lecture coding

8/13/2019 Lecture Coding

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EC 723Satellite Communication Systems

Mohamed Khedrhttp://webmail.aast.edu/~khedr
http://webmail.aast.edu/~khedrhttp://webmail.aast.edu/~khedr


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2006-02-16 Lecture 9 2

Syllabus

Tentatively

Week 1 Overview

Week 2 Orbits and constellations: GEO, MEO and LEO

Week 3 Satellite space segment, Propagation and

satellite links , channel modellingWeek 4 Satellite Communications Techniques

Week 5 Satellite Communications Techniques II

Week 6 Satellite Communications Techniques III

Week 7 Satellite error correction Techniques

Week 8 Satellite error correction TechniquesIIMultiple access

Week 9 Satellite in networks I, INTELSAT systems ,VSAT networks, GPS

Week 10 GEO, MEO and LEO mobile communications

INMARSAT systems, Iridium , Globalstar,Odyssey

Week 11 Presentations



Week 14 PresentationsWeek 15 Presentations


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2006-02-16 Lecture 9 3

Block diagram of a DCS

FormatSource

encoder

FormatSource

decode

Channel

encoder

Pulse

modulate

Bandpass

modulate

Channel

decoder

Demod.

SampleDetect

Channel

Digital modulation

Digital demodulation


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2006-02-16 Lecture 9 4

Channel coding: Transforming signals to improve

communications performance by increasingthe robustness against channel impairments

(noise, interference, fading, ..) Waveform coding: Transforming waveforms

to better waveforms

Structured sequences: Transforming data

sequences into better sequences, havingstructured redundancy.Better in the sense of making the decision

process less subject to errors.

What is channel coding?


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2006-02-16 Lecture 9 5

What is channel coding?

Coding is mapping of binary source (usually) output sequences of

length kinto binary channel input sequences n (>k) A block code is denoted by (n,k)

Binary coding produces2kcodewords of length n. Extra bits incodewords are used for error detection/correction

In this course we concentrate on two coding types: (1) block,

and (2) convolutional codesrealized by binary numbers:

Block codes: mapping of information source into channelinputs done independently: Encoder output depends only onthe current blockof input sequence

Convolutional codes: each source bitinfluences n(L+1)channel input bits. n(L+1)is the constraint length and Lis thememory depth. These codes are denoted by (n,k,L).

(n,k)

block coder

k-bits n-bits


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2006-02-16 Lecture 9 6

Error control techniques

Automatic Repeat reQuest (ARQ) Full-duplex connection, error detection codes

The receiver sends a feedback to the transmitter,saying that if any error is detected in the receivedpacket or not (Not-Acknowledgement (NACK) and

Acknowledgement (ACK), respectively). The transmitter retransmits the previously sent

packet if it receives NACK.

Forward Error Correction (FEC)

Simplex connection, error correction codes The receiver tries to correct some errors

Hybrid ARQ (ARQ+FEC) Full-duplex, error detection and correction codes


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8/922006-02-16 Lecture 9 8

Channel models

Discrete memory-less channels

Discrete input, discrete output

Binary Symmetric channels

Binary input, binary output

Gaussian channels

Discrete input, continuous output


9/922006-02-16 Lecture 9 9

Linear block codes

Let us review some basic definitions firstwhich are useful in understanding Linearblock codes.


10/922006-02-16 Lecture 9 10

Some definitions

Binary field : The set {0,1}, under modulo 2 binary

addition and multiplication forms a field.

Binary field is also called Galois field, GF(2).

011

101

110

000

111

001

010

000

Addition Multiplication


11/922006-02-16 Lecture 9 11

Some definitions

Examples of vector spaces The set of binary n-tuples, denoted by

Vector subspace: A subset S of the vector space is called a

subspace if:

The all-zero vector is in S.

The sum of any two vectors in S is also in S.

