Maximum Likelihood Detection for Detect-and-Forward Relay Channels
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1 Int. J. Communication, Netwok and Sytem Science,, 4, doi:.436/ijcn..4 Publihed Online Febuay ( Maximum Likelihood Detection fo Detect-and-Fowa Relay Channel Abtact Azlan Abd Aziz, Yaunoi Iwanami Depatment of Compute Science and Engineeing, Nagoya Intitute of Technology, Nagoya, Japan Received Novembe 3, ; evied Decembe 7, ; accepted Decembe 3, Thi pape intoduce a imple combining technique fo coopeative elay cheme which i baed on a Detect-and-Fowa (DEF) elay potocol. Coopeative elay cheme have been intoduced in ealie wok but mot of them ignoe the quality of the ouce-elay (S-R) channel in the detection at the detination, although thi channel can contibute heavily to the pefomance of coopeation cheme. Fo optimal detection, the detination ha to account all poible eo event at the elay a well. Hee we peent a Maximum Likelihood citeion (ML) at the detination which conide cloed-fom expeion fo each ymbol eo ate (SER) to facilitate the detection. Compute imulation how that ignificant diveity gain and Packet Eo Rate (PER) pefomance can be achieved by the popoed cheme with good toleance to popagation eo fom noiy elay. In fact, diveity gain i inceaed with additional elay node. We compae thi cheme againt the baeline Coopeative-Maximum Ratio Combining (C-MRC). Keywo: Coopeative Relay, Detect-and-Fowa, Maximum-Likelihood Citeion Detection. Intoduction Coopeative communication ha been developed a a pomiing technique to ealize patial diveity though ue coopeation []. Vaiou elaying cheme have been popoed to exploe the benefit of coopeative communication, mainly divided into thee categoie, including Decode-and-Fowa (DF), Amplify-and-Fowa (AF) [-3] and Detect-and-Fowa (DEF). In DF, the elay alway decode, e-encode and e-tanmit the decoded ignal to the detination. That i, any eo at the elay can be coected and thu, eo popagation can eventually be avoided. On the othe hand, AF imply amplifie the eceived ignal and fowa them to the detination afte powe caling. The diadvantage of AF tategy i that it will alo fowa noie which eceived at the elay. Anothe elaying potocol which i imple in complexity i DEF whee the elay imply detect the ignal (ha-deciion detection), modulate befoe fowaing to the detination. With DEF, in [4] the autho ha hown that the diveity gain can alo be achieved povided the detination know the elay pobability of eo. Recently, many wok wee devoted to impove the elay complexity and yet tive fo bette eo ate pefomance. Fo example, ou ealie wok in coded coopeative cheme [5] adopted DEF at the elay node with futhe enhancement at the detination. The detination employ Maximum Likelihood citeion detection (ML) to combine the diect and elayed tanmiion. Howeve, thi detection at the detination doe not account ufficiently the eo pobability of making eo at the elay eulting in eiou pefomance degadation. In coded elay cheme, the channel decode i initialized with channel log-likelihood atio (LLR); hence, equie high accuacy of LLR computation. The autho in [4] have developed a piece-wie linea eceive appoximating the ML citeion detection that equie knowledge of the aveage ignal-to-noie-atio (SNR) of the fit hop. Howeve, thi cheme cannot achieve full diveity fo moe than one elay. In [6], anothe combining technique namely Coopeative-Maximum Ratio Combining (C-MRC) i intoduced that appoximate the ML detecto. Unfotunately, C-MRC eult in eiou popagation eo unde aymmetical netwok when SNR of R-D link i lage than that of S-D link (o S-R link). Anothe wok in [7], ha tudied the pefomance of the ha-deciion ML citeion detection-baed combining technique unde uncoded coopeative cheme. In [8,9], the autho have popoed a non-coheent combine in uncoded coopea- Copyight SciRe.
