oa Error Contre) Cock ing
EGlEc 613)
H Digital Cowmunscovion skew chace :
(a
Seucce,
ww) |
Pat) = |eskmete of wit)
objechi ve:
—_—_—— HO) S mw) deliver ake message Foon
Be sowete ts “the
Sink we ag Wire
Aiskerbion at Ressiee”
exemple s of boxes
lQ Seucce : Cye) , Brom seucce Ccomgiber deka),
Aexh, Speech Creleplony) oudio, cadio,
Vides | Smate .
(ey Channel.
re Ry
model >| wes) >
\Wa 1
we
& NCH = noise is due TE the channel
(ross WealK on bun ales
Tondom x moise ,
wwkerFerence,
4 telephone coble) +
R noise:
* “VWermel meise ve ts signel processing
Elements wm R
SQ = ML PT TK Vine Lesery Veleghont
at home)
— Coanial carle Comoe Tv)
- Redio Cecommeceiel cadio \arodcosh
~ CEN Phemes)
= Optical Coeprice\ Firece)
Mw came ob Gualsec
Funder sea calle,
Feunks leek. soovtohe
Ts
(cy Re
- Goal Mam
— Lavect ie 4tronsRoemakion en mG)
& he wy ~~ Channe|
> Iw Wher Retloos, We ot be aa wt,
‘Sinan sabboomedion «
my TE MG) 38 omclere | speech | vides | ete.
we
KY convesh ow GT TS BY
S\ceom « Ww
AID =
psa | [Gaeaehen | else
mo wk eee Rak
TO tite bel sample
5, = Bamglien Le ww Sample \sec
wf)
> ¥ % 2B
a
pom So} KE
=> toa 8 RAE
—s b = BB wrs|sengle
R= 64 kK b| Sec
=
not
la) Wk Feackion Re
\N
info \ rl ——r——r——_—.
eee ees Encoder. l — | Encoder [7S Jew
2c
Digtal weedSs
Med. —> 8 — x) collect ow A Ss
8
Gritag a» : of} H-
ae i Seoucce Encodu.
A typice\ Source encode vs ASCTT . uses F bits
sper Wlekten 4 Pacihy weih
VEXT © 8 bits| letter
Emgiish texb
- Soucce . ae Aas
a
Alphabet composed
o§ lethecs,
- Measuce Sowrce 1S Source EwbopPy
sure -\ Sewese SS Se =
Hod =- DL Pwye,
vat 5
Where P= Pree. oo leer in Alphelbe
~s
Since we hove L =26 Vebhecs in Ennion
A\phalbe’ , i
Sauce ony Veo
a WO) = Veg 26 = GF bids|\eHte
y >
Nete. . yo = Minimum number. 4 biks{ lee neeoled
te represent— the Soucce lo dishecen
© TF we use a Sheuckare
WW Beltloust- amd U PeNows
WOO) 3.56 ’ids\\ethee Kmemer 4
one} wi
Goals of source Encoder.
~ Finds a wiwimal Quits| lethu) repesetoi on
cP soucce ie.) remove she cedundanen Seow
Ane source with Source encod cw Conwst
be decorate) . All His encediiy is done
When chownel is eccor Yee.
~ Dow send Beedectorle informalion ts Ten
receive WA dowt Waste Channel eroucces
on YRdickarle mw Oy)
~ a «Maw infecanbies
Ss Soucce.
soucce, \ >
Co —~ _— wrandom outpuks
fas Some order «large Entropy
Ealvopy + DisBrderecl safe.