Example:

.ofsubspaceais)}1111(),1010(),0101(),0000{( 4V

nV

nV

)}1111(),1101(),1100(),1011(),1010(),1001(),1000(

),0111(),0101(),0100(),0011(),0010(),0001(),0000{(4V


12/922006-02-16 Lecture 9 12

Some definitions

Spanning set: A collection of vectors ,

the linear combinations of which include all vectors ina vector space V, is said to be a spanning set for V orto span V. Example:

Bases: A spanning set for V that has minimal cardinality is

called a basis for V. Cardinality of a set is the number of objects in the set.

Example:

.forbasisais)0001(),0010(),0100(),1000( 4V

.spans)1001(),0011(),1100(),0110(),1000( 4V

nG vvv ,,, 21


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2006-02-16 Lecture 9 13

Linear block codes

Linear block code (n,k)A set with cardinality is called a

linear block code if, and only if, it is asubspace of the vector space .

Members of C are called code-words.

The all-zero codeword is a codeword.

Any linear combination of code-words is acodeword.

nV

nVC k

2

nk VCV


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2006-02-16 Lecture 9 14

Linear block codescontd

nV

kV C

Bases of C

mapping


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2006-02-16 Lecture 9 15

(a) Hamming distance d(ci, cj) 2t 1.(b) Hamming distance d(ci, cj) 2t. Thereceived vector is denoted by r.


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2006-02-16 Lecture 9 16

# of bits for FEC

Want to correct terrors in an (n,k) code

Data word d=[d1,d2, . . . ,dk] => 2kdata

words

Code word c=[c1,c2, . . . ,cn] => 2ncode

words

Data wordCode word

dj cj

cj00

1110

01110

101

100

010

111

001011

000


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2006-02-16 Lecture 9 17

Representing codes by vectors

Code strength is measured by Hamming distance that tells how

different code words are: Codes are more powerful when their minimum Hamming

distance dmin(over all codes in the code family) is large

Hamming distance d(X,Y) is the number of bits that are differentbetween code words

(n,k) codes can be mapped into n-dimensional grid:

3-bit repetition code 3-bit parity code

valid code word


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2006-02-16 Lecture 9 18

Error Detection

If a code can detect

a tbit error, then cjmust be within aHamming sphere of

t For example,if

cj=101, and t =1,then 100,111, and

001 lie in theHamming sphere.

Code word

cj

cj 110

101

100

010

111

001011

000


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2006-02-16 Lecture 9 19

Error Correction

To correct anerror, theHamming

spheres arounda code wordmust be

nonoverlapping,dmin=2 t +1

cj

cj 110

101

100

010

111

001011

000


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2006-02-16 Lecture 9 20

6-D Code Space


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2006-02-16 Lecture 9 21

Block Code Error Detection and Correction

(6,3) code 23=> 26,dmin=3

Can detect 2 bit errors,correct 1 bit 110100 sent; 110101

received

Erasure: Suppose codeword 110011 sent but

two digits were erased(xx0011), correct codeword has smallestHamming distance

Message

Code-word

1 2

000 000000 4 2

100 110100 1 3

010 011010 3 2

110 101110 3 3

001 101001 3 2

101 011101 2 3011 110011 2 0

111 000111 3 1


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2006-02-16 Lecture 9 22

Geometric View

Want codeefficiency, so thespace should be

packed with asmany code wordsas possible

Code words should

be as far apart aspossible tominimize errors

2nn-tuples,Vn

2kn-tuples, subspace

of codewords


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2006-02-16 Lecture 9 23


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 blockChannel

encoderCodeword

k bits n bits

rateCode

bitsRedundant

n

kR

n-k

c


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2006-02-16 Lecture 9 24


The Hamming weight of vector U, denoted byw(U), is the number of non-zero elements inU.

The Hamming distance between two vectors

UandV, is the number of elements in whichthey differ.

The minimum distance of a block code is

)()( VUVU, wd

)(min),(minmin ii

jiji

wdd UUU


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2006-02-16 Lecture 9 25


Error detection capability is given by

Error correcting-capability tof a code, which isdefined as the maximum number ofguaranteed correctable errors per codeword, is

2

1mindt

1minde


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2006-02-16 Lecture 9 26


For memory less channels, the probabilitythat the decoder commits an erroneousdecoding is

is the transition probability or bit error probability

over channel.