2 A. A. AZIZ ET AL. 89 tive elaying cheme uing DEF potocol with limited Channel State Infomation (CSI). In [4-], the autho have deived ub-optimal eceive but exploiting effectively pefect knowledge of all link i till an open poblem, pecifically the eo pobability at the elay. Howeve, many of the peviou wok aumed that the detination only know the aveage pobability of ymbol eo at the elay. Since the detection ule in ML citeion detection at the detination ha to conide evey ymbol eo pobability, thi eo model may ubtantially affect the attainable end-to-end pefomance. Thu, to guaantee an optimal ML citeion detection, the detination need to know the eo chaacteitic of the S-R link (pefect CSI) in the fom of elay eo pobabilitie. In thi pape, we aim at poviding the detination with a moe accuate CSI in it deciion citeion. While the idea itelf of having pefect CSI i not new [9,,3], the CSI in the peviou wok i calculated baed on the aumption of aveage eo pobability. Fo binay phae-hift keying (BPSK) cae, the olution i taightfowa, but fo quatenay phae-hift keying (QPSK) and highe modulation contellation, the Euclidean ditance between ymbol become no longe the ame; hence the ymbol eo pobability can alo diffe. To cicumvent thi poblem, hee we develop a imple ML detection algoithm fo the detination node fo QPSK modulation. Fo implicity, we analyze thi ML pefomance with a imple DEF in uncoded coopeative elay netwok and compae againt the baeline C-MRC. Unlike in C-MRC, the intantaneou CSI in the popoed cheme involve Q-function expeion fo each ymbol in the modulation which povide accuate knowledge of S-R link. Ou cheme alo outpefom C-MRC epecially when SNR of R-D link i ufficiently high o low (aymmetical netwok). We how though compute imulation that the pefomance of the popoed coopeative elay cheme can be impoved ignificantly paticulaly in multiple elay node. We obeve that thee i emakable potential to achieve inceaing oe of diveity with bette packet eo ate (PER) pefomance than that of C-MRC. The oganization of thi pape i a follow: Section i Sytem Deciption and the popoed cheme, Simulation Reult and Dicuion ae given in Section 3, and finally in 4, the pape i ummaized. The deivation of the individual SER fo QPSK in Gay mapping i peented in the Appendix.. Sytem Deciption With efeence to Figue, we fit conide a claical elay model in which only one elay (R) ait the ouce y Souce (S) DEF Relay (R) Figue. Block diagam of the coopeative elay ytem with a DEF potocol. MOD denote the modulation of the eceived ignal at the elay. (S) to communicate with the detination (D). We aume each node ha only one antenna and i not equipped with cyclic edundancy-check (CRC) code. In thi pape, we conide a time diviion multiple acce (TDMA) mode whee data tanmiion i plit into two phae, that of the ouce node and that of the elay node. It i aumed that all the eceiving node have pefect CSI. Futhemoe, the detination alo equie CSI of all thee link fo detection... Sytem Model MOD At timelot, the ouce boadcat it infomation, x to the detination and the elay with the aveage powe E. Due to the boadcat natue of wiele channel, both the detination and the elay eceive a noiy obevation of x denoted by y d and y epectively in the following elation yd hdx nd () y h x n whee the ubcipt indicate the node elation uch that h and d h ae independent complex-valued channel gain fo the S-D and S-R link epectively. Fo implicity, all channel ae Quai-tatic Rayleigh fading h, h ~,, channel i.e., d ~ d and whee, denote a complex Gauian andom vaiable with mean and vaiance ; n and d n ae modeled a independent additive white Gauian noie with zeo mean and equal vaiance N at the detination and elay epectively. We aume that the aveage SNR fo all link ae the ame denoted a E N, while the intantaneou SNR i epeented a d d and epectively. The elay pefom a ha-deciion detection (DEF) and e-modulate (MOD) the detected ignal a x with the ame aveage powe E fo e-tanmiion in timelot. The ymbol eceived at the detination i given a y x y x d Sink Detination (D) y h x n () Copyight SciRe.
