Dr B R Ambedkar National Institute of Technology, Jalandhar
Department of Electronics and Communication Engineering
ECPC-306 Digital Signal Processing
Tutorial Sheet: Z-Transforms
Q: 1) Determine the 𝑧 - transform of the following signals:
(a) 𝑥(𝑛) = {3, 0, 0, 0, 0, 6, 1, −4} (e) 𝑥(𝑛) = (𝑎𝑛 + 𝑎−𝑛 )𝑢(𝑛), 𝑎 𝑟𝑒𝑎𝑙
(b) 𝑥(𝑛) = (1/2)n 𝑛 ≥ 5, (f) 𝑥(𝑛) = (1 + 𝑛)𝑢(𝑛)
0 𝑛≤4
n -n
(c) 𝑥(𝑛) = (−1) 2 𝑢(𝑛) (g) 𝑥(𝑛) = 𝑛(−1)𝑛 𝑢(𝑛)
(d) 𝑥(𝑛) = (1/2)𝑛 [𝑢(𝑛) − 𝑢(𝑛 − 10)] (h) 𝑥(𝑛) = (−1)𝑛 𝑢(𝑛)
1 𝑛
(i) 𝑥(𝑛) = −𝑛𝑎𝑛 𝑢(−𝑛 − 1) (j) 𝑥(𝑛) = (3) 𝑛≥0
(1/2)−𝑛 𝑛 < 0
Q: 2) Determine the convolution of the following pairs of signals by means of the 𝑧 – transform.
1 𝑛 1 𝑛
(a) 𝑥 1(𝑛) = ( ) 𝑢(𝑛 − 1), 𝑥2(𝑛) = [1 + (2) ] 𝑢(𝑛)
4
1 𝑛
(b) 𝑥 1(𝑛) = 𝑢(𝑛), 𝑥 2(𝑛) = 𝛿(𝑛) + (2) 𝑢(𝑛)
1 𝑛
(c) 𝑥 1(𝑛) = ( ) 𝑢(𝑛), 𝑥2(𝑛) = 𝑐𝑜𝑠𝜋𝑛𝑢(𝑛)
2
(d) 𝑥 1(𝑛) = 𝑛𝑢(𝑛), 𝑥 2(𝑛) = 2𝑛 𝑢(𝑛 − 1)
Q: 3) Determine the inverse 𝑧 – transform for the following transforms.
1
(a) 𝑋(𝑧) = 𝑙𝑜𝑔(1 − 2𝑧), |𝑧| <
2
1
(b) 𝑋(𝑧) = 𝑙𝑜𝑔(1 − 𝑧 −1 ), |𝑧| >2
Q: 4) Determine the causal signal 𝑥(𝑛) having the 𝑧 – transform
1
𝑋(𝑧) =
(1 − 2𝑧 −1 )(1 − 𝑧 −1 )2
Q: 5) Using long division, determine the inverse 𝑧 – transform of
1 + 2𝑧 −1
𝑋(𝑧) =
1 − 2𝑧 −1 + 𝑧 −2
Q: 6) Consider the system
1 − 2𝑧 −1 + 2𝑧 −2 − 𝑧 −3
𝐻(𝑧) = 𝑅𝑂𝐶: 0.5|𝑧|˃1
(1 − 𝑧 −1 )(1 − 0.5𝑧 −1 )(1 − 0.2𝑧 −1 )
(a) Sketch the pole-zero pattern.
(b) Determine the impulse response of the system.
Q: 7) Compute the response of the system
𝑦(𝑛) = 0.7𝑦(𝑛 − 1) − 0.12𝑦(𝑛 − 2) + 𝑥(𝑛 − 1) + 𝑥(𝑛 − 2)
to the input 𝑥(𝑛) = 𝑛𝑢(𝑛).
Q: 8) Determine the impulse response and the step response of the following causal systems.
3 1
(a) 𝑦(𝑛) = 4 𝑦(𝑛 − 1) − 8 𝑦(𝑛 − 2) + 𝑥(𝑛)
(b) 𝑦(𝑛) = 𝑦(𝑛 − 1) − 0.5𝑦(𝑛 − 2) + 𝑥(𝑛) + 𝑥(𝑛 − 1)
Q: 9) By applying property, determine the 𝑧 – transform of the following signals if
𝑥(𝑛) = {1, 2, 5, 7, 0, 1}
(a) 𝑥(𝑛 + 2) (b) 𝑥(𝑛 − 2) (c) 𝑎𝑛 𝑥(𝑛) (d) 𝑛𝑥(𝑛) (e) 𝑥(−𝑛)
Q: 10) Determine the partial fraction expansion of
1
𝑋(𝑧) =
(1 + 𝑧 −1 )(1 − 𝑧 −1 )2
and also determine the 𝑥(𝑛).
