Performance Evaluation of Cooperative Versus Receiver Coded Diversity

Transcription

1 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh Performance Evaluaton of Cooperatve Versus Recever Coded Dversty Saf E. A. Alnawayseh 1,a, Pavel Loskot,b, Mutaz Al-Tarawneh 3,c and Zyad Ahmed Al Tarawneh 1,d 1 Electrcal Engneerng Department, Faculty of Engneerng, Mu tah Unversty, JORDAN Systems and Process Engneerng Centre, College of Engneerng, Swansea Unversty, UK 3 Computer Engneerng Department, Faculty of Engneerng, Mu tah Unversty, JORDAN a saf.naw@mutah.edu.jo, d zdtarawneh@mutah.edu.jo b p.loskot@swansea.ac.uk c mutaz.altarawneh@mutah.edu.jo Abstract: The amplfy-and-forward and the decode-and-forward cooperatve coded dversty and s compared wth the conventonal recever coded dversty n terms of the parwse error probablty and the overall bt error rate. The dversty systems under consderaton can acheve the dversty order at most two. The performance comparson assumes channel codng wth non-bnary lnear modulatons, ndependent fadng channels wth pathloss attenuatons proportonal to the dstances between the communcatng nodes, and the dversty combnng at the destnaton recever. The expressons for the parwse error probabltes are obtaned analytcally and verfed by computer smulatons. The performance of the cooperatve dversty s found to be strongly relay locaton dependent. Hence, usng the analytcal as well as extensve numercal results, the geographcal areas of the relay locatons are obtaned for small to medum sgnal-to-nose rato values, such that the cooperatve coded dversty outperforms the recever coded dversty. On the other hand, for suffcently large sgnal-to-nose rato values, or f the path-loss attenuatons are not consdered, then the recever coded dversty always outperforms the cooperatve coded dversty. The obtaned results have mportant mplcatons on the deployment of the next generaton cellular systems supportng the cooperatve as well as the recever dversty. Key Words: Channel codng, communcaton system performance, cooperatve systems, dversty methods, fadng channels. 1 Introducton The roll-out of the 4G cellular systems s expected to commence n the near future. Varous forms of the transmsson dversty are one of the key techncal enablers of the 4G systems. The relays deployed about the 4G base statons wll provde the mproved coverage and enable hgher data rates servces by realzng the dstrbuted transmsson dversty. The exstence of relays, however, also sgnfcantly complcates the deployment of the 4G networks, for example, due to the ncreased captal and operatonal expendtures, and the need to allocate addtonal communcaton channels wthn the cell. It s therefore vtal to nvestgate the condtons when the cooperatve dversty realzed by the relays can brng the antennas closer to the user termnals, and thus, outperform the conventonal recever dversty realzed by the multple antennas at the recever. Such comparson can be done n terms of the transmsson relabltes represented by the parwse error probabltes PEPs and the bt error rates BERs.The uncoded cooperatve dversty technques were studed n [1] and n []. The multuser cooperatve protocols are proposed n [3, 4]. An overvew of the coded cooperaton schemes s gven n [5]. The performance of conventonal coded antenna dversty technques s nvestgated n [6]. The performance of coded systems over block fadng channels s analyzed n [7]. An upper-bound of the transmsson error probablty for bnary block codes over slow and fast fadng channels s obtaned n [8]. A specfc twouser coded cooperatve scheme s proposed and analyzed [9,1]. General analytcal expressons for the error performance of the amplfy-and-forward AF and the decode-and-forward DF relayng employng the turbo codes are obtaned n [11]. The coded cooperaton s also studes n [1]. In ths paper, a comparson s carred out between the transmsson relabltes of a cooperatve dversty system employng a sngle relay and a system employng the conventonal recever dversty wth the two recever antennas. Thus, both systems can acheve the dversty order of at most two. We formulate the re- E-ISSN: Volume 14, 15

