**The inverse z-transform & Response of Linear Discrete Systems**

**1. Unit step response of the system described by the equation y(n) +y(n-1) =x(n) is:**

a) z^{2}/(z+1)(z-1)

b) z/(z+1)(z-1)

c) z+1/z-1

d) z(z-1)/z+1

**Answer:** a

**Explanation:** Response of the system is calculated by taking the z-transform of the equation and input to the transfer function in the step input.

**2. Inverse z-transform of the system can be calculated using:**

a) Partial fraction method

b) Long division method

c) Basic formula of the z-transform

d) All of the mentioned

**Answer:** d

**Explanation:** Inverse z-transform is the opposite method of converting the transfer function in Z domain to the discrete time domain and this can be calculated using all the above formulas.

**3. Assertion (A): The system function**

**H(z) = z ^{3}-2z^{2}+z/z^{2}+1/4z+1/s is not causal**

**Reason (R): If the numerator of H (z) is of lower order than the denominator, the system may be causal.**

a) Both A and R are true and R is correct explanation of A

b) Both A and R are true and R is not correct Explanation of A

c) A is True and R is false

d) A is False and R is true

**Answer:** a

**Explanation:** The transfer function is not causal as for causality the numerator of H (z) is of lower order than the denominator, the system may be causal.

**4. Assertion (A): Z-transform is used to analyze discrete time systems and it is also called pulsed transfer function approach.**

**Reason(R): The sampled signal is assumed to be a train of impulses whose strengths, or areas, are equal to the continuous time signal at the sampling instants.**

a) Both A and R are true and R is correct explanation of A

b) Both A and R are true and R is not correct Explanation of A

c) A is True and R is false

d) A is False and R is true

**Answer:** a

**Explanation:** Z-transform is used to convert the discrete time systems into the z domain and it is also called pulsed transfer function approach that is justified only at the sampling instants.

**5. The z-transform corresponding to the Laplace transform G(s) =10/s(s+5) is**

**Answer:** b

**Explanation:** Laplace transform is the technique of relating continuous time to the frequency domain while the z transform is relating discrete time hence Laplace transform is first converted into time domain and then the z transform is calculated.

**6. Homogeneous solution of: y(n) -9/16y(n-2) = x(n-1)**

a) C1(3/4)^{n}+C2(3/4)^{-n}

b) C1-(3/4)^{n-1}+C2(3/4)^{n-1}

c) C1(3/4)^{n}

d) C1-(3/4)^{n}

**Answer:** a

**Explanation:** Taking the z-transform of the given difference equation and solving the homogeneous equation and finding the solution using complimentary function.

**7. If the z transform of x(n) is X(z) =z(8z-7)/4z ^{2}-7z+3, then the final value theorem is :**

a) 1

b) 2

c) ∞

d) 0

**Answer:** a

**Explanation:** Final value theorem is calculated for the transfer function by equating the value of z as 1 and this can be calculated only for stable systems.

**8. Final value theorem is used for:**

a) All type of systems

b) Stable systems

c) Unstable systems

d) Marginally stable systems

**Answer:** b

**Explanation:** Final value theorem is used to calculate the final value as for time infinite and for z = 1 the final value theorem can be calculated and final value theorem is for for stable systems.

**9. If the z-transform of the system is given by H (z) = a+z ^{-1}/1+az^{-1} . Where a is real valued:**

a) A low pass filter

b) A high pass filter

c) An all pass filter

d) A bandpass filter

**Answer:** c

**Explanation:** The discrete time frequency response will be aperiodic and does not depend on the frequency and the transfer function will be representing the all pass filter.

**10. The system is stable if the pole of the z-transform lies inside the unit circle**

a) True

b) False

**Answer:** a

**Explanation:** For the system to be stable in Z domain the pole in the this domain must lie inside the unit circle and for the causal stable region must be outside the circle and hence the locus will be a ring.