Encode
~The bottom line Source coding 15 te Prodluce an average
Code word Jeng € bits] symbo) such thet
& YH) 15 as Close as possible % H(X)
Male that he Source cede outgut is essanhally
random , dey it removes redundancy Prom He Source
message . HOWever) “This makes wTuery sensthi vt
AS Chawne\ Fronswission ercers.
e Mere sa Seoncce aN \ohen . ANY o
— Channel Evceder ,
ebyechve 1 Corteck eccoss made in Chonnel
Ar coms Mi Shien |
Jsc ~ Seucce -te- Sink Chowne)
0
Input 4 isp 4
o _ Plerre) =P
oo o
i-P
Consider Code : 4 5, 1\\ wse The Bsc S Mmes
) For cody impart
o — > 000
Codesords
EF AW coe) = 3 means coderseeds anther
” im 3 Plates
AsSume 4d, No
© —s seo
> az
Note Hrok with His repenkition cade ) cre have added
< Convrolled\® cedundow es
As VO a. S— Porihy bits
ese
e© — s of)
Now if we put the Source are channel encoders
Source Grewal)
[ee |—— [Ee b> Tb
eatrocles wuncentetted Por we Gubeolled
seduandan ca” re duncloun us" %
Mow, let us go back te the simple repeatition code
Decoder ;
. Toke 3 outputs Prom Me BSc od Sind
ve Closest Cin AL) codeword
+ USe Whe encoder Tule ts eshimate wa
Soucce BW Luncoded mLormabien)
ex. A —s» \\\
— Ss oom marty cule
more ‘is then "so
Mer wise > O
Vet soy het we send b —2 Vi cud ceceive
=e Vo\
‘ x
One Aeansenssion eerer
—— > mere NS Tren to's Sede YY
eo
. oN more “O'S *Fhan “As —>"Q"
“
°
*
2 eerors
“he Same con lee obtained inoterms 4 a
= a Ou, \ Silece ch eo!
iN ‘
rent tte cen
SP ALONG) 2 2 1 Ao, GY= A
Then “O" is Sesh
The prob. 4 ecter Yor Aig code is Gotan ey, P Keres) = PL 2 Ty eccocs) 4 PAST, escoes)
- (a) Pees
GSE
+ If pis fw, | Plecr) =3P|
+ Fer ne cocking | P Cevre) = P|
x More on Channel Cooking later.
— Chemnels:
[) (A) AWGN Channel
amplitude fas
Gaussian Pi
ar Re rays SWa00
s a) x
ih message n&
Sijae)
oO
BM s Elo weit) = 7 3 &)
Where Ae = Power spectral dlesihy 4 ase
i 7 Signed sfechron
ea >
£
age SQ) = power sepcleam = FER Cp = 22 yh
(b) Bendiwited AWGN Chonnel :
rene
A a __,® x
5) A,“
nO)
AS Bz bendurdth, | The chawnl gets Norge, we
aer on AWG
Z ‘Syon bol vate
+ Mele that for BT lage Boy L”,
the received Symbols are altecked ‘only toy ae
weise wv) Cue cal\-wis condom error Chanad
» Mosk ercec cocseckima Coden ace designed Pam
Fandom ercem Chamnels (8a ones We shady)
—
with IST , errors are Cocrelated aud Wwe ah
Yours e€creeS Cran
tit
+ Anotur. tre J signe! cshetien is Signal fadiag”
sw os
> Aw) s@ ere burst
deep fade = 6 @ eouss * AWGN chamel (RT ~ large)
Re: Tater the oFerabions im Me Th Plus Channel
oa For Ahe cose Sv ~ ‘ergy he ceceluen
wavecst Whe Gpecalten in
oe
vx Weeiis coded vod ©
_
= Source Gnawa :
\s oe [Pe are re b-| Med b>
a
Se
Re “o™
Sy
=" Seacte mn
—>| Denet [| a > eee G * |
Mo loss in this Strachuce Praviclecl thal Re channel
i$ “memorsless Cue, we ISL) wo Xedimy, condom anise)
his is Called “separokion theorem”
@M VOM : A siskem that comes close
Shonnen Wit such as
ih « 28-8 KePS
nee
V3 modems
ee er Oe oe)
“ Shommon Capatihy
A 2 for AG”
rw 32 Kbps 7
rere
Cretephone ine)
2 More on TCM later. we
Before gona MTs debails of the channel aa Pechaigue
Ver us Res vevied a Smple Communication Peoblem,
% Primm ceceiver
su)
AWGN Channel -e—
Bnoey modulabion, 3)
4 —> PQ, o> -PW
Pa
LV
a Gor)
ex Sewd a neo Pulse execs Sec 4 coresRe lL
i
RB. Oph mam tha Lk gees PL sein
»
Vwo Realreakiong:
@ Makched Filter Realigahen:
CS
ex EE
ca)
Wh = PLT-t) = Pw © Digele a pene
Semple every T
wee Lk key ER
r@ A \ ee
dump pulse
BO)
Ler us Bind Pe Lanceded eccoe Yeub.)
