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Chap 7 & 8 Online Revision Tutorial (Concise) .pdf


Original filename: Chap 7 & 8 Online Revision Tutorial (Concise).pdf
Author: David Li Chung Ping

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Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

i

Mind Map of Overall Picture
Recursive
/ Non-Recursive

Calculate y(n) by
substitution & iteration
Difference Eqn
y(n) = …
Calculate h(n) by
substituting
x(n)=δ(n)

System Block
Diagram

FIR / IIR
Unit Impulse
Response, h(n)

Calculate y(n) by
Z-transform

System Transfer
Function, H(z)

Order of filter

Freq Response, H(ejω)
by substituting z = ejω

Type of filter:
LPF / HPF

Mag Response or
Av at one given
frequency
-3 dB Cutoff
Freq

Mag Response,
|H(ejω)|

Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

Briefing

{A briefing video about using this interactive document}

ii

Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

iii

Contents

Mind Map of Overall Picture …………………………………………………………….……..…………………………………..…… i
Briefing …………………………………………………………………………………………………………………………………………. ii

Contents …………………………………………………………………………………………………………………….………..………. iii
1

System Block Diagram ↔ Difference Equation ..................................................................................... 1

3

System Transfer Function, (𝑧𝑧) ↔ Unit Impulse Response, ℎ(𝑛𝑛) ......................................................... 7

2

4

5

6

7

8

9

Difference Equation ↔ System Transfer Function, (𝑧𝑧) ......................................................................... 7

System Transfer Function, (𝑧𝑧) → Order of System .............................................................................. 7
Unit Impulse Response, ℎ(𝑛𝑛) → FIR / IIR .............................................................................................. 7

Recursive / Non-Recursive System ........................................................................................................ 8

Difference Equation → Calculate 𝑦𝑦(𝑛𝑛) by substitution & iteration ..................................................... 8

Difference Equation → Calculate ℎ(𝑛𝑛) by substituting 𝑥𝑥(𝑛𝑛) = δ(𝑛𝑛) ................................................... 8
System Transfer Function, (𝑧𝑧) → Calculate 𝑦𝑦(𝑛𝑛) by Z-transform ........................................................ 8

10 System Transfer Function, (𝑧𝑧) → Frequency Response, (ej𝜔𝜔) .............................................................. 9

11 Frequency Response, (ej𝜔𝜔) → Magnitude Response, |(ej𝜔𝜔)| ................................................................. 9
12 Magnitude Response, |(ej𝜔𝜔)| → Type of Filter (LPF/HPF)................................................................... 9

13 Magnitude Response, |(ej𝜔𝜔)| → Calculate Magnitude Response at a given frequency. ...................... 9
14 Magnitude Response, |(ej𝜔𝜔)| → Calculate −3 dB Cutoff Frequency .................................................... 9

15 Formula Sheet........................................................................................................................................... 9

Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

1

1 System Block Diagram ↔ Difference Equation
1.1 System Block Diagram → Difference Equation
1.1.1 Worked Example
Example 1
What is the difference equation of the system?
0.5

Z -1

−0.7

Z -1

0.2

Solution

𝑦𝑦(𝑛𝑛) = 0.5𝑥𝑥(𝑛𝑛) − 0.7𝑥𝑥(𝑛𝑛 − 1) + 0.2𝑥𝑥(𝑛𝑛 − 2)
Explanation

Example 2
What is the difference equation of the system?
Z -1

3

Solution

Z -1

0.5

−0.1

𝑦𝑦(𝑛𝑛) = 3𝑥𝑥(𝑛𝑛) + 0.5𝑥𝑥(𝑛𝑛 − 1) − 0.1𝑥𝑥(𝑛𝑛 − 2)
Explanation


Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

2

Example 3
Derive the difference equation of the system.
0.2

Z -1

Z -1

Z -1

0.5

0.6

−1.5

Z -1

Solution

𝑦𝑦(𝑛𝑛) = 0.2𝑥𝑥(𝑛𝑛) + 𝑥𝑥(𝑛𝑛 − 1) + 0.6𝑥𝑥(𝑛𝑛 − 2) + 0.5𝑦𝑦(𝑛𝑛 − 1) − 1.5𝑦𝑦(𝑛𝑛 − 2)
Explanation

Example 4
What is the difference equation of the system?

Z -1

Z -1

Solution

0.6

−0.3

Z -1

−0.1

𝑦𝑦(𝑛𝑛) = 𝑥𝑥(𝑛𝑛) + 0.6𝑥𝑥(𝑛𝑛 − 1) − 0.1𝑥𝑥(𝑛𝑛 − 2) − 0.3𝑦𝑦(𝑛𝑛 − 1)
Explanation


Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

1.1.2 Problem
P1.

Derive the difference equation of the system.
5

Z 

-1

−2

Z -1

P2.