2 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh search problem such that the source and the destnaton are statonary, and the task s to fnd the relay locatons, so that the cooperatve dversty can outperform the recever dversty. Ths s a dual problem to the scenaro where the destnaton source and the relay are statonary, and the task s to fnd the source destnaton locatons, so that the cooperatve dversty can outperform the recever dversty. The locatons of network nodes are taken nto account through the path-loss attenuatons. The results ndcate that, f the path-loss attenuatons, and thus, the mutual nodes locatons are not consdered, then the conventonal recever dversty always outperforms the cooperatve dversty. On the other hand, the path-loss attenuatons may cause the system wth the cooperatve dversty to outperform the system wth the recever dversty, partcularly at smaller values of the sgnal-to-nose rato SNR. All the channels between network nodes are assumed to be ndependent. In both systems, the destnaton coherently combnes the receved sgnals usng the maxmum rato combnng MRC or the equal gan combnng EGC [13]. More mportantly, we assume encodng of the packets usng a smple bnary lnear block codng and mappng to non-bnary lnear modulaton constellatons pror to ther transmsson. For the cooperatve dversty, assumng that tme dvson channel orthogonalzaton and a usual two tme-slot relayng protocol are used n order to avod the nterference of the transmtted packets. In case of the DF relayng, assumng that the relay uses the same encoder as the source and the same decoder as the destnaton. For the decodng of short length bnary lnear block codes, we employ the soft-decson decodng technques developed n [14] that are referred to as the partal-order statstcs decodng POSD. These technques acheve a good BER performance versus the mplementaton complexty trade-off, and, n some cases, the POSD technques can even closely approach the performance of the maxmum-lkelhood ML decoder [14, 15]. The rest of ths paper s organzed as follows. Secton II descrbes the system models ncludng the modulaton and channel codng and decodng for two systems employng the recever and the cooperatve dversty, respectvely. The PEP as a key measure of the transmsson relablty for the two systems under consderaton s analyzed n Secton III. The performance of the two systems are compared n Secton IV the optmum relay locatons for the system wth the cooperatve dversty are determned, so that t outperforms the system wth the recever dversty. Fnally, conclusons are gven n Secton V. System Model We compare the BER performance of two communcaton systems. System I uses a sngle relay R to realze a dstrbuted dversty n order to mprove the transmsson relablty from a source S to a destnaton D. All nodes n System I are equpped wth a sngle transmttng and a sngle recevng antenna. On the other hand, System II acheves the transmsson relablty by explotng the recever dversty. In System II, a source S wth one transmttng antenna transmts nformaton to a destnaton D havng two recevng antennas. Hence, both systems can acheve the transmsson dversty of order at most two. We assume a flat fadng channel model wth an addtve whte Gaussan nose AWGN between any par of network nodes, and also, that all channels are mutually ndependent. Wthout any loss of generalty, we omt symbol-tme ndces n the expressons.for System I usng the cooperatve dversty, we use the followng notaton to descrbe the transmsson from a node X {S, R} to a node Y {R, D},.e., d XY > α XY > h XY C γ XY w XY C y XY C dstance between X and Y path-loss coeffcent channel fadng coeffcent nstantaneous SNR at node Y AWGN receved sgnal at node Y where C denotes the set of complex numbers. For System II usng the recever dversty wth the recever antenna = 1,, we use the notaton, d > dstance between S and D α > path-loss coeffcent h C channel fadng coeffcent γ nstantaneous SNR at node D w C AWGN y C receved sgnal at node D. Furthermore, we make the followng assumptons common to both systems. The channel fadng coeffcents h are complex-valued wde-sense statonary jontly Gaussan random processes havng zero-mean and unt-varance. Thus, the channel fadng ampltudes h are Raylegh dstrbuted, and E[h] = and E [ h ] = 1, where E[ ] s expectaton, and s the absolute value. The channel fadng coeffcents are ether assumed to be constant and then change ndependently durng the transmsson of one codeword correspondng to a slow block fadng channel model, or they change ndependently for every transmtted symbol.e., a fast fadng channel model wth E-ISSN: Volume 14, 15

3 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh deal nterleavng and denterleavng of symbols. All coeffcents of AWGNs w are uncorrelated zero-mean complex-valued jontly Gaussan random processes havng the equal varance σ w = E [ w ] = N where N s a constant one-sded power spectral densty of the AWGNs. In general, the sgnal ampltude attenuaton due to a path-loss at dstance d from the transmtter antenna s proportonal to const d µ/ where the constant s a functon of the carrer frequency, and µ > s the path-loss exponent. Let d be the reference dstance at whch the path-loss s equal to unty. Then, the pathloss coeffcent α XY and α at the dstance d XY and d, respectvely, from the transmtter antenna can be expressed as, α XY = dxy d µ/ d µ/ α =. Snce the nodes S and D are common to both systems under consderaton, n the sequel, we assume that the path-loss between S and D n both systems s unty,.e., d = d SD = d. Hence, the path-loss coeffcents at dstances greater smaller than the reference dstance d are smaller larger than unty. Note that the choce of the reference dstance shfts the SNR values of all lnks equally. Thus, one can choose an arbtrary common reference dstance d wthout basng the BER comparsons of the two systems. Let x denote a modulaton symbol n the transmtted codeword. The modulaton symbols have zeromean and are normalzed, so that the average energy per symbol E [ x ] s equal to a constant E s >. For the cooperatve dversty system wth the AF relayng, the receved sgnals at two consecutve tme slots correspondng to the transmtted symbol x can be wrtten as, d y SD = α SD h SD x + w SD y SR = α SR h SR x + w SR y RD = β AF α RD h RD y SR + w RD where β AF s the amplfcaton factor used at the relay. The amplfcaton factor β AF normalzes the average energy of the sgnal transmtted from the relay to be equal to E s,.e., [11, 18], Es β AF = E[ ysr ] = E s αsr h SR E s + σw where expectaton n the denomnator s condtoned on the ampltude h SR. For the cooperatve dversty system wth the DF relayng, the receved sgnal at the destnaton at the second tme slot correspondng to the transmtted symbol x can be wrtten as, y RD = β DF α RD h RD ˆx + w RD where the relay amplfcaton factor β DF = 1 and ˆx s a re-encoded symbol at the relay. We assume that the symbol ˆx s from the same modulaton constellaton as the symbol x; f ˆx x, then a decodng error occurred at the relay. For the recever dversty system, the receved sgnals at the two recever antennas correspondng to the transmtted symbol x can be wrtten as, y 1 = α h 1 x + w 1 y = α h x + w. At the destnaton, the receved sgnals are coherently combned usng MRC or EGC. In partcular, the MRC output sgnals are wrtten as, y y System I = β α RD α SR h RDh SR β α RD h RD + 1 y RD + α SD h SDy SD System II = h 1y 1 + h y and for EGC, the output sgnals are wrtten as, y y System I = e j h RD+ h SR β α RD h RD + 1 y RD + e jh SD y SD System II = e j h 1 y 1 + e j h y where j = 1 s the magnary unt, and denotes the phase of a complex number. Note that, snce the path-loss coeffcents are tme-nvarant, they can be used as the weghtng factors of the EGC; however, n ths paper, only the phase-compensatng weghtng factors are consdered n the EGC combner. Recall that all the recevers n the network are assumed to have the dentcal tme-nvarant power spectral denstes of the background AWGNs. The nstantaneous SNR of the communcaton lnk between a par of nodes for the system wth the cooperatve and the recever dversty, respectvely, s defned as, γ XY = α XY h XY γ b γ = α h γ b where γ b = E s /N log M s the SNR per transmtted bt assumng an M-ary modulaton constellaton. In ths paper, we assume that all lnks are subject to ndependent and dentcally dstrbuted Raylegh fadng, and thus, the SNR of each lnk s exponentally dstrbuted [13]. Provded that a channel codng of rate R < 1 s used at the source, the AWGNs at the relay and destnaton recevers have the equal varance σw = E [ w ] = N = E s /Rγ b log M. Then, E-ISSN: Volume 14, 15

