
2/51

Hamming Distance

Words of same length

Number of different digits

00 and 11

Distance : 2

000 and 111

Distance : 3


3/51

Hamming distance for Error

detection

t errors to be detected.

Hamming distance: t + 1

Words : 00 and 11

Hamming distance : 2

1 error can be detected.


4/51

Error correction Coding

Data word Codeword

0 000

1 111

Majority rule

1 bit error correction


5/51

Rate reduction in coding

accumulated digitsRate in bits per sec : E

Accumulation over time : T seconds

ET bits coded by FT bits

ET bits sent in t1 seconds

FT sent in t = ( F/E) t1 seconds

t/t1 = F/E


6/51

Error Probability

Fraction = 2 ET/ 2 FT

= 2 (F-E)T

Pe reduces as T is increased


7/51


8/51

Hamming Bound

k : Number digits in the

dataword

n : Number of digits in the

codeword

2 m >= 7 ncj ; j =0 to j = t

m = n k : Parity or checkdigits


9/51

Table for n,k pairst n k

1 3

4

7

1

1

4

2 10

15

4

8

3 10

15

23

2

5

12


10/51

Linear Block Codes

c = (c1,c2,c3 .cn)

d = (d1,d2,d3 .dk)

c1 = d1 ; c2 = d2 ; .. ck= dk

ck+1 = h11d1 + h12d2 + +h1kdk

c k+2 = h21d1 + h22d2 + h2kdk

.

cn = hm1d1 + hm2d2 + . + hmkdk


11/51

Generator Matrix

c = d G

1 0 .. 0 h11 h21 .. hm1

0 1 .. 0 h12 h22 ..hm2

G = . .

0 0 .. 1 h1k h2k ..hmk

Ik(k by k) P( k by m)


12/51

-

!

011100

110010

101001

)3,6(

G

ismatrixgeneratortheCodeaFor

Data word d Code word c

111 111000110 110110

101 101011

100 100101

011 011101

010 010011

001 001110

000 000000


13/51

? A

216;325;31433;22;11

011100

110010

101001

321

ddcddcddcdcdcdc

dddc

!!! !!!

-

!

The Example


14/51

Code word property

C = dG

= d [ Ik , P]

= [ d, dP ]

PH T =

Im

C HT = [dP+dP]

= 0

HT

: n by m


15/51

Error Vector

d : 100

r : received word 101101

c : Code word 100101

e : Error vector 001000

r = c + e

c = r + e


16/51

Syndrome Vector

S = r H T

= ( c + e) H T

= e H T

1 by n n by m

S : 1 by m


17/51

Decoding table

-

!

!

100

010

001

011

110

101

T

T

H

Hes


18/51

Decoding Table Continued

e s

000000

100000

010000

001000

000100

000010000001

100010


19/51

Decoding Table

e s

000000 000

100000 101

010000 011

001000 110

000100 100

000010 010000001 001

100010 111


20/51

Decoding

(1)Calculate syndrome by

S = r HT

(2) Obtain the error vector by

using the lookup table

(3) c = r + e


21/51

Algebraic Codes

k

kk

k

n

nn

n

dxdxdxd

dddd

cxcxcxc

cccc

!

!

!

!

2

2

1

1

21

2

2

1

1

,2,1

)(

),(

)(


22/51

Examples

246

3

)()0010101(

)()0101(

4,7

xxxxc

c

xxxd

d

kn

!

!

!

!

!!


23/51

Modulo2 addition

56

45

46

xx

xxx

xxx


24/51

Generation of Cyclic Codes

cg tH nce

xgxdxc

xdcalculatedeachFor

xoffactorkn

polynomialGeneratorxgFind

nth

)()()(

)()2(1)(

)()1(

!


25/51

An Example

1

.,1,1

3,4,7

3

33

!!!

xtryfirstWe

etcxxxfactorsorderThird

knkn


26/51

Division

1

1

(1)1

4

4

47

473

x

xx

x

xx

xxxx


27/51

Trial Division

1

1

23

7

xx

x


28/51

The Result

1

1

1

23

234

7

!

xx

xxx

x


29/51

Code generation

)0100111(

)1)(()(

)().()(

)(

)0101(

456

233

3

!

!

!

!

!

!

c

xxxx

xxxxxc

xgxdxc

xxxd

d


30/51

Systematic cyclic code

The first k digits of c

are same as that of k

)(

)(

)(

)()()(

xg

xdx

ofremainderisx

xxdxxc

kn

kn

!