GY Assume Ao is sent Boe RTS SANT
we interval
cWs PR) an)
want v +
d= formar. Pemarsleoa
RT . .
x
Be AT a Qi. Ghee 5 (wr de = Ganssiey
‘s 8 ev
~N (268)
Error Even¥ Ay ill Repgen When co
PleLLd= Pass)
3, (ret
e
2
fo : alse Since Lago} = Ld e- ee 1 Be [Fd -T
Assume O is sent , © GQ) =- POA) | oct eT
Show Ker piers) = Ple\ | QWs)
Now, What is AT ?
o™
2 2 2
a ects E Pia) = 47+ AT = AT
-~ € rot ~_ AT. E&
2 os lead alt &
“ples @ (He)
Mote: Somekmes We vse @ Ramckien approximation
Veue Bon con ek)
4 auceion T (no
SRN over
=
eM: ples €=2 For S92
ae
Some besks Use
e
ee
22
efc ds BT 2" an
“JQ Gs 4 ete (he)
) o Fas @ (EE = & ente CE)
Metice Mat, smart p (geor 4
Means \acae .
Un pedi chal Ve
ocucance of eveat d)
wilpomalbien since the event dis
Pel = TUR) = small wo logis
lo je
let Ra sencce Rave on = 2 eawally Vikely symbols
and LANs Veqn sk wits] sqm.
in this cose Me source ovbputs ee Shs | sonese sy
(i) Enbeopy:
Cet the sencce Symbols A,
when, a
ene PAw\= Q@, yi
a roa: ELI] = - x, a4 @, |
ws} 94 16
56
Meaninsy es For ans, entoder.., Ye. ~auecose
—o Abs Pec —— saptscl. AS coco
- GQ = PAK)... :
Thin, you can shows that. 2% Hoe).
(prove i+) «Mate that then
so
Ad
wae Bowe Hae matinal-
means... P-(%,>%, i QQ,
Ves inde yl 58
©, for_pro, 2 elyp a0
a
Roe 7 +
@, foc p=l, plyps cee
OD fee pal 3 hes
a ease Dyn OQ represent completly Pred icbsble
i ource oukpurs, ence , Zero inbormebien ire
_n(p)_3. 2). On Aha Phen. yaad, CuSe—S>—_—
reSenNs |
_ - ihe mosik sande Case, pale 59
Soucce.:
——_
o) be. He pdb for xEX
p60 Rag. GA_dX.
09
tap Empactant Defi
8 A peedsx 36 a q a 60
ie
ie
~&s
Cc de 2. Mab a palix. Condi tien.
le. pa propix. cordition eode, but
_clecodable .Alsa.. st #S-
“instantanous..code"
Bri eOron! bo KX KX
3 o-Ll.o. a
cannes “be -
deco cle new, th =
Aas. te..wait.fec.0-
be. decade A$ X pono
“Cade. bt
is a.tree- Code —> Unique gz
instantaneas. eel Pee.
cade letter Fe ech —— 63
Alote Bat, in this. aanner, se assignment ech
Code. leHer.is..aith equal... prob. .avel! France
AK 1 EAPO nn
(TS eade So alsiped a teee cade =
_SorisSies “peekix comd’ Mem. Fis. wskaukenows.