Answer

0.7

What is the difference equation of the system below?
Answer

Z -1

Z -1

P3.

3

0.5

What is the difference equation of the system below?
1.5

Z 

Z 

-1

-1

2.3

Z -1

P4.

−0.4

Answer

Z -1

What is the difference equation of the system below?
Z -1
−0.3

Z -1
0.6

−0.1

Z 

-2

Answer

Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

4

1.1.3 Related Past Examination Question
15/16 s2 B5, 15/16 s1 B5, 14/15 s1 B6, 13/14 s2 B5, 12/13 s2 B5, 10/11 s2 B6

1.2 System Block Diagram ← Difference Equation

1.2.1 Worked Example
Example 1
The difference equation of a dicrete-time system is
𝑦𝑦(𝑛𝑛) = 2.5𝑥𝑥(𝑛𝑛) + 0.5𝑥𝑥(𝑛𝑛 − 1) − 0.2𝑥𝑥(𝑛𝑛 − 2)
Draw the system block diagram of the system.

Solution

2.5

Z -1

Z -1

0.5

−0.2

Explanation


Example 2
The difference equation of a dicrete-time system is
𝑦𝑦(𝑛𝑛) = 0.2𝑥𝑥(𝑛𝑛) − 2𝑥𝑥(𝑛𝑛 − 1) + 0.5𝑦𝑦(𝑛𝑛 − 1)
Draw the system block diagram of the system.

Solution

0.2

Z -1
−2

0.5

Z -1

Explanation



Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

1.2.2 Problem
P1.

P2.
P3.

The difference equation of a system of a system is
𝑦𝑦(𝑛𝑛) = 0.5𝑥𝑥(𝑛𝑛) − 0.7𝑥𝑥(𝑛𝑛 − 1) + 0.2𝑥𝑥(𝑛𝑛 − 2)
Draw the system block diagram of the system.
Answer

The difference equation of a system of a system is
𝑦𝑦(𝑛𝑛) = 0.2𝑥𝑥(𝑛𝑛) + 0.6𝑥𝑥(𝑛𝑛 − 1) + 0.5𝑦𝑦(𝑛𝑛 − 1) − 1.5𝑦𝑦(𝑛𝑛 − 2)
Draw the system block diagram of the system.
Answer
The difference equation of a system of a system is
𝑦𝑦(𝑛𝑛) = 𝑥𝑥(𝑛𝑛) + 0.5𝑥𝑥(𝑛𝑛 − 1) − 1.2𝑦𝑦(𝑛𝑛 − 1)
Draw the system block diagram of the system.
Answer

1.2.3 Related Past Examination Question
08/09 s2 B3, 07/08 s2 B3

1.3 Answer

1.3.1 Answer of Section 1.1.2
Go back to Problem
P1. 𝑦𝑦(𝑛𝑛) = 5𝑥𝑥(𝑛𝑛) − 2𝑥𝑥(𝑛𝑛 − 1) + 0.7𝑥𝑥(𝑛𝑛 − 2)
P2. 𝑦𝑦(𝑛𝑛) = 𝑥𝑥(𝑛𝑛) + 0.5𝑥𝑥(𝑛𝑛 − 2)
P3. 𝑦𝑦(𝑛𝑛) = 1.5𝑥𝑥(𝑛𝑛) + 2.3𝑥𝑥(𝑛𝑛 − 1) + 𝑥𝑥(𝑛𝑛 − 2) + 𝑦𝑦(𝑛𝑛 − 1) − 0.4𝑦𝑦(𝑛𝑛 − 2)
P4. 𝑦𝑦(𝑛𝑛) = 𝑥𝑥(𝑛𝑛) − 0.3𝑥𝑥(𝑛𝑛 − 1) + 0.6𝑥𝑥(𝑛𝑛 − 2) − 0.1𝑦𝑦(𝑛𝑛 − 2)
1.3.2 Answer of Section 1.2.2
Go back to Problem
P1.
5
Z -1

−0.2

Z -1

0.7

Alternative Answer
5

Z -1

Z -1
−0.2

0.7

5

Chapters 7 & 8 – Online Revision Tutorial (Concise/Moderate/ Detailed)

P2.

6

0.2

Z -1

0.6

Z -1
0.5

−1.5

Z -1

P3.
Z -1

0.5

Z -1
−1.2


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