V

V


31/51

Systematic code continued

)().()()(

)(

)(

)(

)(

)()(

)()(

)(

)(

xgxqxxdx

xq

xg

x

xg

xdxxg

xxq

xg

xdx

kn

kn

kn

!

!

!

V

V

V


32/51

Decoding

Every valid code

polynomial is a

multiple of g(x)

Ifthe received wordpolynomial r(x) is not

a multiple of g(x) an

error is indicated and

remainder polynomialis used as syndrome

)(

)(Re)(

xg

xrofmainderxs !


33/51

Error Polynomial

)(

)(.Re

)(

)()(.Re

)(

)(

.Re)(

)()()(

xg

xeofm

xg

xexcofm

xg

xr

ofmxs

xexcxr

!

!

!

!


34/51

Syndrome-ErrorTable

For various errors syndrome is calculated

and stored in the form oftable.

D

ecoding steps

Get s(x) as remainder in dividing r(x) by

g(x)

G

etth

e error fromth

eta

ble Get c = r + e


35/51

Cyclic Code Generation

1)( 12

2

1

1

xgxgxgxxg knknknkn

D D D D

g1g2gn-k-1

a

d b

c

s1

s2

n-k number of flip-flops

and modulo2 adders Output


36/51

An Example

fl sfliT us

xxxx

xxx

C e

!!

!

!

!

33;0

1

1.0.1)(

1)(

)4,7(

1

23

23

D1 D2 D3


37/51

Verification

)(

)()(

)1()(

1)(

)11(

xg

xdxdivisionthedowexfindTo

xx

xxxdx

xxd

d

kn

kn

!

!

!

!

V


38/51

Remainder Calculation

1

1

1()1

23

23

34

24

245

5

356

233623

x

xx

xxxxxx

xx

xxxx

xxx

xxxxxxx


39/51

Remainder polynomial

0111)(

!

!

V

V xx


40/51

D1 D2 D3

Clock D1 D2 D3

0 0 0

1 1 0 02 0 1 0

3 0 0 1

4 0 0 1

5 1 0 1

6 1 1 1

7 1 1 0

D= 1001


41/51

Interlaced Coding for Burst errors

x1 x2 x3 ------- x14 x15

y1 y2 y3 ------- y14 y15

z1 z2 z3 ------- z14 z15

15 , 8 code t= 2


42/51

Convolution Coder

N stage shift register

V modulo 2 adders


43/51

An Example

S1 S2 S3

v1 v2

Datain

11010

s1 s2 s3 v1 v2

0 0 0

1 0 0 1 1

1 1 0 0 1

0 1 1 0 1

1 0 1 0 00 1 0 1 0

0 0 1 1 1

0 0 0 0 0


44/51

Number of output digits

n = (k+N-1).v

= (5+3-1).2 = 14


45/51

Channel coding in GSM

50 132 78

50 + 3 132 + 4

Parity bits Trailing bits

Convolution Coder

189 bits

378bits

456 bits


46/51

0

1

00

11

00

11

10

01

a

b

c

d

00 a

11 b

10 c

01 d

11 a

00 b

01 c

10 d

Code Tree

a


47/51

S1 S2 S3

v1 v2

Data

in

a

b 01

c 10

d 11

a

b c

d

1111

00

10

10

0101

From to

a a & b

b c & d

c a & b

d c & d


48/51

a

b

c

d

00 00 00

11

01

11 11 11

10

10

0101

10

00

Received Digits : 01 00 01 00

Paths into node a 00 00 00 HD : 2 Survivor path

11 10 11 HD : 3

Paths into node b : 00 00 11 HD : 2 Survivor path

11 10 10 HD : 3

Trellis

Diagram


49/51

Error Probability in Un-coded

system

FunctionGaussianFor

xXobabilityxQ

worddatainbitsofNumberk

noiseofPSD

PowerSignalS

rateBitff

S

QkP

i

i

EU

)(Pr:)(

:

:

:

:

)2(.

"

!

!

L

LP

P


50/51

Error Probability in coded system

err rsec rrect blf Nu bert

w rc eibitsf Nu ber

QtP

t

EC

:

:

2

1

1

!

P


51/51

An Example

6

4

1096.1

101.1

1

12.9

11

15

v!

v!

!

!

!

!

EC

EU

P

P

t

k

n

P

channel coding new

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