(Also noke. thad. this..method amy nob give e=
min. as we..force. Max..Satcopy..at each. £062...
node.,.ane nace ata time. The lestcase
must. treak..the nodes Jointly . coe
Wp 66
|
TB) Xn Phe isk oh
= Such —cthak— a 67
Let us. See..Gome -excamples..on Shauenm Ean.
au el Huffmon. Codea———————— :
Celiac ae
= =
S).e.-A0e _
Pp Ge a
= ;
i
) P.+ BP,
Co :
af i. |
; 3)
Dp _
| oe
oy eM: = |
err a
e ay ;
he - 1. For cdiscrehe- mamoraless = Clremnels
Cenc onal LntsoPyis.Qe
73
O. Poskerecs
PLXW) = PONIR 2 Gp) eR
VON PAY OP PPL
omc x
» KN olues.
a Yolues
-»). aint Exkrapys : Eh
Wixiy de re ~ ple sy) Lege yay) -
ACY) = = (ley Pa}
. ae (es) | Gd)
| HA%,Y)os. A YIX) 4 HX)
- >) Cprove. ob)
a HN) 2 7
We AK) ond WW) ose called A magn
“Mote Rat 2X Oe
nelateck ule ae. sraconal
apd) .beR sce. Loomsmnission:. The claseeven wn certainly
_ _eowk % 1S. WO...
—> ») ARK Aromsemissien:. ( Galleck, all ¥)
Ko closecven..Uuncembeiaky = WAY)
ew if we. Leb a in Qoemabion...sapphy
he eawneh equals ZUG) Fain 3
They) 2A 00 = ANY)
— : A x
befere. Se betes
Frans Leos. | How ines aa a Shao Ane
15
Alse ARe. aa sat oy Ake aia
“Uasks| sec)
Condit ome
Sek copy- Ass ve
w capacity of BSE:
_FOGY eH HL) 6 oo
= Ein. AL a Se |
76
AX) = EL Laon | As mts
“Wate tat ute Cannot.
ole pen deut..n. Hre..claa
(Ooy).2 AON = hPL
aos eh. Finally =
Ci ok. f HON} COOP)
PoP, Se :
jie bio. Het AM amen ad When POPPA
thichoceucs. lath... PaMe
TTT Tee AS WLPY |. . bas chand use Ser 5c aw
7
3) aR Ate Shsenoed ae ARSEE (rendon
eccecs )... 1
al
|
receivers
_choices..of te Same 2 jroup
6 |.cbecader..
to —» Channel maker x= uf
The. enckti ene | EnkcoPy = A
A AYA )2.= x PLD Ney. Pay).
= hte)
¢ aS
H(YIX\ = Fy MOP
= hhp)
imited. to the
I hee cea. available...
at tee. transmitter. “Re ..soltt- decisian eceiue
ae -cen...expancled..... Sleetien “ef Choicea Fe.
gels Ly of Hee... rece ved..Symloo| te Pa nme
Te simplest. foc i seth
_ ES in.cli.cadte. net =
Gu... S69 00, 0 Se. Cores pendian recerved.
-. 4G a9 a oa
Communi cole,
“Sion.
aymbel is 79
Woe WU GP) PY.
