This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Z Transform”.

1. The Z-Transform X(z) of a discrete time signal x(n) is defined as:

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Explanation: The z-transform of a real discrete time sequence x(n) is defined as a power of ‘z’ which is equal to , where ‘z’ is a complex variable.

2. What is the set of all values of z for which X(z) attains a finite value?

a) Radius of convergence

b) Radius of divergence

c) Feasible solution

d) None of the mentioned

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Explanation: Since X(z) is a infinite power series, it is defined only at few values of z. The set of all values of z where X(z) converges to a finite value is called as Radius of Convergence(ROC).

3. What is the z-transform of the finite duration signal

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Explanation: We know that, for a given signal x(n) the z-transform is defined as

Substitute the values of n from -2 to 3 and the corresponding signal values in the above formula

We get,

4. What is the ROC of the signal x(n)=δ(n-k),k>0?

a) z=0

b) z=∞

c) Entire z-plane, except at z=0

d) Entire z-plane, except at z=∞

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Explanation: We know that, the z-transform of a signal x(n) is

Given x(n)= δ(n-k)=1 at n=k

=> X(z)=z

^{-k}

From the above equation, X(z) is defined at all values of z except at z=0 for k>0.

So ROC is defined as Entire z-plane, except at z=0.

5. What is the z-transform of the signal x(n)=(0.5)^{n}u(n)?

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6. Which of the following series has an ROC as mentioned below?

a) α^{-n}u(n)

b) α^{n}u(n)

c) α^{-n}u(-n)

d) α^{n}u(n)

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^{n}u(-n-1)?

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8. What is the ROC of the z-transform of the signal x(n)= a^{n}u(n)+b^{n}u(-n-1)?

a) |a|<|z|<|b| b) |a|>|z|>|b| c) |a|>|z|<|b| d) |a|<|z|>|b|

Explanation: We know that,

ROC of z-transform of a<sup>n</sup>u(n) is |z|>|a|. ROC of z-transform of b<sup>n</sup>u(-n-1) is |z|<|b|. By combining both the ROC's we get the ROC of z-transform of the signal x(n) as |a|<|z|<|b|.

9. What is the ROC of z-transform of finite duration anti-causal sequence?

a) z=0

b) z=∞

c) Entire z-plane, except at z=0

d) Entire z-plane, except at z=∞

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Explanation: Let us an example of anti causal sequence whose z-transform will be in the form X(z)=1+z+z

^{2}which has a finite value at all values of ‘z’ except at z=∞.So, ROC of an anti-causal sequence is entire z-plane except at z=∞.

10. What is the ROC of z-transform of an two sided infinite sequence?

a) |z|>r1 b) |z|<r1 c) r2<|z|<r1 d) None of the mentioned

Explanation: Let us plot the graph of z-transform of any two sided sequence which looks as follows.

From the above graph, we can state that the ROC of a two sided sequence will be of the form r2 < |z| < r1.

11. The z-transform of a sequence x(n) which is given as is known as:

a) Uni-lateral Z-transform

b) Bi-lateral Z-transform

c) Tri-lateral Z-transform

d) None of the mentioned

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Explanation: The entire timing sequence is divided into two parts n=0 to ∞ and n=-∞ to 0.

Since the z-transform of the signal given in the questions contains both the parts, it is called as Bi-lateral z-transform.

a) Entire z-plane, except at z=0

b) Entire z-plane, except at z=∞

c) Contain unit circle

d) None of the mentioned

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Explanation: A discrete time LTI is BIBO stable, if and only if its impulse response h(n) is absolutely summable. That is,

13. The ROC of z-transform of any signal cannot contain poles.

a) True

b) False

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Explanation: Since the value of z-transform tends to infinity, the ROC of the z-transform does not contain poles.

14. Is the discrete time LTI system with impulse response h(n)=a^{n}(n) (|a| < 1) BIBO stable?

a) True

b) False

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Explanation:

Given h(n)= a<sup>n</sup>(n) (|a|<1) The z-transform of h(n) is H(z)=z/(z-a),ROC is |z|>|a| If |a|<1, then the ROC contains the unit circle. So, the system is BIBO stable.

15. What is the ROC of a causal infinite length sequence?

a) |z|<r1 b) |z|>r1 c) r2<|z|<r1 d) None of the mentioned

Explanation: The ROC of causal infinite sequence is of form |z|>r1 where r1 is largest magnitude of poles.

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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