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~I··.'.

D at a a nd V ol co C od in gJnformation Coding Tecl"1niqucs~----.

The sliltr:Olent of above theorem can ,,150 be written as,

A continllOUS time sign,,1 can be completely represented in its samples lind

recovered bilek if the sarnpling frequency j, ~ Z W _ Here j. is the s am p li ng f re qu en cy

and W is the m ~xim um frequency present in the signal.

Q.5 i\k ll linJ i I ll" merits o f D PCM .

Ans.: i) rlilndwidth refillirement of DPCM is less compared to PCM.

oi) Qua t iznt ion error i s redu ,: ,' d beC,lUSC of prediction filter.

iii) Number of bits useci 10 represent one sam ple v alu e are also reduced

con-pared to PCM.

Q. 6 W Il li ! i s l is e m il i n cf.:rJC'rCllCC ill D Pe l" l a nd D l v1 . ,

Ans.: DM encodes the input sample by only one bit. It sends the inform<lt ion about

I c; or '.' i: i , i.c, step rise or l td ). DPCM can have more than one bit for encoding the

_,,,mple, It sends the infonnnlion about difference between ncrun] sample value and

I'ft'dicted sample V<l[U(. . _ - "

Ans.; The message can be recovered f rom PAM by passing the PAM signal through

,'(:conslrtlction filter. The: reconstruction filter integmtes nmplitudes of PAM pulses.

;\El1plitud~ ~fI1o!)thing of the reconstructed signal is done to remove ampli tude

di~C"or1tintlit;t's due to pulses.

Q .S W r; lc 1 1 1 1 , 'x)'TcssioH for lJ(]1JIlwidlll o f b im l nj rCM wilh N m e ss ng es Mel! uiith a moximum

!reql lelil :Y oj 1 ", Hz.

Ans. If 'v' number of bits m-e used to code each input sample. then bnndwidth of

I'C~l is g iven :~S~

1 3 - r ~ ~ N v " j ; "

Here vL; i s the bandwidth required by one message.

( 1. 9 H o u: i~PDA1 WI!!I~ c on ve rt ed i nt o P PM s ys te m s?

AI'''.' The PDM s ign<ll is given (15 11 c lock s ign111to monostable mul tivibra !or. The

multivibr<ltor triggers on fulling edge. l-Ienee a PPM pulse of fixed width is produced

aft er I"Illng edge of rDM pulse. PDM represents the input signal amplitude in

form of width of the pulse, A PPM pulse is produced after this 'width;' of PDM pu ..

In other words, the position of the PPM pulse depends upon 'input signal amplitude.

Q 10 A1<-"l iOIl th« 11", o f n d l !) ! li v c quall/;ur i n n d npJ i v~ dig,Mi w~ ve !o r r u c od in g 5c/reml'S.

Ans.: Adaptive quantlzer changes its step size according to variance of the

signid, Hence quantization error is Significantly reduced due to adaptive quantiza,

ADPCM uses "d"ptive quantiz"tion. The bit rate of such' schemes is reduced due ..~

,.

&"1l&J£iiltll:t¥..4ii41!BA

Information Coding-Techniques 2 ·33 Dataand Voice Coding

Q ,11 W hnt do yo" u nd er st an d f ro m a d~ )! lh >( c(]{lillg ?

Ans • I d· ti di h ..h n II ap ive co:ng~ t e G.uantization step size an d predict ion fil ter coefficients

are c anged ,as pel' properties of input signa!. This reduces the quantization error and

nUdl1_ lberf b i ts :Jsed to represent the sample value. Adaptive coding is used for speech

co mg at low bit rates.

Q.IZ Whn t i s " 'r lm l b Y 'I "" "l im / io n ?

Ans.: While converting the sign~l value from analog: to digitilt qu;mtiz.<ltion is

perfo~:ne:1. The analog .value is as~igned to the nearest digital level. This is called

.q~~~,_~zat~on. The quanll~ed value IS then converted to equivalent binary value. The

ql anlization levels are fixed depending upon the number of bit Q tizatid . 1$, uan izahon isper orrne in every Analog 10 Digital Conversion,

Q.13 TI.( sign"i to ' I lI lmlizntion noise rntio ,'11 {I PCM sy51~m depends on ... % '"

Ans.; TIle sir.;nal to quantizarion noise ratio in PCM is given as,

( ~ 1 0 s (4.8 + 6 v) dB \

Here V is the number of bits used to represent samples in PCM H' . I, . ' svv. enee Slgnn to

quanttzation noise ratio in PCM depends upon number of bits or quantization levels

Q,14 For I l r r transtnission of normal speed: s ig l la i i n t 11~peM chmUlrl ..e~rlsthe B,W _ 0 / _ .

Ans.· Sp' '"' . I I 'eM: cdecIl Slgnl1 s tave the maximum frequency of 3.4 kHz. Normally 3 bits

. IS us e ( 0 1 ' speed. The transmission bandwidth of PC~-f I' , 'IV S given as,

~ ~ vW .

; : > : 8x3.4 kH z i .e, 27 .2 kHz

Q.15 /1 is rrf}lIin-rf 10 transmit sprt'clr O1~'rPCM dtannel with 8/]'/ -L_ 1_ I ' _ • ' - I tlCdrrrcy, Assume the speccl: in

""SCw'" lu",'ra II;3.5 kHz. Determine I"~ bit ral, ?

Ans.: The sign~ling rate in PCM is given as,

r := vi,

Here v number of bits i.e. '8The maximum sign~1 frequency is W = = 3.6 kHz. Hencewill be, minimum sampling frequency

I. 2 W

= 2x 3.6 kHz

8><7.2xl03

57.6 kbits/sec:

I.



Information Coeing Techniques 2 -.34

Q.l6 ~ 'I JI /I 11s meant by adaptive delta modulatlon ?

Ans.: In adaptive delta modulation, the step size is adjusted' as per theslope of

input signal. Step size is made 'high if.slope of tho- inpUt & i 1 5 , t r a l is hlgh:11'iI<'J

s lope overload dis tort ion.

Q,17 W;',1j is t li z n dW l ll ag e o [ delia mo du la ti on o ve r p ul se m o du (n U ol I s c h em e s 1

Ans.: Delta modulation encodes one bit per sample. Hence sIgnaling, rate is redllced

in DM.

Q .1 8 W im ( slwilld Ite 111(:minimum bairdtvidtll r eq uir ed 1 0 t ra ns mi t a P CM f i l l l l m 1 c J 1

Ans.: The minimum transmission bandwidth i r : peM is given as,

Br = = vW

__ . .- .. .. . .. .v !. _~.. .. .. . . .. .. ..__ . .__~_~_ __ " . .. .~ , . .• -~- " . •• ~-~ - - . . ~- I---....,.~~_, .~~~

Error Control Coding i- - ' ~ - - ' - - - " - - ' - ' - - " - - , _ . _ - . . . . . _ . . . - j

.. 1IJI_a------'-----'---~'--.-----.-'-.....-_._1

1 Introduction

Hen, v is number of bit s used to represent one pulse.

W is t he maximum s igna l f requency.

Q,19 ~l'i"'l is (/IC oduantogc Df d e lt « mo d ul a ti on o , 'c r peM ?

000

1": Errors at( '~ intro.d uccd in the data whr-u il p~·I~;:':;'(~:~hi~()U,~:hthe {'h,l1H",Pf. 'r ·tl{~ ch._. lJ)n l~j

·i~1{gj~cnlericrcs the siBnllL The signaJ pO\,~J{,."r j , , - ~ il1.:"0 fe,-jtl~'ed 11Lltjf~l l'n·or'"j .-'~'1_'.~,."'"fn,troduced. In this chapter we w m study various tl'PI~; (11 CI,[I'f ril'i(-,:;inn oJl".1

• . - '1 ; o r i - e c ti o n techniq ues.

)~3.1.1 Rationale for Coding and Types ami Codes

TI,e tr ansmission of th e data over the channel depends tlpO1 two parameters They

are t( ;JVls .mil ted poyver and channel band wid th . The power spt -c tr . t ut.·l1si I, ["01,.1,.11\1\(.'[

noise and these two parameters determine sisn01I to noise power ratio, The ~i;.)r,:d 10

.:;oiSe power ratio determine the prob;lu iiilY o f erro r o f th e mod ul.u ion ~il"h"'Jl'" I'or ft.",

h>lvcn signal to noise ratio, th ... errol' p roiJ.1 l>liity can be r ed uced fu rther r- .: Inl1;';

c & ' U ; , g t echniques. The coding techniques - ' I b n reduce s i g 1 1 < 1 1 to 11(1;:,e jJO\V(~!' ,dtio lu r

f ixed probabi li ty of e rror .

Fig. 3,1.1 shows the block diagi'<l111 01 t il l, d igl l,,: (l ,mIl lU:li (,,",) :l . ·,'.~kf" l I"/l 'Li1

m;cschannel ccding.i-> .--

Ails.: Delta modulation l lSOCS one bit to encode one sarnple:-Nence bit rate of delta

modulation is JOI': compared to reM. _-

Inpul

mussagt! --lo-

bits

No·$Y$tg ~aIfroolChannel

{l·ry. Z:L 1 Dig-i ta l corr rr uun ica ti ou sys tem w i th c ha nr rc I encuding

(:l _ 1)

I

I•



/

Information Coding Techniques3-2

Error control CodIng

h messlIge bits. Theextril bits (redundency) to t e

The channel encoder ndds. over the noisy channel. The chann'e,l d~coder

coded signal is then transmitted d t t and correct the errors In theen , , . d uses them to e ec 1 . reid ntifies the redundnrt t bits an . d ced due to channe noise a1 e h ber of errors intro U d I

'age bits if any- Thus t e num ,d d t bits the overfill ata ra emess, , I d, d r Due to the re un an, 11 t )115-

mininliz.ed by encoder nno. ceo e ' d t this increased data rate. ie sys e. cases l-i"nce channel has to ~ccommo a e 'mer' ~.• ". . I' re s

S'\\'gL.tlv conlJ)lex because of codtrll; tee lnlQI,. .

become ,",

3.1.2 Types of codes k" d convolutional codes.. I '1 ified as bloc .cooes an

T l 1 O " eocles t ire main y c ass b f bits in one block orists of 'n' nurn er 0

j) mOCK codes : These codes conSb. .• 'a re bits and (n - k) redundant_ d 1 This codeword consists of k mess gco ewon.. k) bl k codesbits. Such block codes are called (n, oc '., - - 1 ' convolution of

din opera tion is, disHcte irne ."J COJ1volutional codes; The co g der The convolutiOnal

11 I' Ise response of the enco e . - od dinput sequence with t 1£ nnpu . . I and generates the enc eencoder accepts the message bits contmuOus y

sequence continuously., _, ssifi zd as linenr or nonlinear codes.

The cooes can ilI50 be cla,d ie , dd d by modulo-2

i) L' r code' If the two code words of the linear code are a e1 lJ1ca· . d I th code.

ari thmetic, then it produces thIrd codewor m e., th c8dewords can b ef the codes smce 0 er

This is very important property 0 '

obtl.ined by addit ion of exishng codewords. t necessarilyof thenonlineM codewords does no

ji) Nonlinear code : Addition

produce third codeword.

Informatfon CodIng TechnIques 3 - 3 Error Control Coding- - - - - - - - - - - - - - ~ ~ - - - - - - - - - - - - - - - - - - - - - - ~

3.1.4 Examples of Error Control Coding

Let us conslde~ihe error control coding scheme which transmits 000 to trnnsmit

symbol '0' and 111 to transmit symbol T. Here note that there are two redundant bits

in every message (symbol) being transmitted. The decoder checks the received triplets

and takes the decision in [avour of majority of the bits. For exarnp le if the t rip le t is

110, then there <Ire two L's. Hence decision is taken in favour of 1. Here note If);lt

there is centainly error introd uced in the last bit. Similarly if the received triplet is 00 1

or 100 or 010, then the decision is taken in favour of symbol '0'. The message symbol

is received correct ly if no more' than one bit in carh'triplet is in error. If the message

would have been transmitted without coding, then it is difficult to recover tn.e origin~!

transmitted symbols. Thus the redundancy in the transmitted me$sa~ reduces

probability 01 error at the receiver. Error control coding has following impor tant

aspects: \,

i. The redundancy bits in the mrssage are called check bits. Errors can be

detected and corrected with the help of these bits.

ii. It is not possible to detect and correct all the error in the message. Errors up tocertain limit can only be detected and corrected.

iii. The check bits reduce the data rate through the channel.

3.1.5 Methods of Controlling Errors

There a re two main methods used for error control coding: Forward acting error

correct ion and Error detection with transmission.

I)Forward acting error correction

In this method, t lie-errors are detected and corrected by proper coding techniques

at the receiver (decoder). The check bits or redundant bits are used by the receiver to

_ d e t e s t .and correct errors. The error detection and correction capability of the receiver

upon number of redundant bits in the transmitted message. The forward

,acting error correction is faster, blot over all probability o E errors is higher. This is~ause some of the errors cannot be corrected.



messflge bits and redundant bits,Block length; The number of bit s '11 ' after coding is. called the block length of. til·,'; : : ~ + . j - .

" .- - ~ L _

code. ,;~, ~!I\~

Code rate: The ratio of message bits (k) and the enc.oder output bits (n ) Is callcg;~ ;.

cod e rate. Cod c ra te is defined by 'r' i.e., . _j'k : {I!} - i , . .

i r •.. (3.1.1) if'. " .J\ ,'" ~~ F" q 1 2 Cd'

.~,'e find th;J,t 0 <" r <" 1. _ , ' , ~ - ~ 4 TO: " " " . 10·.... . •_.0 e vuctors r"' lll"senun 9 3·bi t code words

:5 Channel data rate: It is the bit rate at the output of encoder. If the: bit rate at ~/,~: Hanuniug dlsL,lHC.C : I he h,1[l:ul1mg distance between the two q,d(' vcd',), '" j, " '); " i

i~put of encoder is R" the~ channel data rate wi l i be , /o,'~,_'f~ the number of elemen~s in : v hich they differ. For e .xilm~l( ,~,let X "'(It)'!) ,~nd Y~ (J](Ii.~~ , .j,The two code vectors differ in second and third bits, Therefore hamminj; distanc«

Channel data ra te (Ro) '"I, . __ , ", .:" ( 3 . 1 1 ~ : ~ . ' . I ' '! -bc tween X and Y is 'IW(". Hamming d,istance is denoted as d (X, Y J or ~inll - - ':_) ' .s. i.e.. ,,,t~ j. '(X 1'\ d 2

Code ve ctors ; An ']1 bi t code word can be visualized in an n-dirncnsional sp;~&;' _" i1 , ,'" "

as a vector whose elements or co--{)rdinates are the bits hI -the'cUde word. It is £in'ij,ler.;: i Thus we observe iron\ Fi)~. :3.1.2 eMI !,,~ bl.,,"w;ng di,i\;l;1U' hch,·" .' " ' .ii,'

to visualiz e the 3-bit c ode words.' Fig. 3.1.Z·shows the 3-bit code ve ctors. There wi l l ~i~' '1< JOll) is ntaxlmum Le . .3. T1~~\ss indicated by the vector dia gram also.

distinct '8' . code words '(sinc~ nurn.be'r ot" c ode ,;ords "', 2k): If we, ~l' bi~~o on ,~~~~ i' ! ( : 1 1 Minimum distance (d mill 1 : ' It is the smallcs t harnm mg dis rane e iJe lw e l ' n the valid

u ) on y-axIS a!,d.~2 _o~ z-aXIS, then the following table sw = vano;us pcmtsas c~~ > r : _ ; C o d e vectors. '

vectors -in the 3-dirnensional space. ~ .r - , " ' 1 ~ : , " . . " " , . - , -, ,.{I'; ;,' ., __Error detection 1S possible If (h.- ["('ren'cd vector 15 not eqllai In some orhnr code.~ -),( -,;·:iXector. This shows tha t the t ransmission errors in the reccivvci cude \,,,,;1,,1' should be'

~iJ~ '~~~~J

3.1.6 Types of Errors

r,'

(

There ara rnainly tWo types of errors'introduced. dUri'hg U:altsm1:ls:1oll Oil

random errors and bursterrors, ' , .. ,.,. '.-;-,;' ' 4t

i) Random e ~ : o r S< ' : · . ' ~ e ~ e ' . - ~ 1 0 ~ . ~~ ~ ~~:~~~fi~u e . : , .o -~hil~ ' , i / u s . s l a . n ' ·riolse : q ~ . : ~ ~channel. T I le e rr or s generated due 10 whIte -gausSIan noise in the parlii:iff'JI

interval docsr io] a ffect 'ih~ 'p~rfo~n;~;lc~:'bihe- system in-;subscque~_i i I1f~h;1~In other words, these errorsare t?taiiy uncorrelatcd. Henc~ they arc a![ ;o--!~~l!ii,'; randomerrors. '. ", , -, , , '1;

. ·~i~

ii) Burst errors : These errors are generated, due to impulsive noise in,i~

channel. These impulse noise _(bursts) are generated, due to lightning ;ui:

swi tching t rans ients. These noise burs ts a ffect seve ra l succe ss ive sY! flbo ls. s _ ~ T h 'errors are called burst errors. TIle burst errors are dependent on each other,tn,:

successive message intervals. -;. : - - , r : : ,

3,1:1 Some of the Important Terms Used in Er_r9rGdritfcil Cocling

The terms which are regularly used in error 'control coding are defined next.

Code word : TI1e encoded block of 'n' bits is called a code word . It contains

!

7

1- .---- __

Table 3.1.1 Couo vector-s in 3·dimen~ion,,1 s uace

8



... ,'-~---~---,---~---~Error Control Codingnformation Coding Tcchnique:s~~ ..:3~.:.:::6 ~ _

1'11('following table lists some of the requirementsess than min irnurn dista nee d mn'

of error control CJP~bilj ty of the code.

d Dlstance r equirementa me of e r ro r s do te de d. _ I_ c _o ! _T_e_c t_ e_ + ~_

d min:?: S + 1

$r.No.

r - - - - - - - - - - - - - - - - - - . - - - -1 Delee! upto 's' errors per word

Correct upto '/' errors~p~er:_w=or~d ~L-:d~m~inC'...::2::--2-/_+_1' 1

dmln 2:1+s+1

2

3 Cor rect uptc <r ~ errors :.: ;~ dclec: s > r errors per

word

Table 3.1.2 Error control capabilities

For the ( /I,k) block code, the minimum distance is given as,

drn;n"; II-k+

. , . . f age bits in a block to theode efficiency: The code efficiency IS the ratio 0 mess.~

transmitted bits for that block by the encoder i.e.,_._ --.--

message bi ls in' 11 block

Codee fficiency "' transmitted bits for Iii'/! block

there are 'k' message b its and '/1' transmittede know that for an (II, k) block code,

bits. Therefore code eff ic iency becomes,

Code efficiencyk

. .. ( 3 .1 . 4)n

If w,' cornprirc the "hove expression with the (ode rate (r) of equation (3.1.1) we

fi nd that,

kCode efficiency '" code rate =0 Ii ... (3.l5)

. 1 t in the transmitted codeWeight of the code: The number of non-zero e ernen s d t For). '. d b (X) here X is the coc e vee or. -ve ctor is called vector welgh!. It ~s denote.y w. w. . =5.

,~x;]mpi!' i( X '" U I 1 1 0 1 0 1, then welg h t 0 f this ~~dc vector will be_T_v_(_X_) -

Review Questions

1 . B rie fl y d is cu ss t il e da s si j tt l l i i on o f c o de s .

2 . E xp la in th e f ol lo wi ng te rm s.

I) iI Il / /! mw !{ d i st a nc e ii) COOt rate ill) Free distnnce iu) WdxlJ l of code. _;

" ,r Ii'alioll $ysl"". Willi , IS error control codl>lg ? Winch are 11i~[II/Ic/IOHli/ bloch 0 a (0111111111 l

tti«: "Cc'OI'''I'!r':;/r this ? 1'llli(1)lo Ih , ' fmc/ion of eact: block.

. .. ( 3. 1. 3)

Inrorinallon COd/rig Techniques3-7

.~ITor Coritro! COlling

3.2 linear Block Codes

'j, Principle of.·block coding:

For the block 'of k mess,lge

bits, (n - k) parity bits or checkbits are added. Hence ;the to].:1i

bits at the output of channel.

encoder are "n', Such codes are

called (n, k) block' codes.

• Fig:3.2.1 illustrates this concept.

__Code block

output

Fig . 3.2.1 Functional block diagram of blockcoder Systematic codes : In Ille

sys tema tic b lock: ,f ' code, themessage bits appear at the beginning of the code word, Thus as shown,'iil: Fig. 3.2.1 ,

the mess~ge bits appear first and then check bit~ are tninsmitte~ in a block. This type

of code is called systematic code. In 1l0nsystemalic code it is not possible to identify

message bils and check bits. They are mixed in the b lock.

In this section we wHI consider binary codes. That is all transmitted digits are

binary.

Lin ea r code: A code is 'linear' if the 'sum of ilny two code vectors prod uces

another code vector. This shows that any code vector can be expressed as a linear

combination of other code ve ctors. Con.sider that the particular code vector consists of

ml , 1112, m,), ... , Ink message bits and c!, CZ , CJ, . •. . r cq check bits. Then this code vectorcan be written <IS,

Here

x = (ml,m~, ... ml: CI,C2, .• , C1)

q' = .n........ . , . . . . . _ . . . . .

. .. (32.1)

.. ~(3.2.2)

i.e. q are the number of redundant bits added by the encoder. 11)e above codeYe-:tor can also be written as,

X (MIC).~ (32.3)

M k - bit message vector and

C '" q - bit check vector

The check bi ts play the role of error detection and correction,. The iob of the linear

bloc k code is to ge nerate those 'check bits'. The code vector can be rep~esented <IS,

X = Me .~ . (3 .2 . .1 )Here

x '" Code vector of 1x n size or n bits

.;



o J - "

and

M "', Message vector of 1x k size or 1) bi ts

G '" Generator matrix, of k x n size.

,I , 'J

- .I~., ,

To Qbtaln P' sub matrix:j

Prom equat ion (1.7.. 6) we know I)'JI,

G = '[h :I\.ql

Comparing' this cquotion with th e given matri« , we fi!'ld lha!,

P O O lI. = [3.3 l~ ~ 'l

P , . . , ~ P ,. , e [ ; : : ]

Thus equation (3 .2.4) above repr esents matrix form l.c.

I [X1I•I "" [Mh~k [Glkxll IThe generator matrix depends upon the lineal'"block code used. Ge!letally, it

represented as,

: T he g cn em to rc ma tr ix f or II (6, 3) b l oc k e o d~ is glum l !t 'l ow . F i nd a il

this code.

G ~ [~

G

t,

U,I P k x q 1 k K "

k x k identity matrix and

P k x q subrnatrix

The check vector can be obtained as,

' - - - - - -_ c _ " " _ M _ p _ ~ 1

I _h ~oh·jng the above matrix equation , check vector can be obtained. Le,

C) IllI PI! ill 1112 1'21$713 1'31 ill ....ff i /1Ik PH )

(2 JIll Pn E E l 7111[>21 eJ nl3 ?31 E D • • • • E E l /Uk P H '

(3 := n il 1 '1 3 ff i /1 1 ] . 1'13 ij) 7113 P 3 3 ( 1 1 . . . . E E l Jrll Pu

.. .. and soon,

H,'l'Cnote that ail the additions are mod-2 additions.

im>j>- Example 3.2 .1

c od e v ec to rs o f

o 0 : ° 1 1 ]10:101

01:110

SolutIo,,: lhe code vectors can be obtained through Iollowing steps:

i) Determine the P subrnatrix from generator matrix.

ii) Obtain equation s fo r check bits using C",MP.

iii) [)ct~i'millc check bits for every message vector.

'iI) To obtain the equations for check bits:

Here k '" 3, q = : ' 1 and 7! -= - 6.

. .. ' ( 3 .. 2 .9 )



3·10Error Control ~odrng

From- the above matr ix mul tipl ic at ion we obta in,

C1 (O x m d @ (n tz) fD (m3)

C, (mil EEl (0 x mz ) @ (m3)

and CJ (mJl ED nlz) ® (0 x m3 )

From the above three equations we obtain,C 1 7112 E D m3 )

C2 1/71 illnl3

C3 1/11 ill ITJ2

1 1 10 ' above three equ<ltions give check bits for each block of ml, In] ,m~ message

. .. ' ( 3 . 2 .1 0 )

bits-

iii} To determine check bits and codcvectors for every message vector:

Consider the fi rs t block of (n i l , In" In~) '" 000 w e have,

C ! OED 0",,0

o m O~O

Dfl)O"Q

For second block of (iii), m2, nI}) '" 001 w e: have,

C1 '" OED]o.,}

C1 om1=1

The following Table 3.2.1 lists all the message bits, their check bi ts and code

vector s calcu la ted a s above.

Sr. BTl, of",.mgo I Check bits Complete code vector

No. vector In oneblock

-___. ., . . . . . .

m, r n 2 1 ' " 1 1 1 C, '" C, ~ C,~ m, m, ml Ct C2 C,

m ll1 J m , ml ill rn) ml(JJm2

1 () 0 0 () o () ·0 0 0 0 0 o

-2 o o 1 1 1 () 0' 0 1 1 1 o

3 o 1 0 1 0 1 0 1 o 1 0 1

4 o 1 1 0 1 1 0 1 1 I 0 1 1

-5 1 0 0 0 1 1 1 0 0 0 1 1

6 1 0 1 1 0 1 1 0 1 1 0 1

---~-~ , + - - - - . - - - - - - -

7 1 1 0 1 1 0 j 1 0 1 1 0

1----- J--\

6 1 1 1 I o 0 0 1 1 1 0 0 0..~.. -

T<lblo 3.2.1 Code vectors of (a, 3) block code of example 3.2.1

lnformatlon Coding Techniques 3 . 11 E rr or C o nt ro l Codin!

Parity check matrix (H)

For every block code there is .1 q x 11 p~rity check rna trix (H). It is defined J5

..l-i = [ f ' T : I, JI IJ)o:II --. (:3.211

Here pT is th e tran spo se ' r I' bo su -rnatr ix. The J' subrnatri«

equation (3.2.8) as, u, is ddined

. .. (3.2.12)

With the above e ql j' .. ra ron, w e call wnte equation (3.2.11) as

IPI1 [',I P.l] 1\: 1 0 0 0-'

I PI!. (in PJ2 P;, 0 0 0

1 l ' 1 J pc) py! PI:) 0 o 0

j, '

l~l~1 1 P.l1"" r; : n 0 0

.:

(J2.i4j

If w e compare eq t' (3 2 6) . .• . - . - U~ 10'1 . _ ...• 1t1d equation (3_2,11\. \ 1 1 / . . Iii ! 1 - " • •matrix (G) isglvnp then p~r'h' I. k _ ,~ .. nn .... " If generator

, -', -, J,) (1ec. rnatrr» (H) (,H1 be obta ined and '(I .....(1 V ~ ce- \ . ! 'F2 rsa.



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ ~ . ~J .2.1 Hamming Codes

Hamrniru; codes a re (II.!.:) linear block codes.

Those codes satisfy the following conditions,

II) ,\lIl11tl<:f o r check bits q 2 3

(2) Ellod length II '" 2'! - 1

(3) :"uJI1bcrof Il\CS.I<lgC bil, k '" II - q

(-lJ ~11ni;nUt11 JISlIHKcdnlln =3 1

.., (3.2.1')

\\'(? know that the code rate is given as,

kr '"

/1

II-q

II

for hamming code k == I!-q

". (3.2.1t)

, IIr : : .= I~-

2'1 -1

, " (3.2.17)

ho:]] [he above equation IV e observe that r ",1 if q»1 .

3.2.2 Error Detection and Corroction Capabili ties of Hamming Codes

Since the minimum dist ance {d"'inl ot Hamming code is 3, it can be used to detect

double errors or correct single'! errors. This om also be obtained from the generalized

T;,ul,_; 3.22

f.n:- dctc(til~S double (2) errors : :; . r Im iL l 2 :2+1 i.c. dmin 2:3

and fo r correcting upto one (1 ) errors ::;> doni" < ! 2 ( 1 ) + 1 i.e. d mi n < ! 3

"">, 1 ; ) ) ' ' ' ' ' Ex~mp!e 3.2.2: Tile parit!! c hec k: m atrix of a particulM (7, 4) l in ea r b lo ck c oJ e is

si1. 'ell I , ! !

[ :

1 0 1 0

~lH] 1 0 0

. .. ' ( 3. 2 .1 8 )

0 0 o IJ

" :,::i. ltd·~t)~/-~:.:-. aior innirix (C J"

Ii) List al i til e code ve ctors

iii) what is tire mininunn distance be/ween code vectors?

iv) H ow mlllly errors C(lil b e d c /e c tc d ? Ho w 1lI(IJ1Y e rr or s c n n b e c or re c te d?

3- P

.', Number of check bits arc n - k '" 7 ~ 4 L('. q "" 3

Thus 1/""" 2 q - ,J '" 23 -1.>; ,7

This shows that the given code is h.1tiliHing cocle.

-cc. . ; To lI;)lcrrnlno the P submatri :. - ;., The parity check matrix is uf j1x 11 size and is given ;by equation (3.2.i~;). It can be

, wr itt en as, (with q '" 3 and n == 7 and k " '1)

o 0 '10 1

]jI ) . , . J . VI! ()

P» 1'. 3 ... U f)

) row. ~quaticn 3.2.1 i

O n comp.1ring pMily ch ~t'k m atr iccs ( )( c'jlWl-ir.>:1 (3 . 2 , 1& ) ;'lId "'i\~"tl"ll (3) II') WI' )~"i,

['l'" T'!I r -. i'li -l "1 i Ol

p'i' '" 1 1'1 1 P ,-, I< ,;~ i - ' . : J . . 1 1 1 (l 1 J I! P 1 3 Pl'l /':\3 Pn J l_ l 0 1 J

Therefore the P sub!l1J Irix (ill' l;,' obtained .a3 J

[ ' 1 ) 1 t I'I~ / " , \"

1" 1JI I

:01' " I ' : , _ - I , ; o I

I' j ; ,~ : II \.'! ,j. I . , ", I "

P}2 P1J ' JP ,

1 ' I

( il~! /'" /',n ; t o .~i1 x].s

i ) To ob ta in the generator mat rix (G) !

From cquMiol\ q.: :.6) the g(~nCr' llol ' m.urix G is g;iv( 'n i\~;,

G ". [ l J . . : l'hQL."

witl~ k , '' :'~Lq '" ~3 aud u;, .: 7 the above eq~t Jt lon becomes,

G '" ft4 : ['.;.:,111 .-. - "';

PI1Hing t h~ i de nt it y 'rr.. ni ....I,. I ~ : -" -t (d-ll.i lj.ll): subm.io ,;~ / 1. 1. . . : ~!;,Il.l~). ..

<IS <J u t ;, iM l l i n I . '< j ua t{on (J.:I.I~;)w e l~l'(·



Information Coding Techniques3·14

Error Control Coding

G

p 000

1 0 1 0 0

! a 0 1 0

L a 0 0 1

. .. ( 3 .2 . 19 )

1 1 1

1 1 0

1 0 1

o I 1 ~ ',7

This i s the requ ired genernto r rnnlrix.

ii) To find all the code words:

To obtain equations for check bits

The check bits can be obtained using equation (3. 2.7), i.e. ,

C ;; MP

. ", _ c nation (3.2.8) i.e. (with II '" 3, k '" .1)J 11 t he more l ;unt .?r i: lI f orm \ \ C C ;1 n LIsee 1

lCI Cc C ) 1 1d ["1m, I.'I} III~\d l P l .l,:1

Solving the above equntion with rnod-2 addition we get,

[I ( lxmd E ll (lxml) ffi (lxm: d E ll (OX/I!4)

(2 "" ( lX II/I) H) ( Ix T I l , ) ffi (O xm3) (!) (lXIII4)

C, '" (1 X m l) (j) (Ox nJ2) ill (1)( m:d ill (1)( nJ4)and

' I'hus the above equ<l tion~; are ,C

1~ 11 1 1 ci) III) (tj Ill]

C 1 1111 (j) 111 ). E D rrt4

and C3 till (£l Ill] ill 1Il4 1

.., (3.2.20)

' .. . "

10 ctetermine the code vectors

1 ( ) - 1 0 1 1 we get ,Consider for examp e In\ m'). 1T13 1114 -

C1 1 (£) 0 ED 1 = 0

(2 '" 1 ED 0 EE l 1 '" 0

and C3 = 1 (f) 1 (] 1 '" 1 . _ 0 1.f (1 0 1 1) the check bits are (C1 C2 C:d - 0

'Thus for message vector o· - d) b written as-n'crdore the :::ystelllilti<.: block code of the code vector (code wor can e r

(m l nlz nl3 m4 C1 C2 C3) (1 0 1 1 : 0 0 1)

I~

Information Coding Techniques 3·15 Error Control Coding

Using the same procedure as given above, we can obtain the other code words or

code vectors. Table 3.2.2 lists all the code vectors (code words). Table also lists the

weight ofeachcode word.-

Mes$age vector Check bits (el Codo vector or code word

IWeight of

Sr. by eq, 3.2.20

M X codo

No.~ ".

m, r o t mJ m. C, Cz c) m, m2 ,m, m. C. C, Cl vec to r w(X)

-.1 0 0 0 0 0 0 o. a 0 0 0 Q a 0 0

2 0 0 0 1 0 1 1 0 0 Q Q -1.'

1 1 ~. :l

:I 0 0 1 0 1 0 1 0 0 1 0 1 0 1 3I--

"0 0 I I I 1 0 o· 0 1 1 1 \1 0 <1

r-' ,5 0 1 0 0 1 1 Q 0 1 0 0 1 t 0 :l

6 0 1 0 ,1 0 1 0 1 a 1 1 a 1 4

7 ·0 1 1 0 0 1 1 0 1 1 0 0 1 1 4

8 0 1 1 1 0 0 0 0 1 1 1 0 0 0 3

9 1 Q 0 0 1 ; 1 ,0 0 0 1 1 1 4

10 1 0 0 1 1 0 0 1 0 0 1 1 0 0 3

11 1 0 1 0 0 1 0 1 0 1 0 '0 1 0 3--

12 1 0 1 1 0 0 1 1

°'1 1 0 0 1 4

13 1 1 0 0 _0 0 1 1 1 0 0 0 0 1 3

14 1 1 0 ' ' ' - 1 0 1 0 1 1 0 1 0 1 0 4l- -------

15 j I 1 0 1 0 0 1 1 1 0 1 0 0 4

16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ./,

Table 3.2.2 Code vectors of Ex. 3.2.2

Ill) Minimum distance between eodevectors:

The Table 3.2.2 lists 2 , 1 , = 24 ",16 code vectors along with their weights. T1"l"

sma ll es t weight o f any non-zero code vector is 3. We know that the minimum distance

is dmin =3. Therefore we can write:

The minimum dis/ana of a linear bloc k code is equal to flle'minimum weight of any llOTI

Z e ro c ode v e ct o r i.t.

dmJn '" [W(X)JlMiR; X 'oF (D 0 ... 0) ... (3 .2 .2 1 )

Iv) ErrOf'detection and correction capabilities

.. . Since II.....",3,

dmn ~ s+l



Information Coding Techniques 3·16EI'<-orC<mtrol COdit.\,)

or

3 z s + 1

5 :;; 2

Thus two errors will be detected.

and dmin ?: 2t + 1

3 ?: 21 + 1

or

Thus one error will be corrected.

The hamming code (d min ,,3) always two errors can be detected and single error

can be corrected by its property.

3.2.3 Encoder of (7, 4} Hamming Code

Fig. 3,2,2 shows the encoder of (7, 4) Hamming code. This encoder is implemented

for generator matrix of the example 3,2.2. The lower register contains check bits

C1, C 2 and C3 , These bits are obtained from the message b'ris by mod-Z .;lddilioJ1s,

Thes e additions are performed according to eqLl, 'tion (3.2.20). The mod-? addition

opera tion is nothing but exclusive-OJ< Ope r<1lion.

Inpubt_

seqlJ.80Ce

s

Code words

output

Check blts Icgjslef

Fig. 3.2.2 Encode for (7. 4) hamming code or (7, 4) linear block code

The switch 'S' is connected to mcssilgr: rcgist<;,' ,.met a n rpcs sage bits are

transmitted. The switch is then connected to the check bit register and dwck bits arc

transmuted. This forms a block of '7' bits. The input bits are then taken for next block.

3.2.4 Syndrome Decoding

In Ihis section we will see the method to corn'''' "rrors· in linear block coding, Let'

the transmitted code vector be 'X' and corresponding received code vector be

represented by 'Y'. Then we can write,

1{,fottt!Jllion Cooing Techniques 1·17 Error Control Codin(J.--P_...----."~-.,........------~~.~- ~,~----

X"'" Y if Iere <Ire no transmission errors

TI\\' ,k.:,xiN ,:kiecL, or corN~ctS !hf)~(,'errors in Y by "5i~,g the stored L>ltr<I t1C: I~: in

the d< :'c ode T a bo ut th e code, F or l.uger h l.')c !:. le ng th s, m ore ~ nd m ol', ' 1 > \ 1 ' ; Me' rt-',pJir, '1

10 bl! ~t'l!l'd in I .he . .<:cod(·r. '11', i:; ino'f'a,<ls the memory n~1ilir·"IlL.'j)1 ;,11'] ••t. t-, '" I ' " ,

cOflll'I"s~ty .\I\J cost of the system. To avoid Il.C!W f,,.,.bICLI1:., ""n.J.-onw <,,"collinl; h

used ill linear block codes. This method is illustrated in the s\lb"~-,.("1l',1grapib,

Wc know that with every (n, k) l inear block code, then: C'xl,;l~ CI p"lily cheCK

matr;); ( 1 1 ) of size q x II. From equat ion (3.1.11 ) i t is de fined as ,

H "" [ P T : [qll'"

Tlw II':>n';1'05eof the above matrix c.m be o!,U ,ini;.',l hy ipf<:r,iungill).~ tilt' rows <111 ( ;

lh~ (ohuun.:;., i.e.

Ii.er(' PI : , thl' ~a.lbl\li\tm: of size b< ',' O1 i )( l / '1 i s : 11 (' identity nldll';:>: 0: ~;if,V 'I' ' i ' :" ', '

h~ve ddined I' $UbllUltrix in equation (3.2.12) CJriWf

linpcrt:.nt properly' used in syndromo dec c) ding

The I rM.spose o f par ity check ma tr ix (lJ l ) has very Imp0rl,\IIt pillpCl'ly as fO ! !OW5 .

x 1fT" "" (0 0 (J ." I))(J22:'1)

r" .....- . .. .. ...---.-.------.~~-- -----------i

or I ill!"" l } /' L . , " {U 0 It ,,")'1 !L ." .:L. .._ .._., ~.- •. _- - _ ... ,

This is true fur ill! code VCCWf~

E xp la na tio n w ith eXiI.rnple

For p~<1mF"l" ro nr, id er /1 \( , i'Mt:V "l"',.:k i t· ,. dri :, MH. ! , -u , i (> \'<,('101'5 .,H,nl1

\,.l ill

cxampi( ' 3 .22. The pari ty clwck m.Hr;" ie, !.~i\" '" i J y ' '' ii I: ' tiou (:L2.li·;). Th,~ (i :", ..,pm;e fit

this matrix em, be , 'c (ldi ly obt<l il1c ,i . '" fo]tow" ..

Pi1

1

1 ., o iJ. , I

: 0 II~o1 0 ()

1) 1 0

L O G J J , , ~

J



Information Coding Techniques 3 -18 Error Control Coding

, ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Table 3 ,2.2 lists all the code vectors for this parity check matrix. Consider th e th ird

code vec tor in this table.

x = (0 0 1· 0 1 0 1)

Now let's apply the p roperty of equation (3 .2.23 ),

r ~I

1 10,

[00]0101],., 1 10 1

X I - - { T 1 1

0 0

1 0

a 1.7xJ

Solving the above two matrices with the rules of mod-2 addition (Exclusive-OR

operation) we get,_ .-

x [(r ... 0 mom 1 (J) 0 <D 1 (1) 0 < 1 1 0 0 E 9 0 (D 0 Q :I 0 (j) 0 (l) 0 E D0 0 (f) 0 '6) 1 ffi 0 ffi 0 E ll 0 E ll 0 (!l 1)

'""'(0 0 0)

This rroves the property. It can be proved for other code vecto rs also. Thus . f C . ; ? : 'belongs to the valid code vector at the transmitter, At the receiver, the received code, ..

vecto r is Y Then we can write,

'(i-iT = (0 0 _,_ ,0), if X", Y i , c _ no errors or Y is valid code vector

YI_(r" Non-zero, if X;< Y i.e, some errors ..

Definition of syndrome (5)

When some errors are present in received vector Y, then it will not be from-

code vecto rs and it will not satisfy the property of equation (3.2.23), This shows

whenever YH T is non-zero, some errors are present in Y. The non-zero output ofproduct YH"i' is called syndrome and it is used to detect the errors inY. <::,"~,h-n""e-

represented by '5' arid con be wri tt en as,

5 :: YH T

or

Detecting error with tho holp of syndrome and error vector (E)

The non-zero e lement s o E 'S' represent ('nor in Iheoh tput, 'VVhen all

1

»«.,·1- &sMa(I_:p;~

, InfonnatJon Coding Techniques- - - ~ ~ ~ = = ~ ~ ~ ~ ~ ~_ ~3~-~1~9 ~~. Error Cont ro l Codingi) No error in the outpur and Y ., X

I i) Y is' , some ' o tl1er_, :: ,a Iid code vec tor other than

errors are !lndelectable. X. This means the trilnsm ission

Lets consider on n-bir error vector E L hiI nInsmission errors in Y Fa I : et I IS vecto r repreSent the ',)05ilion or

. r examp e consIder,

and.

X (1 0 0) ben II'JnSmillet.i vector

r ty (1 0 0 1) bea rectived vector

i tThen

E ,; (Q, 0 1 a 1)reprcsenrs' lhe e rror vec tor

The non-zero entr ie s represent e rrors in Y..

