wireless networks ch3

7/30/2019 Wireless Networks Ch3

1/37

Ali BAZZI

Chapter 3

Channel Coding


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Introduction Block Codes Cyclic Codes CRC (Cyclic Redundancy Check) Convolutional Codes Interleaving Information Capacity Theorem Turbo Codes ARQ (Automatic Repeat Request)

Stop-and-wait ARQ

Go-back-N ARQ Selective-repeat ARQ


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Information to

be transmitted Sourcecoding

Channel

codingModulation Transmitter

ChannelInformation

received Sourcedecoding

Channel

decodingDemodulation Receiver

Introduction

Channel

coding

Channel

decoding


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The key idea of FEC is to transmit enoughredundant data to allow receiver to recoverfrom errors all by itself. No senderretransmission required.

The major categories of FEC codes are Block codes, Cyclic codes, Reed-Solomon codes (Not covered here), Convolutional codes, and

Turbo codes, etc.


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Information is divided into blocks of length k rparity bits or check bits are added to each block

(total length n = k + r),.

Code rateR = k/n Decoder looks for codeword closest to receivedvector (code vector + error vector) Tradeoffs between

Efficiency Reliability Encoding/Decoding complexity


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The uncoded k data bits be represented by the m vector:m=(m1, m2, , mk)

The corresponding codeword be represented by the n-bit c

vector:

c=(c1, c

2, c

k, c

k+1, , c

n-1, c

n)

Each parity bit consists of weighted modulo 2 sum of the databits represented by symbol.

=

=

=

=

=

++++

knknnn

kkkkkk

kk

pmpmpmc

pmpmpmc

mc

mc

mc

...

...

...

...

2211

)1()1(22)1(111

22

11


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Linear Block CodeThe block length Cof the Linear Block Code is

C= m G

where m is the information codeword block length, Gis the

generator matrix.

G= [Ik|P]k n,

wherepi= Remainder of [xn-k+i-1/g(x)] fori=1, 2, .., k, and I

is unit matrix.

The parity check matrixH= [PT| In-k], where P

T is the transpose of the matrix p.


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Example : Find linear block code encoderG if code generatorpolynomialg(x)=1+x+x3 for a (7, 4) code.

We have n = Total number of bits = 7, k = Number of information bits = 4,

r = Number of parity bits = n - k = 3.

[ ] ,

100

010

001

|2

1

==

kp

p

p

PIG

where

kixg

xofmainderp

ikn

i ,,2,1,)(

Re1

=

=

+


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[ ]11011

Re3

3

1 +=

++

= xxx

xp

[ ]0111

Re2

3

4

2+=

++

= xxxx

xp

[ ]11111

Re2

3

5

3++=

++

= xxxx

xp

[ ]10111Re2

3

6

4 +=

++

= xxx

xp

=

0001101

0010111

0100011

1000110

G


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e HT = e HTe ) * HT =c HT

e be the received message where c is the correct code and e is the error

Block Codes: Linear Block Codes

Let x = c

Compute S = x * HT =( c

If S is 0 then message is correct else there are errors in it, from common known error

patterns the correct message can be decoded.

The parity check matrix H is used to detect errors in the received code by using the fact

that c * HT = 0 ( null vector)

Generator

matrix

G

Code

Vector

C

Message

vector

m

Paritycheck

matrix

HT

Code

Vector

C

Null

vector

0

Operations of the generator matrix and the parity check matrix


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Consider a (7,4) linear block code, given by G as

Then,

Form = [1 0 1 1] and c = mG = [1 0 1 1 0 0 1].

If there is no error, the received vectorx=c, and s=cHT=[0, 0,0]

=

0001011

0010101

0100110

1000111

G

=

1011001

1101010

1110100

H


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Let c suffer an error such that the received vectorx=c e

=[ 1 0 1 1 0 0 1 ] [ 0 0 1 0 0 0 0 ]

=[ 1 0 0 1 0 0 1 ].

Then,

s=xHT

=

=(eHT)

[ ] ]101[

001

010

100

011

101

110

111

1001001 =


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Cyclic Codes

It is a block code which uses a shift register to perform encoding anddecoding

The code word with n bits is expressed as

c(x)=c1xn-1

+c2xn-2+cn

where each ci is either a 1 or 0.

c(x) = m(x) xn-k+ cp(x)

where cp(x) = remainder from dividing m(x) xn-kby generator g(x)

if the received signal is c(x) + e(x) where e(x) is the error.

