Coding For Wireless Channel

Coding For Wireless Channel

Presented by: Salah Zeyad Radwan

Presented to: Dr. Mohab Mangoud

Outline1. Introduction.

2. Linear Block Codes.2.1 Binary Linear Block Codes.2.2 Generator Matrix. 2.3 Syndrome Testing & LUT.

3. Convolutional Codes.3.1 Mathematical Representation.3.2 Tree diagram 3.3 Trellis Diagram. 3.4 Viterbi Algorithm.

4. Interleaver 5. Concatenated Codes

6. Conclusion7. References

1. Introduction

Block Diagram of a general Communication System

Information to be

transmittedSource

encodingChannel coding

Modulation Transmitter

Channel

Information received

Source decoding

Channel decoding

Demodulation Receiver

Channelencoding

Channel decoding

1. Introduction• Channel encoding : The application of redundant

symbols to correct data errors.

• Modulation : Conversion of symbols to a waveform for transmission.

• Demodulation : Conversion of the waveform back to symbols.

• Decoding: Using the redundant symbols to correct errors.

1. IntroductionWhat is Coding?

• Coding is the conversion of information to another form for some purpose

• Source Coding : The purpose is lowering the redundancy in the information. (e.g. ZIP, JPEG, MPEG2)

• Channel Coding : The purpose is to defeat channel noise.

1. Introduction

• channel coding starting from 1948 Claude Shannon was working on the information transmission capacity of a communicationchannel, He showed that capacity depends on (SNR)

C = W log2 (1 + S/N)

C : Capacity of the channel W : Bandwidth of the channel

1. Introduction• Channel coding can be partitioned into two study areas

1. Waveform Coding: Transforming waveforms to better waveforms “detection process less subject to errors”

2. Structured Sequences : Transforming data sequences into better sequences, having structured redundancy ‘redundant bits can be detect and correct”

• In this study I’m going to describe codes designed for AWGN channels, Which requires a background in block & Convolutional codes

2. Linear Block Code• Linear block codes are extension of single-bit-parity

check codes for error detection. characterized by the (n,k) notation

• This code uses one bit ”parity” in a block of n data to indicate whether the number of 1s in a block is odd or even.

• Linear Block Code using a large number of parity bits to detected or correct more than one error

b1,b2,……,bn-k m1,m2,……….,mk

Parity bits (n-k) Message bits (k)

Code word (n)

2. Linear Block Code• The information bit stream is chopped into blocks of k

bits.

• Each block is encoded to a larger block of n bits.

• The coded bits are modulated and sent over channel.

• The reverse procedure is done at the receiver.

Data blockChannelencoder Codeword

k bits n bits

2. Linear Block Code

• Encoding Process

Block Encoder

Coded Data Stream

k-symbol block

n-symbol block

Uncoded Data stream


GMessagem

Codewordc

• Generator matrix

This system may be written in a compact form using matrix notations

m=[m0,m1,m2,…..…mk-1] massage vector 1-by-kb= [b0,b1,b2,………bn-k-1] parity vector 1-by-(n-k)c= [c0,c1,c2,……..….xn-1] code vector 1-by-n

Note: all three vectors are row vectors


b = mP (1)Where P is the k- by-(n-k) coefficient matrix

p11 p12 … p1,n-k

p21 p22 … p2,n-k

P = . . . . .. . . .

pk1 pk2 … pk,n-k


c = [b m] (2)

From (1) in (2) we getc = m [P Ik] (3)

Where Ik is the k-by-k identity matrix.The generator matrix is defined as:

G = [P Ik] (4)

From (4) in (3) we getc = mG (5)


• Syndrome decoding

The received message can be represented as:R = c + e

The decoder computes the code vector from the received message by using what we call the syndrome, which depends only upon the error pattern.

s = R HT


Then we write the error matrix e = (2n-k×n) to get the LUT : e × HT

LUT help us to locate the bit corresponding to the bit error in the codeword then we can correct it by changing the location of the bit error “if the bit error 1 change it to 0 and if we receive 0 change it to 1” to get the correct codeword.

3. Convolutional Codes

• Convolutional codes are fundamentally different from the block codes. It is not possible to separate the codes into independent blocks. Instead each code bit depends on a certain number of previous data bits.