Using the mod-2 addition rules we can write ,

Y '" X E fJE

=(1$00$0 . . , ( J, 2 _2 8)1 if) Iif) 0 0 (j) 1)

Bit by bit mod-2 ildditiGf1'" (l 0 0 ] 1)

or we can Write,

x Y @ E

'" (1 (£) O DE D 0 0 ttl 1 1 (j) 0 1 @ 1)

_" "_ .(L 0 0)

Ip betwee~-~yndrome vector (S) dFrom e . an error vector (E)

quatlOn (3.2.26) We know that syndrome veeta' ._- r IS given as

S '" YHT "

Put ling the value of Y - X ffi E- . From equation (3.2.28) above

. S = (X mE) /{T

- -. ( 3 ,2 . 29 )

= XHT E E l Eftr

From the properly of equation (3.223)w i ll b e, 'We know tha IXHT -- 0 th b. -, en a ove

S = EH T

relation slulU'S tlmt -"''"d'd' ". (3.2.30)"" ti. ' -;1" rome ependsupon IIlI! error patt {r··r icular m(5snge. Synd.rome vocto '5 _ f' ern Ony. I t does / /0 / dcpcru!

re" r IS0 Size 1 x fJ Thus bi (p re se .r :J 1 2 9 syndrome vectors. Eoch s . q Its <) syndromc'

error pallem. -- yndrOlll1! vector corresponds to ;)



Information Coding Techniques 3 - .20 Error Control C f t t 1 1 t 1 . ! J

»* Example 3.2.3 ;

Y " · ,. . . _ : . [ollotus

The parity chec k matrix of il (7, '0. H'twwii l1g code is given

[

1110:100JH= 011 1: 0 10

1 1 0 1 : 0 0 1 3.7

'r;!".,_'", _ 1M:syndrome vector lor singic bit errors .

Solution: This is a (7, 4) linear block code.

This is /I 0= 7 and k '" 4

q ~ l1-k=3

i) To determine error pattern for single bit errors

"lie know that syndrome vector is a q bit vector. For this example syndrome will

be:l 3 bit vector. Therefore there will be 23 -1=7 non-zero syndromes. This shows

th,1t '7' s ingl e b it er ror patt erns will be rep resen ted by_i:]1ese-7 non-zero syndrO!TlC5.

Error vector E is a 11 bit vector representing error pattern. For this example E is '7' bit

vector Following Table 3.2.3 shows the single error patterns in a 7 bit error vector

(No te t ha t only single bit error patterns are shown).

Sr, No. Bit ln error Blts of Error vector{~),Npn-zoro bits shows error

1 1,t 1 0 0 0 0 0 0

2 2nd 0 1 0 0 0 0 0

-I

3 3r.1 0 0 1 0 0 0 0

4 \ I 4" 0 0 0 1 0 0 0

5 5th D 0 0 0 '1 0 0

6 61n 0 0 0 0 0 1 0--7 7'" 0 0 0 0 0 0 1

Table 3.2.3 Single error pat tern of ibit error vector

Ii) Calculation of syndromesfrom equation (3.2.30) the syndrome vector is given' .8;",""

5 = E H T

5 is q bit. E is n bit and H T is n x q bi ts s ize . For th is example n ~7, q '"'-3 W e can

write abOVE: equat ion as ,

• .. ( 3 :2 .. 31 )

,-,.~!'

J 1i~~ , : - g

" M : " - . .

.

. ~,;

It'll'cJ"(,'ItI'(ionCod;''fJ T('chniqucs 3 .2 j trror C\~nUt:'~C~"!li'1g.---.------.~-.-.----~.. -_ ,.-- ..,.....---~---. "

F rum tIl< : l;i\" 'l1 p,lli!}" check matrix 1 - 1 , w e CJI1 obtain its transpose ( 1 ·[7 ) by

inINcI·L;;nC'.\g HlWS 10 columns. I .e .

I f

o J II

1

~ /

/i

l

" 1 I (J,~',:i':10 /1 () ,

", ,1 i

l o 0 Ij~"

. . . .

Syndrome tor first bit in error

Let '» ca lcu lat e syndrome fo r f ir st b it er ror vee-or i.c,

r1 U J iII 1 1 iI :

/1 1

c E H T:= 1 1 OOO()Olll/~) 1 IJ

0 0

1 00

'0 ()L

(I (f) 0 (....0(1) (~$Q(fJ 0 t D 0 (JE! ) o (D 0 It > 0 (j) 0(1;' t il e' o 1;;, I, ' .! .' ( ) : ,) (j':'.' liC, ." e,'

:= [1 0 1)

This is r io" syndrome vector for firstbit in error.

S,'fHlfome for second bit ;< 1 error

Lcf'~ crlicuiate $y!\l.jr()nH~ fnf' )JI-1

;).{ Iii ~~cror. In i~lh!(· : i, , .' ~ liu- (tl'·1 ',"t', iI, 1.\;

er ror in 2 '' '/ bi t is g iven. 'N(' Ciln wr uc.

(Ii1:) O !) 0 0)10

I11

I() l ()1 .0 (I

.J

I) t.l

'" { O I D I E D Q E D 0 $ 0 tB O ( { \ o i ) ( b I (110 C D u m ()(j) l iB ) ( ) lit,) i (J; (}C'11;

:=. [1 1 1

This is the syndrome vector (or , ; , :, '1)11, '1 bit in error.

------.-~"



Information Coding Technlqul:'s Error Control-Coding-22

Syndrome vectors are rows of HT

The Table 3.2.4 lists the error vector with single bit error and corresponding

syndromes. Other syndromes can be calculated using the same procedure as above,

T1)(:,table also lists the syndrome [or no error vector i.e. E '" ( 0000000). Observe that

the co rresponding syndrome is S ",(000).

Sr. I Error vector 'I:' showing single bit error ! Synrlrome Vector Ipatterns ' '5'

No.

:----

f f i 1I 0 0 O. r) 0 0 0 0

---2 1 0 0 o 0 1 0 1 t-~ pr nWI,'of,., r

3 0 '1 o 0 0 0 0 1 1 1 4 znt! row uf H T

4 0 0 1 I 0 0 0 0 1 1 0 , . - 3 " ' '' row or H r

,~---, 5 0 0 o I 1 0 0 0 0 1 1 ~- 4 ' " fO\" o r }-IT

---.---_ 1---

6 o 0 0 0 I 0 0 1 0 0 ...5" row or H'_- '---.7 0 0 0 0 0 I 0 0 1 0 4- 6 fh ro w of Hr

8 0 0 0 0 0 0 1 0 0 1 ... 7 '0 row or HT

Table 3.2.4 Syndromes for (7, 4) Hamming code of single bit error

The foilowing table shows that error in the first bit corresponds to a syndrome

vector of 5 '" (lOl). This syndrome vector is same as the first row of HT, The syndrome

vector ( S " " 1 1 1 ) for error in second bit is Sitme as second row of H!'. This is same for

remaining syndromes.

3.2.4.1 Error Correction Us i ng S yndr ome Vector

Let 's see how single bit errors can be corrected lI sing syndrome decoding. We will

see this for (7, 4) block code. LeI the transmitted code vector be,

~X = = (1 0 0 1 1 1 0)

L et there be error created .;n t r . , : , 3" ( bit in the received code vector Y. Then Y will be

_.y = = (1 0 C D 1 1 1 0) encircled bit Sl10IVS it is in err or.

Now error correction can be done by adopting fo!lowing steps:

i) Calculate the syndrome 5 =y[-{T

ii) Check the row of HT which is same as '5.

iii) For p!" ruwor NT,pill bit is in error, Hence write corresfionding error vf'ctor(E). ~': ,: '

, Hero nota that the assumed code vec tor X is derived lor the parity cbeck matriJ( 01 exarrop/6•..

3.2,3. Students can verily this using the regular procpdure explained In example 3,2.2. TM cede:

vectors depe",l upon the parity check matrix 'It'. Therefore in S ~ YH' , We have used the $< ' l f l l 8 ' : ' "

Informat/on CodIng TechnIques3 -23

Error Control COding

iv) O bta in c or re ct v ec tor by X ~ Y E tl E

Above procedure is ilfustrated next ;

I)To obtain syndrome vector (S)

. Let'~ use the pari!} check matrix and syndrome

dltlstratlOn. TIle {I'Cciv('r c£1ku!atcs 5", YHT i.e.

0 1 1I

1 r 05 YfJ1" =[1 () I 1 1 1 O J 0 1..___--...---

yI 0 0

0 1 0

From eqUi1tJon 3.2,26~.

o 0 lJ~

li T

= ( I illO Q; J 1 ill o a n (l) DID 0 o m Offill ill 1 €f) o o 1 I D 0'" [I 1 0 1 1 E D 0 fD 0 ® 1 Ef) 0 ® 0 m 0)

From equa Ii0:] (3.2,26 ) and equa tion (3 " 30) . ..J. we can write

5 '" YHT = EHTHere 5 =Yl-JT =£1-[ T == (1 I 0)

Ii) T o determine row of HT whIch'IS same as'S'

and (W ) To determine 'E'

On cOmparing this syndr lrh THT F T bl ,- -' orne Wit H, we _observe that (S", 1 1 0'\ .

, rom a e 324 We can obtai h . / 1S the 3',. ' row 0:

~S, ", ' am t e error pattern corresponding t o t hi ss y nd romC?

E '" (0 0 1 0 0 0 0)

. 1 1 1 1 s shows th t th .a ere 1S an error in the third bir f'( W

synd rome vecto r is equ~l to 3,d f [-{T 1 0 • e h:!Ve a[so ver i ( ied

I row o , then third bit of Y is in eITO

v) To obtaIn correct vector r.

The Correct vector c a b btai ., n e 0 ained from equation (3.2.29) as,

X "" '(IDE

thar, if

i.e,

X = [I 0 1 1 1 I 0] EEl0 0 1 0 0 0 OJ

., ..,_. '" ' (I 0 0 I I 1 0) which j.~ same JS t ransmit ted code vec tr r'flUS a Sit) I bir c ).

" . g, e I err ors can be co rrected using syndrome decod lng.



F , o J1 1 T a bl e 3 . 2 ,4 w e o bs er ve that the syndrome 5'" 101 co rr e spo nds to an error

f'"ttern of £=1000000, This shows that there is an error in the first bit. Thus the

error detection and correction goes wrong. The probability of occurrence of multiple

errors is \CS$ c ompared 10 single errors. To correct mult ip le e rrors, extended hamming

codes are used. In these codes one more extra bit is provided to correct double errors';

We KrlOW that for ( /1 , k) block code, there Me lq -1 distinct non-zero .syndrorn~. "

Then: are "C1

~ 11 s ingle e rror pat terns "C, double error piltterns, " e . 1 t ripple error

patterns and so on. Therefore to correct 'I' errors per word the followir'lg relation

should be sa t isf ied,

Information Cading Techniques 3 - 24Error ConirQI

'Nhat happens if doub le erro r occu rs in Y ?

Let 's sec the case of double error in Y. Consider the sarrie message V(~I(II' X. i.e..

X = 1001110

Let's consider that error is prescnt in 3'.1 and 4'" _bits . Then Y wil l be

y 1 0 C D @ i r o enci'tded hit'S are ill error.

Then 5", y / _ ] T gives,r i 0

J 11 1 I

1 01

5 YW;[1 o 1 o 1 1 O J 0

0 0

0 0

0 0

5 :; 1 0 1

3.2.5 Hamming Bound

Vv e can write equation 3.2.33 as,

2 '1 ;:: 1 + "e l + "C < + ... . . ;. n C r

I2: I. i J C ,

i~O

We know that q '" 1 1 - - k Then the a bove equ at io n l>CCO!l1e~ ,

I

2"-* ~ L, "C ii~O

By taking logarithm to ba~e 2 on both sides we get.

. . , ( 3 .2 .3 1 )

Jlltotln:otr()n,(,:oding Techniques 3 "25 Error Contf{J1 (;odinu

~ - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - ~ ~ ~ ~ ~ ~ ~

1 ,~1- r <: - lor', i._," C o

II . . . : ; ' 1 0 " j:n: 0

".. (3,2.34)

This equation relates code rate' r' with the error corr r. cl io;l c<lp.l bi lHy "I 'j' C<TDi~:

per code vector ina block of 'n' bits. We know Ih"t the error correctil;)', (Jp,lbil: rv C'(

the code (i.c. 'I' errors pCI' CO(\(' v,'c\e>r) is 1'('l:lt,..d to lh,; minimum JisliHl(·('. i' ,j·,

minlrn\llll distance is a lso called ilMllt1d!:g rii'o\illK<! Fq""tiC'ln n · 2 . .14) !;iw'"'' U". ","[;l\;un

bel ween code rille,' flU mber of CJT'OIS 10 l . > < , corn,cted and l1umi,(r , ' ,1 I.'il.' LI ;, iJl . , ,"

111is equation i s al so c al led ha fl ll l1ing UlJLH\lL

I l l ' " Example 3.2.4: Fo,' II l in ea r b lo ck code, p ro ve w il li ~X,111Ip!C$ 1i1(li

i) TIll' sY l l ri r omc dq!~l lds only 011 error pliil.'m alii/ 110 1 Oil trnnsmutcd codeword.

ii) Al l (',tol' /,'Iflrrll~ //1'11 < l i . ! f . . , . " 'f 0'1cadesoord /rliiN 1 / , , ; StlJI i" $1/",1","1/1'

SoluIlorl: Ij) SynJrO! l \C depends only on error pattern,

Equation (3.2.30) gives the !dilt i' )l1.~; il ip bdIVl!C;11 error paW'I" "lid ~',vn,:'-'!:' i'

S '" / : : i · iT

!\h("',~ ""i,,;,lion ~;hows tll::!t Syl \o nmt'- ' ( :; ) dqx:nds only on lite In,H f",(i(:,'n iLT ;L

doe$1101 6, ·p f. :n d O il codeword (X) .

In eX;:lInple 3.2.3 IYC have (,iJi di~H'd II ;l~ i.·nd 1);1:' ol , ', :, ",wJ ,,".', ". r " ' ' ' ' ' I,r.

syndromes fo r il particular code, Table 3.;:',[ l is ts these errol F ·, 'l t' ,r [C ' 11 i,; ,! .. , ,, /. "" ,

this table that the syndrome depends only on error pattern and not on the codeword

(i i) All error p;ltlerns that d if Icr t , : , ' ;1c '[Jl lcword have ti,,~salllP syndronv,

Lc"' . ."",j,!,_.I the: tw o ro cle VI~cIO(:; X : , m e : X1, Lei. ,11' (';','(',,' : . " J ' . ;'11:',:",)",,'; ,\,

Iirst uil (MSlJ), ' 111 ( ' 1 "1 th", errol' p,i11""n fe;./,i},·,th of l j\ l'~ " • .ocie VCOUI"'; w)li t,,~ :.",n·",. i.c.,



,Information Coding Techniques3·26

Then the syndrome for first received code word wi l l be,

SI Yj r{f

(Xl (l) E) HT

X!HT @EHT

EHT

Simil ady syndrolYie Ior second received codeword wi ll be,

Y1HT

(X 1 (j) E)HT

X 1H T E D EH T

E I - J T

Error control coding

Si nee X d ·{T "' 0

Here Y z "X z 0) E

Since X2HT '" 0

Here observe that, S 'l ",S2 =in,This shows that the error pattern differs by the

codeword have the same syndrome. TIlls confirms that syndrome is independent of

the codeword.

Example;

III ex,unplc 3 .2 .7 ., t he pur it y check mat rix i s,

rIll 0 1 (J 0]

H = = \1 1 0 1 0 1 °101 1 001

For this parity check matrix, codewords are given in table 3.2.2. Consider the two

codeworcls,

Let an error be introduced in first (MSB) bit of above codewords. Then we get,

Y2 G) 001 011

b""G)010101

Here encircled bH is in error. Let us calculate syndrome for Yl' i.e.,

1 1. 1

X2

0001 011

andX3 '" 0 0;' 0 1 0 1

1 0

1 0 1

51 Y2 I - I T ' " (1 0 01 0 1 1] 0 1 1

0 0,.,-.~

0 1 a

l o 0

lnformatlon Coding Techniques 3 - 27 Error Control Coding

• =0 [111]

Similnrly let us calculate the s yndrome for Y .. i.e.,

r : 0

53 .- 'l3 NT '" [1 0 1 0 I ° 1 J i 0 1

, l~ 0

o 0

f ]