4 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh the nstantaneous SNR at the output of the MRC combner at the destnaton for the system wth the cooperatve and the recever dversty, respectvely, can be expressed as, System I γ SR γ RD γ = γ SD + γ SR + γ RD + 1 System II γ = γ 1 + γ. In general, dependng on the relay locaton, the average SNR at the combner output at the destnaton can be larger or smaller for the cooperatve dversty than for the case of the recever dversty. However and mportantly, f the path-loss s not consdered.e., the average SNR values are locaton-nvarant, then the average SNR of the recever dversty s always larger than the average SNR of the cooperatve dversty. In addton, note that, for a far comparson, we assume that both the source and the relay transmts wth the average energy per symbol E s, so that the total average energy per transmtted symbol s E s over the two tme-slots whereas the total average energy per transmtted symbol for the system wth the recever dversty s E s..1 Modulaton and Channel Codng and Decodng We assume that the transmssons between nodes are realzed usng a lnear memoryless modulaton and usng a lnear bnary block code of short block length. The encodng of nformaton bts by a bnary channel code s performed by multplyng the vector of K nformaton bts by a bnary generator matrx n order to produce a bnary codeword of N encoded bts. The bnary channel codng C s denoted as a trplet N, K, d mn where d mn s the mnmum Hammng dstance between any two codewords, and R = K/N s the code rate. The codewords are possbly nterleaved and mapped to ether bnary phase shft keyng BPSK symbols or to 16 quadrature ampltude modulaton QAM symbols. For the 16QAM modulaton, we assume a natural mappng of the consecutve sequences of 4 encoded bts c 1, c, c 3, c 4 to the modulaton symbols x = x I + jx Q such that the encoded bts c 1, c 3 are mapped to x I {±1, ±3}, and the encoded bts c, c 4 are mapped to x Q {±1, ±3}, as n paper 4 and [19]. 3 Analyss of Transmsson Relablty The theoretcal analyss s mathematcally tractable provded that we assume a block fadng channel model,.e., the channel fadng coeffcents are generated ndependently and held constant for the transmsson of each codeword. Recall that the channel fadng coeffcents between the network nodes are assumed to be mutually ndependent, and they are perfectly known at the recevers. For notatonal smplcty, the path-loss coeffcents α are merged nto the channel fadng coeffcents h, so that the varances E [ h ] are scaled by α. We denote as g = h the ampltudes of the channel fadng coeffcents h. In our analyss, we consder the performance of the EGC at the destnaton recever for the case of BPSK modulaton. For BPSK sgnalng, we denote the codewords =,,, a = a 1,, a N and b = b 1,, b N correspondng to the transmtted sequences x = 1,, 1, x a = x a 1,, xa N and x b = x b 1,, xb N, respectvely. We assume that all codewords are equally lkely to be transmtted and that an all-zero codeword has been transmtted. Note that the latter assumpton may slghtly bas the analyss for System II due to non-lnearty of the DF relayng. The ML detector at the destnaton recever selects the most lkely codeword correspondng to the transmtted sequence wth the smallest Eucldean dstance from the receved sequence y. In general, the probablty of transmsson error for coded systems can be upper-bounded usng a unon-bound [1]. Thus, the BER of coded systems can be upper-bounded as [], BER a C a w H [u] Pr a 1 K where w H [u] s the Hammng weght of the nformaton vector u correspondng to the codeword a of a bnary lnear block code C = N, K, d mn. The PEP Pr a s the probablty that the all-zero codeword was transmtted, and the recever decdes between the codewords and a that a has been transmtted. Provded that the PEP Pr a can be expressed as a functon of the Hammng weght w H [a], the unon bound 1 can be evaluated more effectvely usng a weght enumerator of the code C []. More mportantly, note that the unon bound 1 s domnated by the largest PEP Pr a. Thus, n the sequel, we evaluate the PEP Pr a rather than the overall unon bound 1 as a key measure of the transmsson relablty for the coded communcaton systems. 3.1 System I wth the AF Dversty Assumng System I wth the AF relayng, the output sgnal of the EGC at the destnaton recever can be wrtten as, E-ISSN: Volume 14, 15