The decoded bit error probability is

jnjn

tj

M ppj

nP

)1(

1

jnjn

tj

B ppj

nj

nP

)1(

1

1

p


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2006-02-16 Lecture 9 27


Discrete, memoryless, symmetric channel model

Note that for coded systems, the coded bits aremodulated and transmitted over channel. Forexample, for M-PSK modulation on AWGN channels(M>2):

where is energy per coded bit, given by

Tx. bits Rx. bits

1-p

1-p

p

p

MN

REMQ

MMN

EMQ

Mp cbc

sin

log2

log

2sin

log2

log

2

0

2

20

2

2

cE bcc ERE

1

0 0

1


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2006-02-16 Lecture 9 28


A matrix G is constructed by taking as itsrows the vectors on the basis, .

nVkV

C

Bases of C

mapping

},,,{ 21 kVVV

knkk

n

n

kvvv

vvv

vvv

21

22221

11211

1

V

V

G


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2006-02-16 Lecture 9 29


Encoding in (n,k) block code

The rows of G, are linearly independent.

mGU

kn

k

kn

mmmuuu

mmmuuu

VVV

V

V

V

2221121

2

1

2121

),,,(

),,,(),,,(


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2006-02-16 Lecture 9 31


Systematic block code (n,k) For a systematic code, the first (or last) k

elements in the codeword are information bits.

matrix)(

matrixidentity][

knk

kk

k

k

k

P

IIPG

),...,,,,...,,(),...,,(

bitsmessage

21

bitsparity

2121 kknn mmmpppuuu U


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2006-02-16 Lecture 9 32


For any linear code we can find anmatrix , which its rows areorthogonal to rows of :

H is called the parity check matrix andits rows are linearly independent.

For systematic linear block codes:

nkn )(HG

0GH T

][ Tkn PIH


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2006-02-16 Lecture 9 33


Syndrome testing: Sis syndrome of r, corresponding to the error

pattern e.

Format Channelencoding

Modulation

Channel

decodingFormat

Demodulation

Detection

Data source

Data sink

U

r

m

m

channel

or vectorpatternerror),....,,(

or vectorcodewordreceived),....,,(

21

21

n

n

eee

rrr

e

r

eUr

TT eHrHS


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2006-02-16 Lecture 9 34


Standard array1. For row , find a vector in of minimumweight which is not already listed in the array.

2. Call this pattern and form the row as the

corresponding coset

kknknkn

k

k

22222

22222

221

UeUee

UeUee

UUU

zero

codeword

coset

coset leaders

kni 2,...,3,2 nV

ie th:i


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2006-02-16 Lecture 9 35


Standard array and syndrome table decoding1. Calculate

2. Find the coset leader, , corresponding to .

3. Calculate and corresponding .

Note that

If , error is corrected.

If , undetectable decoding error occurs.

TrHS

iee S

erU m

)( e(eUee)UerU ee

ee


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2006-02-16 Lecture 9 36


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

010110100101010001

010100100000

100100010000

111100001000

000110110111011010101101101010011100110000000100000101110001011111101011101100011000110110000010

000110110010011100101000101111011011110101000001

000111110011011101101001101110011010110100000000

Coset leaders

coset

codewords


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2006-02-16 Lecture 9 37


111010001

100100000

010010000

001001000

110000100

011000010

101000001

000000000

(101110)(100000)(001110)estimatedisvectorcorrectedThe

(100000)

issyndromethistoingcorrespondpatternError

(100)(001110):computedisofsyndromeThe

received.is(001110)

ted.transmit(101110)

erU

e

HrHSr

r

U

TT

Error pattern Syndrome


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2006-02-16 Lecture 9 40

Example of the block codes

8PSK

QPSK

[dB]/ 0NEb

B

P


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2006-02-16 Lecture 9 41

Convolutional codes

Convolutional codes offer an approach to error controlcoding substantially different from that of block codes.

A convolutional encoder:

encodes the entire data stream, into a single codeword.

does not need to segment the data stream into blocks of fixed

size (Convolutional codes are often forced to block structure by periodictruncation).

is a machine with memory.

