The system function H(z) of a stable LTI system has three poles and three zeros, as shown in the plot below. Also, the system response to the input x[n] = (-1)" (all n) is given by y[n] = 20(-1)" (all n). (М 3 2) z = 0 (M = 1) z = 1 z = -2 = 2) z = 1/2 а) Determine H(z) and its region of convergence. b) Determine the impulse response h[n]. c) Give one value of a (other than zero) for which the sequence x'[n] = a" (for all n) %3D is not an acceptable input for this system. 2)
The system function H(z) of a stable LTI system has three poles and three zeros, as shown in the plot below. Also, the system response to the input x[n] = (-1)" (all n) is given by y[n] = 20(-1)" (all n). (М 3 2) z = 0 (M = 1) z = 1 z = -2 = 2) z = 1/2 а) Determine H(z) and its region of convergence. b) Determine the impulse response h[n]. c) Give one value of a (other than zero) for which the sequence x'[n] = a" (for all n) %3D is not an acceptable input for this system. 2)
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![### Problem Statement
The system function \( H(z) \) of a stable LTI system has three poles and three zeros, as shown in the plot below. Also, the system response to the input \( x[n] = (-1)^n \) (for all \( n \)) is given by \( y[n] = 20 (-1)^n \) (for all \( n \)).
#### Pole-Zero Plot:
- Zeros:
- \( z = -2 \) (multiplicity 2)
- \( z = 0 \) (multiplicity 1)
- Poles:
- \( z = 1 \) (multiplicity 1)
- \( z = \frac{1}{2} \) (multiplicity 1)
This can be visually represented with the pole-zero plot:
- Zeros are marked with an "x".
- Poles are marked with a "o".
```plaintext
(M = 1)
o
-------------
| |
x o (M = 1)
z = -2 z = 1/2
(M = 2)
x
z = 0
(M = 2)
```
### Questions:
a) **Determine \( H(z) \) and its region of convergence.**
b) **Determine the impulse response \( h[n] \).**
c) **Give one value of \( a \) (other than zero) for which the sequence \( x'[n] = a^n \) (for all \( n \)) is not an acceptable input for this system.**
---
### Solutions:
**a) Determining \( H(z) \) and its region of convergence:**
The system function \( H(z) \) can be expressed as the ratio of the polynomial of zeros and the polynomial of poles.
Given the zeros and poles:
\[
H(z) = K \frac{(z+2)^2 z}{(z-1)\left(z-\frac{1}{2}\right)}
\]
where \( K \) is a constant to be determined using the given response.
**b) Determining the impulse response \( h[n] \):**
The impulse response \( h[n] \) is the inverse Z-transform of \( H(z) \).
**c) Finding a non-zero value of \( a \) for which](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd07e8233-789e-4760-ba71-e70869af7b54%2F26bc63ef-ff92-4967-be67-2ac43d03dab2%2Fuqgpg7h_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
The system function \( H(z) \) of a stable LTI system has three poles and three zeros, as shown in the plot below. Also, the system response to the input \( x[n] = (-1)^n \) (for all \( n \)) is given by \( y[n] = 20 (-1)^n \) (for all \( n \)).
#### Pole-Zero Plot:
- Zeros:
- \( z = -2 \) (multiplicity 2)
- \( z = 0 \) (multiplicity 1)
- Poles:
- \( z = 1 \) (multiplicity 1)
- \( z = \frac{1}{2} \) (multiplicity 1)
This can be visually represented with the pole-zero plot:
- Zeros are marked with an "x".
- Poles are marked with a "o".
```plaintext
(M = 1)
o
-------------
| |
x o (M = 1)
z = -2 z = 1/2
(M = 2)
x
z = 0
(M = 2)
```
### Questions:
a) **Determine \( H(z) \) and its region of convergence.**
b) **Determine the impulse response \( h[n] \).**
c) **Give one value of \( a \) (other than zero) for which the sequence \( x'[n] = a^n \) (for all \( n \)) is not an acceptable input for this system.**
---
### Solutions:
**a) Determining \( H(z) \) and its region of convergence:**
The system function \( H(z) \) can be expressed as the ratio of the polynomial of zeros and the polynomial of poles.
Given the zeros and poles:
\[
H(z) = K \frac{(z+2)^2 z}{(z-1)\left(z-\frac{1}{2}\right)}
\]
where \( K \) is a constant to be determined using the given response.
**b) Determining the impulse response \( h[n] \):**
The impulse response \( h[n] \) is the inverse Z-transform of \( H(z) \).
**c) Finding a non-zero value of \( a \) for which
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