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Consider the following statements regarding a linear discrete-time system:
H (z) = z2+1/(z+0.5)(z-0.5)
1. The system is stable
2. The initial value of h(0) of the impulse response is -4
3. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequency
Which of these statements are correct?
H (z) = z2+1/(z+0.5)(z-0.5)
1. The system is stable
2. The initial value of h(0) of the impulse response is -4
3. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequency
Which of these statements are correct?
- a)1,2 and 3
- b)1 and 2
- c)1 and 3
- d)2 and 3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider the following statements regarding a linear discrete-time sys...
Explanation: Characteristic equation is (z+0.5) (z-0.5) =0
Its root are z =0.5, -0.5
Since both roots are inside the unit circle, hence the system is stable.
Its root are z =0.5, -0.5
Since both roots are inside the unit circle, hence the system is stable.
Most Upvoted Answer
Consider the following statements regarding a linear discrete-time sys...
Explanation:
Stability:
- The given system H(z) has poles at z = -0.5 and z = 0.5.
- For a system to be stable, all poles should lie within the unit circle in the z-plane.
- In this case, both poles are inside the unit circle (|z| < 1),="" so="" the="" system="" is="" stable.="" />
Impulse Response:
- The impulse response of a system is the output when the input is an impulse function (i.e., h(0) = 1).
- Given H(z) = z^2 + 1 / (z + 0.5)(z - 0.5), the inverse Z-transform of H(z) gives the impulse response h(n).
- The initial value of h(0) is the coefficient of z^0 in the inverse Z-transform of H(z).
- By calculating the inverse Z-transform, the initial value of h(0) is found to be -4, not -43 as mentioned in the question.
Steady-State Output:
- For a sinusoidal input at a frequency equal to one-fourth the sampling frequency, the system will exhibit a steady-state response.
- The steady-state output of a linear system to a sinusoidal input at a certain frequency depends on the system's transfer function.
- In this case, the steady-state output will not be zero for a sinusoidal input at one-fourth the sampling frequency due to the system's transfer function H(z).
Therefore, the correct statements are:
- The system is stable (Statement 1)
- The steady-state output is not zero for a sinusoidal discrete-time input at one-fourth the sampling frequency (Statement 3)
Stability:
- The given system H(z) has poles at z = -0.5 and z = 0.5.
- For a system to be stable, all poles should lie within the unit circle in the z-plane.
- In this case, both poles are inside the unit circle (|z| < 1),="" so="" the="" system="" is="" stable.="" />
Impulse Response:
- The impulse response of a system is the output when the input is an impulse function (i.e., h(0) = 1).
- Given H(z) = z^2 + 1 / (z + 0.5)(z - 0.5), the inverse Z-transform of H(z) gives the impulse response h(n).
- The initial value of h(0) is the coefficient of z^0 in the inverse Z-transform of H(z).
- By calculating the inverse Z-transform, the initial value of h(0) is found to be -4, not -43 as mentioned in the question.
Steady-State Output:
- For a sinusoidal input at a frequency equal to one-fourth the sampling frequency, the system will exhibit a steady-state response.
- The steady-state output of a linear system to a sinusoidal input at a certain frequency depends on the system's transfer function.
- In this case, the steady-state output will not be zero for a sinusoidal input at one-fourth the sampling frequency due to the system's transfer function H(z).
Therefore, the correct statements are:
- The system is stable (Statement 1)
- The steady-state output is not zero for a sinusoidal discrete-time input at one-fourth the sampling frequency (Statement 3)
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Consider the following statements regarding a linear discrete-time system:H (z) = z2+1/(z+0.5)(z-0.5)1. The system is stable2. The initial value of h(0) of the impulse response is -43. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequencyWhich of these statements are correct?a)1,2 and 3b)1 and 2c)1 and 3d)2 and 3Correct answer is option 'C'. Can you explain this answer?
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