Digital Signal Processing Multiple Choice Questions on “Z Transform”.
1. The Z-Transform X(z) of a discrete time signal x(n) is defined as ____________
A. (sum_{n=-infty}^{infty}x(n)z^n)
B. (sum_{n=-infty}^{infty}x(n)z^{-n})
C. (sum_{n=0}^{infty}x(n)z^n)
D. None of the mentioned
Answer: B
Clarification: The z-transform of a real discrete time sequence x(n) is defined as a power of ‘z’ which is equal to X(z)=(sum_{n=-{infty}}^{infty} x(n)z^{-n}), 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
Answer: A
Clarification: 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 following finite duration signal?
A. 2 + 4z + 5z2 + 7z3 + z4
B. 2 + 4z + 5z2 + 7z3 + z5
C. 2 + 4z-1 + 5z-2 + 7z-3 + z-5
D. 2z2 + 4z + 5 +7z-1 + z-3
Answer: D
Clarification: We know that, for a given signal x(n) the z-transform is defined as X(z)=(sum_{n=-infty}^{infty} x(n)z^{-n})
Substitute the values of n from -2 to 3 and the corresponding signal values in the above formula
We get, X(z) = 2z2 + 4z + 5 + 7z-1 + z-3.
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=∞
Answer: C
Clarification: We know that, the z-transform of a signal x(n) is X(z)=(sum_{n=-infty}^{infty} x(n)z^{-n})
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)nu(n)?
A. (frac{1}{1-0.5z^{-1}};ROC |z|>0.5)
B. (frac{1}{1-0.5z^{-1}};ROC |z|<0.5)
C. (frac{1}{1+0.5z^{-1}};ROC |z|>0.5)
D. (frac{1}{1+0.5z^{-1}};ROC |z|<0.5)
Answer: A
Clarification: For a given signal x(n), its z-transform X(z)=(sum_{n=-infty}^{infty} x(n)z^{-n})
Given x(n)=(0.5)nu(n)=(0.5)n for n≥0
So, X(z)=(sum_{n=0}^{infty} 0.5^n z^{-n}=sum_{n=0}^{infty} (0.5z^{-1})^n)
This is an infinite GP whose sum is given as
X(z)=(frac{1}{1-0.5z^{-1}}) under the condition that |0.5z-1|<1
=> X(z)=(frac{1}{1-0.5z^{-1}}) and ROC is |z|>0.5.
6. Which of the following series has an ROC as mentioned below?
A. α-nu(n)
B. αnu(n)
C. α-nu(-n)
D. αnu(n)
Answer: B
Clarification:
Let x(n)= αnu(n)
The z-transform of the signal x(n) is given as
X(z)=X(z)=(sum_{n=0}^{infty} alpha^n z^{-n} = sum_{n=0}^{infty}alpha z^{-1}n)
=> X(z)=(frac{1}{1-alpha z^{-1}}) and ROC is |z|>α which is as given in the question.
7. What is the z-transform of the signal x(n) = -αnu(-n-1)?
A. (frac{1}{1-alpha z^{-1}});ROC |z|
B. (-frac{1}{1+alpha z^{-1}});ROC |z|
C. (-frac{1}{1-alpha z^{-1}});ROC |z|>|α|
D. (-frac{1}{1-alpha z^{-1}});ROC |z|
Answer: D
Clarification: Given x(n)= -αnu(-n-1)=0 for n≥0
=-αn for n≤-1
From the definition of z-transform, we have
X(z)=(sum_{n=-infty}^{-1}(-α^n)z^{-n} = -sum_{n=-infty}^{-1}(alpha^{-1} z)^{-n}=-frac{1}{1-alpha z^{-1}}) and |α-1z|<1 => |z|<|α|.
8. What is the ROC of the z-transform of the signal x(n)= anu(n)+bnu(-n-1)?
A. |a|<|z|<|b|
B. |a|>|z|>|b|
C. |a|>|z|<|b|
D. |a|<|z|>|b|
Answer: A
Clarification: We know that,
ROC of z-transform of anu(n) is |z|>|a|.
ROC of z-transform of bnu(-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=∞
Answer: D
Clarification: Let us an example of anti causal sequence whose z-transform will be in the form X(z)=1+z+z2 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
Answer: C
11. The z-transform of a sequence x(n) which is given as X(z)=(sum_{n=-infty}^{infty}x(n)z^{-n}) is known as _____________
A. Uni-lateral Z-transform
B. Bi-lateral Z-transform
C. Tri-lateral Z-transform
D. None of the mentioned
Answer: B
Clarification: 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.
12. What is the ROC of the system function H(z) if the discrete time LTI system is BIBO stable?
A. Entire z-plane, except at z=0
B. Entire z-plane, except at z=∞
C. Contain unit circle
D. None of the mentioned
Answer: C
Clarification: A discrete time LTI is BIBO stable, if and only if its impulse response h(n) is absolutely summable. That is,
(sum_{n=-infty}^{infty}|h(n)|<infty)
We know that, H(z)= (sum_{n=-infty}^{infty}h(n)z^{-n})
Let z=ejω so that |z|=|ejω|=1
Then, |H(ejω)|=|H(ω)|=| (sum_{n=-infty}^{infty})h(n) e-jωn|≤(sum_{n=-infty}^{infty})|h(n) e-jωn|
=(sum_{n=-infty}^{infty})|h(n)|<∞
Hence, we see that if the system is stable, then H(z) converges for z=ejω. That is, for a stable discrete time LTI system, ROC of H(z) must contain the unit circle |z|=1.
13. The ROC of z-transform of any signal cannot contain poles.
A. True
B. False
Answer: A
Clarification: 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)=an(n) (|a| < 1) BIBO stable?
A. True
B. False
Answer: A
Clarification: Given h(n)=an(n) (|a|<1)
The z-transform of h(n) is H(z)=(frac{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
Answer: B
Clarification: The ROC of causal infinite sequence is of form |z|>r1 where r1 is largest magnitude of poles.