This fundamental difference in approach imparts a

different nature to the design and evaluation of the code. Block codes are based on algebraic/combinatorial

techniques.

Convolutional codes are based on construction techniques.


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2006-02-16 Lecture 9 46

A Rate Convolutional encoder

Encoder)101(m )1110001011(U

0 0 15t

1u

2u11

21 uu

0 0 06t

1u

2u00

21 uu

Time Output Time Output(Branch word) (Branch word)

Eff i d


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2006-02-16 Lecture 9 47

Effective code rate

Initialize the memory before encoding the first bit (all-

zero) Clear out the memory after encoding the last bit (all-

zero)

Hence, a tail of zero-bits is appended to data bits.

Effective code rate :

L is the number of data bits and k=1is assumed:

data Encoder codewordtail

ceff RKLn

LR

)1(

E d i


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2006-02-16 Lecture 9 48

Encoder representation

Vector representation: We define n binary vector withK elements (one

vector for each modulo-2 adder). The i:th elementin each vector, is 1 if the i:th stage in the shiftregister is connected to the corresponding modulo-

2 adder, and 0 otherwise. Example:

m

1u

2u

21 uu)101(

)111(

2

1

g

g


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2006-02-16 Lecture 9 49

Encoder representationcontd

Impulse response representation: The response of encoder to a single one bit that

goes through it.

Example:

11001

01010

11100

111011:sequenceOutput

001:sequenceInput

21 uu

Branch wordRegister

contents

1110001011

1110111

0000000

1110111

OutputInput m

Modulo-2 sum:

E d t ti td


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2006-02-16 Lecture 9 50


Polynomial representation: We define n generator polynomials, one for each

modulo-2 adder. Each polynomial is of degreeK-1or

less and describes the connection of the shiftregisters to the corresponding modulo-2 adder.

Example:

The output sequence is found as follows:

22)2(

2

)2(

1

)2(

02

22)1(

2

)1(

1

)1(

01

1..)(

1..)(

XXgXggX

XXXgXggX

g

g

)()(withinterlaced)()()( 21 XXXXX gmgmU

E d t ti td


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2006-02-16 Lecture 9 51


In more details:

1110001011

)1,1()0,1()0,0()0,1()1,1()(

.0.0.01)()(.01)()(

1)1)(1()()(

1)1)(1()()(

432

432

2

432

1

422

2

4322

1

U

U

gmgm

gm

gm

XXXXX

XXXXXXXXXXXX

XXXXX

XXXXXXXX


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St t di td


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2006-02-16 Lecture 9 53

State diagramcontd

A state diagram is a way to represent

the encoder.

A state diagram contains all the statesand all possible transitions between

them.

Only two transitions initiating from astate

Only two transitions ending up in a state

St t di td


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2006-02-16 Lecture 9 54

State diagramcontd

10 01

00

11

Currentstate

input Nextstate

output

00

0 00

1 11

01

0 11

1 00

10

0 10

1 01

11

0 01

1 10

0S

1S

2S

3S

0S

2S

0S

2S

1S

3S

3S

1S

0S

1S2S

3S

1/11

1/00

1/01

1/10

0/11

0/00

0/01

0/10

Input

Output

(Branch word)


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T ellis contd


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2006-02-16 Lecture 9 56

Trelliscontd

A trellis diagram for the example code

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

6t

1t

2t

3t

4t 5t

1 0 1 0 0

11 10 00 10 11

Input bits

Output bits

Tail bits

Trellis contd


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2006-02-16 Lecture 9 57

Trelliscontd

1/11

0/00

0/10

1/11

1/01

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

0/00

0/11

0/00

6t

1t

2t

3t

4t 5t

1 0 1 0 0

11 10 00 10 11

Input bits

Output bits

Tail bits


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Soft and hard decision decoding


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2006-02-16 Lecture 9 59


In hard decision:

The demodulator makes a firm or hard decisionwhether one or zero is transmitted and providesno other information for the decoder such thathow reliable the decision is.

In Soft decision:

The demodulator provides the decoder with some

side information together with the decision. Theside information provides the decoder with ameasure of confidence for the decision.

