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E

E Data Communications andNetworking

ource o e es gn

E Dr Salem Aljareh

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Source Code Design

Decision Tree10

0 11 0

0 10 10 1 0 1

x1 x8x3x2 x7x6x5x4

111110101100011010001000

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Weather Info System

Very Poor Poor Good Moderate

10

0 0 1Moderate

0 110Poor

110 111GoodVer Poor

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Shannon Fano Encoding

1. List symbols in descending probability order

. v e a e n o equa or near y equa va ues o pro a y

.

4. Divide both u er and lower section in two

5. Allocate binary 0 to top half and binary 1 to bottom half

6. Repeat (4) and (5) until cant go further

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Shannon Fano Code Example

x1: P = 0.25 x2: P = 0.25

x3: P = 0.15 x4: P = 0.1

5 . 6 .

x7: P = 0.06 x8: P = 0.01

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Shannon Fano Solution

0.25 0

0.25 0 1.

0.1 1 0 1

.

0.08

1

1

1

1

0

1 0

0.06

0.01

1

1

1

1

1

1

1

1

0

1

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Shannon Fano Encoding (con.)

X1 X2 X3 X4 X5 X6 X7 X8

. . . . . . . .

00 01 100 101 110 1110 11110 11111

Calculate the following:-

The code length

The code Efficienc

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Shannon Fano Encoding (solution)e co e eng s g ven y:

lxPLii

i

i )(81=

=

=

3)10.010.015.0(

..

+++

5)01.006.0(408.0 ++

sym og ts.=

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Shannon Fano Encoding (solution)

The entropy H(X) is given by:

i 18= =

xPi i

11

)(1=

1115.0

.

25.0

. 22

1lo01.011lo06.0

08.0.

10.0.

+

symbolbits /67.2

01.006.0

=

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Shannon Fano Encoding (solution)

The efficiency of the code is :

100)(=

XH

67.2

72.2

.

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Huffman Encoding (The Algorithm)

List the symbols in descending order of probability.

Combine the 2 smallest probabilities to make a newprobability.

Repeat the process until all the symbols have been.

.

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Huffman Encoding (Example)

x1

: P = 0.25 x2

: P = 0.25

x : P = 0.15 x : P = 0.1

x5: = . x6: = .

x7: P = 0.06 x8: P = 0.01

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x1 0.25

0

000

02

x3 0.15

.

0.55 1.00

x4 0.10.45

0

1

x5 0.1

0.2

0.301

1

1

x6 0.08 0

x7

x8

0.060.07

0

1

1

101111

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0 0

Huffman Encoding (Solution)

x2

1 0

x3

x

0 1 0

x5

1 1 1

x6

x

0 1 1 0

0 1 1 1 0

x8

0 1 1 1 1

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Huffman Encoding (evaluation)

For the above Code,

Calculate the following:-

The code length

The code Entropy

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Huffman Encoding (Evaluation -result)

The code length = 2.72 digits/symbol

The code Entropy = 2.67 bits/symbol

The code Efficiency = 98.2%

Compare to those for the Shannon Fano Coding

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Practical Example

circuit.

At encoded 2.72 bits/symbol this is equivalent to 882 symbols/sec

Improvement = 82/800 = 10.25%

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Implementation of Huffman Code

A4 page scanned from top left-hand corner to the right-hand bottom corner

Each line is subdivided into 1728 icture elements els .

Each pel quantized into either black ple or white pel.

The total number of pels is approximately is just under 2 pels.

If a binary 1 is used to represent a white pel and binary 0 represent a blackpel and

If the signalling rate is 4800bps the time taken to transmit a page using theabove scheme is:

6

The scanned data contains considerable redundancy.

the transmission speed can be reduced by a factor of about 10.

the probability of occurrence of each run length.