~ J[11 1 J

Thus the syndrome 5 2 =5 3 ~[111J even if two codewords

~~~::n:.hat for a particular er ro r pat te rn syndrome is same ev~n<l~~

d iff er en t. Th is

codcwords are

3.2.6 Syndrome Decoder for (n, k) Block CodeFig. 3.2.3 shows the block diagram of a s d dyn rome ecode f r b

correct errors. The received n-bit t 'Y " . er or inear lock code to, 1 vee or IS stored In an n-bit '\ e ctor a syndrome is calculated using, reglster. From this

5 '" YHT

Fig. 3.2.3 Syndrome decoder for l inear block code



fnformatio.n Coding Techniques 3- 28

Thus HT is stored in the syndrome calculator. Theq-bit ; :oyndronlc vcJ .: i()r is

appl icd to " look up table of error patterns. Depending upon ll-.e particular "-VJli1r,,,·~.

an error pattern is selected. This error pattern is added [mod - 2 at1.dWotl . )

vector Y. T he output is thus,

Y®E = X f rom equa [[em

The block diagram shown ~,l"w<;! can correct only single errors in the

Maximum likelyhood decoding for l inear block codes:

VVc know that there are 2q d ifferen t synd romes. These syndromes can only

represent 2'J -1 error patterns. But in an n-bit vector, there can be 2" erro r patter ns,

Hence syndrome dcesnot uniquely represent error vector (E). With the help of'

syndrome we can correct only 21 -1 error patterns and remaining patterns are

uncorrcctable. Single errors are more common than double and higher errors.

Thcrefcrc single error patterns arc ntosi l ikely compared to double and h.igher error

P" t terns. Therefore synd rome decoding corr ects sf !1g ie-erfo ;~· which are most likely.

1-[<:;1(; syndrome decoding is called maximum l ik e ly /w o d d e co d in g . The' maximum

Iikelyhood decoding selects the code vector that has the smallest hamming distance

fr om received vecto r. Such code vector is obtained by

X '" Y + E .~ Here Y is received vector

and 'E' is the most likely error pattern. This error pattern is selected based 011

calculated syndrome. The maximum likelyhood decoding minimizes the word error

P roba b i Iity.

3.2.7 Other Linear Block Codes

3.2.7.1Single Par it~ Check Bit Code

If ther e are /111, I>lz .ins i.> nlk are the bits of the k-bit message word; then,\

In the above equation C1 is the parity check bit added to the messagc_bit. The

above equation shows that if there are even number of Is in the rnessilge word, thenpa rity c he ck l ; 1 i . t C I ""O. If there are odd number of Is in the messageword, then parity

..'1(.·d: Ln("'~ 1. Thus for this code,

n k+l } . .. ( 3 .2 .3 5 ).

q

Note that this code only detects single error but d 9 , e . , s . not correct it.

3 - 29 Error Controf C()dH'~i

.2 R~pllale(l" C o d ! ' : !> " , ) . \ \ . _. J ~~. /"- . .

'In \hL .. C01,..1r',;c ':Yirtgl(~ n, .eS5ngl '~ t :d { ~~ (T;nVin"litlt.t;;~ ~tnd ~ ! . r . . . : ! J i\~~ l -lT~' :~"l1~' t·,:!·_ I I

the transmitted bits are,

II "'. 21 + 1 l .' ,. :' " "

'This code is called repeated 0 3 . 1 1 ' since 1r1,~ny redundanl , : i 1 o : > , · k bir- , J r " ,,',I ",.,ni; t , . . :

with a single message bit. This cede can C,)~CCC!_ '! " C:T () rS iX' 1" :,l,yk ~; 'I · "· ' · I t .. . ,'

uses many r ed un da nt check b its, it re qu ire s a 1 , 1f ?, crbandwidth

'!lle hadamard code is derived Ir(llll !l;1(i<lJ1lard rnn hix.. TJw ho1[j;lI""',j ('.,.,,,,\ (' !:'"n·x'n·square matrix. Rows o f this hadamard matrix repr esen t ,.,)(1" Vl't tdl·:,. TI111:' " II X/I

- c- , h'a 'damard matrix, represents '/1' code vectors of 'II bils each. Jf Ill" bl( l("i.: or Ilh·,;,.if~('

vector contains' k ' bits, then

This cqua l i o n r..hews rho l'~!,'i jl)r!~.·rllp 1 " : ' 1 ( - '1 \ . . .r(;f.~p t u in ~lHI' ,It Il;1:; "1

and. number of bits in the message vector. VV'~ know th:L: I~L]JtI~.)l..f n: ,: :~l k !II .~ (,: i

(11,k ) block code are

q '" n - k

q '" 2< - k11l,i5 p.qOill!\)tl shd.\vt\ th;J! Zl~ nnrllLl'~ c ~ riJjf", ill rIH "('I:-,-,!~~'! ·hh ~ /J:'

p~ l'd) ' bi l~ ; become vcry J~rf,p. This rN:JIJC",; Ill<.' co<l" "'Ii".J I"., c'" t.' r ,I .' " .

kr '"

II

111is shows tha t with inc rease in 'i(, the code rate becomes \ 'l !'Y "I\ l: ,jL

Hadamard MiltrlX and Code WOlxh; :

There a rc some (ol lowi ll l; irnp'Jrl.llIl puint . .. a s Iol lows .

i) One code vector represented b y h ;. d, 1 1 1 1; ll 'd Ill;': II' "'I\t,,iJl~ , [11 . '. ,, : '. , , .: . " ,) ,: .

That is one row of hadamard matrix contains all zero clements.

r'ii) T t , , , , other code vectors contain ·; i

',i - .• .: /1 ('.1j .:;. dIlU ; ) ~""

hadamard matrix contains hilU number of l's and hall I!lil,,[;6 ui L :~ .

. niii) Every code vector differs from other code vectors at '2 places.

every row of hadamard matrix differ'S with other rows a t u pl;)(('5 (i.e. half2

number of p laces). Consider th~ hadamard matrix with single message bit i.c.

k =1. Hence,



, i f -

-~ ,

~

~{~~;

~". The code turned by such parity' h k ..r code wil l be , descr ibed a s (n + 1 k) r c CC

bmatrix IS called extended code. Thus the

~:' m . . ; mear lock code The ne I f)$ affix H < will be of size (q + 1) by ( 1) C .' cw y armed parity check

. ~ hamming rode d equation (3218) It ~I+ . onsider the parity check matrix of (7. 4).' _ . ~" _.. , IS reproduced below

3.'.:::~ :f;r : ~ . ... ~~:,

i~ ..·

Information Coding Techniques 3·30 Error Conlro! Coding

~ - - - - - - - - - - ~ - - - - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -n = = 2 cc 21 == 2

Thus hild~rmlrd matrix for single message bit will be of 71 x II (i. e. 2" 2) size. The

f ir st rOW wil l Go>~ II z er o del l- Ient s- This ]!lalrix i s shCl\vn below.

' " [ 0 0 1 ( -o 1'---v--~

r

All zero row ... (3.:2.40)

These e .!emcnts sat isfy the points

(ii) and (iii) mentioned "bove.

_' Here H ~ is 2 x 2 size hi1dilmard m<ltrix. Observe th<lt the second row cont"ins hntf

number of elements as zero and half as 1'5. We know that the code words arc the row

of h<ldamard matrix. Here the code words are 00 and 01. Consider the hadnmnrd

matrix for two message bits (i.e. k <= 2). Then we have,

II := 2~~2 ". ! =1

Thus the h"rl~m~rd rna lrix will be of size 4 x 4 . T ),i5 rna t rix is shown be low.

.., (3.2.41)_ [ l - i 2 } - h ]HI - -

H2 Hz

He- re H 1 is the marrix given in equation (3.2.40) above H : . is the complement of

']lilt.-ix I'b will be,

- [ 1 1 Jz =

'1 0

.., (3.2.42)

Thus in the above matrix every element of I - l J

t , given by <"quatiul l 3_2.41 above wil l be,

i s complemen ted. Then the ma tr ix

.., (3.2.43)

The above matrix is of size 1\ x 4 and it gives four code words. These code vectOrS

.re (DOUO), (Ol 01), (0011) and (0110) obserVe that the above code vectors and

»damard matrix sali.sfies all the three points (!iscussl'd earlier. Since every code word

liffers by ~ places with the olher code worus, these code woids are orthugonal to ,-

2 AI&LCCI1: f

Information Coding Techniques 3 - 31 Error Control Coding

each other over comple.!e 'II' bits. Equation (3.2.41) can be g , rnumber of bits <IS follows. enera iz ed further (0 higher

H_ T H " H I I ]

2~1 - l -H" H "

Here H I I is the complement matrix of H".

Since every code word in hadamard matrix differs every

the minimum distance between the code words will bc,

" (3.2.44 )

other code word by i.

d min =: II

2 '

2 k

= =2

( .3 .2 .15)

'" 2<- \

We know that, the correct upto I errors per words,

dmin < ': 2 1+ 1

Plltting the value of rimln =;2k-I' bn above equation,

... (3.2.47)

Drawback : Since hadamard code uses man che '.-Hence to transmit the signal '~1' h . . Y ·ck bits, Its code rate is very low

s.>: < a... Hg e r r at es , h igher bandwid ths are requir ed.

3 . 2 . 1 . 4 Extended Codes

We know that every (n, k) linear block code has a a-column of zero elements (ex tIl c < p nty check matrix H. One

. . cep ast e ement) and one f"par:ty check matrix as' shown below. row 0 1 s IS added to the

. . _ ( 3 _2 . 48 )

[

Parity check 1 9 ]H~- I t

matrixH I I1"11 1 (q + 1)x (n + 1)



mrcrrnauon 1.;Ooll1g If!Chnlqucs

H[

111010

1 1 0 1 0 1

101100L

The pnrity check matrix for extended code will be (using C(juation (3 .2. '18»

H ="

\\'ilh the above extended parity check matrix, the minim_l!m distance for extended

coc~ will be.

d. I '{n tin) : ;; . d min + 1

_._(3.2.49)

Therefore for extended hamming code d dUll,,) ,,4.

Advantage:,his code can detect more number of errors compared to normal (n, k) block code.

. ~_'\Gut it does not have ;lny advantage of error correct ion.

. - , > ~< - /3.2.7.5 DualCode- Y \,\'C know that for (n, k) block code,

, Generator matrix, [G]h! '" [ h - ' : k I P t < q ] _ \ . "

Parity check matrix, [HJp"

. .. ( 3 . 2. 5 0 )

. .. ( 3 .2 . 5 1 )

Here: q = 11 - k. Now consider the mat rix product ncr. i.e.,

i-iG T ' " [ P ; ' k jI~q ] P J ~kT'~1L ' I' '' ' . ,. ;

i.e.

.., (3.2.52)

COdinJJTuchniquos J" JJ Error Control Codin[j

illustrate above property, consider the generator and pilrily che-ck fn:il ri~~!.~:~j)

l.c.'J

~ - f ~'I. 0 0

O J1 0 1 o

~h7b 0 1 Il 0," \'"

I o 0 (J qG

0 1 0 0 1 1

~ I0 1 0 0

L O 0 0 0; ( ' t . ; :

1 J~.7

Then lIC1' will h.',

1 0 0 o 'j

0 1 ()

~ !."

[ :

1 0 1 00 0

0

IGT ] 0 0 0() e 0

0 0 0 1 ~.,1 1 0

1 1 ()

1 0 _7x4

[ ] ( D 1 l i D I I C D ! o m 0 11 ~])1 H I. J 1 O (- f l ( ) HD 1 o

1ill) ()ill 0 1 C D ) 1 < D 1 i. 31(,1

OefiniUOllof dual co~~~~--'~

.Consider an (n, k ) block .code. As illustrated above, this code satisfies,

HO T =;; 0

10('1"\ the (n, ,, - k) i .e. (n, q) block ..:"d(~1" c:..lk.,J (1 . 11:11 nIue , T:,,,,, f<\,' (!Vl'l), {iL, k)

block code, there exist s a dual code ot . s i ll : l~ ( , . , 'I)'

We have defined the generator matrix {I( (n, ") block c,xl", : . 1 \ t~tV,\~jtHI (J,.: '>,S(j),

S~lllrly for (II~ q) block code gel\{:r<Hor matrix wi l l be. ..• ,(Gll''' '= (lp"lr.,,,t.,, , " , 1 . 3 ) . _ - · : : : \

SimllMly from 'equatlon (J_2..52), We can write fhe parity. dl\!c:.k 1IIi1ITix .1..- ;'''' '1)

. hlock code OlS,

[111."_ . , ( 3 . 2_ 54)



Information Coding Techniques3 ·34

Error, control Coding

We know that (n, q) code is dual code of (n, k) block code. Then parity check"

matrix (H) and generator ~atrix (G) given above are matrices of dual code.

m ' " Example 3.2.5: CO ll si de r t in ( 71 ," 'k ) l in ea r b io ck c od e will! g l ! rl t !r i ltor ma tr ix G an dparity c hec k m nlrixH . 11 11 ::n , II_) c od e g e ne r a fed Vy H i s c al le d Il !c d ll al c od e o f

(n, k) code. 5110<1)ho! , t il e ma tr ix G 1 $ I / le parity check m atrix o f d un l code.

So!ution: (i) Consider (n, 1c) block code

For this (vue r;eneralor matrix ,md panty check matrix are defined as,

( GJ "" '" [lb kIPkKq]t.1I

[H11'~ '" [ P ~ " k P ~ ~ q J p l I

. . , ( 3 . 2 .55 )

. " ( 3 .2 .5 6 )

The genera.tor rna tr ix and par ity check matr ix sa ti sfy followil1g property,

IGTll'k '" [pr.• I Iq~? ] [ ~ T · k . ]q xk ".k

.----~

. , . ( 3 . 2. 5 7 )se . [ P T (D p T J '" 0

q. k

(ii) Consider [n, q) block codeWe )<nP,W that (n, q) code is dual code of (11, k): block code. The gc'!1crator and

parity check matrices of this (n, q) code can be writhm from equation (3.2.53) and

eqlJation (3,2.54) as,

(G 1 1 1 , " 1 1 ] x n

[H,I"'I J k . n

. .. ( 3 2 .5 8 )

Here we have written G d " , , / and H <fwd so that they can be differenti~ted frort}",y

and H of (n, k) code.Now 'let us check whether (0, q) code abo satisfies the property of equation .

(3.2.52), l.e.,

i.e.

[ P T €I) pT] '" 0k"

[01.,

Taking t ra .nspose of both the sides of above equa tio! ).r , __

[H d"ol e ~ " . , ] : = (0 f. ' XI ? I'lIff

Informallon Coding Tech Iques 3·35 Error Control C di. 0 mg

Here let us Use the 'property of rna tr ix : [ABf _BTAT .

[G,~,""f H S , . " ~ . . [0] . r.e.,pk

Here [G T JT_" ,f - G i l , , , , ' and [ot will be .ero matrix. Hence abov[ G i l " " H~] _ [OJ aoove equat ion become

... ...,, 1"* - p' "

(!II) Conclusion f .0 equntion (3.257) and .)We derived t\ 1 ' auovc equation" vo resu IS,

For (n, k) code ;

[c1.,.1 H I I ] , = [O JIIlt,j ,r U ; 1 . . I]xk

q .k

From above eq tia Ions we can conc lude ,

re T] '" [fiT ]T~k' ". k ,/",1 ,,'d

, Illg transpose of both the sides,

(GJ", " "" [H d"., ]hn

A Vow eql in tion Sh01l)5 th t (''f;I< d ua! c ode (;.,. (n , ,) zr': matrix 'f (n, k ) b lo ck : cod , is th e p ar i! . J " , 3.2 .60)Th' ..' lin y c iecr nuurtx

IS rs the required proof of the '.) ) 1 * E given statement.

xarnpro 3.2.6 : ..For" a IiI1mr block code tohichprove that, corrects single error per c ode v ec tor,

For (n, q) code :

n ~ k =dog2 (n + 1)

And hence desi .' !!SIgn II linear block cod - . ., block SIZI! of r i g . ' ! ! bits e ioit]: a /nll!lmllm d is ta nc e o r If, Solut. " ~ tree lind 17 mcssllg,e

ion " I) PProof of the equation

For linear block code we kn -ow from equation ( 32 " '3 ) til t2~-1 .. ~ , a ,

< : IrC I+ "C 2 + .. . + "C/

equation g' h,~VJrrp<-";r.- ,1yes t, e condition fof single error per cod or correction of 't' errors

on RHS ' ie e vector (I ," 1) the abov . per word. For, " e equation will have only ftrst

2,-1 ~ ftCS. 1mee q = n - ~ and l OC I . . n

2("-~) -1 ~ II



I.'lformation Coding Techniques3·36

. Error CoM t ri )[ C b 'l li n g

(n - k) ? lot: l (1/ + 1)

" ;: k~l(ll:;:(JI ...1)_ ,. ( . 1. 2 -6 1 )

which is p r ovcd.

ii) To determine linear block code

We 11<1\ 'eto design -a l inear blcxk code w ith d

mi"

code with d min ",3 is a single error correding coJ·~-

t;:\'Cn by equation (3_2_61) above.

i .e. for single error correction (d , , , i , , = 3)

II> k+log2(11- ' -1)

Putting k '" S <IS given.

/I ~ 8,· log2 CIl-r 1)

0" S(1lving we get II"" 12 . Thus the code will be (12, 8). Therefore q "'"1 7 . - 8" 4..

The par i I)' check n1J lrix wil l be of siz« q x 11, i.e. pari ty t ;peck- -r ri .a tr ix si ze wi ll be 4" 11

0 3 and k '" g . We know that the

Hence we can use the relation

i.c

[f--I]~.I' '" [pr:I;Jj-l~

The m~t r i x pl wi I I be of size 4)( S a nd I ~ will be an identity matrix of size 4 x 4.

Select the p T mat ri x Sti ch t hat,

i) Its size should be 4 x 8. TI\lS is P submatrix will be of size 8 x 4,

i i i ' ) I'dI rows should be distinct.

Such m~trix is gi\'8t\ below

[

1 0 0 1 1 0 1 0 1pr = 1 1 0 0 0 1 1 1

01101011

l a o 1 1 0 1 0 1 4.8

Here note rhar you have the freedom of which combination should form a

:,),;hwl:tr .:011')1'n., Therefore parity check matrix wil l be,

P 0 0 1 1 0 1 0 : 1 0 a 0 1111000111:0100

H \01101011:0010

~O0 1 1 U 1 0 1 : 0 0 0 1 4.12

~____..... ~pT 1..·- .__

, . - - ~ ..

From the above matrix g~ncrator matrix and code vector s can be obtained .

3·37

c. ""«I h ) d,_ C) tfJ

L~= if, (E l d~

C6 ",dl (f)dJ

I

J

i) COilS rn/C[- gcw,lllior matrixii) COlls /mcl code gellemted hy this matrix

iii) 0 -C(J1I11W error (o)r,.,~dill.~cdJl"hi li ly

i u) P rep a re 11 sui tn b lc d e co il in g !lIIlle

·v ) Decode Ie rec eived words 10/100 arid G OOL'()-

Solution: i) T(I obtain the gCllNiltor matrix :

'.

The above equation can be written as

r il l I

le i C s C d '- [ill d1 s • 11p- .n

[ _ I ' l l

P I2 P jJ

P/l P:J

1 ' : 1 1 1'33

Hence.

CJ",Jll'll@d} 1'21 (fJ d) PJ1 l5 "'rl1 1'12 6 'j , I ). P 2 1 (f) I i : l F3!

Cb ", ti l P1J (j) d, ['21 m dJ PJ) J

( .' " .J~ r : 1..

ComIMri ll g t l~~ above l'quatlOJlS with ' ' ' : r . . .,liol1 (3.2.62) we 1' ,. '1

I'] J 1 - [

p:.. 1 1 n

b D l J

The gcncmto r matrix is given JS,

G

,...~

f t .1 !'J < , I

p Ll 0 ~l: II I 0

t oi

() 0 !_ j



/ ~:-., , j ,

,~~.

Information Coding Techniques 3 - 32 Error Control Coding

ii} To obtain the code vectors :

In this code, there are 3 message bits and 3 check bits. Hence this is (6, 3) block

code. Table 3.2.5 shows the message bits, check bits and code vectors for this code,-~~.

Sr. Message vector M Check bits as per Code vector or code word X Weight of

eqtJ~110n 3.2.51 C cod" vec tor

No, w(X)

d1 d, dJ C. Cs Cs a, d, d) C, Cs C6

1 0 I] I] I] 0 0 0 0 0 0 0 0 0

2 o , 0 1 1 0 1 0 0 1 1 0 1 3- -

J 0 1 o 1 1 0 0 1 0 1 1 0 3

-----..-~~r---,_--!-4 0 1 1 0 , 1 0 1 1 0 1 'j 4

,~

5 1 0 0 1 1 1 1 0 0 1 ._J- 1 4r-

H i :1 0 1 0 1 0 1 0 1 0 3,

)----

7 1 1 0 1 1 1 0 0 0 1 3

8 1 1 1 ~- '00 1 1 1

, () 0 4

Table 3.2.5 : Codo vectors

iii) To obtain e rror cor rect ing capabil ity:

The minimum distance between the code vector is,

dmin '" !W(X)Jmi"

From table 3 ,2 .5 , it is den!' thai

dmin 3

dmin ;:: s + ]

x '" (0 0 0)

3 ;:: s + 1

s _,; 2Thus two errors will be detected

and

3 ?: 2t +

::;

Thus one erro r will be corrected.

iv) To prepare the d...,coding tab Ie ;

The parity check rnatrtx (H) is given as,

H '" I f'T : f")~.,,

Informallon Coding Techniques3·39


Hence tri lnspos~ ,of above m<1tTix becomes,

HT I P 1 '[ ~ q " . q

From equation 3.2.65, i 'lbove rna trix wiII be,

'1

1

1

1

o

o

o 0

o 1 0

001

Th e syndrome vec tor (S ) can beequation (3.2.30) as,

. A · · ·~.

(3.2,67)

calculated from error vector (E ) and }-fT bv)

S '" EH T

_ Here E is the 1 x 6 size error vector L t I111( E will be, ' e us ca culate syndrome (or 2 '''' bit in e r ror

. E '" fa I 0 a 0 OJ

Hence syndrome wi l l be (from S =: EHT),

1 1

0

S "" [0 1 0 0 a OJ0

1 0 0

0 1 0

0 0

'" [1 1 O J

Thus the above synd 'rome vec tor cor responds to 2'''/ row of HT S- .) J

can be ob tain ed directly from rows of H", Table 326 . h irru a~ yand corresponding syndrome vectors .. s OW$ t ie

other

error

Sr.Error vector 'E' showing slrlgl ..

No. Syndrome vectorCommentsIt error p~Uem5

'S',

1 0 0 0 0 0--__

0 0 0 02 1 0 0 0 0 0 1 t 1 +- Flr:strow o r H T3 0 ! 0 0 0 0 1 1 0 +- Second (r:tW ofHr4 0 0 1 0- 0 0 1 0 1 , Third row of Hl'



Information Coding Techniques 3"' 40

5 0 0 0,

0 0 1 0 0 ,_ '-;ounhrow illNt

---l--- --~----. . . . .._6 0 0 0 0 1 0 0 1 0 l'~Pllih mw ·of II',_ ._........". _ _ _ . . , . . . . . _ . . _ _ _ . . . . . _ _ _ _ ..

7 0 0 0 0 0 1 0 0 1 ~- l ' l. i xt h ro W O J H I

Table 3.2.6

v) To decode received words

T o d ec od e 101'luO :

Ottcodiflg ta'b'le

Here observe that the received w ord 1 01 1 00 is not standard codevector (m rn

Table3,25- Hence there is an error in received word. Let,

Y '" [1 0 1 1 0 0)

From equation (3.2.26), the syndrome can be calculated for this word. i.e.,

S '" YH T

Putting the values,

S

r : : 1

/1 0 11 0 OJ[~ : ~

[1 E 8 0$ ( [) 1 E 9 0 E9 0 1 c D 0 E 9 0 E 9 0 E!) 0 ill0 1 ID 0$ ffi 0 Ee 0 1] 1 G )

[ 1 1 O J

J'\Jo!i:,1hilt [I 0] is second syndrome in Table 3.2.6, and the correspond ii1g error

pa tt er n iS I

E '" [0 1000 O J

TJ;e correct word is obtained as,

x Y(fJE

'" (1 0 I 1 0 0) fEj (0 1 0 0 0 0)

This ;5 che cor rect word.

To decode 0 0 0 1 1 0

This also contains an error since it is not valid codeword from table 3_2.5. LeI.

y '" [00 0 1 1 0]

Hence syndrome can be obtained as

S ; niT

Information Coding Techniques J "'41 Error Contro] Codirlg

~ - - - - - - - - - - - - - - - - - - - - - - ' - - - - - - - - ~ - - - - - - - . - - - - - - - -f J q! ! tJ_ 1

Ij ] a

s '" [0001 1 O J :

l~

' 0

1 o

0

= = [1 1 O J

From Tab le 3,2,6 , the co rresponding err or pattern is,

E '" (0 1 0000)

The correct codeword is given as,

x = YffiE

[0 0 0 1 1 0 ] (i) [0 1 0 0 o O J

o 1 0 1 1 0

Thi« is the WJTI:Ct word,

) I ) " Example 3.2.8: ; , l l I CITor c on tr ol c od e lUIS the _fo l lowil l '; : _ P ( I / - i l _ l / check nuurix :

r J 0 (J 0'

II ~I _ ( ;

() l)(li. ,

0 () I J

i) Dc/Crillilli' i l 1 < ' Sfll('l'/Iiar IIInlrn G ,

II ) F il ld t he co rl ,~ iJ - "Hd Ol,-/! he,i,;ills 'I'/Ih )(1. / ,

iii) D,'cod.' IIII ' / 'cCC' lVcd codeuord 110110 . Canuuen! O il en or ' - ' , '11"-/10/1 1111: / correction

(ilimb/lill} oithi« code,

iii) W/IIII is the n'/o/loll i)dw~CII G alld II ? Ven!!! the sall/~.

Solution; This is (6, 3) code, Hence n = 6, k c= .; and '~

i)ro obtain lhe g~lte!,,"tor matrix :

The par ity check marrix is given Cl5 equation (3.2.].1)

H '" IIJT : 1 (1) 1 ' "

}-kilL~ .cqu;ltioll 3.2..6:> C;Ul be ·.,·!l\kf' \ 1 . : : . ,

1 '1 0 : 1 0 o

J-l -" ,1 1 0:0 n,

I I I 'il 1i

""

!'. '.\ '; .. . \



Information Coding Techniques3 ·42

Error Control Coding'"

_---_.-1 0 1 1

p'f " l l 1 0o l dHence the matrix r wil t be,

r x: r ~ ~ ~lo I

The generator mat ri x is given as (eq'wti on 3.2.6)

G == [I x Pk.q)hll

[

1 0 0: 1 1 0

11

]

G '" 0 1 0:0 1

00 1:1 0

... (3.2.69)

This is the required gener<llor matrix.

ii) To obt.lin the code word tb~t hegins with 101

H the codeword begins with 101. This means first three bits of the, c~d~wordfere .' 3 1 - t matic code first k bits a

arc 101. l.ength of the message bits IS k",. n sys e .; . bits.codeword are message bi ts . Hence fir st '3' bit s i n eve ry codeword w il] be message

Thus 101 are message bits. i.e.,

M",P01j

ThIS is {6, 3) code. The three check bi ts can be obt ained by the cquiltion,

C '" M P

Putting <lpprOpriilte matrices.r] (fl

C = [1 0 1) 1 0 1 1 \

Ll 0 1J'" [HDO G-)1 l(DO ffiO O (f)O (Dlj"IO J II

Ht 'nct: the code vector 's,X "" (m t "'2 I1lJ CI C2 C3 ) = (1 0 1 0 1 1)

Thus the codeword that begins with 1 0 1 .... is X =0 1 0 1 0 1 1.

iii) To decode 1 1 0 1 1 0 :

Let the received codeword be,

Y=ll0110

Then the syndrome is given as,

Information Coding TechnIques Error Control Coding

: ? ? _ , " " Y'H"

Putting the matrices in above equation,

1 1 0

0 1 1

1 1 1 0 1 1 O J0 1

S0 0

0 1 0

0 0~

= (1 $$ E 9 1 f fi 0 ff i 0 IffilffiOEeOffilEl10 O (B 1 (B ilID0E90ttJ 0)

. = [01 1J

This is the second row of HT . Hence th~rc is an error inall ~rror in the second 0it.

Hence the codeword is ,

X=100110

Here note that the second bit is made 0 to remove an error.

Error correction capability

It can be verified hr this code that dmin = 3. And we have seen in the earlier

examples that such codes can detect upto two errors and correct single error . This is

supported by following equations:

dmin <! 5 + 1 for d min '" 3, S 5 2

and dmi,,~~2t+ 1 for dmi" ' " 3, t 51. ~.-- '

Here s is the number of errors detected and t is the number of errors corrected.

I I I . , . Example 3.2_9: Tilt! g en er at or m a tr ix o f a (6, 3) s y st em a ti c b lo c k c o de is given by

.~0 0 0 1

: ]1 0 1 0

0 1 .1' 1

i. Find the code ~CIOlS

ii. F in d I 1r t: parity check matrix

iii. F in d th e n : ro r s yn dro m e _

SOlution: j) To obtain code vectors :

Refer to example (3.2.1) . The code vectors are obtained in this example.

_-..,~~:=.,.".,.._~_,.. __. , _ ~ . " '_ , ! , ,_ " ' _ , .. _~.,..._~._~"'!" ' _ " " _ . " " " ' . " " _ . . . . " " " ' . . . . . . . . _ . . . . . " " _ . . . . . . . . . . . . . . _ . . . . ,.~~---.----*~.---- ..._ . , . , .. , . ,. _ . . - . -' _ 1 ' " , . . . . .. - - - . .. . . ,. . . . . . - - -



Information Coding Techniques

ii) To obtain parity check matrix (H) :

The P submatrix is obtained in example (3.1,1) 11$,

P 0 [: : i ]Hence the transpose of this matr ix will be,

pT [:: i ]The parity check matrix is given by equation (3.2.11) as,

H;Wf;JqJ

Hen' r is the fJ x q identity matrix. 10 this exa¥rlple n '" 6, k "" 3 and q '" 3_

above equation becomes, .

[

0 1 1

J - j" , IOI

1 1 0

ii) To find error syndrome:

Vie obtained parity check matrix just now. TIle transpose oC the parity check

matrix wil l be,

a 1

0

H T1 1

1 0

0

L a 0 ! lThe syndrome is q = 3 bits in length. TI1e error pattern vector will consists of n n

oiIs The syndrome vector is given by equation (3.2.3Q)as, ' ..

S = E H T

Ul there be error in the first bit. Hence error pattern will be, E. '" [1 0 0 0 0 OJ.

IOul:ng values in above equation,

liotl li<.ldi...O Icclullqucst:rror Control Coding

iii '. ,

'j (} I',

i 1 \ Uc: ;. [l00000]

I

'" i 1 0 0

i[ 1 IJ

:( 1 (ll

Table 3.2.'1 Error SYlldroll'cs

I....ftolation betw~OfJ G and H

'hie rcli\tion between G and H ls gh'('n by equation (3.2.(,()) 1.(',

ncr :e: 0

1'akh\g I r < l I t l l p O s e of both the sides

U f C T / " " (0)'

Here (0/ ""O.Therefore above c'111<1tionwil] be.

G/P' '"' 0




Error C-ontrol Coding3 -46

11'U5 the relation between G arid His,

) ) I , . . Example 3.2.10: T li e. pa ri ly c h ec k bifs of (l (8,4) block code nre generated by ,

Cs .- d 1+i!2+ti{

C6 dl +il2 +tfJ

C7 d,+ti)+d.

e ll ri 2+d)+d4

Where ~11 d 2, d 3 an d d ~ n rc the m essage b its.

i) Find Iht ' generator n ml rix a nd the pn rity c he ck m atrix for tilis code.

i/} L is t ni l code vec tor s

iii) F in d t he e rr or s d e! cC/ i! 1, '5Ilnd correcti ng capabil i ti es OfJ ! li s ..c-odC~

i1 ) Shaw through (111 example I/ml t ll is r od e d e! ec ls l Ip ta 3 errors.

Solution: i) To obtain the generator matrix and parity check matrix

We know that the check bits, message bits and parity matrix are related as,

Hence,

P r2 d l E D P xx d2 G) Pn d3 E D P42 d4

P IJ d i E D PX3 d2 ED Pn3 E D P43 e,

PH 'd i 0) P 2, d 2 EDP: l i d~ ID P 4 4 d ~

Comparing above equations with the given check bit equations w e find that,

Pu ",,1 P2J=1 P31 =0 P41 =1

- Pn =1 Pll..1 P si ",,1 P~3 .= 0

P\3 =1 P'J ",0 P J . J =1 P4:J- . .<;'1

-»: PH ",0 P2 .~ =1 P34=1 P4~=1 r

" j,

I nf orm ati on C od in g Techniques 3·47 Error Control Coding

Hence P matrix can be formed as,

Generator matrix is given " 5 .

G~>n =: f l k : r\", J

[ I 0 0 01 0 1

Ga 1 0 0 : 1 o 1

o 0 1 0 : a 1 1

o 0 0 1 : 0 1 1J S <~

The parity check matrix is given as,

H - - [ P T : f q J

~ ( i

1 0 1 0 ;

~l)

1 1 0 0 1 0

0 1 1 0 0

J 1 0 0 0 1J

1 1) To O btain all Code Vectors

All the code Ysclorsln' systematic form can be obtained (rom message bi ts and

check bit s. The check bits are given as,

Cs d] tBti2 ®d~

C6 = d) @ d2 ffi .13

C 1 - dl 6;) d) E E ld ~

c, d2 E B d ]"E D d ~

TIl en the code vector can be expressed as,

X :: (I f I d! d J d 1 c~C b C7 ( ~ )



Information Coding Techniques 3.-48

Table .3_2.8Fs ts <Il l the messages, check bit s, codevec tors and weights .: . : : . : : . . . . : . . . ~

9r. Message vector Check bits. GdjJ& vector X

no. ' of Cod ..

dj d2 d; d. es CG C1 Cs d1 di' dJ d.. Cs C6 Cr CaI--

0 0 0 0 0 0 0 0 0 0 0 (} 0 0 0 0 0 Q

- -0 0 0 1 '1 0 1 1 0 0 0 1 1 0 1 1 4

210 0 , 0 0 1 1 1 0 a 1 a a 1 1 1 4

_:_li._o 1 1 1 1 0 0 0 0 1 1 1 1 a 0 4

!1! I 0 1 a 0 1 1 a 1 0 1 a 0 1 a 1 4

.5 I 0 1 0 1 0 1 , 0 0 1 0 1 0 1 1 a 4

I - G I 0 1

~

0 1 o 0 1 1 0 1 0 1 0 4

I - ~ - k - 0 0 0 1 0 1 1 1 0 0 0 1 4

I_ o

a 1 0 0 0 1 1 1 0 1 0 0 ,0'--' 1 1 0 <1 -

-Q-'Il 00 1 I a 1 0 1 1 0 0 1 0 1 0 1 ':

---- ------lD I 1 0 1 a , 0 0 1 1 0 1 0 1 0 0 1 I 4

-----)

I : : i :0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 4

.,0 0 0 0 1 1 I 1 a a- D 0 1 1 4

1~ll 1 0 1 1 0 0 0 1 1 0 I 1 0 0 0 1\

-l ;-----

.,t ! 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 4-----'

.._I

1~ I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 a

Table 3.2.8 Codevectors of Ex. 3.2 .-g

iii) Error detecting and correcting capabilities

From table 3.2.8 it is dear that minimum' weight of the code is 4. Hence minimum

distance is dtnin = = 4 . Hence '5 errors are detected if,

4 ?; 5+1 or

Thu,; three errors (ill) be detected.

Sirnilarl y 'to error will be cor rected if,

i.e. 4 Z 2t + 1

Thus one error car. be corrected by this code.

.J ·49 Erro r Con trol Cor tin rJ

U\b e e . .d't",Ot~tf...r X ~ 0 t) (J '! 1 . ~j J 'l'f\J~n Libh' ~~~f. L'~1hh.'r. ~.~I.)(~t't:

,UJ;.'I':d m Ihi's codcvcctor as r o: r uw s :

, ' " \ o ! 1 IJ 0 'I

' )

L et us calculate the syndrome ["~Ithis received vector i_c,

5 -= YHT

r: ( 1 " 1 ~III

IJ (J 'J!

I I

0 1 Iu 1 I

[1011100l]I () (I il j

0,

0 o !I

(l (J O !I

L (J'0 0 I

i[1 (j) 0 (f) n O J (0 (" 0'1 :I~ I., ..

"1 $ 0 (1'1 IlJ -0 (rl 0 Oi I' ; ' 1 ' " . ' i) 'I

1 ill 1 1 H) (() {I) U t}, Ii ((- to ! ,;

0 ill 0 (i l I.~') '~r. U ~~~.. J ! '. r)~ j

. ~ _ " l 1 o 1 1

The syndrome is nonzero. This !fIL'iLll,' t'!i(' cod.> rli-'tCCl~ up '(' t i ' J( "' _' • ) , ' ",

111* Example 3.2.11

[

1 (I (1

G", 0 1 o 1

o II I (I

[)('1ttrJllhll~ t l u : le /nl1n

/0(' 1 1 , ( ,d'u....' ';f).fl C1 l f . •~.k·UI nu /."!(" (~~j;·,r.:::·,'. ',.I,, I . ' .:

, , ·nJmbi l it l i .~: : . {( th... r'~"lliLI(. 'd 3~'j,II;('Uc-(' i-. t d : 1 n t , ,h·l(1 .n:«: t ' . :: 1 ~ . - . :'h·, ;) ;,j, 1, . "

For lhis code n '" (. ;LJ\df; z.. '1



- r~·

Error cont ro l cod in4J' : .. r~~'nfDnnation Coding Techniques

;) - 50

(i) To obtain' d min' for this code

To determine' d min' we have to findout all the codewords.

To obtain P stlbmalrix

We knOW that, e (i)::Pb'l]

'"[I ~,3 : PJ , 3 ]

[ :

0 0 : Pi l P;2

P" 10 P21 Pn Pl.J

0 1 : FJI PJI P3J

To obtain equ~lions fo r check bits

WI' YIV-W that the check bits are given as,

C -- MP

i.e.

" " [ m \ Ei) Tn2 ml EE l 7112 EE l In

Thus the equations for check bits a re ,

C1

Inl E D nI2

C 2 r1l, E 8 I 7112 E D m

C ] 1 1 1 1 E D 1 1 1 3

To determine the codevedors'11b' essege vectors Hence

Since there are three message bits, there WI I'nme m _ -,' " ck

will be nine codevectors. Table 3.2.9 lists the codev_ectors of this tnble, m e chec

are calculated as per above equations.

Information Coding Techniques 3 - 51 E - C t !rror on ro Coding

Sr. M'n;$a:ge v'eclor M Check bits C Code vector X

No.Wt. of

m, m. m,t ho code

C C, C, m, m2 m, C, Cl C, W(X)

1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 0 0 1 0 1 1 0 0,

0 1 , 3

3 0 1 0 1 I 0 0 1 0 1 1 0 3

4 0 1 1 1 0 1 0 J 1 1 0 1 4

'.5 1 0 0 1 1 1 1 0 0 1 1 1 4

6 1 0 1 1 0 0 1 0 1 1 0 0 3

7 1 1 0 0 0 1 1 1 0 0 0 1 3

!

8 1 1 1 0 1 0 1 1 1 0 "1 0 4

Table 3.2.9 : Calcutatlons of Ex. 3.2.11

As shown in above table, consider the message vector of ml'11:! mJ = D01. Tben

check bits Me calculated as,

c)

C2 :=: IlIl E E l III: E D 1113 = 0 E El 0 E E l 1 = 1

C] :=: 1/11 E E l m 3 = 0 E E l 1 =1

Hence C 1 C2"'3 "011, and the code vec tor wi ll be,

X = ( 1 7 1 1 1 1 1 2 1 1 1 3 CIC2e3) = 001 01

Weight of the ' code~~d d .nn

As shown in table 3.2.9, the minimum weight of the code is 3. Hence,

Since dmln 3

d mi n :=: [ W (X) J min :=: 3

ii) Error correc tion and detect ion capabil it ie s

dmi.n < : 5+ 1

3 < : 5+1

s s 2

Thus Iwo errors wiH be detected.

an d d min < :: 2t + 1

3 ~ 21+1

!: 1

Thus one error wil l be cor rec ted.



5 = = [101101J

pI1 0

o

i Lon Codl !)!) Techo lque snformation Coding Techniques 3 - 52

This is hamming' code (dmir . =3) and it always detects double errors and

single errors.

(iii) To obtain message bits, if Y '" 1 0 1 1 0 1

We have to determine whether the received vector is a valid codevector. 1'his

be done by calculating syndrome.

To obtain syndrome (5)

Syndrome vector is given as,

5 '" YHr

.l\.'l':f v !. cl vr C l: n he obtained o y

X - \'$};

'" (1 0 I 1 0 I) i J .l ( 0 ':0,',0 ,;0,.0, 1 )

'" 1(1 )006 )0 lIDO 1$0:OG)0 ;1'$}

101100

',:'

."

[ ~ " l1 II Kq

x '" ( lI ljm2m3 C ' ( 1 " 3 )

X "" (1 0 1 I 00') as calcula ted above.

O n com p arin g ab ov e t 1 1 10 r.:: qua lions. lh (! 1<)( :5$ ;) , I ;e bi ls arc,

11111I 12111J = = 1 0 1PUliing the v alu e o f P su brn atrix in a bo ve e qu at io n,

Example 3.2.12: Th e p.7riiy Jigi!> l{ II (6, 3) l ine ar bloc k c ode a re X Ii"'" u»,

C. =I/If to 1112, C!i '" 1111 (B !Il2 El l III] 1 1 1 1 1 1 C6 = rill ( j - ) TIl3

i) Determine the g e ne ra to r a n d parity c h ec k m a tr ic e s [ or l ir e s y s te m at ic c o de .

~

, :

. ' / J ? " ii) Commen t 0/1 e rr or d et ec ti on a nd e rr or c o rr ec ti on , c a pa b il it ie s o f t he c o de .

iii) If tile recciued s eq ue nc e is 1 ( ) 1 l Ii I, d e/e rm in e th e mc::w.r;e uiord,: " J .

):SoJution: (i) To obtain the generator matrix (G) and parity check 1ll,11rix (H),

: We know that the check bits, the l'-~lIb matrix and 1l1.es~age I iits an; {el:tk'J as,

[ C 4 ' 5 ' 6 ] 1. .> '3 - -=- [1111 /1/2 I I: J . J[ P IJ . ~

The above equation can be Written i1~,

Hence syndrome becomes ,

eo [lfDOtDO®lEDO@O lE"'OID1@0E90$O io ne iee o e ij

[001] i.e.,

Here tile ,;yndrome is nonzero. Hence. t \: ." r" ; " ''1 erro r in the received codevec tor . . ' . :

C.J - inIAl ill /111J)21 ED,!tjPJ1

C s : : ; m d.lu (D 1 1 12 I '2 J 0 :> 1lI.1 P n

C6 .. m,j'!J tti /II~ [>2.1,D IrIJP3l

; COnlpPlri!ll~ ..bove cquatlons with llit'. given equations of c" (c. 'n,d ,",; w , ~ tJ'oj

' J, va lu es o f P rnatrix.l.e.,

"~,L.. P ll " "1 P 1 2=1 PCI '" 1

P l1 '" 1 Pn ",0

PJ2 =1 1\.1 =1

f

To locate the error

On C0mparing syndrome S '" 0 0 1 with the rows of HT , we find that 6'/'

m<11c11<:swith syndrome. Hence 6 , 1 , bit is in erro r. E rro r v ecto r can be written as,

E",{OOOOOl)

, .:,

:;.:.,.;: ...;}~ "~.



Information Coding.Techniques 3 -54 ~:.'"."

[

1 1

l~ '" 1 1

o 1

The gener~tor matrix is given as,

e [h: P k " 1 ]

[13:P3d]

[ - ~ 0 0 1 1

n . ._"

0 1 1

< l o 0 0 1

And the parity check matrix is given as,

H C P T : [ ] »->:.J q xn

[PT: 1 3 ] 3,..6 , . - - ' . -

r !1 0 1 0

: 1 . .1 1 0 1

I I 0 1 0 0

(in and (iii)

The generator matrix obtained in part (i) is same as that given

pMt (ii) and (iii) MC same <1$ that g iven in Ex. 3 ,2 .11.

Review Questions

1. W h(lt is tll~ diJJerena b dw c cr r s ys tm r al ic c o de s m i ll nOll system (ltic codes. .,. . I'.' Ii e ar b fo ck c od es ?

1. What Me II,e [unct ions o f pari ly c/reck /IIaln.'· 1l1II! genera/or nm rI.\ 111 In, ..

Ho w 1 I 1 c y <1( used 10 g e rr e rr rt e c od e v .: ( "/ or s J r om m e ss a ge Mo ck ?

.1 W lwl nn: HIHTlll!illg codes? Wltlll !Ire tlre;r ! ,roperlics ?4. H ow error corree/iOiI a nd d et ec ti on c a ~' ll bi li li es o f b lo c k c o de s ( Ir e r r l nl ~d 1 0 minimum

d " Ji r " r5. WI ",I is III,· usc of "yndro"'6 ? Explain sy"dromc decorfi"g. ,

6. Wlml arc 1/,(" Iwd'!IImrd and 1'l:leniled block codes !

Unsolved Examples

1, COJl5idu /I (6. 3! l iurar code U ' h 0; 5 ~Kme r a! o r ma t ri x i s

3 -55 Error Control Coding

[

1,., 0 0 . :.1 ~. 1]C 0 1 0 1 1 0

,001011

Find theil codeoectors

if) F in d n T! h m nm in g w ei gh ts a nd d is ta nc es

iii) Find minimum w d gl 'l p nr il y check matrix. i u) D r aw ! II e e n co de r c ir c ui t.

2. Consider If (7, 4) linear block code ",/rose genua/or ma~rix i~ ,,(il'ell belous

- - [ 1 0 0 0 : 1 0 1 1G '" 0 1 0 0: 1 11]

.0010:110

o 0 0 1': 0 1 1

(1) F in d a ll c od e r ec to rs o f Ih is c od e.

h) F in d t he piJrty c he ck m at rix o j t hi s c od e.

e) F in d m in im um tl J( ig M o f th is c od e.

3 . Ths: parity c hec k bils of a (8, 4) b lo ck c od e a " gillen by,

C] .. 1 1 1 1 + m z + In .

Cl znit + 1711.. III)

CJ "'nit + m3 o f-m 4

Ci "''''2 +m,+m.

I-krt In!, ' 1 1 2 , m) and m~arc tire me ss a g" b i ts .

nj Find t he ~naa tor matrixand parify check matrix fo r th is code.- ~ - - ~b) 'Find minimum u.c ight of this code.

c ) F in d e rr or d et ec ti ng ' c n pa bi li ti es o f I hi s c o de .

Cyclic Codes

Cyclic codes are the subclass of linear block codes. Cyclic codes can be in

tic or nonsystematic form. I n systematic form, check bits are calculated

' '' '" ,~ , . .. . ' f <> t. and the code vector is X =(M:g farm_Here 'M' represents message bits

'C' repre-sents 'check bits.

.1 Definition of Cyclic Code

'A linear cod~ is called cyclic code if every cyclic: shift of the codevector produces

other codevec tor. This definlUon i nc ludes two fundamental properties of cyclic

They lire discussed next.

Properties of Cyclic Codes

As defined above, cyclic codes exhibit two fundamentel propert ies:



A U the above codevectors X t, X ; ,X ~\ .. _ .11-(' in uonsvsterna I;,' ~(':';'1;n!;' ' ,"\".' I I , . : \ '

cyclic: property. Nole the pI\l.!r,dU! polYilol'tli:,1 C(i;) r(>lll·i,'''; I ,: ,: , .; " :; ,, i) ;"1

' 0 i ] ; $ ~ t V : : : ~ : ~ ! e 3.3.1 Tile srllN"rur 1 , , , I _ I / I I < J l l u l l if n (1 ' , n cvcltc (,w:',' i" Gil') -:,' ; l' " J

, '; "(, find 11/1JIll': code vectors [or the code ill IIQ/l sy,11'111171 ic jUJJ!1 ,

, ; , ' .Solution: Here 7 1 ' " 7 and k '" 4 therefore i f ' " " - k '" 3.

There will be total 2 k '" 2~ co 16 Pi ( 'ss~6'~ vectors of /' b t t e. , '" r I' ( ",,,,'

. Informatton ,Coding Techniques $r6r Control·56

1 . Linearity and 2 . Cyclic property

3 .3 .2 .1 L i ne ar it y Property

This property states that sum of any two codewords is also a val ld ('0£ il" 'l"Orc.

example let X l and X2 are two code·words. Then,

X 3 = x, (D X2

Here X3 is also a valid codeword. This property shows that cyclic code is

l inear code.

3,3,2.2 Cyclic Property

.very cycli c shift of the valid cod e vector p t C l c 1 uces another valid

Because of this property, the name 'cyclic' is gtven. Consider ..n n-blt codevector

shown below:

x '" {Xn_l ,X,,_2 r- _._", Xl" . to . }

Here X,,- I, X".2 , . .. , XI,.to etc. represent individual-bi:ts~f the codevecror 'X'.

.shift the above codcvector cyclically to left side. i.e., .

Here observe that every bit is. shifted to left by one position . Previously ) .", ,_1

M S B but after left cyc li c: sh if t i t i s at LSB position. Here the new code vector is X ' and,

it ;5 valid codevecto r, One more cyclic sh ift yields another codevector XW, -i.e.,

x~ = ( xn -J , Xu -4 r »••• ,x ] IXO,Xjl-l,Xu-2)

Here observe that x"-J is now at MSB position and . '1 : ,,-2 is at LS13position.

3.3.3 Rcpresent=tlon of Codewords by a Polynomial\

The codewords can be rep resen ted by a po lynomial.

) 'or example, consider the n-bit codeword,

This codeword can be represented by a polynomial of degree less tha,l}.,?r equnllo

( / 1 - 1) . i.e.,X ( p ) = xn-l p,,-I +XII_l p"~2 +...+Xl P +X o

Here X ( p ) is the polynomial of degr ee (n-l).

p is th e arb itr ar y variable of the polynomial.

The power 0[- 'p' represent the positions of th e code word bits. Le. ,

. " . ' . .mes sage vec tor as,

.. : ~ ",~ -..' '

atlon Coding Techniques Error Control Coding

pft-l r,m:,5~nts ~Sll

p ' ll " ( :pnlsef1 ts L S T 3

p 1 represents second bit from LS13side.

Why to rep resen t codewords by " polynomial?

polynomial representation is used due (0 following rC<lSOI1S :

i) These are algebraic codes. Hence algebraic: operations suei) , , ! . addinon,

mul lipl lcat ion, d iv is ion, subtrac tion e tc , becomes very simple .

ii) Positions of the bits are represented with the help ut f,,!'.v('t,'. "I I) II, "

polynomial.

3.3.4 Geno(,atton of Codevectors ill Nonsystcrnatic Form

Let M '" { II I , . .!,I)l_, r ... 11I1 ,1I10} be 'k' bits of message V<.!Ci(1r Then it eL i' [;,~

re.presented by the F'olynorni<11as,

N r ( p ) ' " 1Il1..! pr-o.l + lli)_} ,,1 1 +." -I- III; P + ilL.; \ \ "": -.';

Let X (p ) a'present tlw (ode-word p"lynumi.,,), II is given " l ~ .--.-.--.--.--------Jr X (p ), ;, / '> '1 (1 ') C (p ). j {.L,~.L}

Here C(I )I j , . , ! . I I ' : \' ~l Irm/ irw 1'0!VI/()I/I ia! of dq; ree 'q'. I:()r nn (lI,k) n'di,' (ude,.J o· -1 : .).

q "" II - k rep re sen t th e nu mbe r 01 parity birs. · 1 1 1 ( ;g< .! nc>ra !i n! ; 1 " , '\ i Y 1 \; ' " (:;.1 t-, ,':1' , ' 1 1 ' I ~.

G (I') '" ]1 'J +,Ii q-! J l '1 -1 + , , S I I' +

Here , 1 1 ' 1 . 1 , S q . l , " . . ,(i l are the fOill'il)' bits,

If M 1, M l ,M J .... e re arc lite o th er ilw , c, sag ('

codevectors Gut be calcu lated as,

.,\,1)

XI (I')

Xl (p)

X:'! ( 1 ' )

M ! (p ) G ( J I )

M 2 (,i) G(p)

},,13 (I')C(I') . 1 1 , , 1 so un



E r r o r c o n tr o l C o d in g

Information c o d i n g , _ T : . . . :Q : : c : : . . : . h : : . . : . n : . . : . i q : : : _ u : : . ~_.es:..,_~~_3_._5_8 __~_~~______ --_~. .. .. .. .. - --~, _ ___ .. ., - .f t~--

M '" ( 1"3 1112 ! Il l 1 1 1 0 )= (01 01 )

, Then \he nle~s~ge polynomiid will be (k = 4 in ('qll~tjon (3.3.5)),

M(; r ) In, p 3 +1112 I'" + nJlfJ + rllo

/ 1 . ' 1 ( 1 1 ) = = p' + 1

,\nd gin>J1 genefi'ltor polynornb\ is,

G(p) = r; l + fJ + 1

To obtilin non-systemi1t ic code) vectors

T he non S!'sterr1iltic cyel it: cod e is given by equation ( 3 .3 .6 ) a s,

X(p) "~ IV 1( p) G(p)

(r)~"l) ( 1 " 4or.j-l~ ~

1 ' - ' ,I~ 1 ) " 1 + jJ 2 + I'3 + P + 1

p' -l- pJ -t- 1, 3 + P 2 + P + 1

1 " of- (1 (1)).13 + 1 12 + P + 1

I

f 5 -y p-f' ~1 f)

- } " ( ) ~ !

. .. ( 3 .3 .9 )

... (.~,3.lO)

(since (1 ill 1) 1 ' 3 " . Or~ .. 0)

~

'" . l' 5. j- P Z + I' + "1

opt> + pO + 0. 1'4 +01).1 +pl +p+1

,,~ [vnorni 1 IS 11-1'"6~ The code vectorNote thilt the dcgt '~ 'e or ilclove po ynorma ~

c () rr c~ ;p O i\ di ng 1 .0 ~bove po\ynorni~! is,

Nonsyc,tematlc code vectors

,-, o 000

000 1'2

3o 0 \ 0

O O O O O O ( )

Q G 0 1 0 1 1

0010110

o 0 1 1o 0 1 1. 1 0.1

~_,~ --.-.-.----.1-----~.--. -.-----:;'~--.---o t 0 1 1 0 0 ,

\"D < : : > ~ ~

\

f) \ 0 0

o 1 G 16

G loa 1 1 1

A

~formatlon Coding Tochnlquol> J·59 Error Control Coding

. . . . . . . . ,

~

7 0 1 1 . 0 0 1 1 1 0 1 0

6"··0. o 1 1 1 0 1 .1 G 0 0 1

9 1 0 0 0 1 0 1 1 0 G 0

10 1 0 0 1 1 01 0 0 1 1

11 1 0 1 0 1 0 0 1 1 1 0

12 1 0 1 1 1 0 0 0 1 0 1

-~ 13 1 'I 0 0 1 1 1 0 1 0 0

14 1 1 0 1 1 1 1 1 1 1 1

15 1 1 1 0 1 1 0 0 0 l' 0~.

16 1 1 1 1 1 1 0 1 o _~ 1

~

Table 3.3.1 Code vectors of a (7,4) cyclic code for dIp) ' " pJ + f!'" I ,---~,

To check whether cyclic property is satisfied:

Let us consider cadevertor Xs which is given in above table as,

X~ '" (1011000)

Let us shift this codevedor cyclic<l!Iy to left side by 1 bi t posi tion . ' 111en we gc't,

X' '" 0 1 1 0 0 0 1,

From table, observe that

X' "" x, ",(0110001)TIlU~cycl ic sh it_. of - X ~ rproduces Xa. 111is can be vari fled for other cod cvectors

also,

3.3.5 Generation of Codevectors in Systematic Form

Now let us study systematic cyclic codes. The systematic form of the block code is,

x "" (k message bits; (n - k) check bits)

, '= (mk-1 mk_Z . .. .m\ mo: 'c~- l c~_:! . . "C1 co)

Here the check bits form '(1 pdtynoi1\lal as,

C (p) '" C~- l pq-l + Cq~2 pq-2 +...C1 P + Co

The check bit polynomial is obtained by ,

C(r) '" rem [ l ' q ,M ( P 2 ]C,(p)

... (3.3.11)

'" (3.3.12)

,.. (3,3,13)

... (3.3.14)



..,.v' ""Hlon.t.;ooing Techniques 3 ·130 Error Control

