7. **Stable & causal inverse** → minimum-phase system.
8. **Worst frequency response** → poles close to unit circle.
9. **Shortest impulse response** → FIR system.
10. **Minimum group delay** → minimum-phase system.
---
# (a) Which systems are LP systems?
Linear phase requires **symmetric zeros on the unit circle**.
✔ **(A)**
---
# (b) Which systems are FIR systems?
FIR → **no poles except possibly at origin**.
✔ **(A)**
✔ **(F)**
---
# (c) Which systems are stable systems?
Stable if **all poles inside unit circle**.
✔ **(C)**
✔ **(D)**
✔ **(E)**
✔ **(F)**
---
# (d) Which systems are minimum-phase systems?
Minimum phase → **all zeros inside unit circle**.
✔ **(C)**
✔ **(E)**
---
# (e) Which systems are generalised linear-phase systems?
Requires **zero symmetry about unit circle**.
✔ **(A)**
✔ **(F)**
---
# (f) Which system is an All-pass system?
All-pass → **poles inside circle and reciprocal zeros outside**.
✔ **(B)**
---
# (g) Stable and causal inverse systems?
These require **minimum-phase systems**.
✔ **(C)**
✔ **(E)**
---
# (h) Worst frequency response?
Occurs when **poles lie very close to unit circle**.
✔ **(B)**
✔ **(D)**
---
# (i) Shortest impulse response?
Shortest impulse response → **FIR system**.
✔ **(A)**
✔ **(F)**
---
# (j) Minimum group delay?
Minimum group delay → **minimum phase**.
✔ **(C)**
✔ **(E)**
---
# ✅ Final Answers Summary
| Question | Answer |
| -------- | ---------- |
| a | A |
| b | A, F |
| c | C, D, E, F |
| d | C, E |
| e | A, F |
| f | B |
| g | C, E |
| h | B, D |
| i | A, F |
| j | C, E |
---
✅ If you want, I can also show **a 5-rule shortcut to answer pole-zero questions in exams in under 1 minute**, which is extremely useful for DSP exams.
Question:
Please convert the below content to rendered equations without any errors.In the pole–zero plots:
* **× = poles**
* **○ = zeros**
* **Unit circle** determines stability.
Key rules used:
1. **Causal system** → ROC outside outermost pole.
2. **Stable system** → all poles inside unit circle.
3. **FIR system** → no poles except possibly at origin.
4. **All-pass system** → poles inside unit circle with mirrored zeros outside.
5. **Minimum-phase** → all zeros inside unit circle.
6. **Linear phase** → symmetric zeros on unit circle.
7. **Stable & causal inverse** → minimum-phase system.
8. **Worst frequency response** → poles close to unit circle.
9. **Shortest impulse response** → FIR system.
10. **Minimum group delay** → minimum-phase system.
---
# (a) Which systems are LP systems?
Linear phase requires **symmetric zeros on the unit circle**.
✔ **(A)**
---
# (b) Which systems are FIR systems?
FIR → **no poles except possibly at origin**.
✔ **(A)**
✔ **(F)**
---
# (c) Which systems are stable systems?
Stable if **all poles inside unit circle**.
✔ **(C)**
✔ **(D)**
✔ **(E)**
✔ **(F)**
---
# (d) Which systems are minimum-phase systems?
Minimum phase → **all zeros inside unit circle**.
✔ **(C)**
✔ **(E)**
---
# (e) Which systems are generalised linear-phase systems?
Requires **zero symmetry about unit circle**.
✔ **(A)**
✔ **(F)**
---
# (f) Which system is an All-pass system?
All-pass → **poles inside circle and reciprocal zeros outside**.
✔ **(B)**
---
# (g) Stable and causal inverse systems?
These require **minimum-phase systems**.
✔ **(C)**
✔ **(E)**
---
# (h) Worst frequency response?
Occurs when **poles lie very close to unit circle**.
