DSP 06 _ Sheet Six Digital filters

1. Helwan University
Faculty of Engineering
Department of Electronics, Communications & Computer
ELC 9421 Digital Signal Processing - Spring 2017
Sheet Six
‫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 a direct form realization for the following linear phase FIR filters.
a) h(n) = [1,2,3,4,3,2,1]
b) h(n) = [1,2,3,3,2,1]
Plot 𝛷(𝑗𝑤) for the two filters, considering suitable T.
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2) Determine the response of the system with impulse response ℎ(𝑛) = 𝑎 𝑛
𝑢(𝑛) to
the input signal 𝑥(𝑛) = 𝑢(𝑛) – 𝑢(𝑛 − 10).
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3) Consider an FIR filter with system function.
4321
4.07.14.388.21)( 
 zzzzzH
Sketch the direct form realization of the filter. Find 𝐻(𝑗𝑤).
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4) A linear shift invariant system is characterized by its unit sample
response ℎ(𝑛) = 𝑒 𝑛
𝑢(𝑛)
a) Is the system causal? Why?
b) Is the system stable? Why?
c) Is the system FIR Filter? Why?
d) Find the system function 𝐻(𝑍).
e) Plot |𝐻(𝑗𝜔)| for all 𝜔.
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5) For a digital system with H(Z) = (1 + Z−1
)2
,find:
a) A difference equation realization for the digital filter specified by 𝐻(𝑍).
b) Does 𝐻(𝑍) represent FIR filter? Why?
c) Find H(jw) for all w with T=10−1
sec
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6) Consider a causal IIR system with 𝐻(𝑧) as follows.
321
321
5.08.09.01
2321
)( 




zzz
zzz
zH
a) Check if the system is stable.
b) Realize the system.
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7) Determine H(Z) for the system described by the difference equation
𝑦 (𝑛) − 0.8𝑦 (𝑛 − 1) + 0.6𝑦 (𝑛 − 2) = 𝑥 (𝑛) + 0.2𝑥 (𝑛 − 1)
Is the system stable?
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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) The figure below shows the block diagram of several discrete-time LSI systems.
a) Find the system function of systems H1 and H2.
b) Plot the pole-zero diagrams of H1 and H2.
c) Find the impulse responses for systems H3 and H4.
d) Sketch the magnitude responses of H3 and H4.
e) Discuss the stability of systems H3 and H4.
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9) Given a causal system specified by its system function H(Z) as follow
𝐻(𝑍) =
𝑍(𝑍 − 1)
(𝑍 − 0.5)(𝑍 + 0.25)
a) Find a difference equation realized of the system.
b) Find 𝐻(𝑗𝑤) and then plot the magnitude and the phase.
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10) A digital system has the difference equation:
𝑦(𝑛) = 0.63𝑥(𝑛) − 1.3𝑥(𝑛 − 1) + 0.78𝑦(𝑛 − 1), find:
a) 𝐻(𝑍) of the system and show if it is stable.
b) Draw a block diagram that implements 𝐻(𝑍).
c) Find H(jw) and then plot the magnitude and the phase.
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11) Determine the response of the system 𝑦(𝑛) = 0.83𝑦(𝑛 − 1) − 0.17𝑦(𝑛 − 2) + 𝑥(𝑛)
to the input signal 𝑥(𝑛) − 𝛿(𝑛) − 0.33𝛿(𝑛 − 1) is such system stable? implement
the system. Plot the system function 𝐻(𝑗𝑤) in both amplitude and phase.
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12) A causal LTI system is described by the difference equation
𝑦(𝑛) = 𝑦(𝑛 − 1) + 0.5𝑦(𝑛 − 2) + 0.5𝑥(𝑛) − 0.6𝑥(𝑛 − 1) + 0.5𝑥(𝑛 − 2). Find the system
function 𝐻(𝑍).is the system stable? Plot the poles and zeros of the system.
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13) Given a causal system, that is specified by its system function H(Z) as
𝐻(𝑍) =
𝑍(𝑍 − 1)
(𝑍 + 0.6)(𝑍 − 0.4)
a) Find a difference equation form for the system.
b) Find H(jw) and then plot |H(jw)|, assuming a suitable sampling time 𝑇.
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Best Wishes,
Prof. Elsayed. M. Saad
Dr. Amr E. Mohamed