Above equat ion mqans -

i) Multiply message polynomial by p+ .

ii) Divide pq M (p) by generator polynomial.

iii) Remainder of the divi si on i s C(p).

3·61~~~~~n~C:o~.t~!i~n~g!...T~C::'C:I~H~lj~q~U:::'~:.::S:__ =: ~-:--~.----_-----.-'.'

J ' : ' . . \ 0 . n ' < f . . 0 ~~. ,,)u:.! ii.~nt

;1 " . . . O ll -! , . jf ( 1 ) J ~ s ' : ~ I } ' I ; " - \ - ' · : · ' J j ) - O{1 ' :

1(:'.; (11'-\ ;'1':' + pI

Q. ! @ (f ! (f)

in*" Example 3.3.2: T he g en er at or POlYI IOrl'rU1} oj a (7, 4) cyetic cOl Ic

G ( p) "" p3 + p+ l.

Find al! the code vectors lor the code in systematic [onn.

Solution: Here n -==7nd k '" 4 therefore q '" n - k "" 3.

1110ro wil l be total 2k '" 24 = 16 message vectors

message vector as,

--)

~'d,}iti(~'\":::::.~ () + 0 + I': + O p '. 0---...,,----n~",.:"'IHkr

Error Control C()uinq

. . . . . ,

; ,(i l' ~ [ )

j 1 ' ! '< ,,~ .\ O JI ,,·0 in :h(; divi:;k,n < )1 , ,; ' Min by (:(1'1,lbe ren',ail'" cr po ynomlO< t~ r "

'!'flilCt'(!J()re equ<ltiorl (3.3.14.) can be ·,'.'riItCll (\!;.,

- r p3 M(p1l < , I '.C'I-') ce {em .- -:; : ..- " '- - ' '" P " Op -t- 0. \ L ( '( ) ') J

With q" 3 the polynomhl1 C(p) trorn '''It!lllion 3.3.13 is,

C(p ) C2p1 -I- CljJ + (0

Thus c zp z + C1 P+ CO 1'2 +01 '+0

Therefore the check bi ts <Ir e

C "" (C~Cl(Il)",(l 00)

M '" (m 3 n1z nil m o)=(O 101)

Then the message po lynomial wi ll be (k ",4 in equation (3.35»,

M(p) '" 1rl3 p3 +1712 p 2 + tn lP +mU

M(r) p 2 +1

And given generator polynomial is,

G(p) "" p3 '~P + 1

T0 obt<lln p G M (p )

Since if == 3, pq lv!(p) wi ll be ,

p q M(p) "" p 3 A , r ( p )

'" p 3(p 2 + 1 )

p5 +q3

p5 +Op 4 +p3 +Op 2 + Op +0

G (p ) "" p3 +p+ l

" " p 3 +0 p2 + p+ l

ilnd

. ,. ( 3 3 .1 5 )

."

. ,-~ ~ . . . \ " " " ) ) ,

The code vec tor IS written ill system form as given by CqU(lilOll I,,' ," I~ i.c.

X (fllk_11l1,_2 .• ,!IItHIC :Cq _1 C' l _ _2 ",CleO)

X " ("'13/1l~l/l1Ii;O :[,CILol i (D 101: 100)

form. Th,·, ot !W I u,', l.: v·"iOt.'\

. . . (3.3,J6)

(for message vector 01 0101)

To perform the divis ion P~%~P)

We now have p q M(p) and e(p). Now let's perform the divis ion to find remainder

;,is division.

In lili;; CX.'IIl1pl"

Th is i s t he ! 'equi rcd cyc li c rode ve ctor s i n sys :em: tt ic

can be obtained thing the same proced\ln"

6

(\ o

5 o o 0

o 1 0 !o 1 I 0 I

1__1_ J_ _-__~ __8

-

o 0

o 1

o

o 1



Error control CodIng

Information Coding Techniques3 -62

I1 0 0 0 1 0 1

9 1 0 0 0

1 0 0 1 1 1 0

10 1 0 0 1

1 O' 1 0 0 1 111 1 0 1 0

1 1 1 0 1 1 000

12 1 0

1 1 0 0 0 1 0

13 1 1 0 0

1 1. 1 0 1 0 0 1

14 1 1 0

1 t 1 0 1 0 0

15 1 1 1 0

1 1 1 1 1 1 1

161 1 1 1

= ) + + 1

Table 3.3.2 Code vectors of a (7, 4) CY~!lC ~hOdeO : ~~~cr:tingPpolynomial

. We have obtained nonsystematic codevcctors or e sam

. E 331 1"lCY are listed in table 3.3.1. _.-'In _ x . . . , j ...--

All'n ' t /i ." it c od e p ol yn om in / X (p ) is obtained as ,

I I ' ' ' ' ' Example 3.3.3 : _,

X(p) '" C(r)+p<,,-I)M(p) , ' .. di it da/a and C(PJ IS rcmnmder

what' M (f') r q lr fs e nl :; m e ss a ge polyrromral for k g I Iynomia! G (p ) in. . . ' 'd' ("-k) M (p) by p ro pe r ge ne ra o r p o 1 '

po lY l lomw i ob /l lf l1 e d by diot wg P . li ad ' if G (p) is the f ac lorm od ul o- 2 s en se . p ro ve I/w l X (p ) represents a sys tem allc ey c rc c e

of p" +1i n m od ," lo -2 s en se .b f f ,,~t

SQlution: (1 ) To prove tha t G ( p ) must e a actor 0 P . ..) 0 1 nomial of th is codeword wi l l

Consider the codeword, X " '( .1 ', ,- 1 , .1 ', ,- 2, . .. . ·.1'1'.\0 P Y d

( 1) d it 'an be express!.' as,be of degree less than or equal to n - ,an u cr .... (3.3.17)

. X ( p ) = .1 ',,_ tP ·- t+ .1 'n _2 p~ -2 + • .• + j lP +X O

Now let us shift the codevector 'X' cyclically to left side. We get,

. X' '" ( X , , _ 2 , X II_ 3 , . . . X1,.1'o , X n - l )

111(' polynomial for this codevector can be written as,

X' (p ) == .1'.-2 p n - 1 0 " :, " X n - 3 p n- 2 + . ..+ Xl P 2 + X O P + X , , - l

Multiplying the polynomial of equation 33.17 by p.

. . p X ( p ) ee X.-t p " +X,,-l p,,-l +...+x, p 2 +XoP I

. 3318 as per mod·2 rules. VIe get,Let us add above equation and equatIOn " . .

i'X ( p ) + X '( p ) '" X.-l p " +(.1'.-2 EDXn-2) p ..-t +....+(X1 EDX l) ~~.:(xo ED'1'0) P + X, t»:I

... (3.3.18)

[nforrnctlon Coding Techniques 1·63 Error Control Coding

- - - - - - - - - - ~ - - - - ~ - - - - - - - - - - - - - - - - - - - - - - - - - - -We know that in mod-Z addition, if both the bits are sarne, then result i:; zero. i. c,

.1: , ,-2 ED1 :, ,_ 2 . . 0, x ; ~Xl "2 . and so on. Then above equaiion becomes,

p X ( p ) + X ' ( p ) '" X , , _ I I ' ' ' · j - X , , _ ,

pX (p )+X '(J I) '" .\,,_1 (p " +1 ).e.We know that by mod-2 Ildclil!1)11 'Ilks, there is no addition and subtraction. Tlia t

is if xCDy=l then x=y$;: or y",.\·G!z. This is because mod-2 .uidifion and

subtr action is same operation. Applying this r ule to above equation,

X' (p) = I 'X(p)0 . \ ' , , -1 ( 1 ' ' ' + 1) ... (3.3.19)

Thus flew codevector polynomial X'(I ' ) is obtained with the h e i ·p , ! ! , X(p) and

(f,n + 1). The generator polynomial G (p ) is of the degree rr == II-k. 1t is expressed as,

G ( p ) ' " p 1 + g q-l p 1 - 1 +.....+ g l P + 1 ... ( 3 .3 .2 0)

Let M(r ) be the message vector po lynomial of degree (k -1). It is expressed as.

M(p ) '" IIIk_1 pk-t + mk-2 pk-2 - - • .• + 1 1 1 1 P + 1 1 10 (3.3.21)

Then the product of gener.ltingpolynomirl]

and messagepolyriorn

ialgives

codevector i.e.,

X(p) = M(p) G (p ) . .. (3.3.22)

Important conclusions :

1. Here note that G ( p ) becomes a factor of X(p) .

2. Similarly X ' ( p ) of equation (3.3.19) can be generated with the help of C( p ) and

some other mess~ge vector.

3 . Under thi s condif lon G ( p ) will be a factor of X' (p ) also.

4. Then in equation (3,3,19 ) obser ve that G ( p ) is factor of X(p) as well as X'(p).

Both X ( p ) and X' {p ) are valid cyclic codevectors, For above statements to be

true, G (p } must be a factor of (p " + 1 ) also.

J£ G(p) is a factor of (p " + 1 ), then X'(p) will'be a polynomial of degree Jess

than 'n' and i t will sat isfy cyclic shift property. If G(p) is not a factor of

(I''' + 1). then X' (p) will no t be valid cyclic codevector .

( II ) To prove tha t Xip} .. Cip) + pin-k! Mip) :

111e systernatfc form of a codevector is g iven as,

X '" ('k' message bits l'q' check bits)

Here , , = 11 - k nrc number of check bits. Above codevecto r can also be written as,

X "" (lftt-,I/Ik-~ ... m Jm tllcq .. !c 'I_ ~ ... (ICO)



information Coding Techniques

The above code vector can be writ ten in polynomial form as,

X(p) "" nlt_IP"-J + ml_~pn-2 +,..+ )tIlpu-k+l + IIlopn-k

+cq_tpn-k-1 +C q_ 2p,,-k- 2 + .. ,+ CIP +C O

VVeknow that n=k eq or n ek s- o. Puftlng thosevalues of ' 1 1 ' and'lI-K in

equati on we ge t,

X(p) =: Il1k_lPk~q-l +mJ:_2p k+Q-2 + . .. + lnlP q+ l + mo p.q

+ Cq_IPq -1 + Cq -2P~- 2+... '.;.CIP + 'PQ '

Let's ;"ci1t!'~nge t~ie above equation as, .

X ( F } ~ / - , q f l l l k _ 1 P k - 1 +nIk_zpk-2 + . . , + . TI I IP + 1 T I O J i- C q _ Ip 9 - 1 + C~_2 Pq -2 + ",+ cIP +C O

in the above equation I lI k_ lp k- 1 + TT lI: _2 pk-2 +.. ,+n t lP+mO = M(p).above equat ion becomes,

X(p) => pq M(p)+Cq_lpQ-1 +C9_ 2P 'I- -2 + ",+ CI P+ CO . " (3,3.25)

Let 's define the check bit polynomial of check bits

C ::: (Cq-lCq-2 ",C1CO) as

Cq_lp'1-1 +C

1-2 p4-2+",+CfP+CO . '. ( 3 3 ,2 6 )(p )

The above equation is check bit polynomial of degree less than q. From abovel'ql~.JtioI1 (3.3.25) we obtain, .

X (p ) '" p'l M(p) + C(P ) ' " (3.3.27)

T!1e above equation gives a code word polynomial in systematic form. For this

code vector to be cyclic, then the above equation should be same as equa tion (3.3.22 ).

Thus for ,\ cyclic code vector we can equate above equation and equation (3.3.22) Le.

p '" M (p ) + C(p)

p " /d (p ) C :(p )----EBG(p) G(p)\

M(p) G(p)

M(p)

The above equation has the form of z illt '" 1. We know that rnod-Z addition and

5\!b!;'act ion operation is same i.e, if z illt = , then we can write z =t e : J / or t"" z ill {or

z '3I1; i= 0, Thus there is no mod-2 subtraction a s such. Mod-2 additiorr and

~ubtraC:lOI1 yields same result. With these conclusions we can write above equation as,

,·:jt('. (,M( )'$-1,p G(P)

,.. (33.28)

NI{}Ill'rtlior

Denominator

This equation has the form of

. ~ Rcmaindar ( 14 t :2 \Quoln:lI! e; . roc ~x "' l1 l pl c - = -r + - !

Denominator \.. ..""_ ,3 3)

,Coding Techniques

[

- -. -_. ~"-----~"-- '"" '. l.

[

p'l lv f(p} , I

C(p) =0 rem - - ' - - 1 J(p)

---~-------. -~-'~-C ( J » is check bit polynomial f,)!' "y"krnal,e cock

) is message bit polynomi,ll.

G (p ) is generating polynomial, whirll is the factor of p , . _, I

equa ti on (3 .3 .27) r ep resen ts t he cycl ic code in tystf'Ill.llll' /""" IV.,

X ( 1 ' ) = p 'l M (p ) + C ( p ) (,i ; , ; ( ' i

q => /I - k, hence above equation becomes,

X (I ') . ,. C (p )+ p{ n- J; )M (I ') t ," ,il)

Observe that t hi s equa tion is same as the giVl'l1 equ<llinn, 'iv"" ':\i'j ,. ,,:,,'c'(! :"~,

:equation (3,3,29),

'3.3.6 Generator and Parity Check Matrices of Cyclic Corlo s

.',3.3.5.1 Nons},st.emOltic Form of Generator MatTix

' ,( ' Since cyclic ( 'od< '$ arc f;ubcl,~,:; pi iil1<.'." bl(J(~k codes, gCller.lt(h :ljlt! 1',)1';)" dH:d

' lr iJ tr ice.s can also be defined (or c)'cli<: code s, 1he gencr<lt(H' m:rln,\ hll.'. Ill ,' ' .:," "f

.. ;;X'II. That means there Me •k' rows and '1/' co lumns , Let t he SC!!"I(-I ;!IO,· "".\l(;, G{j'l b\.'

given by equat ion (3.3,7) as,

G (P i ::;: pq +g'l __I,Q-l,c·"+Sl pi]

Mult ip ly both the s ide s - o f this polynomi,li oy p' i.e.

G ; G ( p) " '· pi ·' 1 + g q - : ' ; ' : ~ : ~ . ,· , : ' ~ ~ ; ' ; : ~ - , - : ' ; ) " I __") i)

and i '" (k -1), (k - 2),.",,2,1,0,

The above equation );ivf's' the polvnorniai- for th,: row., ", " ,'.l""'.lio"h

polynomials, This p ro ce du re w ill b e dear a ilc r ( h~ riiscus;;ioll OI"i.'\1 ,", <1"1;".

,Iil>+ Ex;uJlp!o 3,3.4: Obtain liil: S".';c'!,,:,"· "/llIi'i:;_:'.

,1 (7, 4) c ! Jet i c code .

Here 11 "'7, k =0 4 and q =7 - 4 = 3 p i G (p ) wil l be,

( . ' , . "

p i C (p ) '" pi+J + { , , : : ! + I"

k - ' J , : 3; i'", 2 , L 0

{or hil'en C( p )



Information Cod]n!; Techniques3 - 66


four

The generntor mat rix for ( / 1 , k) code is of siz/k Y 71 . For this (7, 4) cyclic code the .;

size w ill be 4" 7. Corresponding to four rows we have obtained four polynomials

given by, above equation. Let's write each polynomial in the fol lQ.wing way.

This is th e r;enerator matrix for given genera'to r matrix.

1))* Example 3.3.5; Filld a u t t he p os si bl e g en er a/ or p ol yn om ia ls (7, 4)' cycliC c ode . F i nd ',

mi l t i l e c od e oe c to r s c o rr e spO I Id in g /0 t he s e g e ne r at o r po ly n om i al s . .

Solution: For this (7, 4) cyclic code,

n ='7 I k '" 4 and q " '" n - k '" 7 - 4 "'"3.

W e know that the generator polynomial is the factor of p"+l. For "this

, generator polynomial is the factor of p 7 +1. The factors·of~p7.,t!·1'He

p7 +1 '" (p E91)(pl I IJpl (1) p j f f ip(1)

The valid generating polynomial is given by,

G(p ) " '" .p1 +g~-l p~-\ + ... +gl p +1 M '"

Thus we will obtain four polynomials corresponding to 4 values of 'j'.' These

polynomi<lls represent rows of generator mat rix,

Ferrow1 : ; 0=3 ~p3 G (p ) "" pb +ps +p3

Forrow2: ; = 0 2 ~p2G (p) "pS+p4 +p2

For row 3 : j",l = > pG (p ) '" 11 4 +p3 +p

ForrQI'!4:i~O~ G(p),,,,p3+T'2+1

Row 1 :0:>

Row2 = : : >

Row 3 ;:::)

Row 4 =>

Let's tr<lnsform the above set of polynomials into a matrix of 4)(7

p 6 p 5 p 4 pJ p 2 pl pO

1 0 1 0 ° ° l

i i::~~Low 1 [1Row 2 °Row 3 0

Row 4 L 0

. .. ( 3. 3. 34 )

. .. ( 3 .3 .3 5 )

(3 ,3 .36)

;'

pij- p ;.

Informal!on Coding Techniques 3·67Th Error Control Cadin

" us the degree of the genera tin' . . ,g..t Therefore the valid geneniror ' pOlynogm~~iy~oml~1 shol. l~d be 'q'. Fo r th is exarn pk' 'i = 1p+l will not b ' s or r: +1 wil) be " ,1 +,:! 1 ' ..

e a generator polynomlals 5. 1 -I- nnd!"' ' ' ' I ' . ..1.

generator polynomials for (7, 4) cyclic co(;c· .~rc,mce It!' dc);r('C' is not '1 (i.c. J). Thus

G\ (p) = pJ +p? +1 '"' (3.338)

CJ (p ) = /, ,3 +/1 +1

Ill' ;' Example 33 6· Fl.'" ind out lil~ gencmtor 1 I I111rh-lind [ind 0111 II,,: code vr.c'or~for (7 4) /' "com'spulltfins 10 G () 3. . eyc rc code. P =I' + P + I

sotut ton : (i) T b .00 tain gcnerilfor matrix

The rows of a generat '. or matrix are given by pj G(I')' Here,

p ' G (p ) '= p' d + pj + 1+ p'

i = 3,2, 1,0

and (3 .3 ,39)

an d since k - J = J

For row 4: i '"0 => )

F or row 1 : i '"3 = J G 6'J (p) '" )J + P' + P 3

p2 G(p ) = )15 + 1'3 + 1'2

pG (p ) '" p - l +pJ +pl

G(p )=p3+p+ l

The above set of pl' .. (ie k > < ) h 0 ynornials IS transformed in to a ge ne rato r mn t ri x• . /I as s own below. " ., of "ize' .jx 7

. ~.

F or ro w 2: i '"2 : : : : >

F or row 3 : i'"1 => , ( 3 3 . .1 0 )

pI . 1 '5 p- l p3 p 2 1 '1 pll

R_gW. l

r ~c " = : ; ~ -Row 2

0 1 1 a 0

~ 10 1 0

R olU 3 a 1 0

Ro w 4 0 0 a J '

cycl ic code is a

J , ,7

subclass o f lin ea r block

by equation (3 .3 .4) i.e.

code, its code "vectors can be

X :;:M G

vector and G is gencriltor matr ix . H ero : k = -4 , I.('t·~

( 1 1 1 3 m " 1 1 1 \ in D ) ' " (I 0 0 I)



Information Coding Techniques :Error Control·68

The code vector co rre sponding to th is message vec to r wi1l be"

X MHa aJ~~~~1

l~0 1 0 1 IJ

0= (1 0 1 0 0 1 1)

(t'.'otc : Here we perform 'matrix multiplication and additions are perfonned

lIlod-2 rules. i.e. 1$1 "'0,1 ())0=1, O('fll ",1and 0$ 0=0).

Thi s code vec to r w e have already obtained in example 3.3.1 and is listed in

3.3 L This code vector i s i n non sys tema ti c form, Also observe tha t generato r mat rix

<1 ::;0 in nonsystematic form. Similarly other code vectors for cyclic code ca!1/be

IJb!~il;,(:d

0.'0 tc : Here note that generator matrix is not in syster~tie form- - - - .check matrix cannot be obtained using direct method. ---

3 .3 , 6. 2 S ys tema ti c Form of Generator Matrix

The svsrernatic form of generator matrix is given by equat ion (3.3.6) as,

G '" [ I < :Phq]"n

The I';' rem: of this matrix will be repres ented in the polynomial form -ns,

f / ' ; ' row of C ; = p" -I -t R/ (p) where t = 1,2,3, . ," k

' " (3.3.42)

. .. ( 3 .3 .4 3)

!_~:'5 divide pHby J generator matrix G (p). Then we can express the resul! of this

division in terms o f quot ient and remainde r. i. e. ,

p":! .' Remainder-- '" Quotient + -.----G (p ) G (p )

. .. ( 3 .3 .4 4 )

Here l e m a inner will be 1 1 polynomial o f deg ree l ess than ' q', since degree _of G (P} is

'f The degree of quotient will depend upon value of l. ' ".

..~ ".., :~+:,Remainder '" R, (p )

,lnJ Quotient = Q, (p)

.~~.

, Coding 'techniques 3· I i ! . l Error Control COding

C 1 . 1 . 0i\lltJf' p.:1A4) wil l be,

p'n-I • {(t ( 1 ' ) ,

(; (p ) ~~ Qr (p) *G(p)

1 '' ' - ' ' " Q, {p )G (p ) G R, (1') an d I'"' L 2 , _.. k

that if z "" y 01, Ihcn z ( {, Y ,~I or z (f) I >= y. ThJI is 'H",d-2 ,lddillUf1 nne!

yields same results. Then wecan write above equation ,IS.

As we have stated in equation (3.]A~) (110 "bmw c~p,~tinn r'~ril','s(>l1t.~ I'"~ ,-"W "I

tic gene~ator matrix. The above pr?cedure is illustrated in " < . : , > , , l:X;In-:pk,

,'- Once the gCf \Cf i l tOl ' matrix in systematic; form is o bta in ed ( }" 2n p il ri ty ,+,f'('jc 1ll,Ii'-lX

. :' can be obta ined as per the p rocedure di scussed tl Iasr section. I'h';d ' > . ' , L ! < l 1 - 1 1(' , : l u " ' r . 1 , < :

. this.

r; (p) <> f.'~ +)i + 1 '. A / so l in d 0111 IiI,' pauly dtcck 1I1111/i,\.

,S(Jlution: (I) To obt"in generator polYll(JDlial

The Ill. row of gene rato r matr ix ,Il glwn u y "'1u<lliol1 (3.3-"» ,,; ;,

p" : + 1 <, (p ) := Q, { p)G (p ) and t '" I, 2 , . . . ,A -

We aregiven that 1I", ,7,b=4 and q"'n-,I:,~3.

11.e above equation-will be,

p 7-1 + R, ([1 ) ' :0 Q, (p ) (p) "Jl i l) and I"l,2, :t .j

t 'JiU\ t '" 1, the above otquation uec(\ I111 '-"

" ,,6+R ,(p) '" Q,(p)(p:l·;,p.f,l) i.LI-;·)}

. - I' To obtain R, (p ) and 0, ( p ) tor 1'1 row..1

, TIle RHS or LKS .oj this equation represents l,r WI.' I!f 'i.\ ', ;u::,~a~l:: ;:('l;"'~-:l!("

: matrix, WI:! have to - find QI (pj. 1 '1 c'ity, .:'ij1s:1tion (~1.3.'l .'i)we know ti",t 0t ip) t-, \('(i

. . dividing p"-/ by C(p). Hem to (,tA.trn Q/ ( 1 " ) we h.'lV ' (" ,.I'\·"~l' i: " by

' ' ' ' '1;3 -+ - p + 1

. t' .,--,. )

.c,



_/~ ..."'--='---------·--------

., . . . . . .

Information Coding Techniques- - - - - - - p 3 + P + 1

3 -70Error control Coding ~'

~ Q uotient

p' +p+l)p6 +0+0

p O +p "' + p~

E f J E f J ffi

Denoles ~ . 4 3 + 0 + 0mod-2 ~ddition 0 T P . . P . ,

p-l +0 +P' +.0

' " e G 1 @$~pl+pl+p+O

p3+0 +p+l

@ @ ffi @

p 2 + 1 f-Remainder

Q, (p)=p3+p+ l

,-'

Here

and R ., (I') sx p 2 + 1

Put ling those values in equation (3.3.49} we get, .".'

p6+p2+1 '" (p J+p+ l)(p3+p+1 ) , 1 > 1

f) above equation representsThe RHS or LHS ( actual ly both are same) 0 tne a ve .

o f generntor ma tr ix i.e.

1,/ row polynomial '" pb + p2 +1

ii) Other row pOlynomialS, . 'd b ther roW polynomials arc ODI:il.LI"~~",!ji

Using the same procedure as dlscusse a o v e , 0 .

and they are given below

t =2: : ::> 2, , , 1 row polynomial"" pS + pl + P + 1 I1=3;:: : .. 3'd row polynomial = p4 + p2 + P

t "" < 1 : : ; : : . 4 ,1 , rOW polynomial ; ; p 3 + P + 1

iii) Conversi~n of row polynomials Into matrix . hfed' tgenerator matrix ass own

TI,e above equation can be trans orm m 0

- - - ~ -( , ~ ~ p 4 ' pJ p 2 . pl p O

Row2 [ O O ~ 1 ° ~ ~ ~ ~ ~l~ : ~ ~ ~ o ~ " ; ~ 4 ~7

~r•.J

R o w l

G

•

Information Coding Techniques 3·71 Error Control Coding

This is the required generator matrix in systematic form, The code vector can be

obtained f:01n equation (3.3A) as

x := MG

Let's take any 4 bit message vector and find corresponding code vector. Let's take

M "" ( 11 13 1 11 2 I nl mo)" (1 1 0 0)

X == MG '" [1 1 0 01 [~ I : ~ 'I ; ~1o 0 0 1 0 IJ

== (1 1 0 0 0 1 0) ~:

This code vector i s obt ained by performing matrix multiplication and mod-2

additions. Observe that the same systematic code vector is listed in \nble 3.3.2,

Using the same procedure other code vectors can be obtained.

. I I) To obtain parity check matrix (H)

We know that G = [ L k : P h q ] J: x ll

from equation (3 .3 .6 )

The P submatr ix can be obtained from equation (3.3,52) ,15

. .. ( 3 .3 .5 3 )

The par ity check ma tr ix i s g iven by equat ion (3.3,11) as,

H == [pr : I ]q if)(1i

pr is the transpose of P submatr ix and

I q i s t he q x qidentity matrix.

'By taking transpose of P subrnat rix of e 'l ;. :a lion ( 3 . J _ : · : : < ' , . he parity check matrix

be, [ 1 1 ] 0 : 1 0 0 ]1-1",0]11:010

1 1 0 1 : 0 0 1 3.7

' -- -- -v-- -- -- -- ~

. V T 1 3IS the required parity check matrix for (7, 4) cyclic code in sy~ematic form.

~f

t

\



Informotion Coding rr ch niquc s '3·72

3.3.7 Encoders for Cyclic Codes

In thls ~('(Iinn '. '·L '" iii d i~ClI~~ the

show« lhe bl(Kk d i.'.,~~1ll of ,1 );l'l1cr,llii'.l'd (11 , 'k) cyclic

d raiv (,11()(kr~ Me ~ll l)wn in Fi! ; ' •. 3. :>.1 .

IT~v!s"" .c ni;).fiops.The,. ~,~ 7'''lCcled in 5aquenli~ Iorder 10",~k~ ~

I> ~ ~ (etjl slcr. The contents (J ihr: 5hift· fegi: j !.er are shined rrorn ' input h)

! outnu ' when c lock pulse is~ppl ied.

They repres Q~Iclosed pa:h if g~ , andopen palh (nooonnec tion) i tg ~ 0,

,.~.Q)-------_ T!'1)5C symbols represent ~1()d·2 acdltion

Fig. 3.3.1 Vilrious symbols uscd in ~ricodcr

Ol'er.,t ion Till" I,","db,lci-:. ~\\"i tell is ( i, 's t dosed. ThO'output switch is connected 10

nV','slg<:: input. ..11 t hu sh: It '('Sister, are inili~ lizcd to < 1 \ 1 zero state. The k :m!5Sagc bits

.'rc sh ifle d lo t he t rrm sr n i t l' l' l ,1 S well as shifted into the regist(?rs.

,\iler 111 , : sh i f t 01 . V IY1CS,lgC bits the fl'gistcrs contain' q' check bits. The ff'edb~ck

';\Ii kh is now op':",ed ~11 0 ou tplll !;,wi tch is connected to check bits position. With the

l'\U:, s il i f t. dw dl(;ci-; L,its M( then shilled to the tr~nsmitter,

l -Ien: we 0\):;(''''''<': Ih"l the block diagram performs the division openlLiol\ :1I\d

",,:~cl';LI('S the n2m~ ind<!i ' (i .e. check bi Is), This r ema inder i s s to red in t he s lr lf t rcgist~r

after all I\WSSi1g~ bits il re s hi f led 011 t

. . ~:,~- ~---.-'q.,om I

yf f i - - . t f : ! J - - f f i - ~.-~.-{t2J

1I C!\oc~Lils DutpU '

__,.,. switch

rfli /~~'_TO transmitter_______--,._..-,;,1

Messat)e bits Messa oelnput bilS·'

Inf~rlnati(lt. Coding Techniques

-3··7:l lorrore'0nlr-ol Coding

l'l~ E ~" mJ lln- 3 .3 .1 1 : D giS :, '7 Ihr cw!..n fi,l :hc(7, 4) t;!j.diC ('C{it X(rIrrJ,'tl" ;'l,'

G (r) : III ")1 +- II.Ilri verify ir s ilJ!"rillirm fr r (1! ' iy m e ss a ge u ec u» :

INovj[)"c.-2003, ~M."k~)

Soh'hon ' f' h~ W 'n er ll to r p ol yr lQH li al is,

and

r; (1 ' )

G (p )

f! l + ()p~ ' I' + I

pJ +g < p t+S<1 !+ 1

On compiuison of the two equation we obtain,

,'\1 find g, ",0

,m d q. . . . . . . . . . .

With these V,IIllCS the block di(!gr~m of Pig., 3 .3.2 will be as shown in Fig. 3.3.3 b,'low.

Fig. 3.3.3 EncodN for (7. 1\) cyc'llc code for C O (p) ""P l + P + 1

Sin,1.' 'I" ?' , !!Wrt' ,.r-,: <Y iiil'·-Il of',' ; in shiH r"~,,isl('r to bold ch~'ck bil~: 'I (l ,11, , '1. <:".

Sln('1! g '~ ' " 0 , its link is not (enne(\",d. Sl e- J, hence its link i$ (fll'lflr.dl'!d. f·!r.-\\-, I"I'~'

V l . ' r H y the operation of this r-nccdvt lor )nll_~5nsC vector t\l nIn:1llil!'r\ i '1(1) ,.,{Il OU).

Table : 1 . " . : ' 1 ~;h,l'.\'~;11«' (\ll'l")'C d .;hil: l,.,.,~is:(>r~;be fore and a fter shi ft s.

T"l,ie (:>,3::1) ~11(l\\"5 tlu l ;d Ihc' "n,; 01 L1:;\ mes$i\ge bit the rcgi~t"r td 0. ;11\)15 , ',rc

r: i . . 0,. ri __1 MId r,; "' O. The k<.'d t>"d; ,,,, ', (c h is opt: "rod an d \Ju:p u t ,,,,, it, h ,".. i,', " ,- "I l u

check bi ts posit ion. The check 'bi:,. il~" U,,"\ ! '11if\ ,'d to the !ra .n~l1.ith~r Tlv: d."d·. bjl~;

, :

; , , ~ ,

- : : ~ ¥

Fig. 3.3.2 Encoder for systcmllt1c (n, k) cy¢llc codo

II. '.



tnforrnatlon Coding Techniques

Input

mC~:1Iiage

o

3-74 Error Control CQI.Ung

" : , _ , :

.. .

bilm

1 2 " I : ' r 1 " ri ro" T O

0 '0 0

R~gi$torbit outputs "ftar 'hlfl

o o o

o

o

0 0 o e o e i-t_ , - ,-

1 WOW1"

0 lID 1W 0"0

U 1 mOUD"1o o o U){)"'0

. - . . . . t . ut message M '" (1100)able 3 3.3 Shift register bits positions or inp .

·.c . , '. '- ' '. =, r' and C o eo r o . The following table ill.ustra tes theare shifted as L1-'2, c] 1 k bit W know that the code vector is,

opera tion of message and checl 5.

eI.

'1 0). X (JJl~III::mlmoc2'I'O)"'(11000

Shift clock Message bit In

Feodb.)ck Output switchTransmitted.switch position

bitso<l loH

on m(lssag(l

0 On m(lssago

message 00 on

message o0 on

off check bits

off check bits

2

j (I

4 0

~.

6 -

off check bits

, ) 'I'c codaancodarable 3.3.4 Operation of (7,4 eve I .

ta ble illustrated how the bits arc transmitted when inputhe above .. ~

( 1100 ) .

shift

;,

lnforpa.ti,?,n Coding Techniques3-75

Error Control Cc.dingI

3,3.8 SYn:dromo Decoding for CycliC Codes

In cyclic codes~'also during t ra{1SlTIission Some e rrors ' may OCcur. Syndrome

decoding can be used to Correct those' errors. Let's represent the received code veclor

by Y.lf'E r epre sents an e rror vector tn en th e correct code vector CJI1 be obt;lin(.'d :l~,

~. X '" Y tB E (from eq ua tion .1.3,.29)

dr we can write the above e quation as,

Y '" X EfJE

.. (D.5'l)

". We ..can Write the abovs equation since it ismod-2 addition.

In the polynorniill form we can write the above equation as,

. . (3J,55)

, Sinceyep) ' " X(p)+E{p) ..

\'X(p)

yep)

M(I')G(p) the abol'e cqwlljon will be ,, \

M(p) G(p) -r £IP)(3J,57)

Let the received polynomial Yip) be divlded by G(p) i.e.

Y {p ) Q f' t Remainder- '" uo lPn +-----_G(p) .. G(p)

(33.SH)

In the above' equation if yep) '" X(p) i.e. if it does nol contain any error then,

X(p) Q . . RemainderG (p )' '" . uotlent + G(p )

SinCe X (p ) '" M(p) G(p ) , Quotient will be: equal to M (p ) , 1 I 1 e l remainde r \viJl be zero.

This shows that if there is no error, then remainder will be zero. Here G{p) is iilC!or of

,code vector polynotriia i:- Let 's represent QU9tient by Q (p) and Remainder by R (p) thenequation (3.3.58) becomes,

J ; ; ; ~ Q (p) + ~ f ; ; " , (3.359)Clearly R (P ) will be the polynomial of qegree less than or equal to q _ 1. Mul fiply

sides of above equation by G (P ) i.e,- -" :.~ r ~: ;: :. ,~ Y(p) " , Q(p)C'(P)+R(p)1 .~ . . . . . .

.On comparing equation 3.3.57 and above equat ion 3 .3 .60 we obta in, .

'. M (p )G '(P )ffJ 'E (p ) "" E (P )G (P )E fJR (p )~.: '

, . '. . . _'.E (p )'" , M (p )G (p )f'D Q(p )G (P )tD R (p ), . - . ~ • • ~ - ' 1 ' " , ' , , ' _ ' , _ • :

Jbe'ahove t .'qUation has all mod-2 additioJ;lS. Therefore 5ublrac!ion ;lnri i ldd it ion is

(3.3,60)

E(P) '" fM(p)+Q(p)J G(p)+R(p)

... (3.3.61)



Information Coding Techniques 3 - 76

This equation shows that for 11 fixed message vector and generator polynomial, ,1n

error pattern or error vector' E depends on remainder It For every remainder' R' there

will be specific errol' vector. Therefore we can call the remainder vector 'f( ~:>

, syndrome vector' 5', or R (p ) = = 5 (p). Therefore; eqU<If!O~ (3.33)) w j IIbe,

y (p) S (p)G (p) '" Q(p) + G (p) '" ( 3 . 3 .62 }

1 i.us the syndrome vector is obtained by dividing received vector} ' (p) by C (1'),

i.e.

. .. ( 3 .3 .6 2( a», I - Y ( p ) ]s (/1) '" rem --

.G (p).

3.3.8.j Block Dinqrarn of Syndrome Calculator

Fig. 3.3. 4 s hows the generalized block diagram of 11 syndrome calculator.

, ;~~~L-- iJ-: :~~.~,"J~Output

syndrome

Fig. 3. 3. 4 ComputnOon of s yndrome for 11 n (n, k) cyclic code

In above figure observe in figure that there are' q' stage shift register to generate' '1 '

hi t syndrome vec t(~r. The operati ons a s [ol lo ,ws -

Ini t ially all the' shift register contents are zero and the switch is dosed in position

.! The n~feived vector Y is shiftcd bit by b tl i nto t he shi ft r eg is te r. Th e contents < t I f nip

Flops keep on dOimging according to input , blts of Y and values of gl, g2 etc. After all

the bits of Y arc shifted, the ' '1' f li p-f lops o f shif t r eg is te r cont ai ns the q - bi! syndrome

vector. The switch is then closed to position 2 and clocks are applied to the Shift

t' (;,gi !;t er .The outuut is a syndrome vector 5 ' ' = (Sq_l , S~-2! . .. . S I 5Q)

.;::,r .:.:Jrnp'" 3.3.9; D~siS" ,i 5y,,,;";,t;e c al cu ln to r f o' r' a (:7, 4) c yc lir. H am min g c ode

!\eJl~f{)!crl i·y liIl' I ' O I Y , ' I I ) . ' I 1 1 1 1 1G (I') x: p3 + P + 1. C n tc uk uc th e s y n d r o m e jor Y == (1 0 0 1

1 (/ I) (NovJDee.-2003, 4 Marks)

Solution; Fur thegivcll code 11",7,k=4,1l>=ll-k",,7-4=3

The given gcncmtor ) 'nlyJlorni .l l is,

C(:» '" ;>3+ 0(12+)1+1

1 10-' '''' ~"'<~"'"

3 - 77 Error COYltrnl (~fJuinD~~,__~----(1formntic:HlCoding Techniques

Jlind C(I') '" J ' : ' 'S: J' ~ " '. \,11 '' '' 1 g('ner~lized equation.

l N il h 1 1 1 : ': .. . .. ; d ut " !l'I': b1 t I ,: k d i agu ' l1 1 of a syndrome calculator f0r (? ' - 1 ) .:ydj( ':1);/('

wilT h.' as ,h(",'" in Fig. ~.:1.5.

Fig. 3.3,5 Block diaqrarn of a syndrome calcu lator for (7, 4) eye l ie cod., witJI

G(p)=pJ+p+1

Qp(!r2ltion and (,xplal'liltion

Til(' swit;:h i~ kept ill position 1 unlil :111the '7' bits of received vector) ill'" siHf[~,i

illln the shin I·~i; j, ;t~r .The flip-flops of the shift rpgfster contain synclr<.1Ilw 1(,([(1! W]"'TI,111bits of 'Y' are shi fred. The swirch I« lhr'n closed to position 2 ilnd dock pul .ses are

:lpplicd to shirt register. This r.;ivcs syndrome vector at the (lutput. The fnl:,, \ v i;;:.~1.,t}I".'

illustrates the operation of this syndrome cnlculator for received vector Y =0 (1 () 0 1 1

o 1). The table shows the contents of flip· n aps with every shift,

The II-,bl(' ,h')\\'$ tll,,[ ,1 1 Ih,' <'lid of 1;'st shi ft t Il <! n>l :~ i: 't f. 'r co.pt"!'\:; arc

( Su 5 1 . 0; 2)~(11 0),

;-

" .

. '

~':~ Tabla 3.3.5 Calcul;rlion of syndrome for Y ox (100'110 I)

... l

. ~ . . >~ .

,'!':' ,;':,



Error Control Codingnfonmitibn' Coding T~e=c~h~n~jq~u~es~._.,_ __:3~-!..7~8_.L_ _~-----.---- Information Coding Techniques

Error Control Codinga -79

3.3.10 Advantages ..and.,_I?isadvan'tages of Cyclic Codes

As we have seen that cyclic codes are the subclass of l inei ]r b lock codes, they havesome udvant.1ges over noncyclic block codes as given below _

Hence. the calculated syndrome is,

S= (525)50);(011)

3.3.9 Decoder for Cyclic Codes d f that

Pattern IS detccte or .. r I ted then an error

Once the syndrome IS ca C'U a, . id '(j to the received vector Y, then rthi' e rror v ector IS ar C f d

particu lar syndrome, W en t'IS. This decoding operation can be per or~egives corrected code ~ect~r at the output.

by the scheme shown 1I1 Fig, 3.3,6.

Ad~nlagos :

i) The error correcting and decoding' methods of cyc li c cud,,: :; a re simpler and

easy to impleme;1i. These methods eliminate the storage needed for lookup

table decoding. Therefo rs the codes becomes powerfu l and efficient

2)-'The encoders and decoders for cyclic codes are Simpler compared to noncycJiccodes.

3 ) Cyclic codes also detect erro r burst that .span many successiv e bits,Input

4 ) C yc lic codes have welJ defined' mathematical structure, Hence Vely efficientdecoding schemes are possible. \I i

In sp ite of these advan tages cyclic codes also have some disadvantages .

Disadvantages:

1) The error detection in cyclic codes is Simpler but error COrrection is l it tl e

complicated since the combillationallogic circuits in error detector are complex,

To avoid such complex circuits some special cyclic codes are used which arediscussed next.,cceived

vector input

Buffer registerCorrected

vector

rI

I



Information Goding Technique!) :3 ..:1)0 Error' Co.\trot Ct'~dlntl-----------_.:...------._;,.--~. - - ~-.-.-.---.-~~- . .

Ulock kngn,

Mess3ge size

Paritv check size

Mini'num distance

II= 2'" .- 1 symbols

: k symbols

: 1 1 - k '" 21 symbols-

: d mift =:: 21+ 1 symbols.

Here observe that the minimum distance Is gn~atE"r than the. number of pittily

symbols. Hence this code is maximum distance separable (ode. These codes prov id~

\vi,~" ';lnt;~of code ra 1135. Efficient decoding techniques are available wlth RS codes.

3.3.13 Golay Codes

Col,,), code is the (23, 12) cyclic code whose generating polynomial is,

G (p ) '" !}II +p9 +p7 .I.p" +p5 +p+l

This code has minimum distance of dmin '" '7 . This code can correct npto 3 errors,

13ut Cola y code cannot be genemlized to other ccrnbtnationsof n' and k.

3.3.14 Shortened Cycl ic Codes.~ . -.~-

FDr the (n. k) cyclic well', the gcnc r~ l·or polynomials nrc divisors of x" + , . The

puiynomini xl[ -l- 1 h8S ver)' fe,\" divisors. Hence there are very rew generator

1)<..11nomiais i\vailah!e. This difficulty can be overcome by shortened cyclic codes. 11'1

sho rtened cyc lic codes, the last 'i' bits out of '1'1.bits of the codeword Me padded withLc',(h This 'i' bits are not rransrnittsd. Only ( 1 1 - i) bits of the codeword are

transrnit ted. The decoder pads 'I' zeros to tile received codeword. Thus for (n, k) cyclic

code, (n - i, k - i) shortend cyclic code is genemlcd. This code has ~ll the

,Iii \',1]1 ti~s:es of original (n, k) code. Its error detection and correction capabilities are

s.une as the original (n, k) cyclic cede.

3.3.15 Burst Error; Correcting Codes

J n the l:>receeciing sections we discussed the codes whic h detect and correct errors

(Jc(urringipdependcntly at different bit positions. Burst errors occur as a duster of

Cn0!5. Cyclic and shortened codes can be used to detect these burst errors.

The burst of length q is defined as the vectors whose nonzero component~, are

confined to ·ct" c onsecutive digit positions with nonzero first and .l~st digits. For

example the vector x '" [0 U 1 0 1 1 1 0 1 0 1 0 0 O J has the burst of length 9. The

q-bursr- d·~·,( ;lr: \. ;, ·. : , (, code is capable of correcting the bUfstslcngth q or less. The

following theorem gives the number of paritybits required by burst error correcting

code.

The q-burst error correcting cod(~ must have at least 2q parity check digits l.e.,

n - k 2: 2q ... (3.3.63)

a!1orlC!'ldlng Techniques 3·81 Error Control t:odin()

we (tin :m)' that the (11, k) burst error correcting code GIn corr ec t the burs ts,# 1 1 - A :

. !¢t!~h upltt " ·2 ~ ' .. ' ll lb 1 "C '! :( > {N~ :; '< he' ~'Pi-~\.'! bound on t h(' bur~! ('trOr c'~n"", (jn~;

bl1t!y of (n, k) (ode i.e. ,

/1 -I:q s -2.- i:; 3.1'-1)

The burs t e rr or corr ec ti ngdfici,-'"c)"

is rknol,..dby

I.: IIis

gi l 'e1> Wi.

2 '1Z '" II _ k

... (J.J.6S)

To detect the burst of length d, then the check bits must be,

n=]: < : d

Thus tl,e check bits must be nt least equ,~l to d.

J),.j>- Extlrnple 3.3.10 Consider / lIe (I5, 9) cyclic code g(·lIaalcd by

G{p) = pf, -I I'~ .,.l,i'l'.l -t- I

T hi s c od !" I I I I ~ (I IlIIrsl ,'!"lOY OHI,·diIiS ni"lily Il l ' '7

t;qidcllcy n J t ui s c o de .

Solution: The given code 1 1. 1$ ' I " " :1It is (15, 9) code. Hence

n '" 15, k ' " 9

11\ (!burs t e rro r cor rect ing dfkiency \ :' i'.'\'€11 by 1·(1\1;1\iO;\ (3.:;.(,:'1 r.c.,

2 '1z ", -. ~-.

n-k

Putting values in above equation,

2>:3z '" ---- '" 1 or lOG%

15 - 9

. Thus Ihe burst error correcting efficiency of this code is] O ( ) " · ; ,

3.3.16Int~~looV!Mg ·of Cod&d D••tator Burst Error Corre c tio n11u! . block ~odeo ~~k,C~ff:l.~ r , t;. i.-y<~t.~\I~I': t . ' : .~n·' .1 1 . ~ f f p c . i f, ,(" \ \ h,:"" d~l..·cn vL· .J ~ t: L~.'

channel are srntistical1y independent. For example, the ~'rrnr5 in ;\dd iiive whi tc

~ussian noise (AWCN) channel are stJtisticl'111y independent But there :In' '.'nrne

channels which produce burst errors. For ('>;nm)'le the Ch~l1i\<~I" I>:,,·;nl\ in "il; i '.1 i t · ,

f~'ding. Because of multlpath pr()l '; ig~t;()n !he <;i'~:I'1"1 f:·,di".I; "("CIJC~ .\n,1 ;\ rrr-.: i, H, ., .cr·" <

at the receiver. This phcnon1cnn ,1('pentL-, upu;; t ,m . . char.1ctcri~:ic: ,,] tilt" c"ti.".I·.~·· r·.·.

burst error can bc produced when the (L,(a is ,;for,·d 1111m~gl1 .. ti,: 1,1("" nr ,!:I. i :"..

d~ie('t:5 In the magnetiC materia! creates clu- tcrs of error." if 1Il<. l·I",~ "'l!'·.' ,'fe



Information Coding Techniques 3 - 82 Error Control CodinJ:j:

optimally designed for statistically independent errors, then they cannot correct ·th~:·

burst errors. In this section we will see how interleaving of coded data is used l e i · < :

correct burst errors.

A burst error of length 'b' is a sequence of b-bit errors. A systematic (n,k) block is·· ..;

capable of correcting' the error bursts 01 length i J : ; } - ( /I - k). The burst errors are

converted to statistically independen t er rors by interteaving the coded data. Then the'

code designed for independent errors Can be used to correct these errors.

rig. 3.3.7 shows the block diagram of the system which uses the interleaving

technique t o cor rect burst errors. The channel encoder encodes the data by some (n,k)

block code, The coded data thus has codewords of l ength "n'. This coded data (rom

the channel encoder , is given to the block interleaver.

The block inter leaver has 'rn' rows and 'n' columns as shown in Fig. 3.3.7.

- - - - , ..--

Output

Fig. 3.3.7 Jnterleaving of coded data to correct burst errors

1 1 1 1 1 s t he codeword b i t s ar e sto red in the interleaver row wise. The numbers

indicate the actual bit numbers as t:hey come from encoder. The interleaver in the

above figure thus stores 'm' codewords of 'n ' bit s length, or total 'rnn ' bits. As shown '

in Fig. 3.3.8, the bits are given to the modulator column wise . The modulator then

transrnits these bits on modulated carrier over the channel. At the receiver the

demodulator gets t hese bit s back by demodulation and so ft or hard decision decoding.

The deinterleaver then stores these bits in the S<liri{"tormat as shown in Fig. 33.8. rh~channel decoder then reads the bits row wise with one codeword of length 'n' at a

t ime. Because 01 this recording of coded data by the interleaver, the error burst' of.

l ength ' rnb' is b roken in to 'm' bursts of length 'b'. The (n.k) block code then has

corr ect these small bursts: of length 'b'. By Inc reasing ' ' rn ', t he l ength oj the bursts

, be further reduced, The block code which uses an interleaver oj size 111)<I, Is

called as interleaved (mn, mk) block code.

''information Coding Tochnlques 3 _83Error Conlrol Coding

Read out bils to modulator

'I t t I t t t r1 8 15 22 29 36 ...

Irnn {)

~ 2 9 15 23 3037

,..mn-5

- a 10 17 24 31 3n ... t-----rnn-4

-+ 4 11 18 25 32 39 - . ."'11-3 .n row- 5' 12 19 26 33 40 '" mn-2

i~ 6 13 20 27 34 41 .. .

mn-l

- 7 14 21 28 3S 42 ...rnn

i--n-k parity bits k data t J i l " - _ _

'S

Fig. 3.3.8 A data r ecord ing in the block inlerle<lvci:

In this section we. discussed a block inter leavor . Another •..• , .., . ... .· convolutional intede<lver S _j • [uti '.. ,}P~ of mlcdl',Jvl'r (;1iI,'d· t h . . UL 1 Lonvo utior ia] Jnle rl eavcrs or ( ,r lCoJc: ," ; n r ce '1'.', , .•. .e next sectidn, -. c ,t 1 .' (: ll .: i~ CL I I II

.17 Interlaced Codes for Burst and Ra ndorn... Error Correction

De fi ni ti on o f i nt er la c ed c o de

Consider an (n,k) block code. Wl1en A number of codc'.vords of this cod" :1]"0

then the code becomes (ArI,AJc) . This code is C"Jllt~d illteri"<;:,,d code.

Ho w al interlaciuo takes.place ?

Consider (15 8) c~~ L t 'Is tlI r- e 1 iree code vectors be as follows.

x = = (XI Xz X3 ••• XJ1 X15)

Y = (Y l Y 2 Y 3 ... Yl4 !lIS)

Z '" (ZI Z~ Z3 "Z 4 ZIS)

: these three code vectors are transmitted directly, then we get the sequence as,

2 XJ " ,X14 XI5 YI Y2 Y3 ' ''Y14 Yl5 ZI Z2 Z3 " 'Z14 LIS

11 individual bits of th th od· e ree c e vectors are interlaced, then we get the sequence

" . ( 3 .3 . 67 )

Y t Zj Xz Yl Z2 Xl Y Z3 3 Xl~ Yl4 ZI4 XIS !In Zl5 (3.3.68)

, is interlaced sequenc H· have JJ. . . e. ere we ave interlaced tJ .Th i <I. c tree co aevectors, Henceere ore UIC interlaced code will be of d" U ,

. unenslOn,

( J . . n U ) ; ; (3XIS,3x8)'"(45,24)



lnformaticn Coding Techniques •. 3 ·84 Error ControlE:r ror Control Goding

:..---~-"""""-----------~-----------""-'-"-'-'"" ~" .. "

atlon coding TechnIques 3" 85

T~1I5 is !Ivaburst

or ~ bits. These

bits arc inerror

CF,C (1'dl"~ ?

lc codes i\ F e yt'rr much ' - ' I . ' \I.1h I~for error detect ion because p r' t",,() f< ' ' '' ' '' ' ', . ; :

. M a ny combinl'ltions like'ly errors (,lin b e d eh xt ed with the hdr o f < : \, < ,, 1 ;, . ''{"i(',>

' fmp!ernenl 'l tion of encoding and error detec tion c ir cui ts i s pt '; '1(lic,lIi If F,qosit)!l.'.

, E r ro r d d ~ cf io l l c lI l' fl l! il il i~ $ of Ilinn"! (II, k) ( : r : . c evi l, , , ,

~MiJt;::,~·f 'CRC C( \de s <Irecapable of detec ting -

1)All . e rror burs ts of length (n - k) or less.

,-.ii) Fraction of error burs Is of lenglh equal to (II - k + 1).

. iii) F raction of error bursts of lenglh grc[lt1Jf than (11- k + 1).

. 1 }o -

.".Y Iv) Al l e rror combina ti (ms of (d n,l" - 1 ) or less.

' ; . : \ I ) If genuntor pol),nolni[l I C (p) hns even number of coefficien Is, tll"n ;,:j vr;.ur

","patterns with odd number of e rro rs can also be detected.

Commoll ly uecd CRe COd6

HO i I. ! l ll lr s! e rr or s r tn : c o rr ec te d by intcrtaced c odes "

Consider the in tcrlaccd sequence of equation :,l,3.M}_.

XI Y T Cj ~ . j N l l Y J z : J " . . . , . , . • . . ' . . ~14 Y1 4 ; :1 4 X 1 S ) ' 1S Z1 S

j

During transmission o f th e in te rla ce d sequence, a burst of error tokes place

< :! lo w n a bo ve . Four successive b its a re in error. A t the receiver, w hen this sf'rjucnce

rcn:i vcd, it is converted to it5 noninterl~(cd natural Iorrn. That is given b y{X l uc ct io l1 ( 3 ,. 3, 6 7) , I t is show n below . .

Three commonly used CIZC codes are given be low:

t . ~ _An error

b u ti s s pl it i nto ~__)

$~ngicor double errors

" Cl{C" 1 2: C(I))"'l+ I'oJ .f'~ '1'1,:1 ,!,iI +1"11,

eRe .. ]6: G (I') '" 1 + , ,1 oJ .1I~ ,.!'~(',

with n-k=11

/'Is shown nbove, till' single burst of 4 digits i s sp li t into single or double e rro rs .

E nO l" r ic ke /i ll S (w d corrn;1illg c a pa hi li ti cs o f ('.II, 1 - k ) wde :

L('t t he (JI ,,~) cod e corrects 't' digi Is. Then the in ted accd code en 1\ C(1rF('Ct

wmbinntiof1 Qf '(' bursts of Jength .1 - orIcss.

Why cyc lic codes are more suitable for burst error correction?

If the code (iIJ) i s cyc lic. then i ts interiaccc i ve rsion (AII,il.k) is also (relic' Ii C (ill

is the p ; E ' l 1 C l " < 1 t i n g polynorninl of ( n , k ) code, then C ( p ! · ) is the generating polynomiar

of (!,I1.\k) code.

Therefore cn ccdi liS find decod ing of inter laced code is also possible uging shift

reg~sters To. obtain the decoder of interbred code, each sh ifl rcgis te r st ' lgr . - . .Ol ' {n,k} r

CYCliC code is replaced wi th iI. stages w ith[1 lll_;~" ..... .; "ther connections, nC'cm.I~;e ,of·,

all the above reasons, cyclic codes arc morc·s\.iitatJ'L ,ur detecting ,lIlU correcting ,I:",

~: Cyclic Rodundancy Chock (CRC) Cod"

\. Definition : A cycl ic code which is used for NUl' dr/allOiI pttrpose 01 '1y i s c al led· ' ,'

\ cyclic red urid ancy check (eRe) code.

with n-k'""lr,

with ".- k "' I G

A ll the above codes contain ) + I' ~3 ~ prime- fac+or. CRee 12 code IS ~"" ,d lor (j·~'i:

'.d\~l'ilcters, CRC-16 and CRC-JTU are used for g·bil char,1ctcrs.

A p pl ic nl io ll s ;

1) eRC codes are used mainly in Al{(l svsterns for error detection.

\ ',)l,")JKY arc also used in digit"l su b~cl,b'~T lim's.

