Control Systems Questions and Answers – The Z-Transfer Function

This set of Control Systems Multiple Choice Questions & Answers (MCQs) focuses on “The Z-Transfer Function”.

1. 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?
a) 1,2 and 3
b) 1 and 2
c) 1 and 3
d) 2 and 3
View Answer

Answer: c
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.

2. The minimum number of delay elements required realizing a digital filter with transfer function
H (z) =
a) 2
b) 3
c) 4
d) 5
View Answer

Answer: b
Explanation: H (z) =
Minimum number of delay elements= (Maximum power of z-minimum power of z)
Minimum number of delay elements = 3.

3. A system can be represented in the form of state equations as:
S (n+1) =A S (n) +B x (n)
Y (n) = C S (n) +D x (n)
Where, A, B, C, D are the matrices , S(n) is the state vector , x(n) is the input and y(n) is the output . The transfer function of the system.
H (z) =Y (z)/X (z) is given by:
a) A(ZI – B)-1 C + D
b) B(ZI – C)-1 D + A
c) C(ZI – A)-1 B + D
d) D(ZI – A)-1 C + B
View Answer

Answer: c
Explanation: Solving both the equations and substituting the value of the output equation into the state equation we get the value of the transfer function as obtained.
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4. Assertion (A): The signals anu(n) and anu(-n-1) have the same Z transform, z/(z-a)
Reason (R): the region of convergence of anu(n) is |z|>|a|, whereas the ROC for anu(-n-1) is |z|<|a|.
a) Both A and R are true and R is correct explanation of A
b) Both A and R are true but R is not correct explanation of A
c) A is true but R is false
d) A is false but R is true
View Answer

Answer: d
Explanation: Both have the ROC as given in the reason is true but the z transform for the second is with a minus sign.

5. What is the number of roots of the polynomial F(z) = 4z3-8z2-z+2, lying outside the unit circle?
a) 0
b) 1
c) 2
d) 3
View Answer

Answer: b
Explanation: Factorizing F (z) and then the factors are the roots which here come out to be 3.
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6. Assertion (A): The discrete time system described by y[n] =2x[n] +4x[n-1] is unstable
Reason (R): It has an impulse response with a finite number of non-zero samples
a) Both A and R are true and R is correct explanation of A
b) Both A and R are true but R is not correct explanation of A
c) A is true but R is false
d) A is false but R is true
View Answer

Answer: d
Explanation: For the system to be stable the value of the transfer function in the discrete time domain must be summable and H[n] calculated is summable hence the system is stable.

7. What is the z-transform of the signal x[n] = anu(n)?
a) X(z) =1/z-1
b) X(z) = 1/1-z
c) X(z) = z/z-a
d) X(z) = 1/z-a
View Answer

Answer: c
Explanation: By definition this is the basic example of the z-transform and the Z-Transform of the equation is calculated is z/z-a.
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8. Which one of the following rules determine the mapping of s-plane to z-plane?
a) Right side of the s-plane maps into outside of the unit circle in z-plane
b) Left half of s-plane maps into inside of the unit circle
c) Imaginary axis in s-plane maps into the circumference of the unit circle
d) All of the mentioned
View Answer

Answer: d
Explanation: S- plane can be mapped into the z plane with certain rules than right side maps into the outside, left side maps into the inside and imaginary axis maps on the unit circle of the z plane.

9. Assertion (A): The z-transform of the output of the sampler is given by the series.
Reason (R): The relationship is the result of the application of z = e-sT, where T stands for the time gap between the samples.
a) Both A and R are true and R is correct explanation of A
b) Both A and R are true but R is not correct explanation of A
c) A is true but R is false
d) A is false but R is true
View Answer

Answer: c
Explanation: T is termed as the time of the sampling instant and z transform is always defined for the instant of the sampling event and this can be as desired by the user.
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10. Convolution of two sequences X1[n] and X2[n] are represented by:
a) X1(z)*X2(z)
b) X1(z)X2(z)
c) X1(z)+X2(z)
d) X1(z)/X2(z)
View Answer

Answer: a
Explanation: Convolution of the two sequences is the combination of multiplication and addition of the two sequences at each instant and convolution in time domain is multiplication in the frequency domain.

Sanfoundry Global Education & Learning Series – Control Systems.
To practice all areas of Control Systems, here is complete set of 1000+ Multiple Choice Questions and Answers.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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