5 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh y = R = { g SD + g SD + β AFg RD g SR β AFg RD + 1 x g RD g g SR x RD + g + w AF SR + c 1 + β AFg RD w SR + w RD β AFg RD w SD } = g AF x + w AF where = 1,,, N s the symbol ndex n the transmtted codeword, R{ } s the real part of a complex number, w AF s an equvalent zero-mean AWGN havng the varance E [ w AF ] = σw = N, and c 1 = N /E s s the nverse of the SNR per transmtted symbol. Note that the sgnal receved from the relay s normalzed by the factor βafg RD + 1 n order to make the AWGN varances of the two dversty sgnals before combnng equal. Gven the value of g AF, the condtonal PEP of System II wth the AF relayng s calculated as the probablty that the Eucldean dstance w E, from the receved sequence y for the codeword s greater than the Eucldean dstance from y for the codeword a,.e., Pr a g AF = Pr w E, > w E,a N = Pr y g AF x N > y g AF x a =1 N = Pr gafx a x + g AF x a x w AF >. =1 Snce, for a zero mean unt varance Gaussan random varable W, the probablty PrW > w = Qw where Q s the Q-functon [13], we have that, N =1 Pr a g AF = Pr W > g xa x AF N [ w E x = Q g a, x ] AF = Q g AF wh [a] γ b N where w E [ x a, x ] s the Eucldean dstance between the vectors x a and x. Then, the PEP s evaluated as, Pr a = where f gaf z s the probablty densty functon PDF of g AF. In general, a closed form expresson for f gaf z s dffcult to obtan. However, snce, always, g AF g SD + mng RD, g SR = g AF, and the channel fadng ampltudes g SD, g SR and g RD are ndependent and have the varances σsd, σ SR and σrd, respectvely, =1 Pr a z f gaf zdz we can lower-bound the PEP,.e., Pr a Pr a z f gaf zdz where, after lengthy manpulatons, the closed form expresson of the PDF f gaf z s shown at the top of ths page, c = σ SDσ SR+σ RDσ SD+σ SR, c 3 = loge and the functon erfx = 1 Q x. E-ISSN: Volume 14, 15

6 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh f gaf z = σ RD +σ SR σ RD +σ SR c 5/ c 3/ 3 σ SDσ SR + σ RDσ SD + σ SR c 3 z 3. System I wth the DF Dversty e σ RD +σ SD +σ SR z / c σ SRσSD c3 ze 1 c c3 σrdσ SRze σ SD z c c3 σsdσ SRze 1 σ πσsd σ SR σ RD e RD + 1 σ SR + σ RD σ SR σ SD c erf σrd σ SR c3 z 4σ SD v σ RD +σ SR σ RD z σ SR + z + erf σ RD +σ SR σ RD z σ SR σsd σrd +σ SR c 3 z 4σ RD σ SR c In order to analyze the PEP of the DF relayng, we assume that the source transmts the all-zero codeword, however, the relay decodes and forwards a codeword b. In ths case, the EGC output sgnal at the destnaton recever s wrtten as, { y = R g SD x = g SD x + w SD + g RD x b + w RD } + g RD x b + w DF where w DF s an equvalent zero-mean AWGN wth the varance E [ w DF ] = σ w = N. The PEP of the destnaton recever condtoned on the values of the channel fadng ampltudes g SD, g SR and g RD s then calculated as, Pr a g SD, g SR, g RD = b C Pr a b, g SD, g RD Pr b g SR 3 where Pr b g SR s the condtonal PEP that the relay decodes the codeword b. The frst condtonal PEP n 3 s agan equal to the probablty that the Eucldean dstance w E, from the receved sequence y for the all-zero codeword s greater than the Eucldean dstance w E,a correspondng to the codeword a,.e., Pr a b, g SD, g RD = Pr w E, > w E,a N = Pr y g SD + g RD x N > y g SD + g RD x a =1 =1 N = Pr g RD t g SD s s + s w DF > =1 N = Q =1 g SDs g RD t s, 4 N N =1 s where we defned, s = x a x and t = x b x a x. Assumng that x = 1 for, we can show that, for any values of g SD and g RD, the argument of the Q-functon n 4 s, n general, ncreasng wth the Hammng dstance between the codewords a and b. The argument of the Q-functon n 4 s mnmzed for a = b.e., the vectors are component-wse dentcal whle a whch corresponds to the worst case scenaro when the value of the PEP defned n 4 s maxmzed. On the other hand, we can show that, for any values of g SD and g RD, the value of the PEP 4 s mnmzed provded that b =.e., the relay correctly decodes the codeword transmtted from the source. Ths also ndcate that the ablty of the relay to correctly decode the transmtted codeword from the source has a major effect upon the overall probablty of transmsson error of the cooperatve system. Denote as w E,b the Eucldean dstance from the receved sequence y SR for the codeword b at the relay recever. Then, the PEP Pr b g SR for the lnk from the source to the relay can be expressed as [7], E-ISSN: Volume 14, 15

7 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh Pr b g SR = Pr w E, > w E,b N = Pr y SR, g SR x N > y SR, g SR x b =1 =1 [ gsrw E x, x b] = Q = Q g SR wh [b] γ b N where y SR, s the receved sgnal at the relay, w H [ [b] s the Hammng weght of the codeword b, and w E x, x b] s the Eucldean dstance between the modulated sequences correspondng to the vectors and b. Usng 3, the PEP averaged over the ndependent Raylegh dstrbuted channel fadng ampltudes g SD, g SR and g RD s expressed as, Pr a = Pr a u, v, r f gsd uf gsr vf grd rdudvdr = Pr a b, u, v f gsd uf gsr vdudv b C = b C Pr a b Pr b Pr b r f grd rdr Let the argument of the Q-functon n 4 be a random varable, where the constants, C 1 = Z = C 1 g SD C g RD N =1 s N and C = N =1 s t. N N =1 s Then, the average PEP Pr a b can be evaluated as, Pr a b = Pr a b, z f Z zdz. The PDF of the random varable Z can be obtaned by condtonng and ntegraton [3],.e., f Z z = 1 1/ 1 k k k1 + k 3 f 1 z z 1/ 1/ 1 1 k k k1 + k 3 k1 k 5 +k f 4 z z < where k 1 = C σ 1, k = C σ, k 3 = C1 4σ4 1, k 4 = C 4σ4, k 5 = C1 C σ 1 σ, and, f 1 z = e 1 + z k 1 k 1 k e z k 1 z + k e z k z k 1 k k 1 k πk k 1 + πk 1 + k k 1 + k z e k 3 +k 5 k 4 z k 5 k 1 +k + π k1 + k 1k k 1 + k z erf f z = e z k1 k k 1 + k k 5 + k 4 π + + π k1 + k 1 k k 1 z e k1+kz k 5 +k 4 k 1 k k z k3 + k 5 k 5 +k 4 k 1 k k z + πk 1 + k z e k 1 +k z k1 z erf. k5 + k 4 E-ISSN: Volume 14, 15