~~~~~: .~/ ., Conca tena ted Block Codes

Z;Z.~l. 0 l 1 bi rm r y c ode s

Till now we have discu~srd linear hlo(1( c()(ies which aHe bir,;u'}' ill n.,t'll,'.

Nonblnary cedes also c:>:is~, The l1orlbin:ny , ;; :'~11<:!onsists of !fw ~,,'1 "~If fl ·" . .. 'r''':~1hrodewordt.. The individual c lements of ('\e. codeword MC ~d{,.Ltq;")_" 'hr:,lil'! ' ,,'l' ,e: of

q symbol~. (0 ,1,2, .. .. q-ll. Normally I ) ' " 2t, means k information bits can H 'CI

'I,(r"tr.' ' < 1 '

.':different symbols. The length of the nunbinarycodeworCi .is repw"'~nted b," N', TMi'

:'. 'number of -information symbols ill nonbinnry codes are represented by K. Thrs!; K

..information symbols axe cncoded into '1\:' number of symbols by the ncnbinary (ode.

· " 1t ~ P'_ZWW?



Information Coding Techniques Error Control Coding- 86

De fi ni ti on o f c o nc a te na te d b lo c k c o de

The concatenated code is obtain ed by combining two separate codes. Notmallyone

code is binary and other is nonbinary to (arm the combined concatenated code.

F ig . 3 .3 .9 shows the block diagram of the system which use .' ; concatenated block code .

OuterNonbinalY

Ino~uCoocaronateo

~

encodercode

encodert,.;(I(t(t

Modulator

(N . K ) ( n, k )

Fi[]. 3.3.9 A cornrnunlcation system which uses concatenated code

1\$ shown in fi1;urc, the nonbinary code (N, K) is the outer code and biliary code

(n. k) is the inner cede. The codewords of the concatene ted code ,HC formed by

subdividing the block of kK information bits into K gWllpS. Each such group is called

symbol and it consists of k bits. The outer encoder encodes the K symbols into Nsymbols. T!1€ inner encoder encodes the k-bit symbol into n bit codeword. Thus the

final codeword is made up of N symbols of n bits each. This is called concatenated

block code of length Nn bits. This concatenated block code consists of Kk number of

information bits. This is equivalent to (Nn, Kk) binary code, The concatenated

codewords then modulate some carrier in the modulator and transmitted over the

channel. At the receiver side, the demodulator generates the transmitted code words

back from the received Signal. The inner decoder then makes hard decision on the

group of every n bits. These "n' bits are then converted 10 k info rmation bits using

minimum distance decoding. These k information bits represent one symbol of the

nonbinary outer code. The group of N such symbols is used by t he outer decoder to

get K information symbols. The outer decoder also uses hard decision minimUJIl

distance decoding. Soft decision decoding can also be used for conca tenated codes if

the number of codewords arc small.

Minil1HUII dis tance and code rate

111eminimum distance of the concatenated block code is dOli" Dmin• Here dmm is

the minimum distance of the inner code and iJm;n is the minimum distance of t J i C

outer code. Similarly the rate of the concatenated code is Kk/Nn. Th is is cquiv.llcnl to

product of code rates of inner code and outer code.

",, ..~

Information Coding Techniques 3-B 7Error Control Coding

m. . . Example 3.3.n; Till: g e n er a to r po l Y li om ia l o r a (15,1 )( ') I Hall1l11illg code is ,"ivL'n UI! 'S .rJ '" 1 + x + x~. DC(ldol' encoder ant! d .'

- 'YII rome c al cu la t a r fo r llti-: cod e 11 . : ; " " , , ,s!IS icmnt ic f11(III, '

Solution: This is (15, 11) block code. Hence,

n 15

k 11

and q = n - k =,15 - II

i) To devalop encoder

The gener ato r polynomia! is,

e(p) = 1+p+p· =r' +0,,3+01'2 +p+1 < ~and G(p) zz p' +sJpJ +g.?P"+,'?I /I+l

Comparing the above two equat ions ,

SJ ;0, S~ =0, Sl =1

Fig. 3.3. J 0 shows the generalized encoder for (n k) r -I -.. _enco i {J . ' eye IC coc c. Based on this the

, c er 0 us CXJmpJe is shown in Fig 3310 In this f" b r

[our (Ji"-fl(lPS to hold f h k b" '" IS 'gun: 0 serve that there are..- our c ec Its Since g -1 't- J' k .

_ ' _ 0 I > " . • 1 - , I~ Ill' IS connected. 6\,(S 1 ._S J - , renee thei r links nrc not connected. .

JCbeck

bits

""---,

_~-Totransml:terMessago m OUlput '.~'

bils switch;, .t:

?1'19. 3,3.10 Encoder for cyclic code G(p) ""1 + p + p4

ii) To del/clop syndrome colJ<;ul.l!O(

n ,Th,e g l'nc('lliZ{'d syndrome calculato r fo r (n, k) cyclic code is shown iF' 3:13

lc ~}ndr~;nll' calculator for this example is shown based on Fig 334 Tl n _ II}' "' ...ca cu la tor IS ~hOWl1 i n Fig. 3 .3 .1 1 . ', ' ". 1C syndrome

.~

Error .Conlroi Co(ling



Information Coding Techniques 3·88

KCCCH..'ed

tHis

Fig. 3_3_11Syndrome calculator for cyclic code O(p) '" 1+ P + " .L1 thl~ above figun~ observe that the links of g~ and g3 are no t connected. Only 81 '

I:; connected The Iou r rlip- f lops contain the four bit syndrome vector. The ~'''';'~1.~,-_,

: c 'i l: ; 11 1'(1~iti(m T til I all the bits of the received vector are shifted into shift re!!L~lrp",,;"

("" _ ,; ,) , TI ,,,,,\ thu ( !il~"f 1ops of the sh ift register con tain 4 -bit syndrome. The switch is

tllen moved to position '2 t o t ransmi t the syndrome vcc tor. i->." , - ' ~ ~

)))* Example 3.3_12: Construe! 11 oyslcmnlic (7, 4) c yc lic co de using tllI~ genera/or

" "I Yl lo li li ni / :( x) ~ ,\"3 -+ v -l- 1. W/iatllrc the error c o rr e ct in g c a pab i li ti e s of i bi s c od e!

COIi, , :mc l I h,' d ec od in g 1 (l lJ le t uu ! J or t ir e r ec ei ve d c od ew or d 1101100 , determine l /r~

Ifnir~Hlil t e· d d a ta ( ('o rd .

Solution: For this cede n '" 7, k '" 4 and q xx 3.

i) To determine gener<Jtor matrix (G)

Tile 1 ,1 , row of t he generotor mil t- fix is g iven as, (equa tion 3.3 .47) ,

p"- ' -" RI ( 1 ' ) = = Q/G(p) and t'" 1,2, k

IV c have obtained the generator matrix based on the above equation in example

3,3 .5 for the same "genemt ing polynomia l: It is given by equation (3.3.52) as,

p 0 0 0 1 0

~ j

0 1 0 0 1 1G

1 0 0 1 0 1 1

t. O 0 0 1 0 1 .-~.;.:.~<.:~:~;"

ii) To construct coocve ctors

,':·:-4~;··-~ .. -, ~

. .. ( 3 .3 . 69 )

The codevector can be obtained from the generato r matrix as,

x '" M G

-, ~'-.-...~.

tton ,Coding Techniques . 3' . . · B . 9

l . . l '!t U~ . t< ; ke t hO ! f!'1(>!;:i<1gevector ;,5 -M "' 0101 Then the code-vector- 1m n · i, '1\". '-;,.:.

1be,

r0 0 0 (J 1 1

(I r a 0

; , 1'" 10 1 0 ~~ o{) 1 0 1

L O 0 0 1 0 1 1

= [0 1 0 1 I 0 01

-rhus tIle ('hcd, hits <Ire 100, We know Iha!,

G = [ f~ : P , • • )

Bence P sub matrix can be obtained from eqtlaHon 3.3.69 lIS,

f l 0 1 1

P " ' l ~ ~ ~o 1 t J

Hence the check bits can be obtained by

C =: MP

ml ffi m2 (j) 1 7 1 : ,

m o (f) 11 11 ~ In

The check hits can be obtained (or < : t i l the coc\evc:,:lors wi~h I:"" h,_,ip 0: "~\(Wl'

equations. Table ~.3.2 li!.rts a il the :>ys temat: l codev(: cto[ '$ , We will 1;(:( lh,~'1~,1JH' (!:'_-( k

bit s of Table 3 .3 .2 f rom above equat ions .

rh) 'trror corrAf'I.ing capa.bllltyIt i s c lear f rom ' fable 3.3.2 Ilt~t,

Ilm!n '" ItI!(X)l~~.. '" J

Hence th is code can' detect upta two erro rs and correct one error-



Information Coding Techniques Error Control Coding

.~.,

iv) To ob tain parity check matr ix

The pilrily check matrix is given as,

H '" [PT: f q 1

HT = [.~]

i,

Hence from equation (3.3.76) we can write above matrix as follows:

0 1

1

1 0/_(T

'" 0 I . .. ( 3 .3 .7 1 )

0 0

0 0

L O 0

. ,.v) To obtain decoding table-

The decoding table can be easily prepared from H1·. For the block code, each row

.,' of N T represents ;j syndrome and unique error pattern. This we have discussed earlier

in linear b lock codes. Table 3.3.6 shows the error patterns and the' syndrome vectors.

Sr. Error vector 'E' showing single bit error Syndrome vector Comments

No. patterns

0 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 + - , ,, row o f H T

3 0 0 0 0 0 0 +- 2M roW olHT

4 0 0 0 0 0 0 0 +- Jill row ofW

5 0 a a 0 0 0 0 ... 4'" row ofHT

5 0 0 a 0 0 0 a 0 . - 5' " rowO(HT-

7 0 0 IJ a 0 0 0 0 ....6'" to« ofHr

8 0 0 0 0 0 0 0 0 ( - 7 1 1 > row o/HT

Table 3.3.6 Decoding table -

.,

Information Coding Technlques3·91


JVi) To decode 1 10 1 1 .P 0

Daterm!ne syndrome

Let the received codeword be,

y = [1 1 0 1 1 0 OJ

Y (p ) '" p6 +p5 +p J +1'2

The syndrome vector is given by equation (3.3.62).

j.e.,

S (P ) :: rem [ ! ( T J ) ].G (p)

We know that G(p) = p3 ~.p + 1, H : Ience c! U.5 perform the d ivisicn f [

equation. yep) can be written as, v <.1" 10\'('

yep) = p6+p5+0p4+p3+p2+0p+O

And C(p) can be written as,

G ( p ) = p:: ' + Op 2 + P + I

The divi:;ion is as shown below:

p3 +p2 +p+l

pl +Op l +p+l )p 6 t-p.'i +Op 4 + p J + p~ + 0,, +0

p b + O p s + 1 '4 + p 3

E l : 1 ffi E f) e

- - fl .5 + p4 + Op3 + p2

pS +O p4 +pJ +p2

E E l E D ill $

p 4 . + p 3 + O p 2 + Op

p4 + Op 3 + pl + P

<: ] ) E D E B iIJ

p3 +pl + P + 0

p3+0p<+p +1

Ef) E D ill (]1

Remainder 4



Information Coding Techniques -Eft-Of Control Coding

Thus t ! 1 C ' rema inde r i s,

S{p) p"-+-Op+l

i.c. [1 0 1

The syndrome is non zero. Hence there is an ('(TN in til\' rl'(L'in'd (nd<:word.

Determine error pattern for S =101 and correct the codeword

Table 3.3.5 indicates that the re is error in the firs! bit, i.c,

F = [1 0 o 0 0 0 0]

Hence correct ccdevector is,

X Y ( D E

(J 1 0 1 1 0 0)$1 0 0 0 0 0 0)

ti l 1 o 1 1 0 OJ

Thus the t i'i lr lsrni tled codeword is X

codcvector in Table :u 2,

o ] 0

:!)~ EXJlllple 3 .3 .1 3 : D cl cl'I JI i! lc tllC encoded l!i,· ,, ' ;,lSC [or lilt' /'o/lo1UiIiS 8-/ ' 11 dnll'l codes

IIslliS I l l e /oiinwiliS C/((: SI_'IIi'l"Illillg pol!Jllonnnl 1'(\') "'.\". + .1 "' + x~

iJ 11001 lOU in 010J1l1l

Solutior i : Here C(I') '" Y" 1 - . \ :\ -l- XU hence q "" 'I

" p -1 +p 3+ pO

a nd l en gt h o f r n es ka g e bits is k = = 8

q n ... k or n r= k + q

B + 4 ~ 12 bits.

i) Consider the first 8~bi t data 11001100

Mess< \g~ F8 ' , "J \orni<\! will be,

Let us find pqM(p). Since q '" 1,

~,'.~-,

. . .wo . . '

Information CodIng TeCrmlqull,~ ... 3_-_9_J ErTO( Control Codlnu

pi + , tJ~ f P I I

!'~+ J ' .lj J ) ; ; ' i 1 : ; r w : ; : ( ) ; ; 9 ' : ; ' i J r s + ~ o i ;-7-;r;-;;-:-iii·,5·~~or7+ Op J - _ ; d r T ~ j ~ - i 1 r ; - ' ~)

I~,':f~_::_0l:'~~,~ J ' ?_, . _

o o o o pi. " ° 1 ' 5 1 ap~ + Op3 + 01' J

pI , +..._: :~J '~+ Op3 + ,p2

O+p~+Op4+0p3+ p2+0p

I'~-, p' + 0(13 -1- Op2 + I'-- - . . . . - . . . . . . .~.~-;----- ---~~--.-. ~-.

o + P 1 + OpJ -1- p 2 + P -+ - (1

p4 + p 3 + Or] + O.c._:~

o + p 3 + p 2 + P +1

Thus the rcma Hider is

C("l '" 1'.1 I- /,. ! -l- 1' + 1

' the re fore check b it s a re ,

c '" (1 1 1 1)

x "" (Message- bib. Check bi ts )

x - " , 1 1 0 0 1 10 0 : 1 1 t 1

Thus there ore 12 bits in encoded rnCS5<lge.

il) th(' glvnn 8-blt data Is 01011111

111e correspond ing message po Iy.,,,m ia Ii::.,

M(p) '" r' o +pt +P' + 1"2 "pl +1

,J

Th,'N'fnre p1M(p) will be,p4M(p) __ p~(p~+P'\ ip :i ,~p~~~I" ; ])

Now let us divide plM(p) by G(p), then we g-f't,

. »



2) Codevector for M '= (11 1 1 J

r0 0 0 0 ] 1

X • [ll "J ~] a 0

f J . [1 J1 0 1 I "j J]

- . ' 0 0 0

3) Codcvecror for Ai = 0001

. r0 0 0 0 q

X - [000 1 J ~I 0 0

~I0 '" [0 0 0 I 0 J 1]

0 0 0

4) Codevector for M =1000

X " [l 0 0 O J [ ]

0 0 a 0

j J

1 0 0

0 1 0

0 0 0

" " [1000101]

Information Coding· Techniques 3· 94 Error Control Coding

::...

p4+pl+ l)p lO +O p9+ pS+ p7+p6+p5+ p4+0p3+0p2+0pl+D

piO+ p9+OpB+.Op1+p6

o + p 9 ~ _ p8+ p7+0p6+ p S

p9 + p8 +Op7 +O p6 +ps

(J 0 p7+0p6+0p5+ P'+OpJ

p1 + p 6 + Op s + Op' +/)3

o + p6 + O p s + p' + pl + op 2

p6 + p5 + ( } P 4 + Op ) + p 2

o .+ p5 + p4 + p3 + p2 + op

p5 + p4 +O pJ +O p2 +p

o + 0 + pl + p2 ~J!~. ->Thus the remainder is,

C(p) = p3 +p2 +p

There fore the ')check bit s a re ,·

C= 1110

Therefore message 10 systematic form will be,

x = = 0 1 0 1 1 1 1 1 : 1 1 ] O.

) )) ... Example 3.3 .14: S ug ge st a s uita bl e g en er ato r p ol yn om ia l fo r 11 (7, 4) systematic

cyclic code and find codeoeciors fo r the fol/owing daia words :

(i) 10]0 (ii) 1111 (iii) {)()()1 (iv) 1000.

Draw an encoder arrangement for the above code find explain its operation. Construct

the decodino table for all single bit error pal/ems and deiermin« 'he data va:lqrs

transmitted fo r the foll()Uljng r e c ei v ed v e c to r s .

(i) 1101101 (ii) 0101000

_. Solution: Given (7, 4) cyclic code. Hence

1 1 '" 7, . k =4

q 1l-k=.7-4 ..3

" (I) To obtaIn tho gonorator polynomial:

The generator polynomial will be the factor of (p n + 1). Le.

(p1 +1) '" (p+l)(p3 +p+l)(pJ +pl +1)

The generator polynomial must lie of the deg ree 'q', As given by above e Q 1 J a i J i o n . , . ' ! f .

two generator polynomials are possible: pl +P + 1 and pJ + p2 + 1. Henel let us-, the .generator polynomial, .

Information Coding Techniques3- 95


.(;(p) '" pJ +['+1

(ii) To obtain the generi.iIOr matrix in systematic form:

We hilv e. ob tain ed the gener ator matrix in systematic form fE 31 .. orC(p)=p:l+p+lin;:'\., ., A. It IS calcula ted (sec equat ion 3.3 .52) as,

p0

°0 0

1 1G = /0 1 0 0 I ~ l j0 I 0 I

o 0 0 1 0

(iii) To delerminethe code vectors:

1) Codcvector for M := 1010 ..:

We know that X '" M C. Tbcrefcre,

[

1 0 0 0 () 1']

X : : [ 1 0 1 O J ~ ~ ~ ~ ] J ~ 1 := [1 01 ( ) a 1 1 J

o 0 0 0 lJ

Ob,scn'c that the same codcvcctor is ohtilincd for M '" 1010 1"1 I bl 1 ~.., ._.. "gene t· J • I J C, ..,,-, ~l"( Hwera mg po ynorrua is same. '



tnforrnation Codin;"! Techniques

i~ Encoderarrangament;

Encoder for G (I ' ) = p J + J7 + 1 is shown in Ex. 3.3.6.Its operation Is 1 I1 w c xp !'1 in ('( .\ ,

il < this example. .

v) Decoding table :

The decoding table for G (p ) = pJ + P + 1 h i give), in Ex 3 . 3 ' . 1 0 and table j.3.4.

vi) To decode the given, vectors

\ .; : '1) decode y",nOll01

T () d ct cr mi ll ~ I ll c s yn dmm c

Hence the polynomial of received vector wilt be ,

yep) = )16 +p5 +p3 +[11 +1

SVI1c[:-(,me can be calculated by equation 3.3.62 (a). i.e.,

S ( p ) ' " rell! [ y e p ) ]Gel') _.--"~ - . . , . -

Ld us divide yeF) by G(p}

p3+(J~+p+l~, ..

----~--/ , ' < -0 ,, 2 -Ip + 1 ) 1 ) 1 , +)1' ' I - O p 4 +p 3 +r' +O p +1

.T ' 6 -t- 0 p 5 + P 1 + P J

C D E D E D

. ; . '

p5 +p4+0p:l+p2

pS +O p4 +p3 -v p?

CD m @ lD

p ,1 +pJ +Op2 +Op

r~+OpJ+p2+p

ffiffi ffiI ffiI

p3+p 2+p+l

('3 +Op2 + p+ l

p2

Thr<; the' : ,\cier is,S ( p ) ' " P 2. It can be written as,

5(1 ') - p2 +Op+O

S 1 00

'.~

:J ·97 ....---.~--~.-~.. ----,-.-.-~--..-.~

1 0 d t' la m il w I /U ' N 0. !! ._ r e ct or

F rom tbc (j(~C(lding j.,blc 3,:~,'\ui'!:'f 'r¥(" ll'i\1 kll ' the s )' l1d, ,, ,, ,W DI S 1!j;I, an ('I ror

E= OOO()IOO

To rlclt'rmul,' the corrvc! (0,1,"1.'('[1(1"

Cor rec t codevector i 's g i\' en as ,

Yff iE

[I I 0 1 1 0 1] (l) [0 0 0 0 1 0 OJ

1 1 0 1 0 o l

Note IhM t his i s one o f t he sys tema ti c codevec tor In table 3.32 for Ivl = t un .

x

2) To decode Y = 0101000

To deienvin« tIll' s,VIHfrOIllI'

Polyr\orn i;1I for this reccivvd vt.'du'· b','con1(':"

Y ( p ) s: I':"- 01)' 'I'c ; (Ji'"j- 01 ' + 0

Let us divide) ' (p) by C (1')' i,,'.,

J I)

pJ +O IJ~ + J ' +1 J P 5 _I,-op~'''~:r'·l-,:il;;~n~.i:j-i;·;o

p O +0/-,1 ~p' +/ , 0

Ij) (1 ; li'

J l " ' " 0 , , + ()

Thus the re~uinder is, S (p ) ,";,2 -lOp - I- ( )

, t, I . . 1 . "" '1 ' ,. , , ", - :; "Jl 4·~j'rOr i n the J~(:I,~t)r;,"c~(~ \.o'~~' ·ttnSince the 5Y'1'.rome I,f, noriz.cro. ~ ...

To dt!lr;f'rrritle CfrOI vedorLI 3 ~ l " t f t' r: Jro ....l ct r:~ " \ \" ) 1 .1 , ; " ~~ , (;rn1f

Frorn the decoding t~"! .J.~),ouserve ,na or ne 5Y·,,- rnmv ,. ~,

vector lb,

E == OUOD100

To , /e le nn ln e t he c o rr ec t c n de ue ct or

Cortett codevector is given as.

X = Y(f )f

",/



'\,I

Information Coding Techniques3·98


= (01010 00) +(0000100)

= 0101 100

Note that this is one of the systematic code vector ~ntable 3.3.2 for M =0101.

. _ , , '

Example 3.3.15: Wily a te c yc lic c od e- s effec tive in d ete ctin g error b ursts ? T~ e

""""'1.~(' W()]OOHJlOis . I~ !Jc transmitted ill (I c yc lic c od e w it h II generator l )ol Y l 1oJ I IU lI

S(x) '" .\' + L

i) How m i lr lY c h ec k b it s d o es t he e n co d ed m e ss a ge ,c o ll ia il l ?

ii) O ! Jl nl ll t he t ra ns nt li tc d c od ew or d.-.-'

~iii) Drlllu e/lcoding a rr an ge me nt t o ob ta i n r ema ind e r b i ts .

i u) A ft er lire re ceive d w ord is d oc ke d into tile de co de r i npu t , tuhat should b e t il e contell t

fif t he r cS i,! < 'r s to re » ?

Solution: gi l To determine number of checkbits:

To dt'lrrl1lin<, s iz e o f I lu ' c od e

The given message is,

M '" (1001001010) i.e. bolO

The generator polynomial is,

G(p) := III +1, hence q:=2

Therefore 1 1 ' " k + q '" 1D.Thus this is (12,10) cyclic code.

Number oj dl~ck bil s

Tho: encoded mess"ge contains '(]' number of check bits. Here q= 2 check bils will

be pn_:' ::]~l1t.

.'. ii) To obtain transmitted code word:

To obtain trilnsmitted codeword we have to perform following steps:

< I) Divide I" l M(p ) by G(p ) .

b) From remainder determine check bits.

c) Transrnitted codeword will be X =(M : C).

a ) T o d iv id e 1 "1 M (p ) b y C (I')

iNe know that I v! ' " (1 001 001 01 0)

Hence message po lynomial wil l be ,

M(I') := r " +O p8 + Op7 +1'6 +OpS +O p4 +p3 +O p2 +1'+0

.' Information Coding'Technlques 3 ·'99 Error Control Coding

Here 1:02 He~cc pq M(p) wi!.l be,

p q M(p) "" p 2 ( p9 '+ Op8.+0p 7 + p 6 +Op5 + O p 4 -I- p:l + 0,,2 + F + 0)

~ pll +Op10 +Op9+p8 +Op7 +01'" +p5 +01'4 +p3 +Op2

The genera tor polynomial is, G ( p) " "P 2 -I- 1

:=p'l.+Op+l

p9 +p? «p» + f, 5 + ;,4 +pl +r+ 1

1 '2 + 01 '+ 1) pll +O p10+0p9 +p8 +O p7 + 01'('+ 1'5 +01 ' , 1 +-1'3 + 01'2 +O p+O

pll + 01'10 + p 9

ill ffi (D

p9+ p B +Op7 .;;.

p~ + 0,,8 + (,7

< D ff i E D

?B+p7+0p6

p8+0p7+p6

1'7 + 1'6 + p5

p7+0p6+p5

p6+0p5+01"

p6+0p' +p~

p1 + pJ +Op~

p " + Op· ' + 1' 2

p3+p l+D I - '

p3+0p2+ p

17"+1-'+0

p" + 0 1 - ' + 1

P +1

From above division, the remainder is,

C(p) := p+ l

b) T o d et er mi ne c he ck bi ts

The remainder is C (p ) := p +]

check bits, C ..(11)

Information Coding Techniques 3 -100 !?,r6t ton'irol

..~_ ,

~ __ n_C_o_d_!n_g_T_c_c_h_n_iq_u_es 3 _- _ 1 O _ 1 __:_E~,_:_r~~.r~ ,~~QI. ~,~,~I.~,~.g.' : p .,



c) To obtai» code 1(',)/',/

The ~y"ll'mJ til.: cvcl ic codc is );i l'cn <15 .

X (M :C)

_, l001(J0101D: 11

iii) To draw encoder for obtaininq rem<linder bits

The! remainder bi ts ~n' chcr k bits, that are gener"ted by the encoder. The generat

~ )( ) 1 Y rHHllj.~1 is gi vcn .:1$/

( ;( ,1 ') p ? 01 '_, _

1

("111c1 G(p) ,, 2 ·i·gt v 1

g, = 0

_".11 cucoder (,111 be' ( " l L , ~ , l ; ned {mm Hg. 3 .3_2 w ith q '" '2 arid gl '" 0_ It is

L l ! , · : : C f1

d!,~;-j r-u~~.:.lilU

III

~ ~ l C h ~ d - ~ -I bits~ - _ - r T O lransrnltter

<)1" 0

(10 c c n n e c t lo n

~.,..~eSS2JgG M C5 sage

bits input bits

Fig. 3.3.12 Encoder for G ( p ) = p 2 +1

The shi !: register T1 ro contains remainder b:l{s-;", -:':ck bits "ftt:r all 10 bits of

:1H!Ssage are entered,

Iv) C onten ts o f reg ister itt decoder:

The syndrome register con til ins syndrome after ,,11 bits of received vector are

clocked into the decode r input . Depending UpO1 the received vector, .syndromc i"

c"lcuJ~tcd_

. ~ K ? l r n t , . ! _ f t " ' ~ ~ ~ . 1n: fi'h ,r .~y:it(in'oti( (/.;1) r.y~Ji( CPt/,' ~pi!Jr gc.tt.-r~~r(H" ib)!ljrr",. . 'lio,r

(xl + xl + 1), d et er m in e I h~ ' d at « l'~(I()r' fr{IJI,;mill.'t! [or I i i" _ : ; , 'lo l l' u 'g 1'<",",,'11;,·,/ ;.~., 'U':,

iJ 11OU01 ii) 0101000 iii) OO()1100

r/,~iflS symlr.rnrrf , r i '{ ' ;{,J i J l .~ /"Chll / '111< '- C o n: /M r . . thr Icdl l l i'il l " Ivill! '"''.1'''''11''' iIi:r.I,'.',','l(',i

dr .c iMOI I r i lle b f fSC. td(co il i llg .

ThIs exarnple can be solved fhr ouqh fol lowing stepsi) Determine gener<ltor matrix (G).

i l ) De te rmine parity chec' : matrix (H)_

iii) Dete rmine decoding table,

iv) Determine syndromes for received vectors ;H,d obtai n (OLT('d II :'f1~." i' ivd

vectors.~.;:

. , . - ': '; ' :: l) To obl1jin w~n(Jrator malrix(G)

The III, row of the sy5t<:m~tio.: gene:r, l \OI ' )'1al;ri>.is givC'1 l as,

1'''-' + R , (I') '" Q, ( I') G ( )') And !". 2" .k

Here . i t i s g iven thnt '!~' 7. k =.J and '. I ~ ;1 . .. ;- ' " 3,

or

Wf) Ci1n write nbOVlt cqu.u lon as,

1 , " 1 i\1 (I')

G(jijl - -C (p ) " ~ Q, (1 ')

p"-I _.. H! ( 1 ' )

G Tp ) - " 0 1 ( I ') -I - ( ; ( ~ / > Y1317;»

~,

'" "

Note t h~ add iIi,m~ in ,,11{)\,!~ t:,! \1 , 1: " )IlS ." I f' n Ii)t!· 7. . HLIKe r H.' '. 1 - , , : '

x '" y$. Here note that R, (p) i~,the rcrn.unrler obtained by J ividing /,,,,

II) To IM,lill I ,olyllonrilli Jar Rou: 1 (I ')

With t" 1,oquation 3J.78 b('COIlW5,

v:' h' (-,)i:.._-z: '0, ( P I' _ ': .: .~ .G(I)) ... (, {;. j

.Wlth II'" 7 and putting for G (,.,) -r r ·1r . ' · I

p 6 "I", (/,1--------,- ::0 QI (I)) < . _ - - - : , - - - -

pJ + P " + 1 r I I:'-

'. ~

i

1I

I




Error ConlrolCodlng

p3+ p2+0p+ l p6

p~ +p5 +O p4 +p3

pS+Op4 +p3

1 '5_+1'4 +01" +p2

p.+1 '3~f j l

p 4 +pJ + O p 2 + p

p 2 + P - i- Renlflinda

Here R, (I') " 'p7. +1'

Hence first row' polynomi~al~w_:_n_l_b_e:__' --- 1

/' '' + RI (p) = 1 '" +p! + 1 ' : : : > p6 + Op s + 0f'4 + 0 1 '3 + p 2 + p +~__

N To a/lini/ l p ol yn om ia l/o r R ow 2 (I'" 2)

Vvith !~2 , equation (3.3_72) becomes,

/",-2 __L-, This division is given below:

( ; ( p ) _ , i- ' J + P 2 +1

p :J + p: ' -r 0 1 ' + 1 p 5

p' 1_ P ~ + 0 p~ + p z

p.j + 0 1 ' 3 + 1 '"

p.J + p3 + 0 1 ' 2 + P

pj +p2 + v

pJ+p2+0p+1---.....-----

P ; 1 (- l<.r llU!i llr i< 'l'

Here i < . " (1')=p + 1

Hence se co nd ro w p oly no rr :.,l -_,-illbe,- ~ - - - - - - - - - - - - - - -flS -I- R, (p) = 1'5 +1 ' +1 ~ _O_c~p5 +_3!~~~~+Op:' + 1 ' ' ' ' : ' :

Information Coding Techniques 3 -103 Error Control Coding, ~ - - - - ~ - - - - - - - - ~ - - - - - - - - - - - -c) T o o bt ai n p ol yn om ia l fo r R ow 3, ( t ' " 3)

With I"' 3, eq\lilti;~n (3,3,72) becomes,

p"-3 1 ' ' '__ = , th is d iv is ion is g iven below:G (p ) pl -j. p 2 + 1

p+1

p' +p2 +DP+'l)?-------

p~ +p~ +O p~ +1 '

p3 -j. Op 2 + P

p3+P2+0F+l

p 2 . . . P + 1 i- Remainder

Here Rj ( p ) = p 2 +p +1

Hence third row polynomial will be,

p ~ + RJ (p) = = p.j + P 2 + P -i- 1 ::;;> ['--O-p-b-+--O-p-5--I-

p-4-+-O-p-3+-1-' -. -+-p-+-l 1

d ) T o o bta in p ol Yl lo mi al fo r R ow 4, (I = 4)With t = 4, equat ion (3.3.72) becomes,

P'1-4 p3

, this divis ion is given below:G(p)=p3+p2+1

1

pJ + p2 + Op+1 ) p . . ~ - . -p 3 - t ' p 2 + 0 p - i - l

P ~ + 0 p + 1 i- Remainder

Here [<4 ( p ) " " p 2 +Op+1

Hence fourth row polynomial will be,

~ - - - - - - - - - - - - - - - - - - - - - - - - ~p J + R 4 ( p ) p 3 + p 2+0p+1:::>

I ·O p 6 _ : 0 ~ 5 - r - O p 4 + p 3 + p 2 + 0 P + 1 1

( e) T o o bta in g en er ato r m atr ix fr om r ow p ol yn om ia ls

Let us write the four row polynomials obtained in part (a) to (d) together- i .e .,

I", 1 =:> 1'1 row poiyncmlal « 'o + O p s + O p 4 + O p J + p 2 + P + 0

I " , 2 : = ;. i"J r ow p o ly n om i al = 01,6 + p 5 + 01'4 +0 ; ;3 + Op 2 + 1) + 1

Information Coding Techniques 3 ·104·3 ·105

Error Control (;udi,1g ':



t '" 3 ::;:) y d row polynomia l : :.: 0p 6 + 0 pS + P 4 + 0 P ~ + P 2 + P + 1

I", 4 : :; :) 411" row polynomial = = 0 p" + 0 p- ~ 4 0 p~ .,_,,3 'I'f'l 4- Ojl ,11

Now let us convert above polynomials into a matrix.

,,6 1 '5 p 4 pJ p 2 p I p O

Ro w 1 [1 0 0 0 1 I 1 0 1

G = Ro~u2 0 '0 a a 1

Ro:» 3 0 0 0 1 .11j

Row 4 lO () 0 0

ii] To obtain parity check matrix;

0) T o o bt ai n i' submatrix

' lYre know th,ll G =[It: Ph,!].""

Here k =4 and I) '" 3 ; ;; ;: ,G ~ [Ih4 : P~ > < . . l ]

Com r~d I1g abo" e eg U.1lion wi th equa lion (3.3.73),"---

Pp = = 1 0 1

l~~~l: I..M To obtai), /-[1'

r» lI J "'-'I J

- 1 - - . ) '

11 n:~'l '~ ' _ H X ' I

I>u'lting values in above eCjtt~tioll,

/ - i " i '

~l

: i0I1 1 ,

I-r' ! 1 0

~j

! 1 oII() 1,!_ V 0

dc'Corling table:

\ ' \1( ' ~l",_rh'l:..:lgcner:l!or matr ix 10 f ., d H r. Becallse row S uf 1 /1 H!!'J";,,'I\~ :;.\'",ir""h·~·

(l'rrOl' plllh)rTlS. The rows I)f 1 fT G!1l .be wr it ten dil'.'cl!y I . , , , , , , "'I'\1.";,,,j,,:,

ials and idenHly !1'1~ trlx. This is illustrated in table J.3.7_ II. IS !,,, ch-n,J .111;

- . ~ - - - .. - - - - - ~ - JError p~ttern (l:,) ,

'-_--I----------t---------ld-"~"-t-I:-r,-m-.,,--I-rt-X+------ ... -'-----.- - - - J0 0 0 0 1

o 0 0 0 0 Io 0 0 0 i

I

o u ij 0 ~

' .:.

~i

1" row of I·{'

3" row or Hr 0

4'" row o r Hr 0

,." rOW 01 1,.,

2'" 10·... o! t" ,

00 i, i.e.L--___j----------"-----.----.--- .....-' "'-'-"

o

o

o

o o o o fI l'

o o o II (1

o o o o-.-,.-.-------~~-----~ ..

Tilhre 3.:U DccC)r! ing tnhle for G ( p ) c- pl -, P7 , 1

Note , Above tahle shows !\- , ,1\ decoding table can be WI it !u, . 1<1L(II;' r 1(;1'1

remai'oder polynomillls.

lv) To obt~in transmitted vectors

1 T o d ( 'c od c 'y'" 1101101

TIle reccive.d-vect()[ pnl} '110111 l ' 1 1 wi]] he,

y( 1- ') ' . ' p b + If' t () l' \ ~.r;i + I'2 + 0 I' + 1

Syndrome can be obtained bv cq\l;luon (3.3.62) as,

r Y (I'llS(p) '" r l ' l ' \ ( " ; ( i ; ) . 1

Hc-nf'f' let \.IS divide Y (r) by (; (I') ~ f'1 + J! 2 - 1 1 -

J,3

p 3 + p 2 + O p +1rr('~p-';1 - 6 : r > " ' - ; · - p ) ' ~pr-;d7:t:l------ . . . . . .._-p 6 + 1' ' ' ",Or'! _q)J

p2 +Op+ l t-f?cl1I l I indcr



3 _106 Error control Coding- 1 1 ' n~ f ~ o ~ r m ~ a ~ t i~ o ~ n ~ C ~ o ~ d ~ i ~ n ~ g ~ T ~ e~ c ~ h ~ n ~ i ~ q ~ u ~ e~ s~ ~ - - - - - - - - - - - - ~ ~ - - - - - - - - - - - - - -~ <.

Thus syndrome polynomial is, S(p) = p.l +O p + 1

5 = 101

. ed vector c o n t a i n sC;ynrlrome is nonzero. H e n c e recerv

5", 1'01corresponds to an error pattern of (sec table 3.3.7),

£=0001000

Correct codevect0 r i s,

X y@ E

( 1 1 01 1 D r) E El0001000)

1100101

2 ) T o d ec od e Y = 0101000

y e p ) ' " p 5 +Op" +p3

Let \IS divide yep) by G(p) i.e.,

)12 + P

p" -l- P 2 -;.0 i' + 1 ) 1' 5 + 0 p~ + p3

p5+p"+Op3+p2

p'+p3+p2

p ' + p3 + O p 2 + P

p2 +1 ' f-Remainder

5 ( 1 ' ) = pZ + P

5 = 1 1 0

From table 3.3.7, 5"" 110 corresponds to

E= ] O D O O O O

Here

Correct codevector will be,

x Y<i lE

(0101000)@(1000000)

1101000

J) T o d ec o de Y = 0001100

yep) = p3 +pl

1

1'3 + fl2 + O r + 1 ) p3+ p2

p3+p1.+0p+l

1 ~ Remainder

errorS. T he s yn dr o me of

3 ·107 Error Control Codingnformation Coding Techniques

HereS(p)=l, Hence 5=001

From tab le 3 .3 .7 ; 5 =: 001 cor responds to an er ro r vecto r of,

E;: 0000 001

. , Correct codevector will be,

x ~ Y ( £ J E

' " ( 0 00 1 1 00 )f D (OOOOOO l)

0001101

Results; In this example the codevector are systematic. Hence mess'iEe vector (Ivl)

and check bits can be separated as shown below.

Sr Correct Message Check bits

nQ code vector, x vcctorM C

1 1 100101

II1100

r -101

2 1 101000

II1 101r ~ooo

3 0001 101

II0001r

•r - 101.- _ --

Table 3.3.8 : Results of Ex. 3.3.14

Maximum like li hood deci sion rul e based decod ing i s d iscussed in nex t sect ion.

I l l " Example 3.3.17 : A biliary message sequence 1001 is coded Iising n generator

polyl lomial G (x ) ",.\"3 + X +}. Assuming a s yste ma tic c yc li c c od in g is u sed , d ete rm in e -

i) Lmgl/r oj coded seql l t ' l ice.

ii) Tile transmitted codeword.

iii) ASSlllllillS ,ph bil ill therccciucd word i s i ll error, workout lire syndrome and draw

t iu : h ar di ca rc b lo ck d ia gr am j or lilt, ~ylldrol l /e generator. I.~

!:'I"'

Information Coding Techn iql)('s 3·10[l

Solution; Cive n (bta

J - 109 Error Contr ol God in! !

' . .. , ,( ) ' [0 oll /l l~U th« codeword



Ccnerator polvnonunl, G ( I - ' ) o.'I'J + P -f- 1

1VfC'%"g'> ~L>'lu"'Kl'td ~ 100]

. 1 ' , 1 = - = - k + q == ~J ·;·3=7

This is (7 ( -1 ) cyclic code.

(Ji:1To de!~rll1il1e transrnltrod codeword

The check b i t po lyr l ( ): l li a ! is given as,

'-,)'1 11/( ) 1e(l'\ '" rin! I~-

" ! G(I ' )~ .J

from eq ua tio n ( 3 .3 .14)

_ - ;) F ( ' " h .' . -, i n 1" ' , A~-(_)))

T!l<' m{',,'~g{' S{ '' 'I '- ' l' nC<2 i s, 1 " '1 ' " IDOl

, , 1 ; / (I') " r· 1 -+ O! ' 2 + 0 r + 1

I'-' t» J + 0 ,~11+ 0p + 1)

' " 1 ' 1 " -l- O p' . , - 0 ] . " + p 3

(1) T o d iv id e p r; ,-\1 ( I ' ) by G ( 1 ' )

The d ivisiori 0 f / .) ') M (p ) by G (p ) is shown below :

/,3 +t'"

p J + 0.1' 2 + I' -;. P (, -l- 0 p " -l- 0 P ~ -t- J ! J

p D - ' - O p 5 + p4+p3

pi + O r "

p4 -Hlp) -1- p' + I'

p 2 -l- I' {- Remainder

Thus the remainder is, C(p) = pl -I- r

C '" (I 1 Q )

1!l~'~~'5~::Il1?i lc form of the codeword is given as,

x sz (n~~ ~": 1 1 , , u r o ; Cl

:;;!;" (1 0 0 0)

11'115s the req uired trilr\~Tn'i!ted (~ldeword .

