T h e probability o f error fo r the 2-user binary C C M A schem e is considered and analysed here. T h e 2-user binary c olla b o ra tive cod e is g iv e n in T a b le 5.1. T h e probability o f rec e iv in g a g ive n adm issible cod ew ord correctly, P^O , is derived in A p p en d ix C , e m p lo yin g hard decision decoding. It is fou n d that;
P ^ O O ) = E r f
2
(0 .5u/on) (5 .3 1 )Pee(0 1 )= P ee(1 0 ) = (Erf(0.5u/oN) ) (2Erf(0.5u /oN) - l ) (5.32)
Pee(1 2 )= P ee(2 1 ) = (2Erf(0.5u/oN) - l ) (Erf(0.5u/oN) ) (5.33)
P cc(ll) = (2Erf(0.5u/oN) - l ) (5.34)
Substituting these p ro babilities in the total pro b ab ility o f correct d ecisio n w e get;
P « (to) = (Poc(
0 0)+
2Pce(
10)+
2Pee(
1 2)+Pce( l l
» / 6(5.35)
and the total pro b ab ility o f error can be calculated by substituting in the fo llo w in g equation,
P « ( t o ) = 1 - P J t o ) (5.36)
S im ila rly, th e derivatio n o f the cod ew ord error rate e m p lo yin g M L S D _ C C M A decoding is g iv e n in A p p e n d ix D . T h e probability o f correct decisio n o f each admissible cod ew ord is,
P M(0 0 ) = E rf
2
(0.5u/oN) (5.37)osw o ,
r,«(01MSrt0.5u/o„)-l)&/{0.5i</o„)* f
(5.38)
(exp(-rJ/2)/v®t) E rfi-t+u/o„ ) dr
J’. P l ) - /
( exp(- t 2l2 )lj2 K )(.E r f (t + u / o - Erf{-t-uJo
(53 9)
-ru w ..
Pce( l l ) = (2Erf(0.5u/oN) - l ) (5.40)
Then, equation (5.35) and (5.36) is used to g e t the total p ro b ab ility o f error.
5.6 Simulation Results and Discussions
T h e sim ulation is carried out to evaluate the relia bility perform ance o f C C M A schem es em p lo yin g various codin g and decoding schemes. V ariou s 2-user C C M A schem es [K asam i, et al., 1975, and Kasam i and L in 1976] are introduced first and used throughout the sim ulation analysis. Th ese collaborative codes are chosen to be sim ple short codes w ith summary rate, in m ost cases, higher than o n e bits/channel use. In addition, they are chosen to have d ifferen t error protection ca p ab ility o f the o verall
2
- user c o d e and it’ s constituent codes. Th ese codes are;(a ) C o d e 1: C ,= (0 0 ,1 1 ), C ^ fO O .O l.lO ), C W ,= 2 , N ,= 2 , R ,=0.5, d ,_*= 2 ,
CWj=3, Nj=2, R2=0.792, <1^=1,
Ri - =1.292. <U,=1.
(b ) C o d e 2: ^ = (0 0 0 ,1 1 1 ), ^(0 0 0 ,0 0 1 .0 1 0 ,0 1 1 ,1 0 0 ,1 0 1 .1 1 0 ). C W ,= 2 , N ,= 3 , R ,=0.333, d l-ta=3, C W j= 7 , N j= 3 , Rj=0.935, < 1 ^ = 1 , R ^ . =1.269, d ^ = l . 126(c ) C o d e 3: C,=<0000,0011,1100,1111),
CaMOOOO,0001,0010,0100,0101,0110,1000,1001,1010),
C W ,= 4 . N ,= 4 . R ,=0.5. d1
-to=2, C W 2=9, 1^=4, R 2=0.792, < 1 ^ = 1 . RMB= 1.292, dM = l . (d ) C o d e 4: C ,= (0 0 0 0 ,0001,0011,1100,1110,1111),C2=(0000,0101,0110,1001.1010,1101),
C W ,=6
, N ,= 4 , R ,=0.646,dimm=l,
C W2
=6
, N2
=4, R2=0.646, < 1 ^ = 1 , R ,^,=1.292, (1 ^ = 1 . (e ) C o d e 5: C ,= (0 0 .1 1 ). C 2=(1 0 ,0 1), C W ,= 2 , N l =2
, R ,= 0.5 , d late=2, C W2
=2, N2=2, R2
=0.5, d2
- . = 2 , R » » = 1 .0 , c U = 2 .It is assumed that the 2-user C C M A com m u nication system is in perfect synchronisation. In addition, the modulation and dem odulation are assumed to be a va ila b le fo r these cod es and considered to b e part o f the discrete channel. T h e simulation perform ance analysis results are presented graph ically in terms o f the probability o f error. T h e com posite cod ew ord error rate (C E R ) and the constituent users sink S E R are calculated fo r each 2-user c olla b o ra tive cod e. T h e com p o site C E R is d efin ed here as the total number o f com posite cod ew ord s in error o v e r the total
transmitted. T h e individual constituent user’ s sink S E R is d efin ed as the total number o f user’ s sink sym bols in e rro r o v e r the total transmitted. T h e channel is assumed to be A W G N o f z e ro m ean and variance o N 2. T h e ratio E/N0, is also defined here as the average sign al e n ergy p e r user to noise p o w e r spectral density g iv e n b y a N
2
= N a/2.T h e c om p osite C E R versus E/N0, em p lo yin g the H D _ C C M A decoding is shown