8 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh The average PEP Pr b can be obtaned by usng the Chernoff bound Qx 1 e x /, for example, as n [8], or by usng the Prony approxmaton Qx. =.8e.971x +.147e.55x as n []. Assumng the latter expresson, the average PEP s approxmately equal to, Pr b. = w H [b] σsrγ b w H [b] σsrγ b + 1 where γ b s the SNR per encoded bnary symbol. 3.3 System II wth the Rx Dversty Assumng the recever dversty wthout relay, the output sgnal of the EGC at the destnaton recever can be wrtten as, { } y = R g 1 x + g x + w 1 + w = g 1 + g x + w Rx = g Rx x + w Rx where w Rx s an equvalent zero-mean AWGN wth the varance E [ w Rx ] = σ w = N. The PEP of System II s obtaned smlarly as for the source to relay lnk n System I. Thus, condtoned on the channel fadng ampltude g Rx, and BPSK sgnalng, the PEP s evaluated as, Pr a g Rx = Pr w E, > w E,a N = Pr y g Rx x N > y g Rx x a =1 = Q g Rx wh [a] γ b. Consequently, the average PEP s calculated usng the ntegraton, Pr a = so that the average PEP s calculated as, =1 Pr a z f grx zdz. The ntegraton to obtan the average PEP Pr a can be carred out usng the Prony approxmaton method []. In partcular, the condtonal PEP s approxmately equal to,.= Q g Rx wh [a] γ b.8e.971grx w H[a]γ b +.147e.55g Rx w H[a]γ b Pr a =.8 where A 1 =.971 w H [a] γ b and A =.55 w H [a] γ b. The PDF f grx z of the channel fadng ampltude g Rx s agan obtaned by condtonng and ntegraton. e A 1 z f grx zdz e A z f grx zdz 5 Thus, assumng the ndependent Raylegh dstrbuted channel fadng ampltudes g 1 and g of the varances σ 1 and σ, respectvely, we obtan the PDF, f grx z = zσ 1 V e z σ 1 + π z V V 5/ rσ 1e z V σ z 1 + erf V σ1 where V = σ 1 + σ s the varance of the EGC ampltude g Rx. Fnally, a closed form expresson for the average PEP 5 based on the Prony approxmaton method s obtaned usng the followng ntegraton,.e., E-ISSN: Volume 14, 15

9 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh I σ1,σ a = e a z f grx zdz = σ 1V 3/ 1 + av + 4aσ V 5/ 1 + av 3/ where a > s a real constant, and V was defned arctan 1 + av σ 1 σ π prevously. The PEP 5 s then computed as, Pr a =.8 I σ1,σ.971 w H [a] γ b I σ1,σ.55 w H [a] γ b. 4 Performance Comparson of System I and System II We use the PEP expressons obtaned n the prevous secton to compare the error rate performances of System I and System II wth the cooperatve and the recever dversty s nvestgated, respectvely. In partcular, the effect of the relay locaton on the performance of the cooperatve dversty, and determne geographcal areas for postonng the relay n whch the relayng can outperform the conventonal recever dversty. Recall that the upper-bound of the BER 1 s domnated by the largest PEP Pr a, so that we can consder the PEP Pr a to be the key performance metrc of the system. More mportantly, assumng our analyss n Secton 6.3, t can be shown that, for System I as well as System II, the largest PEP Pr a corresponds to the codeword a of the mnmum Hammng weght w H [a] = d mn. Denote as PEP AF, PEP DF and PEP Rx the PEPs Pr a of System I wth the AF relayng, System I wth the DF relayng and System II wth the recever dversty, respectvely. The PEPs PEP AF and PEP DF are the relay locaton dependent. The relay locaton s denoted as a trplet d SR /d, d RD /d, d SD /d where d s the reference dstance. Recall that, wthout loss of generalty, we assume d = d SD,.e., the relay locaton s gven by the trplet d SR /d SD, d RD /d SD, 1. For System I, the dstance between the source and the destnaton s a scalar varable d; we assume that d/d SD = 1. Thus, for System I as well as System II, the path-loss between the source and the destnaton s unty. Fg. 1 shows an excellent agreement between the mathematcal expressons obtaned n Secton 6.3 and the computer smulatons for the PEP Pr a of System I wth the DF relayng assumng ndependent slow Raylegh fadng channels, BPSK modulaton, and a codeword a of the Hammng weght d mn for the BCH codes 31, 16, 7 and 3, 6, 4. Fg. compares the PEPs Pr a of System II wth the two recever antennas and System I wth the DF relayng assumng agan ndependent slow Raylegh fadng channels, BPSK modulaton, and a codeword a of the Hammng weght d mn for the BCH code 31, 16, 7. Note that the dstance between the source and the destnaton s normalzed to 1. The relay locaton denoted as 1, 1, 1 corresponds to the case when the path-loss s not consdered. Provded that the pathloss s not consdered, the recever dversty always outperforms the DF dversty as one may ntutvely expect. Relayng outperforms the recever dversty, partcularly at smaller values of the SNR. Ths s further confrmed by the PEP values n Fg. 3 versus the relay locaton d SR /d SD, 1 d SR /d SD, 1 at a constant SNR γ b = 9dB. More mportantly, we observe from Fg. 3 that the relay located closer to the source acheves a better PEP performance than the relay located at the center between the source and the destnaton cf. Fg. 5. Thus, the optmum relay locaton has to trade-off the error propagaton due to the DF relayng and the path-loss attenuatons between the nodes, and t s also nfluenced by the partcular channel code used. Assumng the same parameters and settngs as n Fg. and Fg. 3, the PEP performance of System I wth the AF relayng s shown n Fg. 4. Also a numercal examples are presented for the overall BER performances of System I and System II. We consder uncoded as well as coded transmssons from the source to the destnaton usng the BCH systematc codes 31, 16, 7 and 3, 16, 8 and BPSK and 16QAM modulatons. We employ the POSD decoder at the destnaton and also at the relay provded that the DF relayng s used. The POSD s optmzed to acheve the best possble BER performance for the gven decodng complexty [14]. In partcular, for both BCH codes consdered, the POSD searches two dsjont segments of 6 and 1 ordered nformaton bts assumng at most 1 and 3 errors n each segment, respectvely. We use the notaton Rx to denote the two antenna recever dversty, and the notaton 1Rx to refer to the scenaro where the destnaton s equpped wth a sngle recevng antenna. E-ISSN: Volume 14, 15