Q: 11) Determine the system function, impulse response, and zero-state step response of the system
shown in following figure.
Q: 12) Use the one – sided 𝑧 – transform to determine 𝑦(𝑛), 𝑛 ≥ 0 in the following cases:
1 1
(a) 𝑦(𝑛) + 2 𝑦(𝑛 − 1) − 4 𝑦(𝑛 − 2) = 0; 𝑦(−1) = 𝑦(−2) = 1
(b) 𝑦(𝑛) − 1.5𝑦(𝑛 − 1) + 0.5𝑦(𝑛 − 2) = 0; 𝑦(−1) = 1, 𝑦(−2) = 0
1 1 𝑛
(c) 𝑦(𝑛) = 2
𝑦(𝑛 − 1) + 𝑥(𝑛); 𝑥(𝑛) = (3) 𝑢(𝑛), 𝑦(−1) = 1
1
(d) 𝑦(𝑛) = 4
𝑦(𝑛 − 2) + 𝑥(𝑛), 𝑥(𝑛) = 𝑢(𝑛), 𝑦(−1) = 0; 𝑦(−2) = 1
Q: 13) The step response of an LTI system is
1 𝑛−2
𝑠(𝑛) = ( ) 𝑢(𝑛 + 2)
3
(a) Find the system function 𝐻(𝑧) and sketch the pole-zero plot.
(b) Determine the impulse responseℎ(𝑛).
(c) Check if the system is causal and stable.
Q: 14) Consider an LTI system with output 𝑥[𝑛] and output 𝑦[𝑛] that satisfies the difference equation
5
𝑦[𝑛] − 𝑦[𝑛 − 1] + 𝑦[𝑛 − 2] = 𝑥[𝑛] − 𝑥[𝑛 − 1].
2
Determine all possible values for the system’s impulse response ℎ[𝑛] at 𝑛 = 0.
Q: 15) For each of the following pairs of input and output 𝑧 – transform 𝑋(𝑧) and 𝑌(𝑧), determine the
region of convergence for the system function 𝐻(𝑧):
1 3
(a) 𝑋(𝑧) = 3 |𝑧| ≥ 4
1−( )𝑧 −1
4
1 2
𝑌(𝑧) = 1+(2/3)𝑧 −1 |𝑧| > 3
1 1
(b) 𝑋(𝑧) = 1 |𝑧| <
1+( )𝑧 −1 3
3
1 1 1
𝑌(𝑧) = 1 ≤ |𝑧| ≤ 3
(1−( )𝑧 −1 )(1+(1/3)𝑧 −1 ) 6
6
Q: 16) When the input to an LTI system is
𝑥[𝑛] = (1/3)𝑛 𝑢[𝑛] + (2)𝑛 𝑢[−𝑛 − 1]
the corresponding output is
𝑦[𝑛] = 5(1/3)𝑛 𝑢[𝑛] − 5(2/3)𝑛 𝑢[𝑛]
(a) Find the system function 𝐻(𝑧) of the system. Plot the pole(s) and zero(s) of 𝐻(𝑧) and indicate the
ROC.
(b) Find the impulse response ℎ[𝑛] of the system.
Q: 17) A causal sequence 𝑔[𝑛] has the 𝑧 – transform
𝐺(𝑧) = sin(𝑧−1 ) (1 + 3𝑧−2 + 2𝑧−4 )
Find𝑔[11].
Q: 18) If 𝑥(𝑛) = 4𝑛 𝑢(𝑛) and 𝑋1 (𝑧) = 𝑋(2𝑧) then find 𝑥1 (𝑛).
1
Q: 19) Assuming ROC to be |𝑧| > 3 determine 𝑥(0), 𝑥(1), 𝑥(2) for
1 + 𝑧−1
𝑋(𝑧) =
1
1 + 3 𝑧−1
Q: 20) find 𝑧 – transform of the following signal:
1 𝑛
𝑥(𝑛) = (− ) 𝑢(𝑛) − 4−𝑛 𝑢(−𝑛)
3
