**The Z-Transfer Function**

**1. Consider the following statements regarding a linear discrete-time system:**

**H (z) = z ^{2}+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

**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

**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

**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.

**4. Assertion (A): The signals a ^{n}u(n) and a^{n}u(-n-1) have the same Z transform, z/(z-a)**

**Reason (R): the region of convergence of a**

^{n}u(n) is |z|>|a|, whereas the ROC for a^{n}u(-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

**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) = 4z ^{3}-8z^{2}-z+2, lying outside the unit circle?**

a) 0

b) 1

c) 2

d) 3

**Answer:** b

**Explanation:** Factorizing F (z) and then the factors are the roots which here come out to be 3.

**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

**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] = a ^{n}u(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

**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.

**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

**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

**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.

**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)

**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.