in Figu re 5.3, fo r a ll the f i v e codes. It can b e seen fro m this fig u re that the relia bility o f these c o d e s are v e ry sim ilar, since their correction capability is the sam e under H D _ C C M A decoding. T h e small differen ce is due to the variation in the number o f adm issible and forb idd en co d ew o rd s fro m on e c o d e to another. T h e com posite C E R versus E/Nc, e m p lo yin g the M L S D _ C C M A d eco d in g is also show n in Figure 5.4, for all the f i v e codes. It can b e seen clea rly that co d e 5 g ive s the best perform ance because its (
1
^ =2
, w h ich m eans that under this deco d in g a sin gle error can be corrected.F o r com parison purposes and calculating the en ergy gain achieved b y e m p lo yin g M L S D _ C C M A deco d in g, the C E R f o r each c o d e is presented separately in Figures 5.S-5.9, e m p lo y in g H D _ C C M A and M L S D _ C C M A decoding techniques. A ls o included w ith these Figu res is th e C E R o f each co d e w h en S B S _ H D d eco d in g is em p lo yed . It can b e seen fo r th e first fo u r codes. Figures 5.5-5.8, that the M L S D _ C C M A deco d in g g ives the best perform an ce w ith som e detection gain. H o w ev e r, when the cod e e m p lo yed has som e error protection capability, as th e case in Figu re 5.9, this gain is much higher, as can b e seen v e ry clea rly at high E/N0. T h e gain achieved is m ore than 2.5dB at an error pro b ab ility o f 10"6.
T h e e ffe c t o f e m p lo y in g these cod in g and d eco d in g techniques is also investigated o n the constituent codes and hence th eir user’ s sink data. T h e sink S E R fo r each user is presented in Figu res 5.10-5.14 fo r a ll codes. It can b e seen, fo r exam ple,
Figur« 6.3 HD-CCMA Decoder CER
I —— Code 1 — Code 2 Code 3 C ode 4 Code 8
Figure 6.4 MLSD-CCMA Decoder CER
— - C ode 1 — Co d e 2 Codo 3 - ® - C od « 4 Cod« 6
F ig u re 6.6 COMA D e c o d in g S chem es CER (C o d e 1)
— — SBS-HD — HD.CCM A MLSD.CCMA
Figure 6.6 CCMA D ecoding Schemee CER (C o d e 2)
— SBS-HD — HD-CCMA MLSD.CCMA
F ig u re 5.7 C CM A D e co d in g S chem es CER (C o d e 3)
--- SBS-HD —<I— HO.CCMA MLSD-CCMA
Figure 5.8 CCMA Decoding Schem es CER (C ode 4 )
— - SBS-HD — HD-CCMA MLSD-CCMA
F igur« 6.0 C C M A D ecoding Schem e« CER (C o d e 6)
— — SBS-HD — HO-CCMA MLSD.CCMA
'S m e »~ o o r )
F igu r« 6.10 CCMA D ecodin g Schem e« U ser* Sink SER
(C o d « 1)
— U se r 1 HO.CCMA — (le e r 2 HO.CCMA U se r 1 M L80.C C M A » - U s e r 2 MLSO.CCMA
F igu r« 6.11 CCMA D «c o d in g Schemas U « « r « Sink SER
(C o d « 2)
—— U s s r 1 HO.CCMA U s s r 2 HD.CCMA U s s r 1 MLSO.C CMA U s e r 2 MLSO.CCMA
Figur* 6.12 CCMA Decoding Sch ern*« U * * r * Sink SER
(C o d * 3)
--- U s e r 1 HD-CCMA - U s s r 2 HD.CCMA U s e r 1 MLSD-CCMA *■ U s e r 2 ML8D.C CMA
Figur* 6.13 CCMA D ecoding Schemas U sers 8ink SER
(C o d * 4)
— U s s r 1 HD.CCMA — U s e r 2 HO.CCMA U s s r 1 MLSD-CCMA *■ U s e r 2 MLSO.CCMA
Figur« 6.14 CCMA D ecoding Schem e« U sers Sink SER
(C ode 6)
— U se r 1 HO.CCMA — U s e r 2 HO.CCMA U se r 1 M L30.C C M A * " U s e r 2 MLSD.C CMA
in Figure 5.10, u ser 2 sink S E R is v e ry c lo s e fo r both cases o f M L S D _ C C M A and H D _ C C M A d e co d in g techniques. H o w ev e r, user 1 relia b ility em p lo yin g M L S D _ C C M A decoding is better than H D _ C C M A decoding because d lmil= 2 . T h is gain is also shown in Figures 5.11 a n d 5 .12 fo r user 1 o f co d e 2 and co d e 3, respectively. S in ce co d e 4 is a balanced cod e, th e relia b ility o f each user is v e ry close as show n in F igu re 5.13. C od e 5 is also balanced c o d e w ith d ._ -= d T- - = d _ _ = 2 . T h erefo re, the sink S E R is the same for each user as s h o w n in F igu re 5.14. It can also b e seen fro m Figu re 5.14 that a coding gain o f m ore th an 2.5dB at 1 0 * error probability is ach ievable em ployin g M L S D _ C C M A o v e r th e H D _ C C M A deco d in g technique.
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