.~ i l) To determine syndrome for 41 1 1 bit Ir l error

Here we have to obtain the decodblg table. The table shows error !"dH'rn andcorrcspondiflg syndrome vec tor . h.H \!, ,~ ~_;en( 'r"tor polynomial or G ( I' ) .~I" , ! ' . ,.I 'A', '

h;lI'e obtniTled 1110decoding [able in f>; :l_' 10_ It is iilblc 3.3.6- From th ,'; t;.' ' !c . , ,' 'srrv,-'

th~l for 4'/' bit in error (i.e. E 0= 0 [) () 1 0 IJ 0), the ~yndrorne vector is, S ,c I)! I i.r-. ·1'"

row of J - - r r .

H ere n ote ttwt !here- is no need to obtain complete decoding rnbj·. , I!j:;~ 1 c J: . :': h _'

syndwliie.

How 1 0 oiJt.1ill r!f1rliculllr SlInrlromc r!i,pcrll, ?

In table 3.3.6 observe Iha! fir"t four syndromes are h . L : · ie'")!,, r,. " ",',_",' 1 1

l'-submillrix (see equation 3.3.70). ~,I(jwsee <!<Iuati()n 3.3.52 nncfully "I!"' .•- :".1' " .\."

of p"submatrix arc part of genHiJwr rnatirx (G). i.e.,

r . ~n o 0

C ' oFrom eq\l~"on :1,4 ~:l.(; ~

j u() ')

_ 0 () n

'-- __ -

I : . . fn;J~rl)li;: ~I;.

N<'w f t! . SyJ1dr<lTm~ ~.lf' t - ' " "H·.,iJH'd roy di':iding p": ' by ( , l!') Hr-r r ,- '•. j-ir>ncc

divide r:' by G (p). \\1(, have n ,,,7. )k".-'· "". " '" 1 , 3 "

1\.--.---.-~--------.-.--

p 3 + D p7 +p+l)p~

pJ +Op2 .> p - ' - 1---~.~ .. .-.--.-

The remainder /\4 (p) "" P + 1 [C'pn:g"I1!$ s)'uu rome.

i.e. 5 '" 0 1 1

.'.,



t(~}"~·(!~::r~, ; ' .. : . : . ~ 1_

' . ; Information Coding'Techniques 3 -110 .Error Control Coding

' Important note:

Here k = = 4. Hence first four syndromes cari be obtained by above method. There

are total II= 7 syndromes.

The remaining q = 3 syndromes are simply rows of the 1q matrix. i.e.,

, [ I ° O J +- indicates error in 5th

b i t13 ..:3 ;: . 0 1 ° +- i nd icat es er ro r i n 6th, bit'

o 0 1 +- indic ate s er ro r i n7 th bit

Ha rdware d iag ram for syndrome !1enerator

The g,"nerator polynomial is,

G(p) p3 +O p2 +p+l

and G(p) .=: p3 +g 2 p2 +gl p +1

On comparing above two equations,

g2 = 0 and g"1.

With these values Fig. 3.3.4 can be red rawn as fo ll ows :

- - . -

IReceived IjllPutvecto~

y .

Syndrome

OU l p u t

Fig. 3.3.13 Syn.drome genera tor' for (7,4) cyclic code (orG (p) .. P 3 + P + 1

Review Questions

1. Wh4t IJT/! c y cl ic c o de s ? W hy they ar e c . a J / e 4 sub clII55 o j b lo ck c od es ?

2 . E xp la in how c yc li c c od e s a re generated from the generating polynomials.

3 . E xp la in h ow g en er at or u nd p ar it y c he ck m at ri ce s a re ob ta in ed j ar c yc li c c od es .

4. E xp la in t he e nc od in g a nd d ec od in g m e/h od s for cyclic codes gillillg p r op e r b l oc k d ia g ram s.

5 . Giving b lock d iagram exp la in the opera tion o f syt ld rome calcu la tor for c y cl i c c o de s .

6. Explain what I4.T l BeH codes .

7. Explain lile jollowing

i) G om y c od es

lii) Shortened codes

( ii i) B ur st e rr or - c o rr ec ti ng c od e s1

~tjon CodingTechniques 3- 111Error Control Codjng

Unsolved Examples

)

Ii) Draw lire black dinsr fl ll /~ ~rencoder and sylldmill<' m/(IIi,%r cirrui: 1" II'. I I") F ' '1"( II' co, " /'u itu! th« code polYlI<I.'lIinl [or lhc " "'SSflS" polylloll/inl ,'.

M(p) '"p' + I'~+ r (ill '!lshmillie /orlll) i,'

iii) Is X 'F ) '" p" + II + 1 )6 + J I {\ r I' -.; . a cO, ' - ~polYliomial ? If Hoi 0n II·,·· I I,) li ',_'ynr.FOIllt' ()/ X(p) ,

Ans. ii) X(p) = = pl<.!- p12 + plO + .... 6, 1 .P'P"{''''p+l

iii) Syndrome S (p) =p? .. p " + P 7 ~ pO J

2. USlIIg 1 1 i ( ' gel/em/or poiy"ollll,11 '« ( 1) ~I .l:1 J -t- P + I'/ < • +.\ , genc/III" S,/,IClI1l1i- rl /'

~y~'~ COd~words lor llu: II!l!ssns<' vectors 1011 and 10m. .,: (nn I I0Ih.i' lslmllll ic .

3. .'jAde" I ii . encoder and >I/lId,oll/( cn/eaillior (or I'" ,. I .

allliob/ni" l i l ; ; sYl ld rome jor th« ~ccd1}t!d code 'word ; O ~ ~ ' ~ ' / ; 7r pO!y l/o lil lf ll S( .r) ~ I..' +xJ I

I

1. 11 OS, 5) /illrar1:ydh: code lms n S",lIcral()r "o/yllo/llltd

G(p)' " pl U + p~ + p 5 + P { + P ~ + p + 1

",)4 Convolutional Codes

3.4.1 Definition of Convolutional Coding

A convolutional coding is done b co b' , ,. y m 1I11n" tl'e fIxed ru b f 'mput bits are stored in the fix('d len tl J' 'f. 0 , im er 0 Jnpu t bits. TheI ! ' g 1 Silt register Jnd th . ', > 'ie p of mod-2 adders This ". cy ~!(' COl1lD:ncd with the" .. , . I. operation IS equlvillent to bjn~I" _, .," • .15 called cOIlVO/litlollal codin» T l' . , ) L().hOlUtlOn and nence Jt

example given below. S ' us concep t IS Illustrated with the help of sImple

Fig. 3.4.1 shows a 'convolul -ional encoder .

This bi! represent

curr~nl massage bit,

This bi t is thl .! part

of shift register

Massagebits input --....,j

~revious two SUGces!ijvs message

r=: b,t sarc s tored in t hose two f li p . n apsThose two b it s (m1.m2) represenr '

~ state of shin regisler

OUIPUI

Fig. 3.4.1 Convolutional oncoder with K '" 3, k ::; 1and II :::2

Information Coding Techniques 3 , • 1 12 ~,a"Of' 'Coding ~C'Chni411(':_~, 3_~. 1 _ 1 _ 3 . • _ _ ._E_-f'_r~_r_C~~~_o~_?_~~~~



111eabove convolutiona l encoder opera tes as fol lows .

Operation:

Whenever the message bit is shifted to position 'tn', the J" \( 'W vahlt 's of Xl and x!

are generated depending upon m, ml and /112. m,1 and ml s tore the I"I 'l. vi tm~ two

message bits. The current bit is present in Itt. Thus we can write,

II! ffi ml E D m2

m E D m2

,,, (},·ll)

'" ( 3.4 .2 )

The output switch first samples XI and then X2' The shift register then sh: (ts

content s of 1 1 1 I to T!l2 and contents of m to ! 1 I1 , Next input bit is then taken and stored

in III. Again Xl and."\:2 are generated, according to this new combination nf

1/1, Ill, and !II2 (equation (3.4 .1) and equation (3.4 .2» . The ou tput s, vi tch then snrnples

.1'1tiler! X2' Thus the ou tpu t bit stream for successive input bits will be,

. .. (3.4.3)

Here note tha t for every input message bit two encodedoutput bits x: arid Xl are

transmitted, [n other words, for a single mcsscgc bit, the- encoded code word is two

b,t;; i .c. for t h is c onvo l ut io n al encoder,

Number of message bits, k = 1

Number of encoded output bits for onr InCSS3G;:! bit, n '" 2

3 .4 .1 .1 C o de R ate o f C on vo lu ti on al E nc od er

The cede rate of this encoder is,

kr = - " '-

II 2

. " (3.4.4)

~ .In the encoder bf F ig . 3..1.1, observe that: whenever a par ti cula r message bt enters

<1 shift r~gister, it remains in the shift r eg ister fo r th ree shifts i.e.,

;:~First shift -+ Messag(~bit is entered in position 'm',

Second shift -~ Message bit is shifted in position ms .

Third shift --} Message bit is shifted in position m-i.

And at the fourth shift the message bit is diSCllrd-:-t) or simply lost by overv .,We krr." 'hat Xl .and X2 are combinations o f Ill, 1 11 1, 1 11 ". Since (I single message tnt

remains ;;, III during f irst shift , in ml during second shift and in 1 1 1 1 dur ing thi rd shif t;

it influences output Xl and X2 for 'three' successive shifts,

3 . ,U . 2 C on sl fl li n t lwgth {K )

Th e (omdr.'.lI1t \o.·I1~:!h'\ ; , ) (1,).1.1\·\.11I1bol\ ' < fi d ", , ~ , d d i ne d d:; 1)\0' lHlmber of : ,! i ii :s , " -, , '"

~\ 'hi, \h a :; i!1l~k' 1!I;:';""g" bit ( .\n m lln"nu ' tht! encoder outpu!. It is ex~'rt';;";i·d rn knm

pf tm,s~,'t:eI~jl~,

For th(' encoder of ri!;. 3,-1, c(Jn,slr,linf length K ,., 3, bilfl. This is bCC:;H1S(~ in this

('I1Ce. iN, ;) s it lg ll ' 1 l1<.:! 'S .1gl 'bit i nf lt .H 'I1"Ces l 'n eod! '! o utpu t for three successive shift,;, At

tfle fourth shift, t he me~ '<1g' .' bi t is los t <Inc! it has no effect on the output.

3 .4 .1 .3 D im ens io n o f th e C ode

T h e d im e ns io n of tile cndi,' is gi· . 'en bv nand k. IVe know lhM ' j , : , ' is { li e 01Hdwr ( , II

m (~ s~ a gc b i ts l.lk...1 at a !ilrw bv Ihe Nico(!cr, A nd '[1 is the (,l1('(ld('d Oil-IF ' " I I'i ~" (,,~

Olll! f1H!:~5ng~' l>it5 !J,'PC'J th( ' dill\(ln~'i<ln nf the ('ode i.:; (n. k) . And ~lId1 , , ,\ ( [, , .I < 'r ; ~ .

c"ncd (n, k) convolutional encoder. For example, the cnco(kl' of rig, 3"1.1 hns lhe

d t ll 1 c nS iO I 1 (d (2, 1). .

3.4.2 Time Domain Approach to Analysis of Convolutional Encoder

\", 't till? ~('q1.I""C" 1 . ' 1 ' ; ; ' , , , , ; i \ , gl,11... x ! . ; l l denote !hi? impul se n,spoll'><' ',)! ';1", , , ( : < ' > 1 '

which ~~Cllt.·1 ntt:~ Xi in rig. :~.-1.1 S.ilnU~ilY i Let the sequence !X~;:-n X ~ ; I ) , .:1.') ·.. ·x!.: · ~ 1,I I Fiv

.)'3 \ !deIlC l !" Ihe irl1l'llh.~ n c , ; p n n 5 ~ ' of til,· ; 1 < 1 I< : r "'hi,h generatr's \'7

i rnpuisc responses arc also c.l lied grlJcrt1/ol' s<'l)llmCt's of the code.

Let the incoming message sequence be I m o , ])1 1, 111 2 ....... )· The encoder I:"nf'r;lt·,,,~

,the tll'O Otl!r)tlf .')..'q!l"I"~1!5 Xl ;Jnd x r- These are obtcined by ((\1 voiving th" W·I.... rator

s cquenC l' ;; wi th the J lI c< ;~ :"g, ," $~!q\lpnC ' lI('"c~ the name convoll1ti(ln.~1 (,p(k ,;< gi\,H>.

The $1: '} lI cnce ,1' 1 is ,?,iven .15,

_ I =-~.tI~-~-·- J , - - : ' ~ I ; ~ - ~ - - l.),_" 6."51 ,. i=0,1,2....... ...(3.4.5)

L.....~ I_F,~O ._

Here In; _I '" 0 for all I > i. Similarly t lw sequence X2 is givt~n as,

. ~ ,.0) '" ~!. d\~')' ' ..•2 - .. ; _ L.','1,·1 I - (). 1, 2 .......

- C

Note: AlI additions in above Clju,l\wnS .'rl~;15_permod-2 "ddil inn r tJi r. ";

A, .. s hown in 1111'p;". 3..1,1, the lh",' ~".',:.!,.r!ces Xi and x, HfC :t1u!tiplexed by the

. sw i tc h, H el ic e t he V L < '. t , :, :;equci1cc is g iv en a s,

{1' , . 1 { ( \ ) ,(II Jt1 _I" ,,\1) . (2 ) ~ () ) _Al l 1

Xo ;Ill '\1 \12 ,1-;>. "3 "'3 ',,, .. , J... {J>.~,f)

Here

I

;~m" aUhms dellne ccnstraio: " ! > n g T " a 5 ' i 1 i ; ; ; ) r ) ; ; - r , ) r o l X ~ ; u l bits inltuene.i;:n-:;ya;TiIOlti'·-'-'-·----

mesS3ge bit I.e.

C Q !) ,U llln l l en gth ( kJ ~ ( n , , 1 . 1 , bil. .. . (3 A .S)

w 1 't o1 u n ~ n um bt :~ fj! en co d -ed o u rp u i bit,; !ot e .v · . . 1 inpu! bit

and M ~ number o r stO((~gc. Cf(H1WtfHS u"t I1V.~ MJill leSJr;~er

Pcr tho .",coder or Fig. 3.5.1 Constraint length r 2" 3 ca 6 tJl~.



Information Codjrig Techniques 3.114 Error Control Coding~ - - - - ~ - - - - - - ~ ~ - - - - - - - - - - ~ - - - - - - - - - - ~and

Obsente that bits from above (\"0 :;equel'O::('s are multiplexed in cqutllion (3.4.7)

The sequence { X i } is t he output o f t he convo lu tiona l encoder .

...~mpl' 3.4.1 : F" I/<, ",,,,,'''Ii,,,,,/ encoder 'f Fi".3.42 dd,""in, "" foI',wi'g'

~ DIHiCIISIOII 'Y li«: code

ii) Code ril le

iii) COllslmill1 length

i u) G e ne r at in g s e qu en c es ( im p ul se n ~ po " i5 ~ $J

v) OUlpHI seqllCllcef or m es sa gr ? s e qu en ce o f III '" {1 0 0 1 1/(Nov.lOcc .•2003. S Marks)

,.- -<..-{ ..1-------..

~·Aes5ago

~CqIJonce

Current

message

bt("') \

.. .

1 ,~_../ Oulput

2 sequence

Flip-flopl

~ - - - - - - ~ + ~ - - - - - - ~

Information Coding Techniques 3·115

th,1I of Fig. 3.4.1.bserve that the above encoder is exactly s imil nr t o

i) Dimens ion of ' uie code,

Ob.serve that encoder takes one message bit at atw b f u time. Hence ko Its or every message bit. Hence n <= 2. Hence,

Dimension'='

(n, k) '" (~. 1)ii) Code rate

Code rate is given as,

r '" ~=_!11 2

iii) Constraint length


1. It- genera tcs

,.

~;Here note that' every me a· bit ffence, 55 ge 1, a ects output bits for three successive shifts.

\

. (I).i.e, Xi IS generated by <ldding <lII

generating sequence sl.) is given as

gil) == 1 1 1 1)

Constraint length K =: 3 b it s

,.lv) Generating sequences

In Fig . 3 .4 .3 observe that Xl

1 represents connection of bit m

(I)g 1 =: 1 represents connection of bit III'; I

(,(I) --1'· .ol--~. represents connect ion of_bi t~ ..• ·

i.e.. x : 2) is generated by additU---------------

is given as,

g J 2 ) 11 0 I}

go (l) 1es represents co

gil) = 0 represents tha

82(2) 1.. represents coru

The above sequences are a lso cal led imp

o oblain Qutput sequence

The given message sequence is ,

11 1 = ( 11 1 0 1 11 1 1n2 TrlJ m~)'

Fig . 3.4.2 Convolu ti onal encoder o f Ex. 3.4.1

Solution: In the Fig. 3.4.2 observe that input of f1ip·OOp 1 is the current message' :

bit (rn). The output of flip-flop 1 is the previow, message bit i.e. /11]. The output of

flip-flop 2 i" previous to previous message b it i .e, /Ill' Hence above diagram can be

redrawn as shown in Fig. 3.4.3.

Message

lnpul

Output

Fig. 3.4.3 convolutional encode r o f F ig . 3 .4 .1 red rawn al terna!el y

._..--..----- .... '

t he th ree bi ts . Hence

. .. ( 3 . 4. 9 )

2 ,



information Coding Techniques 3·116 'irror Col"!'roJ Godl

T o o bt ai n aliI pilI rlue 10 adder 1

Then from cquarion (3. '1 .6) we can writ e,

M

x f ' ' " L g ~ l ) m , · _ {I~O

... (3.4.

wit h: '" 0 above cqun tion becomes,

\{i). II

< ' \,(1)", IZ: , Ct ' . ,~ -

'~O

lI1i-1 = 0 [or all l »].

th i= 0 in above egul1tion Vie get.

X~12) g~2)))fa = (1 x 1) '" 1

1 "J =1, Here g~) ' " 1 and 1110 : = 1

i=1 in cqiation (3.1.11), 1 '\ ' ) ~ g~l)ml ED g;J)mo

eo (1 x 0 ) E E l 1 x 1) = 1 ,,,_..,-,.-.~Here note that additions are mod-Z type.

. ",. ,_ .,' (~411) .(1) (1). (I) (I). - - _ m ~'lll,1 < 10n .). . " '2 "" go m2 +gl III, +g2 /lIo

Vlith i.0 2.

:= (1 x 0) (j) (1 x 0) E fl (1 x 1) ' " 1

I ~:; , n cC ]u < ~t io n (3.4.'11). X~ll = s~ )m3 E9 g;l)m2 < D g~)1111 With i"J.

(1 x 1 ) E B ( 1)< 0 ) E F . J (1 x 0)

I ,,'i in equation (3.4.1 i). - < , 1 ) = S~l) 11 4 E f . ) s i l ) 1 11 3 EDS~l) 1 1 1 2

(1 x 1) & (1"; 1) @ (1 x 0) '" 0

r- (l) (1) (I) ffi ('1), in l ''l uM io n (3.4,J J), x5 = g o 1115 E i ' I g 1 1114 \T g 2 1 1 1 : )

(1) (I)2 :, 111·1 E D 8 2 7IlJ

s ince n ls i s no t avai labJe With i::S,

(1 x 1 ) E E l( 1 x 1)

()

t· ("4 'J 1) "(I).,, :;(1) HI .",~.:,.;J.l\,.;:,_..

·'.lil!On ,.. ,.,~ 'J (,":" ").; .

g~\)m4since T1f6 and ms M e not available

lx1

.. ~ . - .

; . - "

(2) .~, , ,

x. ",",,' ":-'111 Ir ~ I I

I•

Here a(2) '" J and ) 11n ' " 11 1 . . . \ 0

·,.(c)

.')

(1" 0) (D(O" 1)

o',.'

(l x 0)$()< :1) (C (l" 1)

'" (1 x 1) @ (0 x 0) E D (1 x 0)

(1 x l) (})(0 x }) (f) (1 >< 0)

1

\.(2)• 5

(Oxl) (fJ (lxl)

')~ll,,2 J

~ 1x l

::: .:

.. :

t.;~§·"-.Yi\i,

Error Control Codll1lJ .:; , f ~ ,nformat ion Coding Technlques 3 -118

J ll 2. 5. i" ~ ~ " , J .." ',-

Information CodIng Techniques 3 ~119: - : = - = - = - : - - " ': ': ': ~ ~ = ~ = : : : " _ _ ~_ : ~ ~ ~ ~E:.:r~ro~r~C~o~n~troloding



To obtain multiplexed seq lienee '!f XI find X2 as per equation (3.4.8)

The two sequences XI and Xl are multiplexed to get the final output i.e. ,} .

X,' il) x(2) X(l\.(2) xP) x{2}x(1 )X(l) x{l).~P)X<1)x P ) il) X(2) -.,0,

o 0 1'1 2 2 3 3 4 ~ 5 5 (, (,

" ~ {I 1,. I 0, 1 I, 1 1, 0 1 " 0 1, 1 1J

3.4.3 Transform Domain Approach to Analysts of Convolutional Encoder

In the previous section we observed that the convolution of generating 'sequence

and message sequence takes place. These calculations can be simplified by applying

the transformations to the sequences. c.et the impulse responses be represented by

polynomials . i .e .,_

(1) (1) (I), (I) Mgu +gl P+g2 p-+· .. · .. +gM p ... (3.4.13)

Simililrly, _ _ -(2) (2) (2) (2), (lJ::r:r-

g. (p) == go +g l T '+ g2 p-+ .. · .. · +gM p . .. ( 3 .4 .1 4 )

Thus the polynomials can be written (or other generating sequences. The variable

'p' is unit deLly operator in above equations. It represents the time delay of the bits in

irnpL ' i.q' response.

Similarly we can write the polynomial for message polynomial i. e.,

lI l(p) = l I !o + /IllP + Jr l2p l +........ + I I IL-1P t-: ... ( 3.4 .1 5) •

Here L is the length of the message sequence. The convolution sums are converted

to polynomin! multiplications in the transform domain. i.e.,

}ll) (p) = g(1) (p ) • m(p)

xI2)(p ) '" gP) (p ) . lIl(p). .. ( 3 .4 .1 6 )

The above equations are the output polynomials 01 sequences x ; ! ) and x j 2 ) .

Note; 1-\11additions in .above equations are as per mod 2 addition rules.

HU.., Example 3.4.2: Repeat part (V) of example 3.4.1 using tran sform domain

ca lcula t i on s ( pol v nom i a l- r n u l ti p li c at i on s ).

Solution: a) To obtain generating polynomial for adder-I ;

The fir st generating sequence is given by equation (3.4.9) i.c.,

g?) = = 1 1 1 1)

Hence its polynomial can be obtain ed (equation 3 . ~ · : ' i 3 ).1s·,{QlIows :

gO) (p) = 1 + 1 x P + 1 x P 2 ;'

(.

h) To ohla_in goneratlng polynomial for adder-2 . ----

The second generat ing sequence is given by eq uatio n (J 4.HlJ _ i.v .,

g,{2)--~ 1101/'

Hence its pclynomia} Can be obtain ed ( equation 3 -1 -J ) . C r II~,.1· ,I» 0 O"/~ :

g(2)(p) '" 1+0xp +lxp2

'" 1 + pZ

c)_To obtain message polynomial

The message sequence is,

(l.l I,~)

m = (1 00 I 1)

Hence it s I ;~ ly~omin1 can be obta ined (equa tion 3,'1.15) ,1$,

m(p) = 1 +Oxp+Oxp2+ 1 xp J+lxp-1

= 1+p3+p4

d) To determine the output due to adder-t

Now x(l)(p) can be ob tained from equation (3.LJ_ ]) i.e.

X(I I(pJ g(lJ(p)-m(p)

(1 +p + 1 ' 2 ) ll-;-pJ -j- 1 " ' )

:=: 1+p+pZ+p3+p 6

i The above polynomial can also be written as,

i Th tl xll)( p) "" I+(1(~Il)+(lXPZ)+(lxp3)+(OxJl"}.j.(Oxp5)+('Ixl'(')

_~_, us re output_,sequence Xi is,

- ,

J ~ x P l = 11 1 1 1 0 0 II

% - .;r - oj To determine the output duo to adder-Z

'~: Similarly polynomial x(2)(p) can be obtained as,

~ X ( 2 ) ( p ) " " g(lJ(p) 'm(p)

~;

(' := (1+p2)(1+p3+p-: ) '

: :: : 1 + P 2 + P 3 + P 4 + pS + P 6

Thus theoutpur sequence x ? ) is,

x , ( 2 ) = II 0 ] 1 1 1 ])

f} To determine tho.mul tipl exed Oli tput ' s equence

TIle multiplexed output sequence will be .:IS follows:

Ix;) "" !1 1, ] 0, I 1, I I, 0 1, 0 1, 1 1 1

Information Coding Techniques 3 ·120 Coding Techniques 3·121 Error COlltrol Coding



Here no te that very few calculations Me i nvolved in t rans form domain .

3.4.4 Code Tree, Trellis and State Diagram for <l Convolution Encoder

Now let 's study tbe' operation of the convolutlonal encoder with the help of cod

tree, trellis and state diagram. Consider again the convolutional encoder of Fig. 3.4 ...

It i s reproduced below for convenience .

Previous two successive message

~ bils am slored in those two nip, flops.,. Those two bits (m"m]l represent

,.------A------ state or ~hjrt register

This bit representcurrern messape l)i l.Thi s b i~ i s !h e part .. .._

o r shifl registerMes58,Jc Ibits input _,------'-...,..----1.-..,__'

U ~_\

~ '---OutPut

Fig. 3.4.4 Convolutional encoder with k '" 1 and n '" 2

3,4,4.1 Slates of t he E n co d er

III above f i g L ! l " < : : ' the previous two successiv e message bits 1 1 1 , and m: represents

state. The input message bit m affects the 'stn'le' of the encoder as well as outputs

X I ,m e!.Y 2 during that state. Wllenever new message bit is shifted to '/II', the contents

of 1 1 1 , and I I I " define new state, A nd outputs XI iln d.\ "2 are also changed according to

new state 1711, m ; \ and message bit !II, Let's define these states as shown in Table 3.4.1

Let . the ini rial values of bits stored in 1 1 1 1 and 7n 2 be zero, That is 7 II 1! nl .. O{)

initially and the encoder is in state 'a'.

Im , I m1 State of encoder

I) I 0 a. _ . . - .J . ' . .~ . • '

1., b

f--__--

1 0 c

1 1 d

Tabla 3.4.1 States of the encoder,,~f Fig. 3,4.4

u'l 1l!,Jco:"~i,1t'1 tJ'l~ dt'~'dopn"'''l c,[ ende (r,·c tur the me~,5~gc ~.('qIlCIH:';' H' ,., : I"

urne th.1! II!! !l!l ",OO"lnilia!ly.

1) Wllfm m .. 1 ie.[irs t bii

The first messilge input is m = 1. Wi til this input XI and ."2 wil! be GdC\il.:t!~d as

New siale

f

l

[iI~li:.Jrn m

l[Il,

Be'or" shift

1, =1 (j) 0 (j) n", 1

.1 '1,"1 m ( ] = I

AI!N shil't

rhe V"hH '~ of .\ i .r) tr II ;1re rTolll~rnj"tt('d to tIl('

output <Inc! regis te r content s a re shi ft ed to r ight by one bit pos ition as. sl ,o·" ,t I

I

l .. , ._ _ ,_ L J Pw t > rC J , . 'fI}Wl"'IiG(!!~"

I11131ne5s~ge bi\ i s rn w 0

... _...----Node or state

'"'I't-.,-_.o·..mwiild anaw indicates

thDt message rJit is m 21 1

Tbi~ ;nijlr.;.tf~f. output . .wtl1k~ ~Jt)ls~11f01r1 nO(e. ~~._. ._~ ._- 1b TtH~ i~ .ftf!"Y ~-,t~.1>

'c ' f a 'b' '-------.,...! ~ I or r)od~~whC"r"1 rn:- I

FIg, 3.4.5 Code tree from node 'a' to 'b'

11"!u~ the P.~i\' .·;t"t(' of '~~·"Hf),jer i.~ 1 1 ' : . u) .- 01 {U~ 'i;~ r,nd iHltp~lt :·(~n~;rnill~.'d i1r~

.\: 1 2 ~~1 1 . T his . t. : ,hc~"", I '!~t-l"iJ! li (!ncl)"'k~r I~i ir, st;l_te ' . , 1 ~'nd if input is m ~. 1 lht:"'!1 fhl ' P(Jx \

!il..,te is '/( and outputs are 1'1.1'2 '" 11. The first row of Table 3..1.2 illu~l rill('~ this

operation.

TI,e 1<'1~tolumn of rhls l :1bl~ shows th\ ' cede tree di~gr;:~m. The co.:\[ ' [n:r, ( ; . : : : ; } · " ' T \

start- . at node or S~a~f·'ot. The d j :; a } ~, , ·· : tn ~f; I"t ': · l": l"! "(xll,lced ;-\5~h("' ) ~\ :. .i: n, ~; .: ; .~ .

d ' " , ,~", ' '" ,r· .- D "h "('OtJ$ . . .. ." i : e lh<H iI /lj :0:.; 1 -, ...:(.!" f)~' : : 'r " , ' ~ · n " l . . ./i.lf J:fOH-:t n n . : . . . . · H . : : r7. ~.. .. .. ..;."lf!n·vl~~e. n - . t ':)

I { \-" 'l iF \V~rd I ro rn nod :. :- 'o', It Cil n be veri fied t hr .t if m,:;:: ~h~;fl l~Xt n.0C.C ~ t. ?t el 1 5 f:i oruy.,

d I . ,-" . , ,J 0., 'I r., J 1 1'1' tI,:" l'n,~(,t)"~Since III = = ' I here- we go o\\~n\ ·." :r r( ,~ to'Vilr~C n"')(~t(::.. ~H"h.,OU~t)!.' IS ~ 4 ,I.' ',. '. ,

state),

Z ) W h ~1 I /II= 1 i.e. se con d " 'I

Now let till' second rnes$agc bil lx' I . The conten ts of shift regit ;i ("t wifh lhi·; ini'llt

wi ll be as SIlO"'''' below,

i Information Coding Techniques

NeF .......

3·123 Error Control Coding



:J .3.122Error control Codii :ig "t

-.Information Codifl9 Technique"!

Xl : : := 1 ttl 1 ffi 0 = 0

X2 == 1 ED 0 = 1

m

second input bits.

3) Wh, 'n Tn := 0, i.e. 3,,1 bitSirnililrly 3(.1 row of the Table 3.4.2 (Refer table on next pilge) illustrated the

operation of encoder for 3" input message bit as III'"O.Now observe in the code tret!

of last colwnn. S ince input bit is m = 0, the path of the tree is shown by upward

arrow t owards node (or state) 'c', That is the next state is 'c' (i.e. ]0) and output is

X 1X 2 ;01.

Complete code tree tor convolutional encoder

Fig. 3.4.6 shows the code tree for this encoder. The code tree srarts at node 'a'. If

input message bit is '1' then path of the tree goes down towards node 'b' and output,

is 11. Otherwise if the input is m '" 0 at node 'a', then path of the tree goes upward

towards node 'a' and outptlt is 00. Similarly depel'1ciing upon the input message bit,

the path of the tree goes upward or downward. The nodes arc marked with their

r.,jah:S a, b, c or d. On the path'between two nodes the outputs are shown. We have

veri!,<:d the part of this code tree for first three message bits as 110.

If you ~i1fefully observe the code tree of Fig. 3.4.6, you will find that the branch

pattern begins to repeat after third bit. This is shown in figure. The repetition starts

after 3rd bit, since particular message bit is stored in the shift registers of the encoder

for thI~e shif ts . If the length of the shift register is increased by one bit, then thepattern of code tree will repeat after fourth message bit. (Pleas,e refer Fig. 3.4.6)

3.4.4.3 Code Trellis (Represents Steady State Transitions}

Code trell is is the more compact representation of the code tree. We know that ill

the code tree there are four states (or nodes). Every state goes to some other stale

depending upon the input code. Trellis represent!; the single, an unique diagram for

such tr'l\\~,itiun5. Fig. 3.4.7 shows code trell is diagram.

,. ';'

These values of XlX2 '" 01 are then transmitted to:

output and register contents are shifted to ribht by • ,

one bit. The next state formed is as shown. ~';'Thus the new slate of the encodt:r is 1 11 11 11 1 : 110(" W ? : : - :

'd' and the outputs transmitted are . \' \. 1 '2 ' " 01, Thus

the encoder goes from state 'b' to state 'd' if input is

'1' and transmitted output x,xz ",01. This operation is illustrated by Table 3.4.2 in

second ro'1,'>'.The last column of the table shows the code tree for those first and

a

u

.~

oII

o

< l J "~0

"0 0

g!l)(N C D u~ ; : : : : ) ' " O 00",0.<::

'S"5'"' roGl';0';-u .. 11

~~

Ql (i)

o

co , " " "

~ § , 1 1~-L:_

.; I I "!

I I ;o

"8 GJ

0 0 ; :: :: .

11 "N>< "

~o

2. !'J

'"S

,~

E 0E

<= >

Table 3,4.2 Anatys is of convolutional unc oder of Fig. 3,4.4

Inform<ltiol1 Coding Techniques3 ·124

3'·125 Errer Ccrur ol Coding



Upward path

iIldic.:'t!r;s

In;::.L'~ru=u

~t;l_l;'~.:"""O(!

I

St~r1 1Ia Outputs

G O ; ; ; I ;, a r d t r I OinJIC<:;"ltcs r ~mpu: rn::: j f 1

I l' i b

_-,,0-,-1_ .•

b G1

,; iod 1' :

Fig. 3.4,G Code: tree for convolutional encoder of Fig. 3.4.5

Current slaIe Output Next slnle

01 ~ b

11 ~ d

Fig. 3 4,7 Codetrellis

ofconvolutional encoder

ofFig.

3.4.1

1 ne IIOl~l~.S on 1. 1.:;. ' r - f t I I f b!. (ef1CJ [> our possib e current stnt~; and those on the: rkht

l \C ": , ' I~ r ~H'\~ :~,1,tc' ; - ~ .. .i t '. I·' \?, ~ , . : - . u! t, rans: lion me represents inp(lt m = 0 and broken line

" . . . .l J n . : ' ~ 1 _ · I H ~ il~l!ut 0 1 ~ ~1 !\J'1f'lO- wit]... e ) t iti I' h, , • ' , c » ,ac 1 rallSl IOn me t e output X IX 2 is represented

durine till! I I J I" ' r,lil!;t 1<'11 'or (X<1Il1P e let the encoder be in current s tate of 'a'. If in ut

III "' (I, thei1 next S("ll ". 'ill be '~' with outputs . 'l:1.\2 .. 11: TI1us code trellis is ~le

COll~Fqc~ rt..'r·)J'l·~"'_'ilLlrionuf cocie tree.

If we cor nb in -: ihl.' i"."~)['tTn~ ;l!II·~

next states, then Wc' <'.,\.1['li" :;I~I<.'

diagram.

For examplecon;;kkr !j .11 Ih('

encoder in state 'n'. I f input /1 1 '" D.

then next state is ~.llne i.c. it (i e. D O )

with outputs - '1:1 . '1:2 '" 00, Tit is i5 ~ ,i" )" 'n

by self loop ,9 t nrwk ',,' in I he ~t,l rc

diagram. If input /II .'" I, then sl~ tc

diagTam shows tha t next st ate i,; 'I"

'with outputs XtX2 := 11

code It(!s and trn Ills dillgram :

Table 3.4.3 shows the comparison between code tree and trellls diograrn JS a

gr aphic structur es to genenlie and decode convolutional code.

c

. FTg. 3 .4 .8 Stale df a g ram fo r co rrvo lu tianalencoder of Fig. 3.4.1

Code Iree Trellis dagrnm-----"l

-----------j-----,---- ..--..........,--.-Code tree indicates now of tile coded sinnal Trellis diilgram indicales trJn"illons f i ( ) f > ' cum,"li

_~IOI:q~:ol tl"'~:_. -_-_ to, n"xt status. _~_., __ ". ._ •. _ .•.. _ . jCode troe Is !el1gthy way or reprcserrtjr!'P ~ode trellis dra9r(lU~ i'j r.J rlJJt\C' ' )r co- I~1·l,Jc..1 w~"Y Icoding process, of representing Ct1dlngprocess

Oecod ing Is IICry s imple usiPlg coda ~ ::___ Decodlrlg is .IIItle C~n;Plex~~;~;~~:~I~<;-~j;':I~,~~~

Code Iroe reoeats after number of stages used Trellis diagram repeats in l've,y,'~'.,~. 10 ' {" "" ; IIII the encoder, ~lnle. treltis~~flr,"n_!_~"~~:,?~'' 'IU !

.code~s c(," 'rnp '; ;-~o-:'PJ~rns~;:----·~'- Tre"i~ dfaaram is sfmpktf tf'; tmp!am~~r';t:1 ,

ptOgr<'lrnmi{l!j, ... >1....- ~:~~_...,_.. __...~._.- ...-,-._-j

T ab re 3 .4 .3 ' Comparison between code tree and trl!Jlis dlagrarn

4"

5

3.4.5 Decoding Methods of Convolutlonal Codes

These methods are used for decoding of convolutional codc~ T!'1l'~,.-;n.' vilN\Ji

",~Qrithm, lwqu!!.nti<tl decoding and fcl:dbntK decoding. LeI's c on$idN then' jn d;:,i;ds

tt l subseqnen t sectlons. '

M.~ , 1 Yl tMbr A!~Jlm r~r t lw;cq1ng t ! 1 r COl'lvo:!iil0I131 CQdcs (f.iaximtU7; l 1 : leHh"oc f GccodingJ

Let's represent the received signal by Y, Convolutiona] e l1codi J; ?, OP(:",,!cs

continuously on input datil. Hence there (Ire no code vectors and blocks ,)S such. lcts

assurne thl1.1,thetransmfssionerroc pf(~b'1billty of symbols 'I's and G's is same. bI'"

define an integer var iab le metric as follows. .

Information Coding Techniques 3 ·126

. ' 'i,-:;T\~< ":~'M 'Error Control Goding::., "',, Information, Coding Techniques 3 .127

-- ........ __ ~ .... __ ~ . ... ~ ~E~r~rorControlcoding



Metric:

It is the discrepancy between the received signal Y and the decoded Signal at

par ticular node. Thi s metric can be added over few nodes ior a particular path.

Surviving Path:

111is is Ihe path of the decoded signal with minimum metric.

In vile!b; decoding a metric is assigned to each surviving path. (Metric of a

p"rlicula,. path is obtained by adding individual metric on the nodes along that path).

Y is decoded as the surviving path with smallestmetric.

Consider the following example of viterhi decoding. Let the signal being received

is encoded by the encoder of Fig. 3.4.1. For this encoder we have obtained code trellis

in Fig. 3.4.7. Let the first six received bits be

y ~ 11 01 11

a) De co d in g o f f i rs t message bit fo r Y = 11

- 'ote that for single bit input the encoder transmits two-bits ( X t X 2 ) outputs . These

outputs are received at the decoder and represented by Y. Thus Y given' above

represents the outputs for rhree successive message bits. Assume that the decoder is at

state G o . Now look at the code trellis diagram of Fig. 3.4.7 for this encoder. It shows

that if the current state is 'a', then next state will be 'a' or 'b'. This is shown inFig.3.<l.9. Two br anches ar e shown f rom 00. One branch is at next .node al representing

decoded signal as 00 and other branch is at hi r epresenting decoded signal as 11.

The branch from ao b l to represents decoded output as 11 which is same as

-" , ...

Bran ch f or m ' " 0

with output00 Cumultive or

Iv", 11 1 I=~ path discrepancy~ 00 \l1 (metric) i s two

B ranch f or mN a

1

Path met ric is zerow ilhoutput11 , )O~W

, Discrepancy or metric '.b1

is zero

Fig. 3.4.9 Vjterbi decoder results for first message bit

received signal at that node i.e, ] 1 . Thus there is no discrepancy between received

sign"! and decoded signal. Hence 'Metric' of thi lt branch is zero. Th is metric is shown

in brackets along that branch .. The metric of branch hom ao 10 (/1 is two . Tho encoded1

number IICJr a nude shows path metric reaclung to the node.

. . . . .' ,

, '..~. .~

'J .

b ) D ec od in g o f s ec on d f l11!$snsc bit for Y = = 01

. _ .. -

When the next part of bits Y = O l

is received at nodes a I and /1 " tlwil

from nodes il, and ", (our pos~.ihlr.'

rrext s t ates «s . > « . C; ,~nd " 2 ~1J (_'

possible. !'ig.3.'1.10 shows ill1 lh{~"c.'

branches, their decoded outpu Is .n« i

b ranch metrics co rresponding to tho se

decoded outputs. The encircled

nurnber near a i' ~ 2 . cZ ~ l 1 d d2

indicate path metric. eli~ICt'glllg 110m

nQ' For examl?le the path metric D rpath 110,111,112' is 'three'. The palh

metric of path nobtd, is zero.FIg. 3.4.10 Viterbi decoder resu lis for so co n d

message bit

c) D ec od in g o f 3 rd IIIcssag(' hi I fo r Y = 11

Fig. 3 .4 .11 shows the t re ll is d iagram for .. ]J the six bi ts o f Y.

Fig. 3.4.11 shows the nodes with their path rnetrics on the right h,1l1d side at Ill('

end of , sixth bit of Y. Thus two paths are COmmon to node 'n' C),,,, p~tl1 -~ " '• ' •• 1,-· ~I 1-;- {I uti I/I,Ii~

With metric ,5. The other pa~h is nOhl.C2113 with metric 2, Simililrly there arc two pn"lh~';

at other noaes .also. According to viterbi decoding, only one p<11h with lower mt'tnc

sO.Quld be retall~:9,jlt-particLllar node. As shown in Fig. 3.4,]], the paths markr-d

With x (cross) are. cancelled because they have higher metrics than other pMh coming

to that pilrtindar node. These four

paths with lower rnetrics are stored in

the decoder and the decoding

continues to next received bits,

01 11

d) Fur/her cxplanntion vj uiterb!

d e c odi n g f o r 12 /llesSIISc bits

Fig . 3.4.12 shows the connnuation

of Fig. SA.1 J for a lm'SSage 12 bits,

Observe tha t in this figure, received

bits Y Me marked at the top of the

decoded value of output i.e. Y + E is

marked <l t the bottom and deCOded

Fig 34 11 P messa<',c' si(',nil! is also marked.. • .. aths and their motriql to"- viterb; "c

qecodlng

Information ,Coding Techniques 3 -128

3·119 ErrorControl GorlinU



Fig. 3.4.12 Vitcrbi decoding

Only ODe path of particular node is ke t wh! h',· .'t here a rc two paths hav o e o • p 1(" L, h<1~II1g lower metric. In case if

, ~ e same metric then n fl'that a node '111" only 011

6

. tl . '. a y one 0 t icrn . is continued, Observe_ c; pn 1rrives with rnctri I' TI v- I'

line, Since this p<1th is 10 t . , . ' JC • \, O. us pat 1 IS shown by a thick_ .> wes met ric Jt IS the S I \' . I '

from this path. A I! the d ~ - i :l .' ' t r ' IVII1g pOll1 an d hence Y is decoded, .ccoc ec \-<1 lUeS of output are t ke f 11

path Whenever this p~"h I" b k . I ,~ n rom 1C outputs of this. ~" " ro en It 5vows message b it /11 -1 d If i .message b it I II ' " 0 betwecr I d ,- an 1 It IS conunuous

~ _ 1 1 \>I/O nO··C$. ~

This is the complete ( 'X l ) I ti . [ vi ,lJ1 v i bi > ,.", . ana Jon 0 vitcrbi decoding. The method of decodin dIter I decoding IS called I1l1lxinllllll tikclihood decodino, g usc

c) Surviving paths <

During decoding yo u ""11 f' d Ij 11 1 t lat a viterbi decoder ha s 1 0 s to repnth:; for four nodes; ,~ four survivity

5" \

lll'vlvmg paths !; 2(K-JJk . . , ( 3. 4 .2 0 )

Here K is-constraint length i I,'.~;~ anc x is number of message bits.