• We can coding using a structure consisting of shift register, a set of exclusive or (XOR) gates and multiplexer


D D2

+

+

+

MU

X

Data

Codesequence

Input: 1 1 1 0 0 0 …Output: 11 01 10 01 11 00 …

Input: 1 0 1 0 0 0 …Output: 11 10 00 10 11 00 …

D1

x2

x1


The previous convolutional encoder can be represented mathematically

g(1) (D) = 1+ D2

g(2) (D) = 1 + D + D2

For message sequence (1001), it can be represented as m (D) = 1 + D3

Hence the output polynomial of path 1 & 2 arec(1) (D) = g(1) (D) m (D) = 1 + D2 + D3 + D5

c(2) (D) = g(2) (D) m (D) = 1 + D + D2 + D3 + D4 + D5

By multiplexing the two output sequences we get the code sequence

c = (11,01,11,11,01,11)

Which is (nonsystematic)


+

+ MU

X

Data

Code sequence

Data

parity

g(1) (D) = 1g(2) (D) = 1 + D + D2

Systematic Convolutional

D1 D2


Codesequence

DataD2

+

+

+

MU

X

D1

x2

x1

Structural properties of convolutional encoder are portrayed in graphical form by using any one of

1. Code tree2. Trellis3. State diagram

Tree Diagram

First input

00

1100

10

0111

11

0010

01

1001

00

1111

10

0100

11

0001

01

1010

11

00

01

10

01

10

11

00

1

0

… 1 1 0 0 1… 10 11 11 01 11

……

First output


Trellis for convolutional encoder

11d

01c

10b

00a

Binary descriptionstate

a

b

c

d

00 00 00

11 11 11

01

10

01

10

11

00

10

01


We may collapse the code tree in the previous slides into a new form called a trellis

Trellis for convolutional encoder

11d

01c

10b

00a


a

b

c

d

00 00 00

11 11

01

1010

11

01

11

00

10

01

For incoming data 1001the generated code sequence becomes 11.01.11.11

11


Viterbi Algorithm

• To decode the all-zero sequences when received as 0100100000

a

b

01 1

1




a

b

c

d

01 001

1

1

3

2

2



a

b

c

d

01 00 101

1

1

3

2

2

2

2

5

3

3

3

2

4



a

b

c

d

01 00 10 2

2

2

3



a

b

c

d

01 00 10 002

2

2

3

2

44

2

34

34



a

b

c

d

01 00 10 00 2

2

3

3



a

b

c

d

01 00 10 00 002

4

3

3

5

3

4

4



a

b

c

d

01 00 10 00 002

3

3

3



a

b

c

d

01 00 10 00 00

00 00 00 00 00


State diagram of the convolutional codes00

11

01

11

00

10

01

10

a

b

c

d

a

b

c

d

A portion of the central part of the trellis for the encoder

The left nodes represent the four possible current state of the encoder

The right nodes represent the next state

We can coalesce the left and right nodes to get “state diagram”


11d

01c

10b

00a


c

d

b

a 10

01

01

0010

11

11

00

4. Interleaver

• The simple type of interleaver (sometimes known as a rectangular or block interleaver)

• Interleaver is commonly denoted by the Greek letter л and its corresponding de-interleaver by л-1.

• The original order can then be restored by a corresponding de-interleaver:

4. Interleaver987654321

4. Interleaver

987654321

2019181716

1514131211

109876

54321

Interleaver (л)

4. Interleaver

987654321

2019181716

1514131211

109876

54321

3171272161161

Interleaver (л)

Interleaved data

4. Interleaver

987654321

2019181716

1514131211

109876

54321

3171272161161

Interleaver (л)

Interleaved data

2019181716

1514131211

109876

54321

De-interleaver (л-1)

4. Interleaver

987654321

2019181716

1514131211

109876

54321

3171272161161

Interleaver (л)

Interleaved data

2019181716

1514131211

109876

54321

987654321

De-interleaver (л-1)

5. Concatenated Codes

• We have seen that the power of FEC codes increases with length k and approaches the Shannon bound only at very large k, but also that decoding complexity increases very rapidly with k.

• This suggests that it would be desirable to build a long, complex code out of much shorter component codes, which can be decoded much more easily.


• The principle is to feed the output of one encoder (called the outer encoder) to the input of another encoder, and so on, as required.

• The final encoder before the channel is known as the inner encoder.

• The resulting composite code is clearly much more complex than any of the individual codes.


Interleaver Outer encoder

Innerencoder

Outer decoder

deInterleaver Inner decoder

Channel

General block diagram for concatenated codes

Serial Concatenated Codes

6. Conclusion

We have reviewed the basic principles of channel coding including the linear block codes, the convolutional codes and the concatenated coding, showing how it is that they achieve such remarkable performance.

7. References

• Wireless Communications “Andrea Goldsmith”• Digital Communications “Simon Haykin”• Digital Communications “Bernard Sklar”

Documents

Coding For Wireless Channel