This document contains a 10-question problem set on digital signal processing. The problems cover topics such as determining the impulse response and transfer function of discrete-time systems described by difference equations, analyzing stability of systems, computing outputs given certain inputs, and realizing systems in block diagrams. Students are asked to find input-output relations, plot frequency responses, solve for poles and zeros, and analyze other properties of linear time-invariant discrete-time systems. The problems progress from simpler analyses to more complex issues like stability determination.

1. Helwan University
Faculty of Engineering
Department of Electronics, Communications & Computer
ELC 9421 Digital Signal Processing - Spring 2017
Sheet Five
‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ ‫ب‬ ‫ة‬ ‫س‬ ‫د‬ ‫ن‬ ‫ه‬ ‫ل‬ ‫ا‬ ‫ة‬ ‫ي‬ ‫ل‬ ‫ك‬–‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ ‫ة‬ ‫ع‬ ‫م‬ ‫ا‬ ‫ج‬ Page 1 of 2
1) Determine the impulse response and the unit step response of the systems
described by the difference equation.
a) 𝑦(𝑛) = 0.6𝑦(𝑛 − 1) – 0.08𝑦(𝑛 − 2) + 𝑥(𝑛)
b) 𝑦(𝑛) = 0.7𝑦(𝑛 − 1) – 0.1𝑦(𝑛 − 2) + 2𝑥(𝑛) – 𝑥(𝑛 − 2)
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2) Find the transfer function 𝐻(𝑍) for the two systems in problem (1). Realize the
systems, in suitable system block diagram.
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3) Consider the discrete time system shown.
a) Find the input – output relation.
b) Apply the input 𝑥(𝑛) = {1, 1, 1, . . . }, and
compute the first 10 sample of the
output 𝑦(𝑛).
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4) Determine H(Z) for the system described by the second order difference equation
𝑦(𝑛) – 1.6𝑦(𝑛 − 1) + 0.8𝑦(𝑛 − 2) = 𝑥(𝑛) – 𝑥(𝑛 − 1)
a) Implement the system, is the system stable?
b) Find 𝐻(𝑗𝜔).
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5) Given a causal system specified by its system function H(z) as follows
𝐻(𝑧) = 𝑧(𝑧 + 1)/(𝑧 + 0.6)(𝑧 − 0.4)
a) Find the difference equation.
b) Find 𝐻(𝑗𝜔), Plot |𝐻(𝑗𝜔)|, assuming a suitable sampling time T.
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6) A system has H (Z) =(z + 0.6) (z − 0.6)⁄ , if an input x (n) is applied to that system,
with 𝑋(𝑧) = (𝑧 + 1)(𝑧 − 0.6) (𝑧 − 3)(𝑧 + 0.6)⁄ , find the output 𝑦(𝑛).
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7) A system has H (Z) =(1 + 0.2Z−1
) (1 − 0.2Z−1),⁄ if an input 𝑥(𝑛) is applied to the
system, with 𝑋(𝑧) = (1 − 0.2𝑍−1
) (1 − 𝑍−1
)(1 + 0.2𝑍−1
)⁄ find the output 𝑦(𝑛).
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2. ‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ ‫ب‬ ‫ة‬ ‫س‬ ‫د‬ ‫ن‬ ‫ه‬ ‫ل‬ ‫ا‬ ‫ة‬ ‫ي‬ ‫ل‬ ‫ك‬–‫ا‬ ‫ج‬‫م‬‫ة‬ ‫ع‬‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ Page 2 of 2
8) An LTI system is characterized by the difference equation
𝑦(𝑛) = 0.25 𝑥(𝑛) + 0.125 𝑦(𝑛 − 2) + 3 𝑥(𝑛 − 2)
a) Find the transfer function 𝐻(𝑧).
b) Implement the system.
c) Determine the poles and the zeros of the transfer function.
d) Is the system stable?
e) If the input to the system is 𝑥(𝑛) = 5 𝑢(𝑛), find an analytical expression
for 𝑦(𝑛).
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9) Consider the discrete system shown in fig.
a) Find 𝐻(𝑍).
b) Discuss the system stability.
c) Compute the first six values of the impulse response of the system.
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10) For the shown digital circuit, find H (Z) then discuss the stability of that circuit
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Best Wishes,
Prof. Elsayed. M. Saad
Dr. Amr E. Mohamed
𝑧−1
𝑧−1
𝑧−1
ΣΣ
𝑋(𝑍) 𝑌(𝑍)
𝑐0
𝑎1
𝑎0
𝑐1 𝑐2
𝑏1