Soft and hard decoding


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2006-02-16 Lecture 9 60

Soft and hard decoding

Regardless whether the channel outputs hard or soft decisions

the decoding rule remains the same: maximize the probability

However, in soft decoding decision region energiesmust beaccounted for, and hence Euclidean metric dE, rather that

Hamming metric dfreeis used

Transition for Pr[3|0] is indicated

by the arrow

0ln ( , ) ln ( | )jm j mjp p y x

y x

Decision regions


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2006-02-16 Lecture 9 61

Decision regions

Coding can be realized by soft-decoding or hard-decoding principle

For soft-decoding reliability (measured by bit-energy) of decision regionmust be known

Example: decoding BPSK-signal: Matched filter output is a continuosnumber. In AWGN matched filter output is Gaussian

For soft-decoding

several decisionregion partitionsare used

Transition probability

for Pr[3|0], e.g. prob.that transmitted 0

falls into region no: 3



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2006-02-16 Lecture 9 62


ML soft-decisions decoding rule:

Choose the path in the trellis with minimumEuclidean distance from the receivedsequence

ML hard-decisions decoding rule:

Choose the path in the trellis with minimum

Hamming distance from the receivedsequence

The Viterbi algorithm


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2006-02-16 Lecture 9 63


The Viterbi algorithm performs Maximumlikelihood decoding.

It finds a path through trellis with the largest

metric (maximum correlation or minimumdistance).

At each step in the trellis, it compares the partialmetric of all paths entering each state, and keepsonly the path with the largest metric, called thesurvivor, together with its metric.

Example of hard decision Viterbi decoding


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2006-02-16 Lecture 9 64

Example of hard-decision Viterbi decoding

)100( m

)1100111011(

U)101(m)1110001011(U

)0110111011(Z

0

2

0

1

2

1

0

1

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0 2 3 0 1 2

3

2

3

20

2

30

ii ttS ),(

Branch metric

Partial metric

Example of soft decision Viterbi decoding


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2006-02-16 Lecture 9 65

Example of soft-decision Viterbi decoding

)101( m)1110001011( U

)101(m)1110001011(U

)1,3

2

,1,3

2

,1,3

2

,3

2

,3

2

,3

2

,1(

Z

5/3

-5/3

4/3

0

0

1/3

1/3

-1/3

-1/3

5/3

-5/3

1/3

1/3

-1/3

6t

1t

2t

3t

4t 5t

-5/3

0 -5/3 -5/3 10/3 1/3 14/3

2

8/3

10/3

13/33

1/3

5/35/3

ii ttS ),(

Branch metric

Partial metric

1/3

-4/35/3

5/3

-5/3

Free distance of Convolutional codes


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2006-02-16 Lecture 9 66

Free distance of Convolutional codes

Distance properties: Since a Convolutional encoder generates codewords with

various sizes (as opposite to the block codes), the followingapproach is used to find the minimum distance between allpairs of codewords:

Since the code is linear, the minimum distance of the code isthe minimum distance between each of the codewords and theall-zero codeword.

This is the minimum distance in the set of all arbitrary longpaths along the trellis that diverge and remerge to the all-zeropath.

It is called the minimum free distance or the free distance ofthe code, denoted by

ffree dd or

Free distance


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2006-02-16 Lecture 9 67

Free distance

2

0

1

2

1

0

2

1

1

2

1

0

0

2

1

1

0

2

0

6t

1t

2t

3t

4t 5t

Hamming weightof the branchAll-zero pathThe path diverging and remerging toall-zero path with minimum weight

5fd

Interleaving


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2006-02-16 Lecture 9 68

Interleaving

Convolutional codes are suitable for memoryless

channels with random error events.

Some errors have bursty nature:

Statistical dependence among successive error events(time-correlation) due to the channel memory.

Like errors in multipath fading channels in wirelesscommunications, errors due to the switching noise,

Interleaving makes the channel looks like as amemoryless channel at the decoder.

Interleaving


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2006-02-16 Lecture 9 69

Interleaving

Interleaving is done by spreading the codedsymbols in time (interleaving) beforetransmission.