10 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh 1 M BCH31,16,7 S BCH31,16,7 M BCH3,6,4 S BCH3,6,4 1 3 PEP DF γ b [db] Fgure 1: The PEP Pr a for System I wth the DF relayng, the BCH 31, 16, 7 and 3, 6, 4 coded BPSK sgnalng over slowly Raylegh fadng channels, and the EGC at the destnaton M-mathematcal expresson, S- smulaton. 1 PEP RX, PEP DF Rx DF.5,.5,1. DF.5,.8,1. DF 1.,1.,1. DF.8,.8, γ b [db] Fgure : The PEP Pr a for System I wth the DF relayng and the BCH 31, 16, 7 coded BPSK sgnalng over slowly Raylegh fadng channels, and the EGC at the destnaton. E-ISSN: Volume 14, 15

11 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh 1 PEP DF d SR / d SD Fgure 3: The PEP Pr a for System I wth the DF relayng and the BCH 31, 16, 7 coded BPSK sgnalng over slowly Raylegh fadng channels, the EGC at the destnaton, the normalzed dstance d RD /d SD = 1 d SR /d SD, and the SNR γ b = 9dB AF.4,.8,1. AF.6,.8,1. AF.8,.8,1. AF 1.,1.,1. PEP AF γ b [db] Fgure 4: The PEP Pr a for System I wth the AF relayng and the BCH 31, 16, 7 coded BPSK sgnalng over slowly Raylegh fadng channels, and the EGC at the destnaton. Fg. 5 compares the BER performances of System I wth the AF relayng and the conventonal recever dversty assumng MRC at the destnaton. We observe that, for some relay locatons, the AF relayng outperforms the conventonal recever dversty. The best BER performance of the AF relayng s acheved when the relay s located n the center between the source and the destnaton. On the other hand, as ntutvely expected, the BER performance of the AF relayng deterorates sgnfcantly when the relay s located at larger dstances away from the source and the destnaton. In addton, we observe that the channel codng benefts sgnfcantly from the avalable dversty gan due to the relayng and all relay locatons or due to the multple recever antennas. The BER performance of the DF relayng s shown n Fg. 6 assumng the same parameters and relay locatons as n Fg. 5. Unlke for the AF relayng n Fg. 6, we observe from Fg. 6 that the BER performance of the DF relayng s much more relay locaton dependent than the BER performance of the AF relayng, and such dependence s even more pronounced for hgher order modulatons. In addton, as already ndcated n Fg. 3, the optmum relay loca- E-ISSN: Volume 14, 15

12 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh ton for the DF relayng s found, n general, closer to the source than to the destnaton n order to suppress the detrmental effect of error propagaton due to erroneous decodng at the relay. Further examples of the BER for the DF relayng over fast and slow Raylegh fadng channels are shown n Fg. 7 and Fg. 8. We can agan observe that there exst geographcal areas of the relay locatons where the conventonal recever dversty outperforms the DF relayng for all SNR values. On the other hand, also t s observed that, for suffcently large SNR values, the conventonal recever dversty outperforms the DF relayng for all relay locatons consdered. Furthermore, we observe from Fg. and Fg. 5 Fg. 8 that, partcularly for hgher order modulatons and the DF relayng, System I does not acheve the dversty order of System II. BER uncoded, 1Rx uncoded, Rx coded, 1Rx coded, Rx coded, AF.5,.5,1. coded, AF.7,.7,1. coded, AF.5,.9,1. coded, AF.8,1., γ b [db] Fgure 5: The BER of the BCH 31, 16, 7 coded BPSK and the AF relayng and the recever dversty wth the MRC at the destnaton for several relay locatons for fast Raylegh fadng channels. 4.1 Optmum Relay Locatons The performance results n Fg. 1 Fg. 8 ndcate that the relay locaton sgnfcantly affects the BER performance of System I wth the cooperatve dversty. We determne the optmum relay locatons n the sense that System I wth the cooperatve dversty outperforms System II wth the recever dversty. In partcular, we evaluate the PEP dfferences, PEP Rx AF = PEP Rx PEP AF PEP Rx DF = PEP Rx PEP DF. 6a 6b Hence, f PEP Rx AF > or PEP Rx DF >, then the cooperatve dversty wth the AF or the DF relayng, respectvely, outperforms the second order recever dversty. The relay postons for whch the PEP dfferences 6a and 6b are greater than zero are obtaned numercally by samplng the two-dmensonal space of all possble relay locatons. Examples of the PEP dfferences 6a and 6b versus the relay locatons d SR /d SD, d RD /d SD, d SD for the SNR γ b = 9dB are shown n Fg. 9 and Fg. 1, respectvely. More mportantly, f the SNR exceeds a certan threshold value, then, for any relay locaton, the PEP dfferences 6a and 6b wll always be negatve,.e., the recever dversty wll outperform the cooperatve dversty. In general, determnaton of the exact boundares of the geographcal areas of the relay locatons where System I outperforms System II appears to be mathematcally ntractable, partcularly, when the channel codng s employed. However, by evaluaton of our extensve numercal results ncludng those that are not presented n ths paper, we make the followng proposton. Proposton 1. Assumng path-loss attenuatons of the transmtted sgnals and ndependent channel fadngs between the transmtter and the recever antennas, the E-ISSN: Volume 14, 15

