Subject Name: Digital Signal Processing (030270402)
(Question Bank)
Unit-1: Z–Transform
1. Mention the equations of forward and inverse Z – transforms.
2. What is the importance of Region of Convergence (ROC) for Z – transform? Discuss the
various properties of ROC of Z – transform.
3. Prove linearity and time-shifting properties of Z-Transform. Give one suitable example of
each property.
4. What is the relation between Discrete Time Fourier Transform and Z – transform?
5. Obtain the relation between Z-transform and Laplace Transform.
6. Enlist the conditions for system to be BIBO stable and causal with respect to Z – transform.
7. Write down the ROCs of the following type of finite duration sequences:
(a) Right sided sequence.
(b) Left sided sequence.
(c) Both sided sequence.
8. Draw the ROCs for the following infinite duration sequences:
(a) Causal.
(b) Anticausal.
(c) Two-sided.
9. Determine Z – transform of the following finite duration sequences with ROC:
(a) 𝑥1 𝑛 = 1 2 5 7 1
↑
(b) 𝑥2 𝑛 = 1 2 5 7 1
↑
10. A sequence x(n) with Z – transform 𝑋 𝑍 =𝑍6+𝑍4−2𝑍2+ 2𝑍 − 3 is applied as an input
to a LTI system with the impulse response ℎ 𝑛 = 2𝛿(𝑛 − 2). Find output at n = 0.
11. Determine Z – transform of the following signals with ROC.
(a) 𝑥1 𝑛 = 0.5 𝑛𝑢 𝑛
(b) 𝑥2 𝑛 =− 0.5 𝑛𝑢(−𝑛 − 1)
(c) 𝑥3 𝑛 = 2𝑛𝑢 𝑛 −3𝑛𝑢(−𝑛 − 1)
(d) 𝑥4 𝑛 =𝛿 𝑛 − 𝑘 ,𝑘> 0
(e) 𝑥5 𝑛 =𝛿 𝑛+𝑘 ,𝑘< 0
(f) 𝑥6 𝑛 =𝑎 𝑛 ; 𝑎 < 1.
(g) 𝑥7 𝑛 =𝑠𝑖𝑛(𝜔0𝑛)𝑢(𝑛)
(h) 𝑥8 𝑛 =𝑐𝑜𝑠(𝜔0𝑛)𝑢(𝑛)
12. Using power series expansion method, determine inverse Z – transform of
𝑋 𝑧 =1
1−1.5𝑧−1+ 0.5𝑧−2
When
(a) 𝑥(𝑛) is causal.
(b) 𝑥(𝑛) is anticausal.
13. Determine the inverse Z – transform of
𝑋 𝑧 =5𝑧−1
1−5𝑧−1+ 6𝑧−2
If
(a) ROC: 𝑧 > 3
(b) ROC: 𝑧 < 2
(c) (iii) ROC: 2 < 𝑧 < 3
14. If 𝑦 𝑛 =𝑥1 𝑛 ∗ 𝑥2 𝑛 , where 𝑥1 𝑛 = 1
3 𝑛𝑢 𝑛 and 𝑥2 𝑛 = 1
5 𝑛𝑢 𝑛 . Find 𝑦(𝑛) by
using convolution and multiplication properties.
15. A liner time invariant system is characterized by the system function
𝐻 𝑧 =3−4𝑧−1
1−3.5𝑧−1+ 1.5𝑧−2
Specify ROC of H(z) and determine impulse response for the following conditions:
