Probability of error for LDPC OC with one co-channel Interferer over i.i.d Rayleigh Fading

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1 IOSR Joural of Electroics ad Commuicatio Egieerig (IOSR-JECE) e-issn: ,p- ISSN: Volume 9, Issue 4, Ver. III (Jul - Aug. 24), PP Probability of error for LDPC OC with oe co-chael Iterferer over i.i.d Rayleigh Fadig Rekha Rai (Departmet of Electroics ad Commuicatio Egieerig, Pathakot, Pujab, Idia) Abstract: A aalytical approach is used to derive bit error rate for LDPC coded-optimum combied (LDPC- OC) system i the presece of oly sigle co-chael iterferer. A probability desity fuctio has also bee derived usig momet geeratig fuctio for optimum combiig with oe iterferer. This paper cosidered that sigal is BPSK modulated ad trasmitted over i.i.d Rayleigh fadig chael. The aalytical result shows that for BER -2, the LDPC-OC system provides a additioal gai of 4. db over OC system aloe. Keywords: irregular low-desity parity check (LDPC) codes, optimum combiig, rayleigh fadig chael I. Itroductio The optimum combiig is popular diversity combiig techique while workig i the iterferece scearios. As compared to MRC system, OC has achieved a large output SINR thereby providig a powerful meas to ehace sigal detectio i the presece of co-chael iterferece []. O the other had performace of LDPC codes has approached the Shao limit o the additive white Gaussia oise (AWGN) chael [2]. I the fadig eviromet, a exact aalysis of performace of chael coded ad diversity combied system is usually quite complicated ad computer simulatio is ofte used to study system performace. Computer simulatio has derived the threshold SNR of regular/irregular LDPC codes for SIMO system [3]. For LDPC coded-optimum combiig (LDPC-OC) system, oly simulatio work has bee doe by Surbhi Sharma [4]. The aalytical efficiet BER expressio for LDPC coded ad SC/MRC combied system has bee derived [5]. The bit error rate of LDPC-SC, LDPC-MRC system is derived for BPSK modulatio over a idepedet ad idetically distributed (i.i.d) Rayleigh fadig chael by varyig the umber of receivig atea elemets from 3 to 6.[6].The aalytical probability of error for LDPC-OC has bee derived whe umber of iterferers are greater tha or equal to umber of receivig ateas [7]. Usig momet geeratig fuctio approach, Wu, Yogpeg et al.[8] derived BER from SINR at the combier output for multiple arbitrary-power iterferers. I this paper probability desity fuctio has bee derived usig momet geeratig fuctio for optimum combiig ad exact BER expressio for LDPC coded optimally combied system (LDPC-OC) i the presece of sigle iterferer over Rayleigh fadig chael has also bee derived. II. System Model The system model which is cosidered i this paper is as show i Fig.. I this model, irregular LDPC coded sigal is trasmitted ad at the receiver side, M-elemet atea arrays receives LDPC coded sigal which is operated i the presece of oe co-chael iterferer. Suppose a array of M ateas is used for receptio of desired sigal corrupted by AWGN ad sigle co-chael iterferer. The optimum combier combies the received sigal vector after weightig which mitigate the effect of iterferece. The received sigal at the output of combier which is fed to the LDPC decoder is give as y(t) =a*s (t) + (t) where a is the optimum combiig chael gai over i.i.d rayleigh fadig. s(t) is trasmitted BPSK modulated sigal ad (t) is additive white Gaussia oise with mea zero ad variace ς 2.The this received vector further decoded by LDPC decoder which based o the message passig algorithm. Geeral idea of such a decodig algorithm is to pass messages i a cycle. Each cycle has two phases. I the first phase messages are passed from variable ode v of degree j to check ode c of degree i of the factor graph. I the secod phase, messages are passed back from the check ode c to the variable ode v. 59 Page

