bài 4 Mã hóa kênh_ Compatiblity mode

Facuty of Electronics & Telecommunications, HCMUNSNội dung trình bày • Mã hóa kênh Channel coding • Mã hóa khối Block codes... Facuty of Electronics & Telecommunications, HCMUNSBlock c

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Facuty of Electronics & Telecommunications, HCMUNS 1

BÀI 4:

MÃ HÓA KÊNH (Channel coding)

Facuty of Electronics & Telecommunications, HCMUNS

Đặng Lê KhoaEmail:danglekhoa@yahoo.com

dlkhoa@fetel.hcmuns.edu.vn

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Nội dung trình bày

• Mã hóa kênh ( Channel coding )

• Mã hóa khối (Block codes)

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Pulsemodulate

Bandpassmodulate

Channeldecode

Demod

SampleDetect

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• Tín hiệu truyền qua kênh truyền sẽ bị ảnh hưởng bởi nhiễu, can nhiễu, fading… là tín hiệu đầu thu bị sai

• Mã hóa kênh: dùng để bảo vệ dữ liệu không bị sai

bằng cách thêm vào các bit dư thừa (redundancy)

• Ý tưởng mã hóa kênh là gởi một chuỗi bit có khả

năng sửa lỗi

• Mã hóa kênh không làm giảm lỗi bit truyền mà chỉ làm giảm lỗi bit dữ liệu (bảng tin)

• Có hai loại mã hóa kênh cơ bản là: Block codes và

Convolutional codes

Channel coding là gì?

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Định lý giới hạn Shannon

• Đối với kênh truyền AWGN, ta có

C: channel capacity (bits per second) B: transmission bandwidth (Hz)

P: received signal power (watts)

No: single-sided noise power density (watts/Hz)

Eb: average bit energy

Rb: transmission bit rate

C/B: bandwidth efficiency

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Galois field

• Binary field :

– Tập {0,1}, thực hiện phép cộng và phép nhân 2 thì

kết quả cũng thuộc trường

– Binary field còn được gọi là Galois field, GF(2)

0 1 1

1 0 1

1 1 0

0 0 0

0 0 1

0 1 0

0 0 0

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Galois Field Construction

14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 0

3 2

3 2 2 3 2 3

3 2 2

3 2

1 1 1 1

1 1 1

1 0

1 1 0 1

1 1 1 1

1 1 1 0

0 1 1 1

1 0 1 0

0 1 0 1

1 0 1 1

1 1 0 0

0 1 1 0

0 0 1 1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0



 4  1

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Block codes

• Block codes là loại forward error correction (FEC) -> cho phép tự sửa sai ở đầu thu

• Trong block codes, các bit parity được thêm vào bảng tin

để tạo thành code word hoặc code blocks

• Khả năng sửa lỗi của block code phụ thuộc vào code

distance

• Block codes có tính chất tuyến tính => kết quả phép tính

số học giữa các phần tử luôn là một phần tử thuộc

trường

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Block codes (tt)

• Luồng thông tin được phân thành từng khối k bits

• Mỗi khối được mã hóa thành khối lớn hơn có n bits

• Các mã này được điều biến và gởi qua kênh truyền

• Quá trình sẽ thực hiện ngược lại ở đầu thu

Data block Channel

encoder Codeword

rateCode

bits

Redundant

n

k R

n-k

c 

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Repetition Code

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Repetition Code (tt)

• Truyền:

• Nhận :

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Repetition Code (tt)

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Linear block codes –cont’d

– A matrix G is constructed by taking as its rows the

vectors on the basis,

, , {V1 V2  Vk

k

n n

k

v v

v

v v

v

v v

2 22

21

1 12

11 1

V V G

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Linear block codes – cont’d

• Encoding in (n,k) block code

– The rows of G, are linearly independent.

mG

U 

k n

k

k n

m m

m u

u u

m m

m u

u u

V V

V

V V

2 1

1 2

1

2 1

2 1 2

1

) , , , (

) ,

, ,

( ) , , , (

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• Example: Block code (6,3)

0 1 0

0 0 1

1 1 0

0 1 1

1 0 1

3 2 1

V V

V G

1

1 1

0 0

1 0

1 1

1 1 0

0 0 1

1 0 1

1 1 1

1 0 0

0 1

1 1

1 0

0 0 1

1 0 1

0 0 0

1 0 0

0 1 0

1 0 0

1 1 0

0 1

0 0

1 0 0

0 1 0

Message vector Codeword

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• Systematic block code (n,k)

– For a systematic code, the first (or last) k elements in the codeword are information bits.

matrix

) (

matrix identity

] [

k n

k

k k

I P

G

) , ,

, ,

, , ,

( )

, , ,

(

bits message

2 1

bits parity

2 1



U

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• For any linear code we can find an matrix ,

such that its rows are orthogonal to the rows of :

• H is called the parity check matrix and its rows are

linearly independent

• For systematic linear block codes:

n k

n )  (

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error

) , , ,

(

or vector codeword

received

) , , ,

(

2 1

n

e e

e

r r

T T

eH rH

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• Standard array

– For row , find a vector in of minimum weight

which is not already listed in the array.