13 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh BER uncoded, 1Rx uncoded, Rx coded, 1Rx coded, Rx coded, DF.5,.5,1. coded, DF.7,.7,1. coded, DF.8,.8,1. coded, DF.8,1.5, γ b [db] Fgure 6: The BER of the BCH 31, 16, 7 coded BPSK and the DF relayng and the recever dversty wth the MRC at the destnaton for several relay locatons for fast Raylegh fadng channels uncoded, 1Rx coded, Rx coded, DF.5,.5,1. coded, DF.7,.7,1. coded, DF.8,1.5,1. BER γ [db] b Fgure 7: The BER of the BCH 3, 16, 8 coded 16QAM and the DF relayng and the recever dversty wth the EGC at the destnaton for several relay locatons for fast Raylegh fadng channels. cooperatve dversty wth a sngle relay outperforms the two antenna recever dversty provded that the relay locaton d SR /d SD, d RD /d SD, 1 s constraned as, d SR /d SD < 1. d RD /d SD < 1. d SR /d SD + d RD /d SD < A γ,c where the parameter A γ,c > upper-boundng the path-length from the source to the destnaton va the relay s a decreasng functon of the SNR and a functon of the channel codng C. Specfcally, for small to medum SNR values and the path-loss exponent µ =, and bnary lnear block codes of d mn < 1, A γ,c. for the AF relayng, and A γ,c 1.5 for the DF relayng. In addton, for suffcently large SNR or when the path-loss attenuatons are not consdered, E-ISSN: Volume 14, 15

14 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh coded, DF.5,.5,1. coded, DF.3,.7,1. coded, DF.7,.7,1. coded, DF.7,.3,1. coded, 1Rx coded, Rx BER γ b [db] Fgure 8: The BER of the BCH 31, 16, 7 coded BPSK and the DF relayng and the recever dversty wth the EGC at the destnaton for several relay locatons for slow Raylegh fadng channels. PEP Rx-AF x dsr / d SD d RD / d SD.5 Fgure 9: The PEPs dfference PEP Rx AF of System II and System I wth the AF relayng and the BCH 31, 16, 7 coded BPSK sgnalng over slowly Raylegh fadng channels, and the EGC at the destnaton for the SNR γ b = 9dB. the parameter A γ,c < 1. and the recever dversty always outperforms the cooperatve dversty. Note that Proposton 1 mplctly assumes the trangle nequalty constrant, d SR /d SD +d RD /d SD 1.. Thus, f the parameter A γ,c becomes smaller than 1, then, for no relay locaton can the cooperatve dversty outperform the recever dversty. A sub-optmum 5 Concluson The transmsson relabltes of System I wth the recever dversty and System II wth the cooperatve dversty were nvestgated. Both systems can theoretcally acheve the maxmum dversty order of two. decodng scheme that s used n our numercal examples, and subsequently, used to formulate Proposton 1 appears to nfluence the threshold SNR value when the parameter A γ,c becomes smaller than 1.. Fnally, t s straghtforward to show that f the pathloss attenuatons are not consdered, then the recever wth K ndependent recever antennas wll always outperform a cooperatve system wth K 1 relays. However, partcularly the performance of System II suffers from the error propagaton due to sgnal processng at the relay. Path-loss attenuatons of the transmtted sgnals, ndependence of the channel fadng coeffcents and the use of channel codng wth nonbnary lnear mctepodulatons were the man assump- E-ISSN: Volume 14, 15