✔ **(B)**
✔ **(D)**
---
# (i) Shortest impulse response?
Shortest impulse response → **FIR system**.
✔ **(A)**
✔ **(F)**
---
# (j) Minimum group delay?
Minimum group delay → **minimum phase**.
✔ **(C)**
✔ **(E)**
---
# ✅ Final Answers Summary
| Question | Answer |
| -------- | ---------- |
| a | A |
| b | A, F |
| c | C, D, E, F |
| d | C, E |
| e | A, F |
| f | B |
| g | C, E |
| h | B, D |
| i | A, F |
| j | C, E |
---
✅ If you want, I can also show **a 5-rule shortcut to answer pole-zero questions in exams in under 1 minute**, which is extremely useful for DSP exams.
This is AI generated Response.
Asked by: Salaar
Created at: 2026-03-06 23:29:34
Final Answer:
RenderedRaw Text
Did you like this answer?
# Notation and Key Rules
- \( \times \) = poles
- \( \circ \) = zeros
- **Unit circle** (\( |z| = 1 \)) determines stability.
## Key Rules (Math Notation)
1. **Causal system:**
Region of Convergence (ROC) is \( |z| > r_{\text{max}} \), where \( r_{\text{max}} \) is the largest pole magnitude.
2. **Stable system:**
All poles satisfy \( |p_i| < 1 \).
3. **FIR system:**
No poles except possibly at the origin (\( z = \)).
4. **All-pass system:**
For each pole at \( p \), there is a zero at \( 1/p^* \); all poles inside unit circle (\( |p| < 1 \)), zeros outside (\( |z_| > 1 \)).
5. **Minimum-phase:**
All zeros inside unit circle: \( |z_| < 1 \) for all zeros.
6. **Linear phase:**
Zeros are symmetric with respect to the unit circle.
7. **Stable & causal inverse:**
System must be minimum-phase.
8. **Worst frequency response:**
Poles very close to unit circle (\( |p| \approx 1 \)).
9. **Shortest impulse response:**
FIR system.
10. **Minimum group delay:**
Minimum-phase system.
---
# (a) Which systems are Linear Phase (LP) Systems?
Linear phase requires **symmetric zeros on the unit circle**.
- **Answer:**
\( \boxed{A} \)
---
# (b) Which systems are FIR systems?
FIR: **No poles except possibly at \( z = \)**.
- **Answer:**
\( \boxed{A,\, F} \)
---
# (c) Which systems are Stable?
Stable if **all poles inside unit circle** (\( |p| < 1 \)).
- **Answer:**
\( \boxed{C,\, D,\, E,\, F} \)
---
# (d) Which systems are Minimum-Phase?
Minimum phase: **All zeros inside unit circle** (\( |z_| < 1 \)).
- **Answer:**
\( \boxed{C,\, E} \)
---
# (e) Which systems are Generalised Linear-Phase?
Requires **zero symmetry about unit circle**.
- **Answer:**
\( \boxed{A,\, F} \)
---
# (f) Which system is All-Pass?
All-pass: **Poles inside unit circle, reciprocal zeros outside**.
- **Answer:**
\( \boxed{B} \)
---
# (g) Stable and Causal Inverse?
Requires **minimum-phase system**.
- **Answer:**
\( \boxed{C,\, E} \)
---
# (h) Worst Frequency Response?
Occurs when **poles lie very close to unit circle (\( |p| \approx 1 \))**.
- **Answer:**
\( \boxed{B,\, D} \)
---
# (i) Shortest Impulse Response?
Shortest impulse response means **FIR system**.
- **Answer:**
\( \boxed{A,\, F} \)
---
# (j) Minimum Group Delay?
Minimum group delay = **minimum-phase**.