The reverse in done at the receiver by

deinterleaving the received sequence.Interleaving makes bursty errors look like

random. Hence, Conv. codes can be used.

Types of interleaving:

Block interleaving

Convolutional or cross interleaving

Interleaving


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2006-02-16 Lecture 9 70

Interleaving

Consider a code with t=1 and 3 coded bits.

A burst error of length 3 can not be corrected.

Let us use a block interleaver 3X3

A1 A2 A3 B1 B2 B3 C1 C2 C3

2 errors

A1 A2 A3 B1 B2 B3 C1 C2 C3

Interleaver

A1 B1 C1 A2 B2 C2 A3 B3 C3

A1 B1 C1 A2 B2 C2 A3 B3 C3

Deinterleaver

A1 A2 A3 B1 B2 B3 C1 C2 C3

1 errors 1 errors 1 errors

Concatenated codes


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2006-02-16 Lecture 9 71

Concatenated codes

A concatenated code uses two levels on coding, an

inner code and an outer code (higher rate). Popular concatenated codes: Convolutional codes with

Viterbi decoding as the inner code and Reed-Solomon codesas the outer code

The purpose is to reduce the overall complexity, yetachieving the required error performance.

Interleaver Modulate

Deinterleaver

Inner

encoder

Inner

decoderDemodulate

C

hannel

Outer

encoder

Outer

decoder

Input

data

Output

data

Optimum decoding


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2006-02-16 Lecture 9 72

Optimum decoding

If the input sequence messages are equally likely, the

optimum decoder which minimizes the probability oferror is theMaximum likelihooddecoder.

ML decoder, selects a codeword among all the

possible codewords which maximizes the likelihoodfunction where is the receivedsequence and is one of the possible codewords:

)( )(mp U|Z Z

)(mU

)(max)(ifChoose )(allover

)()( mmm pp(m)

U|ZU|ZUU

ML decoding rule:

codewords

to search!!!

L2

ML decoding for memory-less channels


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2006-02-16 Lecture 9 73

ML decoding for memory less channels

Due to the independent channel statistics for

memoryless channels, the likelihood function becomes

and equivalently, the log-likelihood function becomes

The path metric up to time index , is called the partial pathmetric.

1 1

)(

1

)()(

21,...,...,,

)( )|()|()|,...,...,,()(21

i

n

j

m

jiji

i

m

ii

m

izzz

m uzpUZpUZZZppi

U|Z

1 1

)(

1

)()(

)|(log)|(log)(log)( i

n

j

m

jijii

m

ii

m

uzpUZppm U|ZUPath metric Branch metric Bit metric

ML decoding rule:Choose the path with maximum metric among

all the paths in the trellis.

This path is the closest path to the transmitted sequence.

""i

Binary symmetric channels (BSC)


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2006-02-16 Lecture 9 74

Binary symmetric channels (BSC)

If is the Hamming distance between Zand U, then

Modulatorinput

1-p

p

p

1

0 0

1

)( )(mm dd UZ,

)1log(1

log)(

)1()( )(

pL

p

pdm

ppp

nm

dLdm mnm

U

U|Z

Demodulatoroutput

)0|0()1|1(1)1|0()0|1(

pppppp

ML decoding rule:Choose the path with minimum Hamming distance

from the received sequence.

Size of coded sequence

AWGN channels


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2006-02-16 Lecture 9 75

AWGN channels

For BPSK modulation the transmitted sequencecorresponding to the codeword is denoted bywhere and

and .

The log-likelihood function becomes

Maximizing the correlation is equivalent to minimizing theEuclidean distance.

)(mU

cij Es

)(

1 1

)()( m

i

n

j

m

jijiszm SZ,UInner product or correlation

between Zand S

ML decoding rule:Choose the path which with minimum Euclidean distance

to the received sequence.

),...,,...,( )()()(1)( m

ni

m

ji

m

i

m

i sssS ,...),...,,()()(

2

)(

1

)( m

i

mmm SSSS

Soft and hard decisions


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2006-02-16 Lecture 9 76

Soft and hard decisions

In hard decision:

The demodulator makes a firm or hard decisionwhether one or zero is transmitted and provides noother information for the decoder such that howreliable the decision is.