2 Probability of error for LDPC OC with oe co-chael Iterferer over i.i.d Rayleigh Fadig Fig. system model of ldpc-oc with sigle iterferer III. Pdf Of Ucoded Optimum Combiig By Usig Mgf The momet geeratig fuctio (MGF) of oc at give ad Probability desity fuctio for SNR of iterferece [9, equatio ( & 8) respectively] ψ oc (t ) = + + t t M p( ) = M Γ(M) M exp ; (2) Calculate the coditioed PDF of OC over from momet geeratig fuctio from equatio () is p( oc = ψ 2πi OC t e t oc dt ; (3) Put the value of equatio () ito equatio (3) ad usig the itegral [, equatio 3.384(76)], the the PDF coditioed o is calculated as p( oc = (+ ) (M ) oc e () oc M F ; M; oc (4) s Now to get ucoditioed PDF over, the PDF of SINR at the optimum combiig is p( oc ) = p( oc p( oc ) = K M (, oc ) ( + oc ) s p d (5) Put the values of equatio (2 & 4) ito equatio (5), the the PDF of OC becomes + (M ) M oc + M For M>> the above equatio becomes + oc p( oc ) = K M, oc + + M (6) oc s BER for ucoded OC system over the Rayleigh faded chael is 2 erfc( oc ) p( oc )d oc (7) Put the value of equatio (6) ito equatio (7) the probability of error is calculated as 2 + π exp + erfc + k M 2 2k k=o k 4 + 2Γ M M + M 2 2k! k= k! 4 + k (8) IV. Ber For Optimum Combiig With Ldpc The coditioal PDF of chael LLR over the chael gai is give by [7, equatio (3)] p q a, s(t) = + = ς 2 q 2a2 ς exp 2a 2π 8 a 2 (9) ς2 Now to trasformatio the PDF of oc which is i equatio (6) ito chael gai a is give by 6 Page

3 Probability of error for LDPC OC with oe co-chael Iterferer over i.i.d Rayleigh Fadig p a = + M 2a 2(M ) exp a2 ς 2 () M! ς2m + ς2 a 2 The ucoditioal PDF of chael LLR is calculated by averagig the equatio (9) over the chael gai a p q = p q a, s t = + p a da () Puttig the values of equatios (9&) i equatio (), the the ucoditioed PDF of chael LLR is calculated as p q = + M 2π M! ς 2M ς 2 ( )M ( )M M b M M b M 2 π 2 exp ς q b 2 + q 2 (2) where b = 3 2ς2 ad The PDF of check ode message is give by [] p c q = exp (q m c) 2 (3) 4πm c 4m c where m c is the mea of check ode c of degree i. At (l+) th iteratio mea of check ode c of degree i is updated by m c l+ = 4 erfc d c i= ρ i 2P b l i 2 The PDF of variable ode is calculated by covolvig the PDF of chael LLR with the PDF of check ode that is p v q = p c q τ p q dτ (4) Puttig the values of equatios (2 & 3) i equatio (4) by usig the itegral i [, equatio 3.322(2)], the PDF of variable ode is p v q = +M M M M 4 2 M!ς2M b M ς2 M b M exp ς 2 bm c b 2 m c 4 Φq2mc+ςbmc2+exp( ςb2+2)q Φ q2mc+ςbmc2 :5 exp( ς b )q To obtai the probability of bit error P e, itegrate the PDF of variable ode give by equatio (5) from - to, +M M M M 4 2 M!ς2M b M ς2 8ς22ς2b erfc(mc2) :( 6) M 4 b M b 2ς 2 3 b2 erfc ς b m c exp ς 2 b m c m c Now at (l +) th iteratio, averagig above equatio over all the bit ode degrees j becomes +M d v λ 4 2 M!ς2M j=2 k M M b M ς2 M M bm4b 2ς2b32erfcςbmcl+(j )2expς2b(j )mcl+2 (j )mcl+4+8ς22ς2b erfc((j )mcl+2) :(7) Because at (l+) th iteratio mea of variable ode v of degree j is m v l+ = (j )m c l+. V. Result Ad Discussio I this sectio, the aalytical results for the LDPC-OC system have bee preseted. To evaluate the performace, the system has bee cosidered whe the umber of receive ateas M is take as 3, 4,5ad 6 ad umber of iterferer is take as. 6 Page

4 Probability of error for LDPC OC with oe co-chael Iterferer over i.i.d Rayleigh Fadig For the ucoded OC system, show i Fig.2 it is clearly observed that as umbers of receivig ateas is icreased, sigificat diversity gai is achieved. For BER of -2, a improvemet of.db is achieved i SNR whe umber of receive ateas are icreased from 5 to 6. Fig.2 ber for ucoded oc system Fig.3 presets the result of LDPC-OC system obtaied by usig equatio (7). For the LDPC-OC system, at the BER of -4, a improvemet of.3 db is achieved i SNR whe umber of receive ateas are raised from 5 to 6. Fig.3 ber for ldpc-oc system Compariso of ucoded OC ad LDPC-OC system is give i Fig.4. It is clear from the figure that sigificat codig gai is achieved whe comparig both the systems. At BER of -2, a codig gai of 4. db is achieved by LDPC-OC system over OC system aloe whe the umber of receive atea is 6. Fig.4 compariso betwee ber of oc ad ldpc-oc system VI. Coclusio Usig the Gaussia approximatio approach, the BER expressio for ucoded OC ad LDPC-OC over a i.i.d rayleigh fadig chael has bee derived the presece of oe iterferers. From the umerical results, it is show that LDPC-optimum combier gives the more optimistic results as compared to ucoded-oc system. 62 Page

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