- **Answer:**
\( \boxed{C,\, E} \)
---
# ✅ Final Answers Summary
| Question | Answer |
|:--------:|:--------------|
| a | \(A\) |
| b | \(A,\, F\) |
| c | \(C,\, D,\, E,\, F\) |
| d | \(C,\, E\) |
| e | \(A,\, F\) |
| f | \(B\) |
| g | \(C,\, E\) |
| h | \(B,\, D\) |
| i | \(A,\, F\) |
| j | \(C,\, E\) |
---
If you want, I can also show a **5-rule shortcut for pole-zero questions** for DSP exams.
Question:
Please provide me the answer same as it is without any changes and errors
This is AI generated Response.
Asked by: Salaar
Created at: 2026-03-06 23:30:43
Final Answer:
RenderedRaw Text
Did you like this answer?
# Notation and Key Rules
- \( \times \) = poles
- \( \circ \) = zeros
- **Unit circle** (\( |z| = 1 \)) determines stability.
## Key Rules (Math Notation)
1. **Causal system:**
Region of Convergence (ROC) is \( |z| > r_{\text{max}} \), where \( r_{\text{max}} \) is the largest pole magnitude.
2. **Stable system:**
All poles satisfy \( |p_i| < 1 \).
3. **FIR system:**
No poles except possibly at the origin (\( z = 0 \)).
4. **All-pass system:**
For each pole at \( p \), there is a zero at \( \frac{1}{p^*} \); all poles inside unit circle (\( |p| < 1 \)), zeros outside (\( |z| > 1 \)).
5. **Minimum-phase:**
All zeros inside unit circle: \( |z| < 1 \) for all zeros.
6. **Linear phase:**
Zeros are symmetric with respect to the unit circle.
7. **Stable & causal inverse:**
System must be minimum-phase.
8. **Worst frequency response:**
Poles very close to unit circle (\( |p| \approx 1 \)).
9. **Shortest impulse response:**
FIR system.
10. **Minimum group delay:**
Minimum-phase system.
---
# (a) Which systems are Linear Phase (LP) Systems?
Linear phase requires **symmetric zeros on the unit circle**.
- **Answer:**
\( \boxed{A} \)
---
# (b) Which systems are FIR systems?
FIR: **No poles except possibly at \( z = 0 \)**.
- **Answer:**
\( \boxed{A,\, F} \)
---
# (c) Which systems are Stable?
Stable if **all poles inside unit circle** (\( |p| < 1 \)).
- **Answer:**
\( \boxed{C,\, D,\, E,\, F} \)
---
# (d) Which systems are Minimum-Phase?
Minimum phase: **All zeros inside unit circle** (\( |z| < 1 \)).
- **Answer:**
\( \boxed{C,\, E} \)
---
# (e) Which systems are Generalised Linear-Phase?
Requires **zero symmetry about unit circle**.
- **Answer:**
\( \boxed{A,\, F} \)
---
# (f) Which system is All-Pass?
All-pass: **Poles inside unit circle, reciprocal zeros outside**.
- **Answer:**
\( \boxed{B} \)
---
# (g) Stable and Causal Inverse?
Requires **minimum-phase system**.
- **Answer:**
\( \boxed{C,\, E} \)
---
# (h) Worst Frequency Response?
Occurs when **poles lie very close to unit circle (\( |p| \approx 1 \))**.
- **Answer:**
\( \boxed{B,\, D} \)
---
# (i) Shortest Impulse Response?
Shortest impulse response means **FIR system**.
- **Answer:**
\( \boxed{A,\, F} \)
---
# (j) Minimum Group Delay?
Minimum group delay = **minimum-phase**.
- **Answer:**
\( \boxed{C,\, E} \)
---
# ✅ Final Answers Summary
| Question | Answer |
|:--------:|:--------------|
| a | \(A\) |
| b | \(A,\, F\) |
| c | \(C,\, D,\, E,\, F\) |
| d | \(C,\, E\) |
| e | \(A,\, F\) |
| f | \(B\) |
| g | \(C,\, E\) |
| h | \(B,\, D\) |
| i | \(A,\, F\) |
| j | \(C,\, E\) |
---
If you want, I can also show a **5-rule shortcut for pole-zero questions** for DSP exams.
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