For the encoder for Fig :> 4 1 K - 3' d k• :J. - dr\ '" 1

Surviving paths == 2(3-1),1 '" 4

Thus the viterbl decoder has to store four su ..:nl!ssage bits to be did .. rVlvlllg paths always ff the number of

eeoc e are very large the 5tO . ." '' '' !. 'i :od . . h" .,., n r~ge reqluftJ"'''';' .s al so la rge s ince

f.;r:)bi~~ :~trica~i:Oe ~tore ffmtllt. iple (in pres ent example four) paths. To avoid Ihis. rSLOn e ect IS used. .

fl M e tr ic D io er sio n E ffe ct :

For t he two surviving p th ., .less likel na . 11' S originattng from the s ame node, the running metric: of

about 5 (~ ~ l/~r::~c~ t~ increase more rapidly than the metric of' other path withines rom the common rrode. This is called metric divergence effect.

lie {"q!l~ider 1 1 ' .' 9 pO ll hs C O! ll in t; f ro m nod« /)1 in Fig. 3.4.Hl On.: p3th

", t (I~ iU~cl ,)!I}(lI~ 1',,!11 <·(tnl('~ (It d~,. The p~th <I t II~ is le5:, Hkcly arl'tl lwncY' i!~1

i : ' ; mNO ('o1nr)l)red to Uw path (It d5• Hence <tI node "5 only rhe survivor Jr. ,U~ is

nnd trw IIU;\j\S;lgO bits nrc ciee»ded. TIlt' f l"csh paths an' :HaTled from tl«,

of this, the memory s~orage is reduced since complete path need not bf;

S p .q ue nt !a J O e r. nd in g f or Convolutional Codes

•SeQIl1.'l1\i;o 1 decoding uses mdric di"('fl.~cnce e(fect Fig, 3.4.1:1 ( 1 1 ) sJ1C'Wk tJ,.~ ( : < : J d ( ,

;~~ill,>':""";" for the convohlliol1;l1 encoder of Fi!~. 3.4.1. The same code tr,·l Ii s we h'1Vf~"et '11

the l as t subsec ti on fol lowing are t he i rnpo rl. :m t poin ts abou t sequcn ti l' \ decoding .

, 1) The decoding starts < It ITo. It follows .Ule single path by t;lking the branch with

sml lll es t me tr ic. For example a s shown ill Fig . 3 .4 .13 (a), the pGtli for ' f irst three

nodes is lIul:ld2 since its metric is t'lH! lowest.

2) ff there are two or more br"!KI1I's f,om the 5;Jn1e nod" \vi lh ,1a rne I\w!..r.. rhi'll

decoder selects lilly o ne b r; 1{ 1c h fHld C(lIltinucs decodin;;.

3) hom (2) above we know that if !hen' are two branches from one 110d.!s Wil:l

equal mc tri cs , then till) ' 0111' i s selec ted nt random. If the s elected p~ [I , is f()llJ1J

to be unli kel y l vi lh rap id ly incn: :.1s ing mer it. t hen decocl er c<rnccl .~ th :\l 1 ',0111

ilnd gtll''; h.1Ck to that nod,~, 1[ t l1L"o ~dec't~ other p;1th (1)1Crgi,.lg 1.-01" I.h. I nnd".

For c,x,Hnpk observe in the Fj~ ).3 .4.13 (a) that rwo bnlnches with same metric

C!l lcrge Irom node d2• One path is ri]d.1Clfls (or path marked '[\') , , . . . i t l , ,m'lrk

'3' (It n~, ' nwrefOl 'e dccodc r d rops t his p11 th ;1J1dfollows other p;lth,

4) The deci si on abou t d r.?pp in l~ ,I path is based on the expected V;dL,(~(lf runnJl1g

metric at iI givcn··node. Running metric at il pnrticu[Olr J '" I\ode i s ! ;i \' Tl1 ~S,

..' lJ"j·21)

R\lnning 1 l1 cl ric " ' jl! a

where 1 is the nude 11 1 which rne tr lc is to be calculnted.

II is the number of encoded output bits for one TT,eSS'lf,c bit,

<lnd(t i s t he f r( ln srni ss ion er ro r p robab ili ty per b it.

111c sl?"wm t i < ' l I decoder ;lh;nd,')l1s ;1 P'll)-' I·!)wn("\'r:'f itt. rWl)1inr fiwrn( ,",""",:'(1~

(j ria +"I~;):'HN~ i:', the ! .hmdd 111' ;;:J<.l\'(' !I (I ;11 t" nod,!. fi::, .i,;.::; ( \ . 1 " " j " . . . ~ I".running rnerr ic ill a p; lrticu!ilr nod,') with )<':;1'('(1 \(1 nU!l1Pr.r of 111.1t l 1< "k, T jw IW"

dotted lines Sl1QWS the filng(' of threshold' ,\ ' ;lb,)Vc i" (1. ,)1 a par tk ll lM nod, ', Ob",'r'.I[·

that since metric of path '5' exceeds the Ihre::hold <I t 5 '" node, it is abandoned and

decoder starts from node '2' again. Slmilar ly path' A' is also abandoned.

......;.:.4~:i~

Information Coding.Techni(jues . 3 ·130 Error Control Coding

- ,<0 • • h zoes t r th eshold limits then the value5) If the running metric o r every pat goes 01.1 0 r ."

Information .Coding Techniques 3 - 131Error Control Coding



of t hreshold '/ :: ,' is increased and decoder tries back agalll. In Fig. 3.4.1 ,3

d f F 341 we know thai /1 = 2 . Let sb) the va lue of a. '" I/j6, for erico er 0 Ig. .,

Gllculatc [n u at 811 , node.

(a)

4 1 c

- - - - ----_ -

(b)

Survivor. jna +~,__ -------.

2 - - - - - - - i - - - - - - r

i 1 _' ·_-~l

_ _ _-.._-+-- .....::::~"._.....,...--~jna

-

2 10 128 9 11, 4 5 5

Fig. 3 .4 .13 Sequent ia l decoding

At 81/,·node [n [ J. ' " 8)( 2)( 1 / 16", 1. The value of 6 . : " ,2 - Therefore th~eshold will be,

'/1,', + A '" 1 + 2 ' " 3 at 8'" node. Similarly the threshold at,other nodesThreshold", J ~ ' - ' ' . • . 1 d d i g are Jess than

can IJ, ' GlkuiJlc;ci The computations Involved In sequenna ceo In, "1

". '" I d odi z Is comp lex . The ou tpuiterbi dccodinv. But the back trackmg Insequentia ec. mg I . d

o h . bi d ding anerror probability is more in case of sequential decoding. Both .t e vil er I ceo f

' .' 1 t d ith the help of compute r so (wareequential decoding methods can be unp emen e WI

eff icicntly.

_ '

3.4.5.3 Free Distance and Coding Gain

For the block .ind cyclic codes we have seen that the error correction or

, dl b twe the code vectors.ower depends upon the mirumum rstance ~ ween . . '

convolutional encoder does not divide the outpu t encoded signal into dlfierent cO < l e ; ; J : : : ~ ! ' ~ j l Evector, bu t complete tr ansmitted sequence can be consider~_~_as a Single code

kn th t .. um distance beet X rep rexcn I the transmitted sequence. We ow a aurum

t h L ' code vector is cqunl io minimum weight of the code vector. Therefore a

_ _ _ .

. "

.~

dis !. lncc i s ' , - , q U i l l t,o the min imum d istance between the c()de vectors . Since minimuru

di~t.ulce is equal to minimum weiHht of the code vector WI.. ' Ciln write,

Free disl;1nCC ( d f)Minimum distance between code V('(tors

r.e.

Minimum weight of t he code vec tors

(w (X )] , and X is non zerolllU -., r~.-L22)

H('fl' f w (X ) I n : , " i~ the minimum wei[;hl uf code \'0ctOI' IC or COI1\'oIU(iC'I1,ll C(l(~;,\~;

free distance (d I ) represents the errol' control power.

COding G a in ;

Coding gain is used i1S " basis of compilrison for el i rfercnt coding Il1dhods, T()

achieve the same bit error rate the coding gain is defined as, ~:

(~) Encoded

(~I~)odedFo r convolutional cod ing the coding gain j~ given .'1$,

A = rdJ-2-

hcre ",-' is the code rate and

\) rtf is the free dis tance.

{4.6 Probability of Errors for Soft and Hard Decision Decoding

In this subsectiol"\._we 'wlll s tudy the error rate performnnrr, of the vi(erbi algorithm

(or the addit ive whit e gaussian noise channel with soft dec is ion arid hard dec is iondecoding.

_ _ 3 .4 .6 .1 P ro bab ility o f Error w i th S of t D ec is ion Decoding

lei us consider that the all zero sequence is tT<lIlSmitted a [lcl we en lculare Ihe

probability of error [or detector decidi ng in fa vou r of ana IIll' I' scc ll Il 'ncc . Let the cod cd

binary digits for the jill b ranch of- the. convoluuonn t cnd r' .j),:, cepresclltcJ ilS I.",,,m""J,2, n. The input to the ·viterbi dcc~der be the sequence, rim, nl := 1, 2 , ... .n ano

j =< 1, 2, Herr- "'" is given as,'

. .. ( 3. .1 .2 < »

Here ei m r epresent the transmitted bits. j i ndicates the j" " b;'anch and mind iCil k~

ffllli bit I ntha] branch, E o is the transm itted sigl1i11energy {OJ' eilcll cude bi t ;lJld II,.... isthe additive noise.

TIle viterbl soft decision decoder cnlculotcs also branch llldr'ics byrelation: (n!lOWiJlF

"

Information Coding Techn!ques 3·132 Error control



HenC8 i represents the III] path and J . l f , ; , is th.e metric O r r~brtlnchvit crbi decoder then obtains the path metrics as,

C t ' v 1 ('l " , f ,1- 1 ;1 )

I" I

. .. ( 3 .4 . 2 7 )

i= v »==1

Here i is the i''.. path arid n is the branches in tha t path.

TI1 ( ' c0l1"ulul"i011i1! code doC's not have i1 f ix ed l en gt h. Hence w e w ill derive its

r','rlormilllC(' troll1 the probability 01 error for the sequences that merge with the ~Il

/,'nJ sequence lor the first lime' ill <1 given node in the trellis.~- ,?

rlw !_,rubnbility "f e rror whc r, another )Jil:h m(' ,g l' s; :, ;i~h n l! ZC](1 pnth is C(J.l,jVli-e<11

In probnbility thr.r )llctric o f this path exceeds the metric (It an 7.(>1:0 pnth lor the titst

Ilme i .e .,

1' 2 (d) P(CA ' I ( I ! ?: CM( [ l ) )

p(CM(I) -CM(ilJ\

putting (0'- C\,)(I) and eM ( 1 1 ) in above equation from equation (3.4.27) we get,

. .. ( 3 . 4. 2 8 )

Here the d i(f('rcnc(C [PI _lJf)) is equ:l! to 1 at the bit j)o5itions which are in errorI J J I~ lit!

bl'C:l use of incorrect Fil til wif h respect to all zero p;, th o Thus the above cqu;.tiO\l ir. thl!

rlroba~j)lty of such non zero bits at 'd' positions in the incorrect path. ' . 'fet1~ above

equation can be w ritten in the simplified form as,

P 2 (d) '" r { t n : o } " , . . ( 3 .4 .2 9 )

Here ( r J i are t il" s~'l ();;,(),,, '. bits at 'd' p os it io ns . . Irf) have gnt1ssin!1 distribution

r;-- 1wit!l -.yE.- mean and -3 : No variance, This is b('c~use these bits <Ire basically C\'f(lr hit~.

Hence the above probability equation can be written ;]5,

. . . ( 3 . 4. 30 ),

f'~(I f) '" Q ( . j 2 Y ~ R - ; d )_._(3.4.31)

11,b; i~ the prob;\bil i\y of p.1th (If dist~\Ii(c '0' from the nil zero path. A·~tu·'!!Y\1~?1\'

wil l be 1 1 1 : 1 I 1 Y such paths w i t h d if fl 'r en l dis t~t1ces and they merge with <Il lzero p a l h at

given node 1:3 The first event error -pron,1bility can be obtained by summing Ih" '~rrGT

pr()bab ;l ;tk'~' nvcr n il the possilale pMh distnncl's. Hence the upper b(lund , . " t I; t :,!< : " " , , > 1

enn,r pr obabili ty will be,

r" $ t I 1 . rP:t(dj :~ f lI 'IQ(li~J<~~:;),I • ",... ,I" )" ...

1-krc n,1 is the number of p<llh,; or distance 'cl' from the illl zero p.ith wl1• . : 1 1 nWlge

... (JA,32)

with "II zen> p~ti1 [or the first time

t--knn~ II" , ,11.)(,.:.: '-'.\I''',·:,~i(\n J: ... ,lIed (;,:;1 r-vc n t en('" prob",l>iijly, TI,;:; {j:';:( ',',,'n!

,'n'OI' prnb;lt>\lity pn,,,kk;; the n1'~\'Url~ of the p(!rfnrmilllC1' of cGrI\'0Iuli')nni (o'k. -1he

HI ( 'r roT' I ,~ob. 'b ;l it )' i ~ more u sdu l )-'('I'(('rm"I1Cl' me<JSllI't'. The bit error 1'~'(·1Uilit), ,'0

upper bounded by first event error p\·obi1bi!ity.

3 .4 .6 -2 P ro ba bility (If E rr or w i lh H ar d D -! 'l: is io n O ~c Q d! ng

'nw hi1\lIry symmet. r ic : d~itnn( 'l uses hotrd dCclsiop d(>(·odit1~;. Ih,t(' 1<'1 '.l~; c·ow.J,_kr

ihe p<!rformnncc o r viterbl <ltgorithm for hard decision decoding .

Consider Ihitt the all zero path ;< , l r, ,) '.~n ,i!t l~d, ;In (1 the p~th ,~'il',--j\ r~ ,,'k·:~~..! ivs

di~;tallcc 'd' frotn.thc·-;!H 7,CHl pil th . Wi th h~rd dec ision decodiT1(~ , Ihe f'" ' )' ,\hIEt\· l ;" ,t

incorrect path is,selected is gi\,l~n as,

P1( tf) I: t ( l ( ~ )"I (1 - vr:'1.-I~11 kh1-

, _ , ( 3 . 4 .3 . 1 )

Here p is the probnhility of error in the binary symn;cl:k chan''.!.':' 'lh.' ,llnio:\

b,}\ ll"ld (ll\ the fi~t ewnt ('.ror pl't'A.n\bilily thin all the possible p , \t h~ r . .~ : rg ( ' ·.,'JlIt , ' . i 1

zero path at a given node is, .

- . t : ' . . . . .

-- ->: ....:

p ,. < f 1 1 , 1 ; ' , \ 0 1 )

,r n fl'~- ...

Here Ill,,) represents the number of p~ths corresponding to the sd o( o1i ' 1(:111 ' : \ ' : ; , l.

The bit error probnbility can 'be more use fu l than the first event crror proIHbd~t.y, [he

uni()n b'Jund given by equation (3.4.34) is the upper bound on bit error probabt!\ty,

_ . ( 3_ 1.31)

information Coding Techniques Error Control Coding3 ·134

~.

Error'Control Coding



Comparison between hard decision decoding and sof t d ecision decoding:

Wo! studied har d decision decoding and soft decision decoding for block codes and

I r I 1CiES Fo llowing table 344 lists the comparison o f them.OllVO U rona C( > ,

s r · l Parameter Hard decision uQcoulng Soft deciSion decoding

~Jo. I

The demodulator outpu t isPr~r!cj~le The decode r ope/ al es on hard

deci si on s mad" b y tbe quantized '0\0 f f i O f ' " Im.n Iwo

demodulator . This decod ing is I(lvels. Henc9 decoder operates

called haro decision decoding. on soli decislons made by

demodulator. tl ls called soil

decision decoding.

p",ierrcd IYile of channel Binary symmetr ic channel. Gaussian channal.1---.-----..

The decisions c<>nno\ beorrect I lncorrect decisions The dec is ions can be labeled

as correct or incorrect. direcUy_l;,beled as correct orirf.;o;;'ecL

~ Probetnti t 0 ' " " ~ Symbo l er ror pr obabi li ty c an be Likel ihood of symbol types cal'" calculated directly. be expressed. Hence

k -Co ", "' O "' , , ,' 0"_"

conditional probabHil ies are

normally written.

lmpternervtation ol decoders is Irnplell)entalion o r dococers i.

tsimple. complex"

Prdeucd code Block codes and corwotuuonat Co nvo lu lional codes use so/ l

codes use hard dac is io l1 decision decoding,

. decoding.

T able 3,4.4 Compa rls on between har~ de cis ion docod tng and soft declslondecoding

t 1/2 con stra in t length N ""21 ) * Example 3.4.3: The figure belotu depicts a ra c ,

c on uo lu uo na t c od e e nc od er . S ke tc h I hL ' co de t re e f or t he s om e.

Inputbillary--~

seque~ce. L..-.....-J,.._"""'T_..J

Fig. 3.4.14 Convolutional encoder

. '

..,i;;.. ,, !

,"::;' i~

,. .\~I'-~

: J."

"~'~.i~ , ; 1 ' I

;.~. .: :~

Solution: ';i) '.Define .states of encoder . 'J

,The constraint length of the given' convoluliom] encoder is K =2Its rate is 1

"2~ i : , : ; ·'·means [or s ingle meSs.lge bit input , two bits .'\"1 and -'2 are encoded at the output. .Sj'

., ;.;:represents the input message bit and 52 stores the previous message bit. Since only

one previous message bit is stored, this encoder can have states depending upon thiss tored message bit . Let 's represent ,

state 'a

and"-..'state 'b'

b} Outputs of encoder

C'Let's assume that the contents o f 51 and $2 are zero initia) [y o From ('n'(Odef o f

Fig. 3 .4 .14 i t is c lear tha t outputs Xl and X2 are given as,

} . (J.n5)

c) Prepare state diagram

Before drawing code tree, We will first prepare the state dii lgram (or this encoder.

Stale diagram represents the compact version of code tree. Ti1blc 3.4.5 shows the

present and next states couesponding to different inputs of JYl<:s5i1ge signaL Fir st row

of the table shows that if p resent state of encoder (Which is defined by 5:,) is ',7 and ir

input 51 (i.e. / II ) : :: :0, then outputs Xl X2 '" 00 and next state of the encoder will be 'II'

Second row of the table shows that in present state of encoder is 'ii' and input is J

~ then contents of 5J52'" 10 . Then the output XI X2 '" 11 and next stare of encoder is '//

l ' Similarly other two rows represent how encoder operates jf it's present state is 'b'.

(Please refer Table 3 .4 .~ on' r .ex! page. )

Based on the result of Table 3.4.5, we can draw the state ditlgrilll1 as shown inFig:3.4.IS.

00

This t ino shows [ransil ien

.rrom 'a' 10'b' when inpul

/ mAS"")8 bi! is '1' and11 ,,'L[pul IS 11

~b'/-')-10

" - _ ../D 1

Fig; 3.4.15 State diagram for the encoder of Fig. 3.4.14. Continuous

Hna represent Input as '0' and dotted line represents input as '1

Information Coding Techniques 3 - 136J . 137 Error Con(.ol Codillg

Error Control Codln " m;Hion Coding l'(!chnlques- - ~ - - - - - - ~ - - - - - - -



...~

S"l s2

ITl

b

Next stale arter

tronsmil1in9Ollt,paiS

t.e,

Next slate

TIll! rod,., tree j~;derived from the state di.1grilll1 is shown in Fi'~. 3.-I.I'~ 01"".'[""'"

the branch pattern of the code Ire,," repeats IIft..:r two :,u,"C<!ssive rn('.;~:.lg<: L"t<;. This

bCCilu~e ,1I1Y !11('ssllge bit remains slnred ill (he encoder register (U!' !""O ,,,C,,-";-jl '.','

. . " ' :

Input

m e s s eg e

bi: rn

Prese,,1stale

o

o -~ I~ 1 :2

1

b

x , "1Xz = 0

i.e.

I ' ; 1 ~ I c

l.e,

I np ll 1 ~ ( ]

t Slart

1

a

i.e,

i,c.

I.~.

C l Jo

o-··ff]i.c.

luput . .. .i

a

Table 3.4.5 Operation of encoder cf Fig. 3.4.14

As shown in state diagram, above encoder remains in s!~te 'n' if input_!S2;cto. In

the above diagmn1 continuous lines are used when Input message·bit IS"W. and dotted

];'1(;$ arc used when mess<1ge bit is '1'. The arrow on the line shows the. mmsHlon

11n+- Exnmple 3.4.4: For th' (oJllli)iII li')II!1! l'lwoder n rr ll ll[ >t cm cr rf 5 /~ 'H L'1 ; , ,. ; 's' .' '/,j /,dmw tile slnt~ di.ngnll11 Iwd lt~nCl~ InNis t1' ''SrlWL IJdermirlC oll/pul rll~il ," '1':<'1"( l»/JU! . l atR r ii gi fs 1 1 fI 1 0 1 () O . Wlml tlfr 1111 :dimens ions oj 1/; , ' CO dl ' , (r:, .u ,;,;,1

com/minI lmstl! ? Us e vilnbi's II/g(lyilirm 10 decode Ol e s equ en ce , W () '1 10 1 j I 10 1

0 0 1 101 001 010.


Error Control Coding'~!;•

Information Coding Techniques Error Control Coding

b



-

tl)

x·i

(3)

~x;

. . Output sequence

Fig. 3.4.17 Encoder of Ex. 3.4.4 .

Solution: i) To obtain dimension of the code: Observe that one message bit is

taken 'II a lime in the encoder of Fig. 3.4.17. Hence k = L There are three output bus.

for every message bit. I·rence n '" 3. Therefore the dimension of the code is (n, k) '" (3,

1). ._,,--

ii) Constraint length

Here note that every message bit affects three output' bits. Hence,

Constraint length K '" 3 bits.

iii) To obtain code trell is and state diagram

Let the stales of the encoder be as given in Table 3.4,1. Fig. 3.4.18 s hDws the code

tr ell is o f the given encoder.

rn2 mlm2 m ,

0 O= a

O { ) O

~--a~ O 0

- .. . ",~- - - ~.. ~.. - - - Dot ied l ine11 1- indicates

o . 1 ' ;= b----.

b = O 1 input m '" 1-r;;-1~~~

1 O = c : ~--c~1 0

l= d

&!Iid Ijneindicates

input rn > 0

011

" 110

100 ~,____________-----------2'.::. d ' " 1

Fig. 3.4.18 Code tre llis of encoder of Fig. 3.4.17

TIle nodes in the code t re ll is can be combined 10 form sta te dingr•1m .b shu,,'!1 in

Fig. : - \ . 4 . E'.

, t '

c

Fig.. 3.4.19 State diagram of the encoder of Fig. 3.4.17

i v) To determine output sequence

a) Determine .'ICIlaalor polynomials

The gencf i1 ti ng sequence can be writt en for . '. .1 ) f I'", rom ' lg. 3AJ'7 as,

since only 1/1 is connected.

Sirni larly generi1ting sequence (or x;2) wil l be,

g f2 1 := (1, 0, 11 since m and 1112 Me connected-

And generating sequence fo r X;3) w i ll b e,

giP) = 1 1 1 01 are connected.ince III arid m,

Hence the corresponding generating polynomials can be written as,

gtl)(p), := 1

g(2)(p) 1 + p2

g(J J (p) '" 1+ P

b ) D e te rm i ne m e ss a ge p ol yn om i al s

The given message sequence is,

m '" (1 1 0 1 0 1 0 01Hence the mess<1gcpo lynomial wil l be,

m(p) = 1+ P + pl + 1 1 5

c) DMail l outpu! for sf )

Information Coding Techniques :J·H()

The first sequence x i l) is given as,

"

bits f'P1'n "hove three SC'llll'l1C(,S <Ire muhip!exed 10 g ive the ()Ulpili !i'"[,.Il'r,le



g(l) ( p) . m ( p )

=: 1(1+p+pJ+p5 )

1 +p+p3 +p5

Hence the corr espond ing sequence will be,

{ x ? ) } ez 11 101 0 II

. - I ) Obtain Oil/put for g,(2)

TI 1 (2). •H~ soconc sequence Xi IS gIven [1$,

X;2) gPl(p).m(p)

(1 + P 1 ) (1 . . P -l- P J + P 5)

1+p+p l+p7

Hence the cor responding sequence will be.

x i 2) cz, 11 1 i o o 0 0 II

..-., .

t'J OLJia;!! O[{t/Illi for s : . ' )

The third sequence . " < 3 ) is given us,

< I ) ge l)(p ), m {p )

( 1 +1 - ') , (l + p+ p J + 1 ' 5 )

1 + p2 +p3 +p4 .+-pS +p6

Hence the corr cspond lng sequence is,

. 1 .: " ?J ' " 1101 11 1 I}

/i T o m ul tipl ex th re e o utpu ! s eq ue nc es

The three sequences .r,(\j, x i 2) I1 l11 i X,~31 are made equlll I n length i.e, 8 bits.J-ience

7,('ro~;arc apperic.cd in SC'-ll.,,,ncc x ) ' ) and x i 2). These sequences are shown below :

il 1 0 1 0 1 001

1 1 1 I 0 0 0 0 I}

11 0 1 1 1 1 1 OJ

{ X i ' } ='1111 11 0 011 101 001 l01 001 0101

. "

Fig. 3.4.20 Viterbl decodl<l9 for example 3.4.4

" ..

3 -142Error Control Coding

, .1 F .. (~

Infonnatlon Coding Techniques 3 -143 Error Control Coding




v} Viterbi algorithm to decode the given seqU(jnce

Fig. 3.4.20 (See Fig. on previous page) shows the diagram based o n vitarbi" :..~. J

decoding. It shows received sequence .:It the top. The. decoded (Y + E) sequence and

decuded message sequence is shown at the bot tom.

The dark l ine shows m?lximum likelihood path. It has the lowest running metric.i.e. 3."Other path are also shown for rekrcnce. AI any point only (our paths arc

retained. The decoded mesSilge se-q'leoce is.

m ; n 1 0 1 0 1 0 O}

__ . Example 3.4.5: A conw/llliOlltll encoder has single sbift register willI two s il lg e s t hr e e

modli/0-2 adders and an ou/pul muuiptcxcr. T he f ol lo w in g general or seqll~IlCeS ar e

combined by t lu : ·mul l ip lcxer 10 produce the 1 : 1 I (0(1 I : ro i l lp u t -

i) D rm .v b lo ck d ig ra m o f t he e nc od er .

ii) tOT t il e me s sa g e sequence (10011), determine e n c od e d s e q ue n c e.

If a bov e hardware is enhanced by iI/creasing T11l1nber o f s la ges in siliJ I re gis te r and

Tlumber of niod-Z adders respecl ivdy. u sh at is th e effe ct o n

a) Gencra/cd oupul sequence

Ii) P er io di ci ty o f l it e c o dc tr ee .

Solution: i) To obtain block diagram of the encoder

TIle shift register is two st~ge but every output gl.gl and S3 combines three

inputs. Fig. 3.4.21 shows the encoder.

. . . . . .

Messages(.'<juence -----'

m

91 = mii ffi2

92=mwmt

!ll=mwmt wm2

Fig. 3.4.21 Block diagram of the convolutional of Ex. 3.4.5

!i) To obta. ln .ou tp~t sequence for m ::::(1 0 0 1 1)

a) Ob ta iT l·generp to r p o l Y l ! o m i a l s

The polynomials of gl.gl and p can be written as,

St =(101) ~ gt(r)=1+T,2

g1 .. (1 1 0) ~ g 2( P) = 1+ p

gJ =(llI) ~ g J ( p ) = l + p + p ' l

b) O btain, message polynom ial .~

The message polynomial becomes.

111=(10011) => m(p)=l+pJ +p4

c) Output s e qu e nC l ! d u e to gl

Output of sequence gl i s given as,

x l ( P ) , ~ g l ( p ) m ( p ) =: ( 1 + p '2) (1+p3 +p')

~ 1+p2.+p3 +p" +p5 +pb

Xl (1 011111)

d ) O u tp ut s eq !J en ce du e to g i

Similarly output of:;2 is given as.

X 2( P )= g2 (P ) I I I ( p ) " " ( 1 + p ) ( 1 + p 3 +p4)

Hence

Xl (110101)

e ) Ou tpu t s eque ll c e_ jJ . le - /(FgJ

Output of g3 is gillen as,

x J ( p ) = " g ; l ( p ) m ( p ) = (1+p+p2)(1+p3;·P"')

= J+p+p2+p3+pl.

X3 = (1111001)

j) Mu lt ip le x in g t he s e q ue n c es due 10,gl ,g2 and g3

The multiplexer wi l l multiplex the b its of XI -r--, and x -' Iol. ,.," " 3 " " UllOWS:

Output sequence = II1 1 0 1 1 1 0 1 1 1 1 1 0 0 1 ] 0 i o u

Note that X2 contai 1 6 ... ins 00Y output digits. Hence its 7,n output digit

zero m above multiplexed sequence. .

Ifhardware is enhanced by adding shift registers and 'adders

a ) E ff ec t 01 1 gellC'ralrd o ut pu t s c ou c nc c

Hence

Hence

is assumed

i ll fo rm<l lionCod in9 Tc chriiquas 3 - 144 Error Corrtto[

For each input mesf>ngc bil, three output hits arc f,l"nefMecl (See Il 'fg. 3A.2;T).

Me thr.....rnod-Z ildders in the encoder. ~creforl! three outpu t "Its are gnn.t"rutQ.o:t

3 - 14S

Along thf l' bnmd1tC'S between the l1"dr-s, D with proper cxp<-jnents·:j.rc·,,;,·jt(t![I. l l . - . _ · · · ' ~ ' ' ' · ' ' ' '

pIe there is one branch ( roz'tl 'a'to 'c' wHh output Hl.:I-~chce'D~ is wi'ilicn al



m ud -2 a dd cr « a n' in cr ros ed . I f J ( ' 1 I outpu; b its for eTJery f l IC! lS' Ilg<: bit ,7t1! i! l(m"tl~~a. 1

l ength of the coded sequence inc reases.

/!} E~h'l11 period icily o f c od e tr ee

Period icily of codctrce is related to number of stagesJn the shift· . register.

I"ig. J.4.21. observe thilt three message bits are present at any time in ihift ",, '' 'H,P' ' .Hence cod" tree will be periodic after fourth messnge bit. If number of stages

increased, then period of code tree also increases,

3.4.7 Transfe r Funct ion of the Convolut iona l Code

The state diagram of the convolutional code gives information about distance

!)I(lr<'rtics ~ncl error 1-'1e performance. Consider the state diagram shown in Fig 3.4.22.

,\i(' will lise this stare di;1gr~m to obtain the distance proper ties of the convolutlonaj. ~ . - - -

Fig. J . d : 2 2 Tile state dlaqrarn of rat e . !, k == 3. 3

convolutional code

~o-J---+-+~------+-~--~---+~

Fig. 3.4"23 State diaqrarn of Fig. 3.4.22 with

d lstancc labe ls on the branches

Let us label the branches' of this

state ~iagr~m as DO,01,020r D3 .

Here the exponent of D represents

the number of ls .In the output

corresponding to that branch. Thus

the exponent of D is equivalent to

hamming dis tance of the output with

respect to all zero output. Such

diagrams with

D3 is SllOWJ1 in

reorg'nnized stare

labels D, D2 and

Fig. 3.4.23.

The self loop at node 'a' has all

zero outputs, i.e. D° ""1. Hence it is

no! shown in Fig. 3.4.23. This self

loop with all Zero output does not

contribute to t he dis tance propert ie s

: the code sequenrc relative hI ;.11zero code sequence. The node 'a' is

split into two nodes. One node is

called 'a' only and it is like input

node and the other node is 'e' and it

is like, .output node of the state

diagram (See Ffg.·3.4.23).

,lmtnch, Simll l l .r1y the branch from 'b' to 'a' ha s output Oll.in,Fig. 3.4.22. '{his I'

o alj br.rp(h hu m 'b' to Ie ' (output node 'e', which l5'·oblained due ttl splitting 'II')

label 1)1. as shown in FIg. 3.4.23.

. Sitrti!arly all other branches are labeled in Fig. 3.4.23. There ill'{! (;Io<' I ]o<: i"s in

3.4.23. 'The four state equations can be written fot state diagram of Fig. 3.'1.2:'1ag

x, =D3X . +DXb

-x, ",OX, +DXd

X iI ",02X" +D2Xd

X< ~D2XI'

) . . (1.4.31 1)

The ! ' twt< ' equation for the node Is writtr-n (rom incident bml1cht:.<; \11""-" rI ,;l( 1," ,1 . . ' ,

For example node 'c' has incident branches from nodes '3' and 'b'. " 1 1 1 e transfer

function of the code is defined as,

T(D) '" ~: (-'.,1:,7)

on solvInG the ~~tlltP. e<Jt:nlions If! equation (3.4.36) we can w rile »bov« iT.~lls("r

(ullctj~n as,

7'(0) .-1-202

. . . ( : \. ·1 3 .' : l)

'fhe (ir~t term o r the transfer fUl\cliol\ i.~ D". II means tl,,'n' j. , ~inr:!(' 1'1:11 ,. !lml)\min~ disllll'l(,c d c 6 between node 'a' .Hlt.! "c'. As shown in Fig. 3.4.2] tbi~ 1,,:1:, i:'.

acbe. The second term in aoove equation is 20M • It means there are ~<J p,d l,~;01

dlstnnce d "" -8 between nodes a to e in Fig. 3.4.23. These two pil th$ are ,v-ri \w ~~ il

acbcbe. Actually the path from node a to e means the path stilrHng from node 'a' llnd

comrning back to node 'a' only. There is ill! zero ou tput starring ilnrJ end jnr~ ~~n ,) (k

'1\'. Tile dlsrances of the various pnths obtained above arc the h,:lTnm!,\_;.; " ih1" , . .:c<,;;..","

e: :.;}> zero output palh on' node 'n'.

Thus the dlr>~ance propNHes of the cUlwoju lkl fl it l Cl' lr i\ ; can beob:;liTWd f i'L:! l : :" .

transfer {unctIon, The minimum distance of the code is called <1 5 minimum lrl'l '

dlstance. It 15 denoted by 1 1 / , , , , , In this example dlrrr :: 6.

' C o . •

0/'.h 'r Inro.rmaUon Coding Tuchnlques 3.147" • "T" _ : _ E : . : ~ : . _ r o : r : . . . . : : c : . : : : o n t r o )o d j n g(nfannation Coding Techniques 3,146 Error Control Coding

3.4,8 Distance Properties of Binary Convolutional Codes 3} Synchronization problem does not a([l·r! tile [ f. - per ormance 0 convoilltion"J



1Consider the convolutional code of consfralnt Jength K and rate Then the

n

minimum free distance (or this code is given by the standard result as,

l2/-1 . J

dIm: S; min -/-(K+{-l)l11;,1

2 -1

Here the symbol L x J means larg~t integer vcontained in x. Table3.4.6 lists the

minimum free distance and its upper bound for rate j code at various constraint

....(3.4.39)

lengths.

The generator; are also tabulated for these constraint lengths. This code is optimal

in the sense that it has largest possible d [,,-r for the given rate and the constraint

length.

- .-Constraint

Genorators In octal d,,~ Upper bound. outength K d,_

3 5 7 5 5

4 15 17 6 6

5 23 35 7 8I) 53 75 8 8

7 133 171 10 10

8 24 7 371 10 11

9 561 753 12 12

10 1,157 1.;;45, 12 13

11 2,335 3,661 14 14 .

12 4,335 5,723 15 15

13 10,533 17,661 16 15

14 21,675 27,123 16 17

_ ' Table 3.4.6 Rate _! , ma~jmum free' distance code2. .

3.4.9Advantages and Disadva~Jage~of Convolutional Codes

Convolutional codes can be designed to detect _or correct the errors. Some

convolutional "odes available which are used to correct random errors and bursts.

Convolutional codes have some advantages over block codes.

Advantages :

1) The deccding delay is small in convoludorial=codes since they operate on

smaller blocks of data.1

2) The storage h~rdware required by convolutional decoder is less since the blocJi:.

sizes are smaller,

codes.

Disadvantages:

1) Convolutional codes are di((jcull 10 <Ht'llyzl' since their .na lysis i~ <:Ol11plex.

2} Convolutional codes are not developed much as compared to block co, :,':.

! I I . , . Example 3.4.,6r : A 1 ' 1 11 1 '1/3 cmnotut ion I'll 'odcr i I /" .[ , Ii~<'l Iem IIIg ucctor« fi~.\'1 0,(100),

g2 = ( 111 ) an d gJ == (1 01 ) -

i) Sketcl: JIll: encoder crmfiSIi rill ion.

ii} Draw ttu: c o d et re e , s t at e dill,'?rtllll an d /n'lIis d/llgTllIIi.

iii) If input message s eq ue nc e is 10110, dclcmullt' the 011 /1 ' 111 sc ou en cr o f / ) 1 / ' ('!leoti,"/'.

Solution: To determine dimension of the code: -

This is rate l/~ code. We know that

k Iratc ""Ii= = 3 " ' therefore k =1 and II'"3,

i) To sketch encoder configuration:

Her~ k = 1 and 1/0= 3. This means each message bi t generates three ou tput bi I c~.There WI!! be three st age shift regis ter. It wil l contain III, iiiI and III"

First output XI will be gencl'aled due to 5 [1 " " (WO )

Since g J " " (100) , XI = Ill.

Second output X2 .~i!1 be generilted due to S 2 == (111)

Since'>:2 = (Ill);' Xl ;0 III (B I Il l ID 1/12

Third outpu t x} will be genera ted due to gJ '" (101)

Sinc~ :?3 ' " (101), XJ =1/1 ID mz.

Fig. 3.4.24 shows the diagram of encoder based on above discussion.

OUIP IJ :

sl~quC'ncc~

Fig. 3.4.24 Encoder of Ex. 3.4.6

InforrnationCoding Techniques 3·148 Error Con!ml C(}flir.g

'\

ErfCH"Con\r 01C,1d't1g.--~--~_...~~ .. .~-...------



- - - - - - - - - - - - - - - - ~ - - - - - - ~ - - - - - - ~ - - - - - - - - - - - - - - ~ ~ - - - - - - ~ . - - - .~--.~.ii) To draw code tree, sta tediagran' l and trellis dittgr-,nn.

n ) T o o b/ ai rt t re ll is d ia g ram

The two bits "'2 1111in the shift n:glstcr will indicate the $!o'Ilc o i 'ht' \'IK",kr Lt't

these states be defined as follows:

112111 '" 00 state 'a'

1 1 12 . 11 1 1 • == Q 1 state 'b'

1 1 12 1 1 11 di s ta te 'c '

1112 'n ~ 1 1 state 'd'

Table 3.4.7 Lists the sta te t ransit ion cnlcut"t[ons.

SL I Current state Input Outputs Next state

No. m,tnl

J_m , )(1",m m, m

)(2 "m(J)m, t flm,

)(l ~m(1)ml --"

0 0 0 0 00, i .e . a1 \" " 0 0

\1 1 1 1 0.1, t .e , b

2 b " 0 10 0 1 0 1 O. l. e. c

1 1 0 1 1 1, i.e. d

~ \ ..0 0 I 1 00. i .e . a

1 1 0 0 o 1, l.e. b

0 0 0 1 !, l.e. c

1 1 i 0 1 1, I.e. d

Table 3.4.7 Sta te t ranslt lon catcula ttons.

HolU next sInge is wri lrcl t ?

Co~idcr the following diagr:lm . .

Istatus before Shift!

L , . 1 ~ 1 % y t + : ; J~These two

bits represent

state (m2 m1)

_DIscarded

Fig. 3.4.25 Present and next states

As !'h\lwn If! .a J~w ,~fif~llr(', current stl'lt'e is "'1 t)Jj.' W:lcn tile bits , , < ' ( ' -shiff,',!, (hen

!II'\! :'1:,11,bec6,nes 1111_""Table 3.4.7 show: current and next states ac,;ofdint; 10 this

collet'Pt.

A tr( 'l Ii~ dil l1tram is shown In Pig. 3.4.26 based On table 3.4.7.

000a e 00 t JI _: :_ .- -- -- -. - , fI a Co 00

----- .•_ 1·\1 _" , ~

-----'-«:."

"0' .. -~- ------- .•-- b",OI.... I ~~~........ ~ ..."".....~----"-~- , - - - -

"~'7::r ..~ -

/ .:» ~<- ""'---- 010//_-'... =>-«.C" 1 0 . , r - : -- - W o _ _ _ _ > < ; " " " ",>. e~ 1 0

"

001 "'-"""':01

d = 1 1 ~-:::- ...---~-----------:::::.> d" 11

Do lt ed l in e i ndi ca te s i nput m " 1

So li d li ne i nd ic ate s i nput m " 0

-".--

Fig. 3.4.26 Trellis diilgrarn for encoder of Fig. 3 .4.24.. _

iI.l 'fo 0 1 ' 1 ( 1 ; 1 1 ;;1(1/, ' r i i l 1 8 T n J l 1

Jf w,' comb;!", tlw I1Ind,':< in lH'lli~; diH!,',I'i'lm, tlwn We' w il f ,c '! ,;!a:(: d"\;,,,.r;" II '~,

shown below.