15 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh x PEP Rx-DF d RD / d SD d SR / d SD Fgure 1: The PEPs dfference PEP Rx AF of System II and System I wth the DF relayng and the BCH 31, 16, 7 coded BPSK sgnalng over slowly Raylegh fadng channels, and the EGC at the destnaton for the SNR γ b = 9dB. tons adopted n the system modelng. At the destnaton recever, the dversty sgnals were combned usng ether MRC or EGC. A low- complexty softdecson POSD was extended for the decodng of bnary lnear block codes used wth non-bnary modulatons. The PEP was nvestgated as the key performance measure of the system transmsson relablty. In partcular, assumng channel codng and BPSK sgnalng, the PEP expressons were derved analytcally for System I as well as for System II. The obtaned PEP expressons were verfed by computer smulatons. The performance of System II was found to be strongly dependent on the relay locaton as expected. More mportantly, t was found that, for some relay locatons and SNR values, System II wth the References: [1] J.N. Laneman, D.N.C. Tse and G.W.Wornell, Cooperatve dversty n wreless networks: Effcent protocols and outage behavor, IEEE Trans. Inf. Theory, vol. 5, no. 1, pp , Dec. 4. [] X. Zhang, W. Wang and X. J, Multuser dversty n multuser two-hop cooperatve relay wreless networks: System model and performance analyss, IEEE Trans. Veh. Tech., vol. 58, no., pp , Feb. 9. [3] M. D. Renzo, M. Lezz and F. Grazos, Error Performance and Dversty Analyss of Mult- Source Mult -Relay Wreless Networks wth Bnary Network Codng and Cooperatve MRC, IEEE Trans. Wreless Comms., vol. 1, no. 6, pp , June 13. cooperatve dversty may outperform System I wth the recever dversty. The approxmate boundares of such geographcal areas of relay locatons when System II outperforms System I were formulated n Proposton 1 usng both the obtaned mathematcal analyss of the PEPs as well as usng extensve computer smulatons. The DF relayng was found to be more senstve to and more restrctve about the relay locaton than the AF relayng. More mportantly, f the path-loss attenuatons are not consdered, then the recever dversty always outperform the cooperatve dversty. These results have sgnfcant mplcatons for the deployment and desgn of the current cellular systems supportng both the recever as well as cooperatve dversty. [4] S. Chen, W. Wenbo and X. Zhang, Performance analyss of multuser dversty n cooperatve mult-relay networks under Raylegh-fadng channels, IEEE Trans. Wreless Comms., vol. 8, no. 7, pp , Jul. 9. [5] Y. L, Dstrbuted codng for cooperatve wreless networks: An overvew and recent advances, IEEE Comms. Mag., vol. 47, no. 8, pp , Aug. 9. [6] M. K. Smon and M.-S. Aloun, Dgtal Communcaton Over Fadng Channels: A Unfed Approach to Performance Analyss, John Wley and Sons, New York,. [7] R. Knopp P. A. Humblet, On codng for block fadng channels, IEEE Trans. Inf. Theory, vol. 46, no. 1, pp , Jan.. E-ISSN: Volume 14, 15

16 Saf E. A. Alnawayseh, Pavel Loskot, Mutaz Al-Tarawneh, Zyad Ahmed Al Tarawneh [8] E. Malkamak H. Leb, Coded dversty on block-fadng channels, Trans. Inf. Theory, vol. 45, no., pp , Mar [9] T. E. Hunter and A. Nosratna, Dversty through coded cooperaton, IEEE Trans. Wreless Comms., vol. 5, no., pp , Feb. 6. [1] T. E. Hunter and A. Nosratna, Performance analyss of coded cooperaton dversty, Proc. ICC 3, vol. 4, pp , May 3. [11] M. R. Souryal and B. R. Vojcc, Performance of amplfy-and-forward and decode-and-forward relayng n Raylegh fadng wth turbo codes, Proc. ICASSP 6, May 6. [1] T. Wang, A. Cano, G. B. Gannaks, and J. N. Laneman, Hgh-performance cooperatve demodulaton wth decode-and-forward relays, IEEE Trans. Comm., vol. 55, no. 7, pp , July 7. [13] J. G. Proaks, Dgtal Communcatons, McGraw-Hll, New York, 4th ed.,. [14] S. E. A. Alnawayseh and P. Loskot, Lowcomplexty soft-decson decodng technques for lnear bnary block codes, n Proc. WCSP, pp. 1 5, Nov. 9. [15] M. Fossorer and S. Ln, Computatonally effcent soft decson decodng of lnear block codes based on ordered statstcs, IEEE Trans. Inf. Theory, vol. 4, pp , May [16] M. Isaka, R. H. Zaragoza, M. P. C. Fossorer and S. Ln, Soft decson decodng of lnear block codes based on order statstcs n multlevel sgnalng, IEICE IT tech. group meetng, Tokushma, Japan, May [17] S. Allpress, C. Lusch and S. Felx, Exact and approxmated expressons of the log-lkelhood rato for 16QAM sgnals, Proc. Aslomar Conf. Sg., Syst. and Comp., vol. 1, pp , Nov. 4. [18] J. N. Laneman, Cooperatve dversty n wreless networks: Algorthms and archtectures, PhD dssertaton, MIT,. [19] P. M. Fortune, L. Hanzo and R. Steele, On the computaton performance of 16QAM and 64QAM performance n Raylegh fadng channels, IEICE Trans. Comms., vol. E7-B, no. 6, pp , June 199. [] S. Benedetto and E. Bgler, Prncples of Dgtal Transmsson wth Wreless Applcatons, Kluwer Academc, [1] E. Bgler, Codng For Wreless Channels, Sprnger, New York, Teachers ed., 5. [] P. Loskot and N. C. Beauleu, Approxmate performance analyss of coded OSTBC-OFDM systems over arbtrary correlated generalzed Rcean fadng channels, IEEE Trans. Comms., vol. 57, pp , Aug. 9. [3] A. Papouls and S. U. Plla, Probablty, Random Varables and Stochastc Processes, 4th Ed., McGraw Hll,. [4] P. Loskot and N. C. Beauleu, Prony and polynomal approxmatons for evaluaton of the average probablty of error over slow-fadng channels, IEEE Trans. Vehcular Tech., vol. 58, pp , Mar. 9. E-ISSN: Volume 14, 15