– Call this pattern and form the row as the corresponding

coset

k k

n k

n

k k

2 2

2

2 2

2

2 2

1

U e

e

U e

e

U U

i  2,3, ,2 

n V i

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• Standard array and syndrome table decoding

1 Calculate

2 Find the coset leader, , corresponding to .

3 Calculate and corresponding .

) ˆ ˆ

( ˆ

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• Example: Standard array for the (6,3) code

010110 100101

010001

010100 100000

100100 010000

111100 001000

000110 110111

011010 101101

101010 011100

110011 000100

000101 110001

011111 101011

101100 011000

110111 000010

000110 110010

011100 101000

101111 011011

110101 000001

000111 110011

011101 101001

101110 011010

110100 000000

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111 010001

100 100000

010 010000

001 001000

110 000100

011 000010

101 000001

000 000000

(101110) (100000)

(001110) ˆ

ˆ

estimated is

vector corrected

The

(100000) ˆ

is syndrome this

to ing correspond pattern

Error

(100) (001110)

: computed is

of syndrome The

received.

is (001110)

ted.

transmit (101110)

H rH

S

r r

U

T T

Error pattern Syndrome

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• Hamming codes

– Hamming codes are a subclass of linear block codes and

belong to the category of perfect codes.

– Hamming codes are expressed as a function of a single integer

– The columns of the parity-check matrix, H, consist of all non-zero

m k

n

m m

1

: capability correction

Error

:

bits parity of

Number

1 2

: bits

n informatio of

Number

1 2

:

length Code

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Hamming codes

• Example: Systematic Hamming code (7,4)

] [

1 0

1 1

1 0

0

1 1

0 1

0

1 1

1 0

0 0

1

3 3

T

P I

1 0

0 0

1 1

1

0 1

0 0

0 1

1

0 0

1 0

1

0 0

0 1

1 1

0

4 4

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Cyclic block codes

• Cyclic codes are a subclass of linear block codes

• Encoding and syndrome calculation are easily performed using feedback shift-registers

– Hence, relatively long block codes can be implemented

with a reasonable complexity.

• BCH and Reed-Solomon codes are cyclic codes

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• A linear (n,k) code is called a Cyclic code if all cyclic

shifts of a codeword are also a codeword

– Example:

) , ,

, ,

, , ,

, (

) , ,

, ,

(

1 2

1 0

1 1

) (

1 2

1 0

i n i

n i

n

u u

u

u u

U

U “i” cyclic shifts of U

U U

)1011(

)0111(

)1110(

)1101(

) 4 ( )

3 ( )

2 ( )

1 (

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• Algebraic structure of Cyclic codes, implies expressing codewords in

polynomial form

• Relationship between a codeword and its cyclic shifts:

– Hence:

) 1 ( degree

) (

, )

(

1

) 1 (

1 1

) (

1 2

2 1 0

1

1 2

2 1 0

1 )

1 (

X u

n

n n

X

n n

n

n n

X u

X

u X

u X u u

X u

X u X u X

X

n n

modulo )

( )

(

) (



i

X X

modulo )

( )

(

) 1 (



X U U

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• Basic properties of Cyclic codes:

– Let C be a binary (n,k) linear cyclic code

1 Within the set of code polynomials in C, there

is a unique monic polynomial with minimal degree is called the generator polynomials

1 Every code polynomial in C, can be

r  g

r

r X g X

g g

X )    ( 0 1

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• The orthogonality of G and H in polynomial

form is expressed as This means is also a factor of

1 The row , of generator matrix is

formed by the coefficients of the cyclic shift of the generator polynomial

r r

k

g g

g

g g

g

g g

g

g g

g

X X

X X X

1 0

1

) (

1)

()(X h X  X n 

"

1

" i

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• Systematic encoding algorithm for an (n,k) Cyclic

code:

1 Multiply the message polynomial by

1 Divide the result of Step 1 by the generator polynomial

Let be the reminder.

1 Add to to form the codeword

( X

U

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• Example: For the systematic (7,4) Cyclic code with generator

polynomial

1 Find the codeword for the message

) 1 1 0 1 0 0 1 (

1 ) ( )

( )

(

: polynomial codeword

the Form

1 )

1 ( 1

(

: (

by ) ( Divide

) 1

( )

(

1 ) ( )

1011 (

3

, 4

, 7

bits message bits

parity

6 5

3 3

) ( remainder generator

3 quotient

3 2

6 5

3

6 5

3 3

2 3

3

3 2

U

g m

m m

p g

q

X X

X

X X X

X X

X) X

X

X X

X

k n k

n

X (X)

(X)

k n

)1011(



m

3

1)(X   X  X

g

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– Find the generator and parity check matrices, G and H,

0 1 0 1 1 0 0

0 0 1 0 1 1 0

0 0 0 1 0 1 1

) 1101 (

) , , , ( 1

0 1

1 )

row(2) row(1)

row(3) row(3)

0 1 0 0 1 1 1

0 0 1 0 1 1 0

0 0 0 1 0 1 1

0 1 1 1 0 1 0

1 1 0 1 0 0 1

H

4 4

P P

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• Syndrome decoding for Cyclic codes:

– Received codeword in polynomial form is given by

– The syndrome is the reminder obtained by dividing the received polynomial by the generator polynomial

– With syndrome and Standard array, error is estimated.

• In Cyclic codes, the size of standard array is considerably reduced

)()

()

Received codeword

Error pattern

)()

()()

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