Hence, its output is only zero or one (the output isquantized only to two level) which are called hard-bits.

Decoding based on hard-bits is called thehard-decision decoding.

Soft and hard decision-contd


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2006-02-16 Lecture 9 77

Soft and hard decision cont d

In Soft decision: The demodulator provides the decoder with some

side information together with the decision.

The side information provides the decoder with ameasure of confidence for the decision.

The demodulator outputs which are called soft-bits, are quantized to more than two levels.

Decoding based on soft-bits, is called thesoft-decision decoding.

On AWGN channels, 2 dB and on fadingchannels 6 dB gain are obtained by usingsoft-decoding over hard-decoding.



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2006-02-16 Lecture 9 78


The Viterbi algorithm performs Maximum likelihood

decoding. It find a path through trellis with the largest metric

(maximum correlation or minimum distance).

It processes the demodulator outputs in an iterative

manner. At each step in the trellis, it compares the metric of all

paths entering each state, and keeps only the path withthe largest metric, called the survivor, together with itsmetric.

It proceeds in the trellis by eliminating the least likelypaths.

It reduces the decoding complexity to !12 KL

The Viterbi algorithm - contd


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The Viterbi algorithm cont d

Viterbi algorithm:A. Do the following set up: For a data block ofLbits, form the trellis. The trellis

hasL+K-1sections or levels and starts at time and

ends up at time .

Label all the branches in the trellis with theircorresponding branch metric.

For each state in the trellis at the time which isdenoted by , define a parameter

B. Then, do the following:

it

}2,...,1,0{)( 1 KitS ii ttS ),(

1t

KLt


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Example of Hard decision Viterbid di


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2006-02-16 Lecture 9 81

decoding

1/11

0/00

0/10

1/11

1/01

0/00

0/11

0/10

0/01

1/11

1/01

1/00

0/00

0/11

0/10

0/01

0/00

0/11

0/00

6t

1t

2t

3t

4t 5t

)101(m )1110001011(U)0110111011(Z

Example of Hard decision Viterbid di td


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2006-02-16 Lecture 9 82

decoding-contd Label al the branches with the branch metric

(Hamming distance)

0

2

0

1

2

1

0

1

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0

ii ttS ),(


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2006-02-16 Lecture 9 84

decoding-contd i=3

0

2

0

1

2

1

0

1

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0 2 3

0

2

30



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2006-02-16 Lecture 9 85

decoding-contd i=4

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0 2 3 0

3

2

3

0

2

30



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2006-02-16 Lecture 9 86

decoding-contd i=5

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0 2 3 0 1

3

2

3

20

2

30



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2006-02-16 Lecture 9 87

decoding-contd i=6

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0 2 3 0 1 2

3

2

3

20

2

30

Example of Hard decision Viterbi decoding-contd


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2006-02-16 Lecture 9 88

cont d

Trace back and then:

0

2

0

1

2

1

01

1

0

1

2

2

1

0

2

1

1

1

6t

1t

2t

3t

4t 5t

1

0 2 3 0 1 2

3

2

3

20

2

30

)100( m)0000111011( U

Example of soft-decision Viterbi decoding


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2006-02-16 Lecture 9 89

p g

)101( m)1110001011( U

)101(m)1110001011(U

)1,3

2

,1,3

2

,1,3

2

,3

2

,3

2

,3

2

,1(

Z

5/3

-5/3

4/3

0

0

1/3

1/3

-1/3

-1/3

5/3

-5/3

1/3

1/3

-1/3

6t

1t

2t

3t

4t 5t

-5/3

0 -5/3 -5/3 10/3 1/3 14/3

2

8/3

10/3

13/33

1/3

5/35/3

ii ttS ),(

Branch metric

Partial metric

1/3

-4/3

5/3

5/3

-5/3


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2006-02-16 Lecture 9 91

Trellisdiagram for

K= 2, k= 2,

n= 3

convolutional

code.


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State diagram for

K 2 k 2 3

lecture coding

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