To check if received signal is error free, the remainder from dividing

c(x) + e(x) by g(x) is obtained(syndrome). If this is 0 then the receivedsignal is considered error free else error pattern is detected from

known error syndromes.


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Example : Find the codewords c(x) if m(x)=1+x+x2 and g(x)=1+x+x3

for (7,4) cyclic code.

We have n = Total number of bits = 7, k = Number of information bits = 4,

r = Number of parity bits = n - k = 3.

xxx

xxxrem

xgxxmremxc

kn

p

=

++

++

=

=

1

)()()(

3

345

Then,543

)()()( xxxxxcxxmxc pkn

+++=+=


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Cyclic redundancy Code (CRC) is an error-checking code. The transmitter appends an extra n-bit sequence to every

frame called Frame Check Sequence (FCS). The FCS holdsredundant information about the frame that helps the receivers

detect errors in the frame.

CRC is based on polynomial manipulation using moduloarithmetic. Blocks of input bit as coefficient-sets forpolynomials is called message polynomial. Polynomial withconstant coefficients is called the generator polynomial.


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Generator polynomial is divided into the messagepolynomial, giving quotient and remainder, the coefficientsof the remainder form the bits of final CRC.

Define:M The original frame (k bits) to be transmitted before

adding the

Frame Check Sequence (FCS).F The resulting FCS of n bits to be added to M (usually

n=8, 16, 32).

T The cascading of M and F.

P The predefined CRC generating polynomial with

pattern of n+1 bits.The main idea in CRC algorithm is that the FCS isgenerated so that the remainder of T/P is zero.


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The CRC creation process is defined as follows: Get the block of raw message Left shift the raw message by n bits and then divide it by

p

Get the remainder R as FCS Append the R to the raw message . The result is the

frame to be transmitted.

CRC is checked using the following process: Receive the frame Divide it by P Check the remainder. If the remainder is not zero, then

there is an error in the frame.


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Code Generator polynomial

g(x)

Parity check

bits

CRC-12 1+x+x

2

+x

3

+x

11

+x

12

12CRC-16 1+x2+x15+x16 16

CRC-CCITT 1+x5+x15+x16 16


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Encoding of information stream rather thaninformation blocks

Value of certain information symbol also affectsthe encoding of nextMinformation symbols,

i.e., memoryM Easy implementation using shift registerAssuming kinputs and n outputs

Decoding is mostly performed by the ViterbiAlgorithm (not covered here)


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Input Output

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 D2

Di -- Register

x

y1

y2

c


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01/001/1

11/1 11/0

11

10/1

00/0

00

10 0110/0

00/1


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00

1100

10

0111

11

0010

01

1001

00

1111

10

01

00

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 inputFirst output


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00 0000 0000 0000 0000 0000

10 10 10 10 10 10

01 01 01 01 01 01

11 11 11 11 11 11

11

10

11

11

01

11 0 0 1

10 10 10

01 01 0101 01 01

11 11 1111

11

10 10 10

00 00 00


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a1, a2, a3, a4, a5, a6, a7, a8, a9, Input Data

a1, a2, a3, a4

a5, a6, a7, a8

a9, a10, a11, a12

a13, a14, a15, a16

Write

ReadInterleaving

a1, a5, a9, a13, a2, a6, a10, a14, a3, Transmitting

Data

a1, a2, a3, a4

a5, a6, a7, a8

a9, a10, a11, a12

a13, a14, a15, a16

Read

W

riteDe-Interleaving

a1, a2, a3, a4, a5, a6, a7, a8, a9, Output Data


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0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,Transmitting

Data

0, 1, 0, 0

0, 1, 0, 00, 1, 0, 0

1, 0, 0, 0

Read

WriteDe-Interleaving

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, Output Data

Burst error

Discrete error


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The information capacity (or channel capacity)Cof a continuous channel with bandwidthBHertz can be perturbed by additive Gaussianwhite noise of power spectral densityN0/2,provided bandwidthB satisfies

wherePis the average transmitted powerP =EbRb(for an ideal system,Rb = C).