Breer a
cud bbe pechihieud’ whe cuosabs. Such hed
_kremsikicn .mebak. Bor tech Sswlesek Ras
Mouss.. cihetk. ace. Permutation... X each o:
Goncl__Columus..-re att ace Tpecmurs
vo nh Oe
~ RD
> Bsc. Jeausibien..mad cic
— COWws..are. = Pema inn nae it. - . :
Celumus..o..0 “a a” POE)
_effbet i nelse. _loes ne:
a1
Sef iotet symbols
| he above is as. follows. 84 =
a Eccoc.. Control. coding
i= Exror Control coding yas opposed Zé “Source
Co Cacling dds redundancy te infecmadion
in order. te Compensake Por he cftects
Forder ecear. cacceskion LEE
GA)_Awkomakic ~Repeak. Request (ARG)
: ARQ, ecrecs ave _debecked ond the |
eceiver.asks othe Sromsmittea th _seleconsmits!
cehe ecconeens “packebs.—
a> The advonhage A ARQ 5 Ae Sow avecXweadl, |
Youd. rhe dis advantage A$ $e daciclle._delar_
hot makes vk owsukalole Bor saline |
app licariens. = ee —
Uy Mece., we Focus on. FEC coding =
Block Cades map co k-dimensional Veber 13
ee w—_idimensional, Neckor . TY we limite cucsel
Cone 08. binacy.cedea, the number 4 k- dimensicnal_
vectors ~(binaey).is.. 25 So, uke only need 2
A Yeckecs.. im. a= dimensional Space eos. |
= of. 2" ~~ 3 |
85 |
|
|
|
|
Ju!
eelese: 2* Valid. vecters Chinary stings) are Called
_ @odewerds. oc Code veckers, -
5 Since Khe number of na dimensional. vectors
45. 2, amd ure. only Suse. 2% oR them, Ber.
every, codevosrd, “thece. ace.
a eg
Q =) inva lid vectecs
Ls Tf the choice..of the. 2° codemecds is such
that they are matimally apact Prom ont
another, Sraen..the Chance One .Osdevsord
eins, mistaken Kec one wer iS! Swadl
w [seuece| P| ~Gremes\ | _OE
____| block “[_Evceden | black
1 n
“2 passibihity
Ne) TR, 4-18. defined. as
The... code.
OR
©
TDelne Ca Hof Pei) bis = x Chuck bits
Kn -
a Ik.
« Deine Phe..c]p. Codeword. Necker
igpone oder Ry Rove Be) mad. ca “exer bo
a@b a7
Gy ce LS
= a ~ v oO os -
The Coderooeds..ace (006). (O11), Cet), amd.
LA 10). he codewords are Sasd che. ee ere |
Paris. aS. OG OS so... Thus,
eck one. eccec
code_is Said te be _lineac_if the Sum._|
A __Cade coeds. yee OQ . Codsussed hak]
+ The ppacity cheek squabiens.con.
Vector makeix Room:
ieee eae nett!
Cidedkihy maberX is _
Ps kK xe Paris, check. cacto,
“The first. k.Compenents. of. ang..Code vector
Ace =the. Compoments..ob the mess. Necler m+
Such a Code 3S Called TS. \be “sgtenedic 88
epresen bation Por the
(32,2) Pecihy Cheeks code. 89 _ oo
Given Phe aloaus...cdabiaibieny—
— a
EK (000,110).
4
Tecefore, at. the .receiver.,..ore receive.
(Sa Gey) or Ss OG OH
~ - }-—.--
put, SS Sy NO. Levee. accucedl
Ip. Ss fo-> error. occuced
“Moke. thak Por any.generaber mabeiX, Gy thee
exists amateix H Such that:
GH = 0. \
his memes. “thal oon cas. ‘oh. Wois -os¥hogenal
HG COS hee See co _
“then We (Ss
Qe an
On e.croe _Oceneeeh).. Tn General, rhe number of errors “thal Com
Corrected by Code ts Given by:
ca) oe
Mls lacgesk inbegu &X
called. error
Floor Sn.
Whece,
EX.
For the (3,2,2) Code
nin
( — = \Belys Wie
O43. => te apjs et
That is the (312,2) code Con only Correct
ene. Channel ercer.
2 Remember, he numorr. ch ercors’ thet
Con ve dAebected is = du-1.
¥
ele US mors See hous Com we debeck and
Rerenck errars wNinly sew ple Vac blocs
code.
Ex. In Class
ie