,.,ig. 3.4.:.', State dll~"HIof encoder of Fig. 3 .4 .24

c) To ()I,fnirr emit ' I r~ ~

Code tree can be developed \\"Ith the help of stilte diilgrilrn. Foll owing procedure

should be performed:

1. Llcgin with node (m y node i J ' 1 [l I 'l B : ll h ' , ~ )

.. ,,",.,

'.2. \)rtl\" its next states for In ~. 0 ,~~,d '1

_ I n _ f o _ r m_ a _t i o ~ ~ ~ ~ : i n ~ g ~ T ~ e = C h ~ n ~ 1 ~ ' q ~ U ~ e ~ s ~3~-_1~5~O~ E _ r _ r o _ r _ c_ o _ n _ r r _ o _ I _ C _ o _ d _ j n ~ g

3. For every state determine next states for m ,. 0 <l.nd

Informatlon C o d i n g Techniques 3~151 Error Control Coding

. ' r " t Similar ly generat ing polynomia l (or l P ) can be written as,



4. Repeat step 3 till code tree starts repc<>ling.

Assumption : Upward movemen t in code t ree indicates Ill"O.

Downward, movement indicates !II" 1

!3,,~ed on above p rocedur e, th e COOl' t ree is developed ;J.S shown in Fig. 3 .4 .28,

000 a

II00 0

301 0

C

11 1 dO(}()

a

01 0a

m=O I>

b001111

~:~a

01 0l>

C01 0m=l c

d

0 011111 a

b

d101

_~c" : . '~

110 110 d

Fig. 3.4.28 C o d e tree of encoder of Fig. 3 .4 .24

I·. :h[> above fi~lre, observe that code tr ee r epeats after th ir d stage. This is because s»

each inpu t bit ,,(f eets upto throe output bits of every m o d - 2 adder.

iii) To obta in output s equence f o r m = 10110

a) Tv dc/CrIIlllll' i \c !! (' rr lior polvnomials :

The f,cnerillor p o l y n o m i a l cnn be written ior : . P ) as ,

~,t)= { I O O }

.\'(1) (p)

" .. ", , : r, ir

=

liZ) '" {111}.;

g(2l(p) = = 1+ P + p2

And gener ating polynomial f or

x f 3l w il l be,

gil) !" (1 01)

g(J)(p) = = 1+ 0 P -I - p2

( b) T o d et er mi ne t il e m es sa ge p ol yn om ia l

The messilge sequence is given as,

171=10110

c) To o bt ai n o ut pu t s eq ue nc e f or g }! )

The sequence x f) is given as,

x p l g,~!) ( p) m (p )

::. 1(1 +p2 +p3 )

- .,-'~~

Hence the corr esponding sequence is,

X}I) '" {1 0 II}

d) To o b ta in o u tp u t s c qu l !I 1c cf or g f 2 )

ll1e sequence Xfl) can b e obtained as,

:IF) g?) ( p) /l i( p)

: : . (1+p+p2)(1+pz+p3)

!!! 1+ P 2 + P 3 + P + P 3. + P 4 + P 2 + P 4 + P 5

1+p+Op2 +O p3 +O p4 +p5

= gP)(p) =d + p l )

:::>

3- 152

Hence the co rrespond ing sequence is,

x i 2) '" {1 1 0 o o t]

lfj~r'"'litItlb"C.n~I~gIec fmlques . .

T 6 delen'nme ;d1mensiO~l.' l {j( t h e ' rode - ; + :; ~ :

i s~~ ~,n:l"~od(!.We bmw th~r:"ii.. ~ ' .~: .• ;~ .. [.fT i. ~~:. i tt .~:~ I




c) To ol'/ni" OIi/pU/ sequence for gf3) .

The sequence .,;(3) can be obtained as,

x V ) S P 1 ( p ) m (p )

(1 + )J 2 ) (1 + P 2 .j. P 3 )

1 +p' +p3 +1 '2 +p4 +'1'5

1+01 '2 + p.1 + 1'4 +1'5

Hence the co r.r esponding output sequence is,

.t < , , J ) '" { lO O }] !}

/) ' Fo m ul rip lc ;r th re e o ut pu ! s eq ue nc es

':Ill' l:lJ'I~es e C ] ucnccs . A ~ ) , /".1 and ;r(J) are made in equal Je;;',!th, i.e.J I j Q

Z : -05 <HI! appended to X,)I) These sequences arc 115follows ,

{1 0 1 1 0 O }

{110001}

(1 0 o ] I 1}

J: 1 , t , '. " , C o ' , , 'c '" ,.,. _. ;ee - ... - therefore k=l rind 1'1 ",,3.' ,

" n f~:, ;" ; , ., ." " - ~ ' - ' - ' : ~ - -':-~'_--r~:-:c-("(';odert:Pllfog'urntlon ilrtd;loglc t.1blw:i' :) :. ',\,' ': _

f- y w _ m , n _ ' . _ ~ ~ .j ~ ' ~ '- - . .-~----~--r--.ft- ':.Enc1oder cO[llfigur:atfon , (, ,

generator 'sEiq'ue~<.'CS~gi.yc'.'. inJb~_"_~~;ID.1:rIe ,arc similar to thoac : . 1 ' ,v 3.4,6

""""'vl(ms example), OnlY:S2' and g" are,'t'.xch(Inged. Hence encoder will be s.une ~s

3.4.24 with £2 and gg ' exchanged.

.~

,.. .

l3i['; from tlK above three sequences nrc multiplexed to give the output sequence.

Jig. 3 .4.29 Encoder of ex. 3.4.7

~In' above figure .observe that,

gl '" (100), hence Xl <m

Xi '" {111 010 100 101 001 Oll}

This is a ll OU l 'P \ lt sequence of the encoder .

: , " ' + I:::x~mplc 3.4.7 ; /, rate 1/3 conuotutionnl coder with amstmint lengtl: of '3' uses Ill/!

xen;~rf!t;l,g vectors

S I ~ (100), g 1 '" (1 0 l) and S :1 (Ill)

1, 1 Skdd l encoder r . :o !~ ,0glml f io ! l IIIld prep l l re / he l og ic t ab le .

Il) Dmw the 5/111(' dingri.'il! f " " " , , , . '

j· ii) Determine t lse dfr~·L. di : ; / rn1cc 0/ the coder.

(1 D.1). hence X2 '" m EE lm z

( 1 1 1 ), hence .1) " '-m ID ml EDnil

2. Logic l llblo!! :

{,_ Table :t~.8 51'0""$ t he s ti \r e t -: .: u" lS iH Q n calculations, The st<1tes me dcf;" '_'(1 ,\::, lv"ttow:-, :

'~, , , ,, , 11 12 ntl "" 0 0 s ta te ' ;/

01

state ....'n 2 _ 1 1 1 1 '" I 0

T 1 1 2 mill state'd'

~I

Information C Error Control Coding154ding Techniques --Current state Input Outputs Next staler.

k < . .t'~~.. Inf~nnaUonCodJng Tcchniq ues

:3 -155Error Control Coding

d) Lowest order of transfer (unction is d fn r .



No. m2mj x,,,mm,m

Xl "m$l».

x3"'mEj)I1~IDm.

a=:OO 0 0 0 0 00. i.e. a

I

1

o 1. Le. b1 1 ~

b = 0 1 0 0 0 1 1 O.i .e. c

1 1.i .e . d1 1 0

c '" 1 :J 0 0 1 1 00, I.c...

01, I.e. b1 0 0

d = 1 1 0 0 1 0 1 0, i.e. c

11, I. s. d1 0 1

"able 3.4.8 Logic table of encoder ot . ~ lf J _: _1 . !I . 29 .

.. When shift takes place,In ab ov e ta ble n ote th at cu rren t state 15 wntten as m2 m~. . . I ac e o f m I 'then 1lI1 is discarded. Ill) is shifted in pla~e r . JIl2 and "' l~ sh1~t~ In p

Hence next sta te is represented by 1 1 1 1 m. This IS illustated In Fig. 3,. .

ll] State diagram: . . 0 h' the state

With the help of t.. ble 3.4.8 state diagram can be drawn. Fig. 3.4.3 sows

diagram.

b

00 0

-~~~ f 101, I

d ...._...

011 010

c

Fig. 3.4.30 State diagram of encoder of Fig. 3.4.29

lii) To d etc rm ine the d,," of the coder

The Ji:;I, •• lCC d (r... Can be ob tained through follow ing s~~ps :

a) Split the st . . te d iagram with inpu] not .. as 000 output.-

b) Write s late equations for all nodes.

c) Determine t ransfer funct ion.

II) To split the s tale diagram ifl to s ignal flow graph

In the state diagram of Fig. 3.4.30, node 'a' generates 000 OU rpu t when it returns

back 10 itself, Hence we wi l l split th e node a into two nodes il : input nodl' ,111(/

e: output node. TheSiplill

flow graph is then prepared as shown infig.

3.4.31

ea J

o 2

o

D

Fig. 3.4.31 Signal f low graph of Fig. 3.4.30 (Node a is split to 'a' and 'e')

The outputs are marked as D,D2 ,D3 in above figure. For eX:Jmple the output [rom

'a' to 'b' is 111. It is marked as D3. The output f rom 'b' to 'c' is 001. It is marked as U.

Rule for writlna output In terms of D , D 2 and Dl ;

We have considered the reference node with output 000. From '<1' to 'b', the output

is 111. This output differs with 000 in 3 positions. Hence DJ is written on br~nch from

'a ' t o ' 0'. Thi s rule is. !:Cj:>eatcQfor all outputs. 111i5is explained in sec . 3.4.7 a lso.

b ) T o io rit« l h« s ta te equations/aT 0111nodes

3> ; , : ; 0 Xa" 0 x,

t L_his is !tIe outputThis i s lhe output from node ' c' t o ' b'fromnode 'a' to 'b'

. . ( 3 . 4. 4 0)

These equations can be written on the same lines as equat ion 3.4-.36. Consider nodeb', I t' s .equation becomes,

Thus in above equation, the outputs due to branches incident on 'b' are considered.

Sunilar1y at node 'c' branches (rom 'b' and "d ' are incident. Hence its etJuaUonbecomes,

... (3.4.11)

Similar ly a t nude 'd', one br;1!l<.'his in.:id""t 1""t1 it""lf , 1 11d other i s incident f rom

'b', i.e.,

D~+ 7D8 ~ 4DIO of Wil

1 - '}.f)l[i6------·--.--~-"-·~-~-"

{)t -')_{J~ _,



A t node 'e' only one branch is incident frolll 'c', i.c.,

c) To determine Iml1sjc'r JI-III(liolJ

To determine transfer [unction, we have to solve the state equations obtained

above. Eliminating Xb from equation (3.4.40) and (3.4.41) we get,

(1-02)X, -OX,r '" 0 4X "

Eliminating XI, (rom CqU;ltiOI1 (3.4 .31) and equa tion (3.4.32) we get ,

-DXc +X" = ()

Eliminating X d f rom above cquil !Jo llnnd equat ion (3.3 .34) we get ,

,.. (3.4.44)

04X"

D 4--X1-20:/'"

Putting this value of X " in equation 3.1.33,

X, .

x , 0('

X" 1-20z

Transfer function of the code is given as,

d) To determine lowest order oj '0' in T(0)

LeI us determine the polynomial of T ( D ) . .i.e.,

2D~

7D~ _-tom

4DlO

4DIII-8D12

8Du

I .' D 12 - 1 6D 1 4

16DH ... . mill $(I on

DbT(D) '"" ---2 ""'D~+205 +4010 +8D12 +....

1-20

Above equation shows that first term b 06• This means, there is sint~le [,;1(11n/

dist.11l-< 'X" 'f, be"..,{'('n n(1l1t! ' II' and 'c'. This path is "hee in Fig. 3.4.31. Second terrn ill

1 ' ( D ) is 2VI. 11,15means there lire two paths of distance 8 and SO on.

Free dlst.'!nce ( d" ,. .. ) :

Tlre free di~tance Is given byIowest order of 1 1 1 ( 'term in 1 '(0) . Here it is 6. H<'l1c(',

rir . . . . '" 6

dim' can In:obtained /;'1 inspect / ( 1 1 1 (If s~'{nr"fi"W gm,,',

Hore we der ived T(n), then ddcnninc rff'(r' We! know tha~ Iff"'" is ti-" mini,n!fm

distance path between nodes-'a' and 'c' , By I()okh,!~ at Hg. 3.4.31 we can say that trw

minimum distance path. is a-b-c-e.. Along this 1 ' , 1 th - the distances are DJ, D ,md D2 .

They correspond to distances of 3, 1 ans 2respt·dfvely. Hence the distance between 'a'

and 'e' wilt be 3+ 1+ 2", 6. Other palh$ will have definitely more. distances.

Hence,

d"... '" 6

,'* E)(;1mplp. 3.4,8; Determine ilie stat« d ' fl g /r lm f it r 'the conoolutional encoder drtlltllf ;1 /

Fig. 3 .4 .32 , Drmo tlic trrlfis dfllgrJlm IFtrollS!J 1 I 1 e )irst set of sff:!'dy state transit ions, Dr :l it e S < 'c on dt re ll l« tf in gm m , s ha w th e te rm in ati on o j I re ll ls t o a U z er o s ta te .

t

,_

. £t

- < 1

)

tt

t --

1'-

Information Coding Techniques Error Control Codl1l93 -158 Information Coding Techniques 3 -159Error Control Coding

t . · · ..B .ased on' above ta.. le , the' s.tate diagram b



,~

' .. .Fig. 3.4.32 Convolutional encoder of Ex. 3.4.8

Solution: (i) To determine dimension of the code:

For every mcssage bi t (k ""1), two output bi ts ( 1 1 ' "' 2) are generated. Hence this is

rate ~ c(ld~. Since there are three stages in the shift re&~ler, ·every message bit will

affect output for three successive shifts. Hence constraint length, K "" 3. Thus,

k '" 1, 1 1 ' " 2 and K "" 3

ii) To obtain the statu diagram:First, let us define the states of the encoder.

53 52 ,_ 0 0, state '<I'

S352 '" 0 I, state 'b'

S3 S2 = 1 0, state tel

S352. '" 1 I, state 'd'

A table is prepared that lists state transitions, message input and outputs. The

table is as follows :

Sr. Current state Input Outputs t ll lxt state

No. 5352):j~.s;IDSlm$J

5,x,~ ...\Ds,

$25,

1 a = 0 0 0 0 ri 00, Le, a

1 1 1 01, i.e. b

2 b '" 0 1 0 1 0 1 0, i .e. c

1 0 1 1 I, Le, d

3 c" 1 0 0 1 1 00, LII. a

- , 0 0 o I, I.e, b

-I {j" 1 1 0 0 , "'10,1.(1. C

1 1 0 1 1,i.1l.'d

Table 3 .4 ,9 ; S ta te transi tion table ,

- - can e prepared easily. is belowIn F Ig. 3.4.33. .- .

00

c \

F" '34 . IIg. . .33 State diagram of convolutional encoder of Fig.J.4.32

iii) To obtain trellis diagram for steady slate:

From table 3,4.9, rile code trelli d''W IS lagram can be prep' an~d. It is . d

diagram, It is shown below. stca y 51',11('

O J . _

Fig. 3.4.34 Code trellis diagram for steady slate

Iv)Termination of trell is to all zero state:

•... !Fig. 3.4.35 shows the trellis diagram for first four received set of symbols.. U1st.1lc 1 1 ' " 00. . 11 begins

Infoonation Coding Techniques i;

- ~{'J :l.;~~~C()(tI~hf1I~~:::__. _~:.:...:'.~ :---._E(ror5:::~~,~~'!::, ':~~.~~~,~," ,~Soluflon: il ' 1 " " prcp ...rt' corle tl',·i1i~~



Fig. 3.4.35 Code trellis showing termination to_!!!_zero state

There is a path nl -112 - IIJ - II4 - liS which passes'tfU:~ugh n il ze ro s ta te s, Second

path is II) -[It -c) -{I,I' 111 i5 path ha s a metric of 5 and it terminates on state (1 , .

Thi rd pa th is, ilt-bt -liJ

-1:4 +ns with metric: 6. it Icrmi.n<ltes on state (1s· Thl '5<! are

the shortest paths tl1M tcrmin:ltc 1"0<111ero state. ft shows dInt after rct:ept!o!l of .4

ll1ess"ge bits, the !,("Iiis GHl lcn1'li!llltc to all zero state.

:-;. Example 3.4.9: All e nc od er s f1 mv n in F ig , 3.4.36 gmrmles an ( 1 / 1 z e ro s cq r ll ~nc t '

w hic 11 i s s en t o uc r II I J/ ml nj s ymm e tr ic dm rw ri . T he r ec e iv ed s rQ l ll !l 1c e 0 10 01 00 . .. . There

ar c i,(JO e rr or s in th is s eq ue nc e (Ill 2nd lind 5'/' p os it io n) . S ho t» t Jm ! tltis d ou bl e e rr or

detection is p as si Ni ' w it h correct iol l by ( 1 pp ii c n/ io n o f viler-hi a[gorirl,rl"I.

Fig. 3.4.36 Convolutional ':mcoder of-Ex. ,3.4.9

" ~~ '::~ .. .f'ir, st we will prepare the code trd!i5 diily"m. Lei rIll' Slates of 11\(' ('nel)d,·'· t,.'

' , .~ ) , . (de f ined as fo llows :i~..~

: : . ~ 51 ~I - C r t . < - I : . t ( > 4;1

51 51 01, !;b!(' 'h'

$251 10, s ta le ' c

s z Sr n. state 'd'

A table is prepared IHat shows the stMc transitions, nlessi-lgu- inpu: al1d out!',,!

Th is table is as foll ows;

N~xt slaeutputs

x, ~ sllj s,1D '"X z ' £ ' 1 s'fl S_2

Sr. Currcnt state Inf'''!

No. 5,51

s

., ,__ ---,-+--_1 ------ ..--1-------.--21"00 o 0 0 H 0, i.e,:)

1 1 1 0 1. i.c, b~ - - - - . _ . .~.---.--------.'. --.----~---~. . . . - _, .~ . . . . -~o 1 a. 1 n, r.e. C

I 0 ( 1 1, i.e. ,j

c___"__, ._

;) tt,,(t!

--""-f---._---o I 00, i.e"

1 I) 0 0 1, i.n, t>

l----I---,--- I--,--l-----~-- -----------

o

c « t 0

o 1

od '" 1 1

CurrC1I1

01 . - ' - - - - . . e . . 1

d ~~ -- -- - ..- !! -- -- -- --- -- -~ ::::3 dFig. 3.4.31 Code trellis for encoder of FffJ.3.4.36

:~.~ ..

Errore,ontrol CodingJn fonnaUon Coding.Techniquc:s 3·162

i

InfonnaUon Coding Techniques 3-163 Error Control Coding



ii) To prove double error detection: '

Based upon code b - e n i S of above figure, the-trellis diagram is' sbownfor multiple

stages. The diagram begins from node aj. The outputs and metric: are marked alo~g

each branch. The cumulative matric are also marked near every''node. In above figure

observe that th e path ,al -Iii ~a3-a, -a5 ~ metr ic (2) is maximum Hkelyhood path. This path h 4 5

lowest metric, compared' to ~l l other paths, Hence it is maximum Iikelyhood path.

Output due to this path is 00 00 00 00 ... nus shows that two errors at 2 ! u i and SI" bit

posi rions are corre~ted by viterbi algorithm.

,:,

Fig. 3.4.38 Viterb! algorithm f or detection o f all zero sequence

;

m... Exampl. e 3.4.10: fo r t he c Cl 1l r; o/u lio na l e nc od er u ntl : c on str ain t l en gth o f 3 aT ld ~al f_ : : :':

1( 2 as shown in Fig. 3.4 .39 , d raw the stale diag_~am alld trellis dillgfIIIII. Is I ! ' ! , . " ~ ,

gl? ll I :ru tcd code sys temaf ic ? By u s ing . v ; le r bi a l go ri thm , ' d t: ,, ;o J e s e quen c« 0 100010000 , ; < ' ; i. 0~;t

! I

:>'j-r. . ~. -r ,

Input----<>--_.....I

Flip. f lop-

0100·2+ adder

Palll2

Fig, 3.4.39 Convolutlenal encoder o f Ex - 3.4.·IQ

Solution: ('J T d0 etermine dimensions of the code;

k ·1rate = ~ """. Hence k 1 dII .l. ... an /I=2 . For every message bit, there~ ,Ire two bih

encoded at the output.

Constraint length K _. 3 H> .- -. ence output IS influenced b)' three hifts.. S 1 o In the encoder.

Il) To obtain state d iagr am and trellis diagram:

L et us redraw the diagram of encoder as shown below 'J- •

represented by 1112111

1' Input is' 'C f 11 . n e stare of the encoder j_ ,

I. rn . are u y observe that . b fi .

convo utional encoder of Fig 34'1 In b . . , a ove 19Uf(~ IS similar 10'. . '" a ove figure two fJ i n h Idmputs (I.C. 1/11 m2)~-nlird flip-flop i'j . pops 0 previou s two

P. 34 s not S10wn but mput' , .19.. .40, there are three st. ., h·if . ' rn IS used directly, InF . ages inSit register whicl .unctionally both the encoder s are same. 1 con tam /11,1/11 and III"

Fig. 3 .4.40 Convolutional encoder of Fig, 3.4.39

~. Hence code trellis and slat d' ... t:. e ragram of this encode '11b, m .Ig. 3.4,7 and Fig, 348 Thev , . r WI e similar to those givenI ' '" cy MC given below :

In fO{TTlationCadi og Tech niques E;lTbr Control COding

00

!ri1,Omi;lt1r.m Coding Techniques :3 -165 Error Control Coding



10

" ... ~... \

'10,d . .__ . .. .

01

,,

c

(b) Stale diaqrarn

Fig. 3.4.41

ii) Whet~'er the generated code systematic?

For the output code to be systcmaHc, the message bit and check bits must be

identified. But this is not possible in the output sequence of given encoder. Hence

gcm:ratcd code is not svstcmntic,

iii) To decode 01Op01~,~r"A trdlis diagram is shown in Fig. 3 .4 :42 . In this figure obse rve that only survivor

paths are shown dark. Running metrics are marked near every node. The path giving

lower metric is retained at particular node. Thus at the nodes (I; ,bi, C; ,d; only {our

pa ths a rc re ta ined. · I11E~se pa ths a rc eva lua ted Ht every stage of decoding. At the end,

four survivor pat hs <Irewritten along with their metrics.-They are;

Please refer Fig. 3.4.42 on nex~ page.

I Flg. 3.4.42 Viterbl decoding

l'nlh "2 : '"=U:: ! -11~

- 1 > "- C s -b,. wilh metric 3

1 ' ,, 1 11 3 ; III - 1>2 ,-113 - (~ - " 5 ~Cfl with metric 3

Path 4 ; II) -112 -d3 - c . , -/'05 - do with metric 3 .. ,.. : ,,_

Out of these {our paths, first ri,lh has lowest metric. hence it nll,sl be rnn:.id',n:d

for oeeooil ' l f ' ; the output sequence.

!',11l! 1 : .11,.-111 -(l~ -fl4 -115 -116 is shown thick in Fig. :1.'142.

For r)llth '1' m e ou t:putis,

Output; o c r 00 00 00 00 00

f

, n 1 s r t l l " " "H/lni /i$ d il lsmm. Ot'!t·rmi,lw .lil(' oll1p1l1 rla!'fI seqllmcr: f l i t ' l i, t " ':" '.;

~'~I","r.T ill W110, .

..j.:

F i ! J . 3.4 .<13 Convol ut ion;] 1 encoder of Ex. :3.41 1

f n f S ~ r J l ~ t i o n Coding Techniques 3 -166 Error Control Coding

Solution iJ To obtain dimensions of the code

In above figure observe t h a t three bits are geIJ~rat~.' (or every m~ge ,bit. Hence

Inform~tionCodjng Techniques 3 -167 .Error Control Coding

b) To oblain'trellis,d!(lgmm



k'" 1 and n '"3. : n 1 1 s is rate 1/3 encoder. There are ~ stages in the shift register.Hence coristrairit length is K =3 -

jj) To prepare state diagran'l andtrellis diagram

aJ To prepare state transition table

L et !Il~tates of the encoder be defined as follows:

m2m, 0 0 , Le. state 'a

TlJ:;.rnl 0 1 , i.e. stale 'b'

1/12ml' = 1 0 , i .e, state 'c'

11/21n1 ~ 11, i.e. stale 'd'

1 1 ' 1 1 "outputs of the encoder can be represented as,

,

X2 ~ m ffi mz

Xl Xl E D Tnl

[Ill$m2 JE D nl P u t t i n g f o r X2

- m E D m) E D m2

Table 3.'1.'11 shows the transitions between various states along with inputs. and

Sr. Currant stala Inp'ut .Outputs Next stateNo. mlffl1

m Xl",mm,m

X,mmfl'lm.

X, mmWm,Wml

1 a,;OO . 00 0 0 / ° 0, ~.e.

1 1 1 1 01, l. e, I>

2 b ,., 0 1 0 0 a 1 1.0,1.0. c

1 t 1 0 1 1,1.0. d. - -~ ..0

.3 C" 1 0 0 1 1 ,·00.1 .... a

1 ~ 0 0 o 1. 1 . G . b

4 d " 1 1 0 0 1 '0 10,Le.c1 1 d

.~ .. -1

-' - 1 " 1 , La.d.. - '.

Table 3.4.11 State transition tabl~

shows the trellistransitions between current stale and next slates.

CUrren! stales 000 Nex I S lOlles

a --=:-_~~~---J"""--___ a

011

: :: . .. . b. . . . . . -_---- 100

Fig. 3 .4 .44 Tre ll is d iagram for the convolutional encoder of Fig. 3.4.42

c) To obtain state diagram

State diagram can be obtained by combining the currcnr i 'lnd next stares In al.1nv('

figure. The state diag:ram is shown below;

b

000

011 '

,,,\,,,\

010

c

FIg. 3 .4 .45 Sta te dingram o'f encoder of Fig. 3.4.43

iii) To 'obtain output for m = 10110

a) To obtain gcnerator sequences and their polyuomials

The three outputs, their corresponding generator sequences and polynomials <Hegiven in table 3.4.12.

";""

I'lf~~m~i.ionCoding Techniques 3~ 166 Error Control Coding

Solution: i) To obtain dimensions ofthe code .

3 ·167 ',Errqr Control Codinglnformatlcn. Coding Techniques

b ) T o o bta in irc1lis.djagram



In above figure observe that three bits are ge1}.!lrate.:t.O"revery message -blt, Hence

k '"1 and II= 3. This is rate 1/3 encoder. There are three stages in the shilt register.

Hence constraint length is K '" 3.

il) To prepare state diagram and .trel lis d iagram

a) To prepare state transition table

Let the states of the encoder be defined as follows;

, '

- ~ .. "

Tn2 In. " " 0 0 , i.e, state 'a'

--m]. n~l ee 01, i.e, state 0

/lIz Tnj':= 10, l.e. state 'c'

Til2l1l1 " " 11, i.e. state 'd'

TIle outputs of the encoder can be represented as, »->

Xl m

XJ xz Xz E D m

~ [m E E l m2]$ 111 Putt ing for Xl

m ffi In) E D "'2

Table 3.4.11 shows the transitions between various stares along with inputs. and

outputs.

Sr. Current slaw Input Outputs Nsxt state

No. m~t1Jjm

x , . : . ; : mrnlm

xz"mlBmz

x,,, ma> mlm mz

1 ,,=00 '0 0 0 0 I 00.1.0. a

1 1 1 1 o 1. Le. b

2 b=01 00

0 1 1 , . 0 . Le , c1 1 1 0 , 11. Le,d_ ..~

,.

3 0-

c = 1 O· U 1 1 , ,0 0,lO.8

1 1. 0 0 o 1, Le, b

4 d'" 1 1 0 0 1 '0 1O,le . e

1 1 o '. ..'1

. , .'1"1. [II. d

" '_'"

Table 3.4.11 Stata transition tabl~

.\

1

Fig. 3.4.44 shows the trellis diagram based on above table. It indicates the

transitions between current state and next states,

CUrrent states 000 Next slates

a~~----_"_;----?J

111 011~ - . . . . .-~~~

b•.~. b

~/

- 100

, ~-...x...- - ,.. ~ . . . .. . . . . .

c ._.~ X~lO~ 1~1 . __~ -;

dr-----~--__------..__~

Fig. 3.4.44 Trellis d iagram for the convolutional encoder of Fig. 3.4.42

c) To obtain s tate diagram

State diagram can be obtained by combining the current and next stales in above-

figure., The state diagram is shown below :

_ - " ~_ , . . . . . . . . . b

000

/, ,v 1 101d . .. ._ . .. .

olf. DiD

c

Fig. 3.4.45 Slate diagram of encoder of Fig. 3.4.43

,iii) 10 'obtain output for m '"10110

a ) T o o bt ai n g en er at or s eq u en c es a nd tluir poiY"lJmiais

The three outputs, their corresponding generator sequences and polynomials <H C

given in table 3.4.12.

Infon71atlon Coding T(!chniquc>s3 168 E Co t r c- rror n.tQ tJd!:rIg

s-. No_i , -utput ':eqllence Output equation Gencrnlor sequence Genetnot

P 6 f r t ! I : l " " . , t- , ,

1 x,X, =In 1 (I 0 g,(P) .. 1

2 Xl

~

1n1'.trnnl'lfftllt OodJng Techniqucs 3 - '169 Error Control CodinlJ

.:~2) '" (1 0 0 1 1 1)

(:1JJ :.... " " (1 tO O 0 1)

Multiplexing above scqtll!nces, we gel, . . . . .



1rl "'m+ "7_ 1 0 !h(P)" 1+ p~

3 x.JxJ "'m+tr/j+m;, 1 1 1 !b(P)"'tP+pz

Table 3.4.12 Generni'jng polynomials.

b) Determine message pO !Y tlom iaI""" '_-" .The message $(_"<:juences g+vcn as,

/II 1 0 1 1 0

m(p)

() To oinnin output seqllencl'S Xl, X2 mi d Xl

s' ' ) ; 1 ( ' s~>'1uenceof Al_ can be obtained by multiplying'g, (p) and m(p). Similarly other. ( qu~ncc5 can be obl'<lmcd, The:ce calcubtir'mS are listed in table 3413

Sf.

r - ~ o .

,..-

j Output sequence . . . . . . . . - . . . . . . .

Output polynomial Corresponding

Ix, "'g (p) m(p) ", g(p) (1 , . p ? + p l)

sequence

ir - - - - ~ - - -._--------> : / " " y,(p)m(p)

X}IJ -- 1 ( 1 - . . p' ~ p')= -1+ p"l + p:l .},) ~ { 1 0 11}

~ 1+0p+pl+pl

U)x, = Yl(P)m(P)

) 1 , 1"1 = (1 + p2){1+ pl + pl)

~ '1 + p2...pl + p1 + p' + ps )/ ,(2) ' : { 100111}

:: 1+0p+Op'~pl+p'+p'...._-"----" x ,< " J ' " o,(p)m(p)

,[ .11 = (1+ p + p')( 1+P' ~ p')-, 1 ·, .P ' . .. p' .. p + p' ~ p' + p' + p" + fJ~ X,(3) ~ { 110 0 0 1 }

'" 1+ p + Op' . • O pl + Op' + P" -.

Table 3.4.13: Output poiynorn,~:~~Snd sequences for m(p)=1+p2 +p3

10 multiplex X(l) x(·) <lnd xP) ., r i" i'

Fin.,i output sequence can be obta ined by nniltipkx'in); t he three sequences lengtho r r(!) and .(3) . - (; '\ d j 1 (I) .

. , < , . \ 1,,) .• 11 engt 1 of.1', is 4. Hence two zeros must be appended to

1'/1) to mako l engths of nll the sequences equal. 'The ~('qur'nc('s nrc given below ;

1 . ' f I j (1 () ) 1 0 q )

1:1 "" (111 OOl 100 I m ora 011 ,

'TIli!! is fht'! oUTpu t data seq u en c e .

3.4.10 Comparison between Linear Block Codes and Convolutional Codes

.;:.

Till now we studied convolutional codes and block codes. They C;'In be (Om~'~n~"

on t.he basls of their encoding methods, decoding methods. error conL..:t;H!~

qlpablHti(J!l, complexity etc points. T~ble 3 . . 1 . 1 - 1 lists f~e comparison.

Sr. Un!!:!!r block codes Convolutlonal ,;od,,~'---'----"-'_l

No. I-..._ ~ __ .._~_.,....._-_-____ "._-----1

ICoovoIut!orml code s a,, , !~,,~,1lnr I by !~ lf (j on ! :P cM \. .~ n r rh ~: l Sf l. _}t ~ s .' _~ qu (; "n r. .: c < ln LJ I

o r ' " " , : t l n < : i sequence "" . Ix; :M .L : 9 I 'rJ;wl. i I 0. t 2. i

10 __-_[

Eadl message bit is e"e"odt'd ,cp;l("I,>iy. )On' !

. every messape b it , t wo () (' mom encoded blts r

3rc gencrated. .c.---l-·----------------+--~-------------------r. ._ ~ C o d i" g_ ~~ ~ : _ ,t > _Y b k _ > : : : : _ . _ __~~in!2.~~ b i ~ . ~ ? : . , ~ i~ . . " i

"' "

6

For a block 0 r messoce b it s, encoded b lock

(code vec1 or) Is generated.

f : J t o c l : codes I lre genemt ed u y,

X .. MG or

X(PJ- M(pP(p)

,.,.....-+------_._---------._-------2

Synd,om.. dl .'<'Jx ljngIs "s.cd for most Vr tert>i<I,,.<:;od;ngs used lor mosl !i~dy;o<.1

r , ~ , , ! y I 1 O < , l d ,ie codlfil l. _~ .~ ~:~i"(J_. __ ~_~ '.•_~_."______ _,

~1Of m;1trlce$,_l'?rttyt~ ""'~"l" and Code tree. ~ Irellis a nd Slille < 1 " , , ) , " " ' S are Jsyndrome \ I C C f ( l l " $ :1tu used lor anntyms. used for analysts_

. ----~---.~,----, '._.-_. _ _ . _ - _ ...Dislllnefl ~!o!s o f th e code (A1nbe " I, ,,t ie d D is1anr e p ro pe rt ie s 01 the cooe- "'' '0 btl 'l,jUj'!<I/

'from code...ectofS. from '",nSf"" luOO)on.-- .- .-......__-- . . . . . . . . .--.---..-..._~--.- ---_-..... . . . .-..._ __ .. - - . _ . -'---'[

7 ("",nerefing po!yn()fnial an<! !J"ncrnlor nUJloix G<>nernllngsequenc"" [1m,,~,'d to !'I"I r:,>rj., !"... u""<l 1" gel codev(.-c1ors. V1lctors. ' -'-._., .f

Ern: ,; . co n~ 'di <7f l l JfT <1de ter .:t iQn re p; )bi !rt y Em, .. c or re cnon "m) , I \ " " " ! j , , , , e:~""bf';[y I.r.repends uPO" fYl1nfmo'TI <.1ist';lllcc ~..,. depends upon f ree di! i~"":<! dh'~' J

L-_.J...... - ~ __ '''..~~_ ..~_ ~-- .. - -,---.--.~-, .-.----

----------_._----, .. . -_ ..-_.--"---_._-:i

Re~ Questions

1. WlIIlI sr« ctmvoMJonnl C 'OI I t "S ? Hou: art! I"~ diffr:rrnl j ro l1 lb /oek Corf~5 ?

2 . G iT T in g b lo c k t li tr grmn, cl'Frain t h« oJ lun l io r r o f any coniotutionat rnrodcr.

l_ ;' WJI/II is constrain!lm,~/hfor (-"",IP<II/lI;(", , ,1 C'lTrorfl"{$ !

4; IV/I"I art: cod« In.... , < " ( > < I , . rrrl li~ f ind ~Idk d;I1.~rr""" f tl r conuol ut ional 1 'I ,( () ,1"r,, !____,_ .,- ,c ._____ __-~ -"-"-- '

-~ E

... ~. ,

i

' r, \ . Information Coding Techniques 3 -170 Error Control Coding

5. Explain the uiierbi "lgorilhlll ll lld sequctJlial.decodilig of convolutional codes.

6 , Compare l inear block . codes , cyclic cod•,; and conuolutional codes by givhrg their adv,m/ages

and disaduaniugrs:

lu(onnation Coding Techniques 3 -171 Error Control Codinq

.~.

Q.2 Def ine code effiCiency 1

Ans. : 'The code efficiency is the ratio of message bits IIIa block to the transmittedhils for that block by "the encoder Le.,

. m esSilge bits kcode ejJiCJ[WC1j ;; _:",......., __



Unsolved Examples

1. A rate \1 , K = 3, binary c o nv ol ut io na l e nc o de r i s shOHin ill F i g . 3 - 4 .4 6 .

a) Draw I I , , ' t ree diagram. t re ll is diugr"" , and I I" , s tate dil lsml ll for " 0 0 1 1 " encoder,

Output

Fig. 3.4.46

b ) 1 / t i le r e ce iv ed signnl a/ t he d e c od e r J o r right ""'S$ilge bits is,

I' '"(o o 01 10 00 00 00 10 01)Trace 11/( decision 011 a tre ll is or code t ree d;aSflwl alii/find 0111 1 I I , ·s . ' ;os r bit sequC/lu.

2. Fo r l ite c o n uo l ut io n ul e n c od e r s /WWJ l ill Fig. 3.4.47, .,krldl I i i. . co d , ' t ree.

Fig. 3.4.47

3.5 Shut t Answered Questions

Q.1 Wha t is hQ/IIming dis/once ? [Nov.Dec.-2004,2 MMk. NovJDcc·2003, 2Marks]

Ans.; 'I11C hamming distance between the two codevectors is equal to the number

of element '> in which they differ. For example, let the two codewards be.

X :: (101) ~nd Y :: (110)

These two codewords differ in second and third bits. 11~~rCfore the hamming

distance between X and Y is two.

lmnsmitted bits /I

Q .3 W hat is m ea nt iJ y systematic I1i1/I nnrlsyslwlalic codes ?

Ans;: In a systematic block code, message bits appear first and then check bits. If!

the ncnsystematlc code, message and check bits cannot be identified in the cod.:vector,

Q.4 Whal i s meant iJ y linear code?

Ans.: A code is linear if modulo-2 sum of any !wo codevectors produces another

codevector, This means any code vector can be expressed as linear cdinbinntio" orother codeveclors,

I,Q.5 Wlt"t au til,· error detection and correction capabilitie» cf HtllJI11llS cades?

Ans.: The minimum distance (dmu,) of Hamming codes is '3. Hence it can be lIsed

to detect double errors or correct single errors. Hamming codes an: basically linc";ll'

block codes wi th d ,, ,, . ee 3.

Q .6 W i t, ,! is utean! by cyclic code?

Ans.: C~c1ic. codes are the subclass of linear block codes. 1l1Cy have the properly

thnt a cyclic shift of one codeword produces another code word. For eXilmpJe considerthe codeword.

x : : ; (xn - 1, X ,, - 2, ' .. :'1, xo)

Le t us shi ft above code vector 10 left cyclically,

~ , X -; · ' ~ = (Xn - 2 r Xn ~ 3~ ~.. X [ ] ; - XII Xn _ l)

Above codevcctor is also a valid codevector.

Q.7 .Hoto syndrome is calculated iii Hamming codes and cy~lic codes 7

Ans.; hi Hamming c odes th'e syndrpme i s c alcu la ted a s,

S = YW

Here Y is the received and J - l T is the transpose of PMi1y check rnatr!x.

In cyclic code, Ihe syndrome vector polynomial is given a s,

S(p) =: rem [ f ~ t J

Here Y(p) is rec~jvcd vec tor po lynomial and G(p) is generalor polynomiaL

·_._------------~---.---~-_.

Informaion Codrng Tcchrliquns J .1n ;lZrror-aontroJ Coding

(l.S WI",I i, HOI n,I,' 1

Ans _: [ J - C H c '( ') <. 1"S ~ rc most ex tcnsi ve and powcrfu [er ror cor rect ing cycl lc codes , Thl'

cilyudinli ,,( nell ("d('~ is «Imp,' rnrivcly simpk'r, For any pl"'SlIIve hller,cr :ni ' l ind '{"

.: ~.:~. ~' 1l'fforrnation Coding Techniques

.~

J - 173

'!

Error Con~to[Codino~.-,------.------_ .



(where t <: 2"'" ') tl1('n' cx ists . , HCH code with follQwing prr rarnetcrs :

L\lo..:k It-l1gth II '" : :O m. ,

Numbc]' (, r p:tril\' (hCck 'bits : n - k 5 mr

;"!:illimum d;,;I;ln<:(' : d" ,, " ; -, .2 t .- 1

'Ans_. These arc nonbinar y BCH codes, The e(lcoder for RS codes operate on

multiple bits sii1lt11[:1tH!(lU~:ly. The (n,k) RS code t"kes the groups of m - bit symbols of

!Iw incPTIll1lil1!; binary d<l[" stream, It takes such 'k number of symbols in one block.

Tlwn the encoder .idds (n - k) _reclund:mt symbols to Iorm the codeword of 'n'

·~\"nll'l('l:~.

RS cock has .

!" ilriiy check. < ;izl< - n - k= 2 1 symbols

!vfil1imun1 distance . dm", "" 2 t + 1 symbols

Q.Hi >V i r , , 1 is 1 / 1 ( ' dirf'fL'I!C<' / '<:/WCCII 1 , 1 < , ( 1 ; codes lI11a c "" "O /l Il im 'l I/ c ot ie s ?

Ans.: Block codes tilkc 'k ' number of'message bit simultJ.1neotisly and form ·n'·bH

,,,de vector, This code vector is also called block, Convolutional (:ode takes one

me;;s ;~ge bit at <1 tt rne :md gencrille$ two or more encoded bit s. TIlt!S convo!utiona1

nvk~ t;encmte <1 s\cing of encoded bits for input message siring,

Q .II O tiim ' ("or!5trrri~t! l en S/ il i n (.)H.._,Ju:i tmn[ codes.

Ans_: Constraint length is the number of shifts over which the single mcsS<1gl! bil

em influence the encoder output . It is expressed in terms of message hits.

Q.12 D~fim~ free di_.;lnrrcc and C()(!i1!Sgain.

Ails .: Free dis tance :5 !he minimum distance between code vectors. It is <lISt:l_qunl

to minimum weight of the code vectors.

Cz)ding gain is u~ed as <1 basis of comparison for different coding o171dh' )~;-(l

d,hieve the ,,<1111(,it error rate t he coding gain is defined ~.S,

(~I:lrcodcdA '"

(E ~ )CO d e dN o

,- Here ' r' i s the :< :0d( ' rat e

and ' . I t ' is the free dis tance.

1

DDD

itc_chitode_unit_iv

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