Eb isthe transmitted energy per bit,Rb is transmission rate.

ondbitsBN

PBC sec/1log

0

2

+=


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0 10 20 30Eb/N0 dB

Region for whichRbC

Rb/B

-1.6

Shannon Limit

Capacity boundaryRb=C

1

10

20

0.1


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A brief historic of turbo codes :The turbo code concept was first introduced by C. Berrou in

1993. Today, Turbo Codes are considered as the mostefficient coding schemes for FEC.

Scheme with known components (simple convolutional orblock codes, interleaver, soft-decision decoder, etc.)

Performance close to the Shannon Limit (Eb/N0 = -1.6 db ifRb0) at modest complexity!

Turbo codes have been proposed for low-power applicationssuch as deep-space and satellite communications, as well as

for interference limited applications such as third generationcellular, personal communication services, ad hoc and sensornetworks.


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Convolutional Encoder

1

Interleaving

Convolutional Encoder

2

Data

Source

XX

Y

Y1

Y2(Y1, Y2)

X: Information

Yi: Redundancy Information


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Convolutional

Decoder 1

Convolutional

Decoder 2Interleaving

Interleaver

De-interleaving

De-interleavingX

Y1

Y2

X

X: Decoded Information


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Source Transmitter Channel Receiver Destination

Encoder

Transmit

Controller Modulation Demodulation Decoder

Transmit

Controller

Acknowledge


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Transmitting

Data

1 32 3Time

Received Data 1 2 3 Time

ACK

ACK

NAK

Output Data 1 2 3 Time

Error

Retransmission

ACK: Acknowledge

NAK: Negative ACK


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Stop-And-Wait ARQ (SAW ARQ)

Throughput:

S = (1/T) * (k/n) = [(1- Pb)n / (1 + D * Rb/ n) ] *

(k/n)

where T is the average transmission time in terms of a block duration

T = (1 + D * Rb/ n) * PACK+ 2 * (1 + D * Rb/ n) * PACK* (1- PACK)

+ 3 * (1 + D * Rb/ n) * PACK* (1- PACK)2 + ..

= (1+ D * Rb/n) * PACK i * (1-PACK)i-1

= (1+ D * Rb/n) * PACK/[1-(1-PACK)]2

= (1 + D * Rb/ n) / PACK

where n = number of bits in a block, k = number of information bits in a

block, D = round trip delay, Rb= bit rate, Pb = BER of the channel, and

PACK= (1- Pb)n

=1i


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Transmitting

Data

1Time

Received Data

Output DataTime

Error

Go-back 3

2 3 4 5 3 44 5 6 7 5

1 Time2 3 44 5Error

Go-back 5

1 2 3 44 5


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Throughput

S = (1/T) * (k/n)

= [(1- Pb)n / ((1- Pb)

n + N * (1-(1- Pb)n ) )]* (k/n)

whereT = 1 * PACK+ (N+1) * PACK* (1- PACK) +2 * (N+1) * PACK*

(1- PACK)2 + .

= PACK+PACK* [(1-PACK)+(1-PACK)2 +(1-PACK)

3+]+

PACK[N * (1-PACK)+2 * N * (1-PACK)2

+3 * N * (1-PACK)3

+]= PACK+PACK*[(1-PACK)/PACK + N * (1-PACK)/PACK

2

= 1 + (N * [1 - (1- Pb)n ])/ (1- Pb)

n

Go-Back-N ARQ (GBN ARQ)


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Transmitting

Data

1Time

Received Data

Error

Retransmission

2 3 4 5 3 6 7 8 9 7

1 Time2 4 3 6 8 7Error

Retransmission

5 9

Buffer 1Time

2 4 3 6 8 75 9

Output Data 1Time

2 43 6 875 9


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Selective-Repeat ARQ (SR ARQ)

Throughput

S = (1/T) * (k/n)

= (1- Pb)n * (k/n)

whereT = 1 * PACK+ 2 * PACK* (1- PACK) + 3 * PACK* (1- PACK)

2

+ .

= PACK i * (1-PACK)i-1

= PACK/[1-(1-PACK)]2

= 1/(1- Pb)n

=1i

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