3 9 whee h and. The ~, n ~, N intantaneou SNR i. At the detination, the ignal fom the ouce and the elay node ae combined to ecove the oiginal ouce data... Poceing at the Detination... Baeline Scheme Many coopeative elay cheme ue MRC to exploit patial diveity gain. It i one of the implet and pactical appoache when S-R link i eliable. Howeve, thi conventional MRC at the detination cannot guaantee full diveity in coopeative cheme. The detination node equie pefect (intantaneou) CSI of S-R link with effective combining. In [6], the autho have popoed an impoved veion of MRC temed a C-MRC. The combined ignal at the detination node i given by * min * ycmc hd yd h y (3) whee min min,, and ae intantaneou SNR of the S-R and R-D channel epectively. The uual intuitive meaning aociated with (3) i that when i high, the detecto place full confidence to the aiving ignal fom the elay. In cae of low, the confidence i weighted accoing to the atio of both hop, that i S-R-D link.... Popoed ML-baed Combining Stategy In thi ection, we fit deive the popoed ML combining technique in cae of one elay. In the ubequent ection, we genealize it to multiple elay. The main pupoe of thi algoithm i to optimally combine the noiy ignal eceived at the detination node, y d and y. It hould be noted hee that the maximum likelihood detecto at the detination hould alo conide the effect of detection eo at the output of the elay. Such eo ae mainly due to fading event in the S-R link. When thi link i affected by a deep fade, the detection eo committed at the elay ae popagated to the detination. To mitigate thee eo which ae oiginated fom both ouce-elay and elay-detination link, an end-to-end ML-baed detecto hould be employed. Thu, the baic aumption fo thi tategy i that the detination node make a coheent detection fo the ignal fom the ouce and the elay node equiing all thee channel to be known at the detination node, i.e., hd, h, h and the noie vaiance ae available at the detination (only the eceived ymbol y i not known). Thi i a tanda aumption baed on the fact that thee paamete have been etimated a a pioi. Fo a fai compaion, A. A. AZIZ ET AL. we maintain the ame aumption applicable fo the baeline cheme C-MRC. The popoed ML deciion ule at the detination i detemined by taking all the poible ymbol detection cenaio both at the detination and the elay. By applying Baye ule, the deciion citeion can be hown a xˆ ag max P x, x y y, y y d d x d d, Py y, y y d d P x P y y x P y y x ag max x d d P x P y y x Py y, x x Py y, x x x ag max P y y, y y x d d ag max Py y x Py y x d d, x Py y, x, x x x (4) Fom (4), by conideing the potential eo at the elay node, the deciion citeion can be expanded a xˆ ag max P y y x d d x P x x P y y x x P x x P y y x x whee we aume P x M e.g., M = 4 fo QPSK; denote the finite et of the contellation and y i the obevation pace fo the epective eceived ymbol. Fit, we aume that the tanmit ignal ae modulated by QPSK and late, we conide a pecial cae of BPSK. Thoughout the equel, we ue capital P a the pobability function. Fom (5), afte ome implification, the detecto at the detination will find x ˆ, an etimate of x by uing the following citeion xˆ ag max p y x d d x x P x x p y x (5) Px x p y x x (6) whee d d p y x i the pobability denity function (PDF) of the ouce ignal y d conditioned upon the Copyight SciRe.
4 tanmitted data p y x x i the PDF of the elayed ignal y conditioned on the equality of both tanmitted ymbol x x.. Howeve the backeted tem in (6) ha to conide the eo pobability of the eceived ignal y at the elay accounting the SER of each ignal point. Fo QPSK cae, the olution i not taightfowa due to diffeent Euclidean ditance among ymbol. In QPSK, a ymbol with two bit, x b, b take fom the x and 3, 4 contellation et,,. Auming the Gay mapping, we epeent the complex ymbol of a,,,,3,,4,. The detection at the detination i pefomed jointly by the ML citeion and we can expand (5) a p y x x j p y x x j ˆ x ag max pd yd x p y x x x j 3 p y x x A. A. AZIZ ET AL. 9 (7) whee, and 3 denote the ymbol eo pobabilitie fom 3, and 4 epectively; and 3 ae analytically expeed a the Gauian Q() function whee Qx: exp t dt. In the nd tem of x (6), we include the multiplicative eo tem in exponential function, with the following equality x x whee j j,,, denoting the phae change that depend on the ymbol tanmitted fom the elay. Thi mean that (7) take into account that the elay doe not opeate eo-fee. In thi pape, we deive cloed fom expeion fo the pobability of eo fo each contellation ymbol fo QPSK a j Px ( x ) E E efc Q 4 N N j j Px ( x ) 3 Px x E E Q Q N N d y (8) (9) whee efc i the complementay eo function. When y and ae eceived at the detination, by ineting,, 3 o 4 to x and examining how lage the agument value fom the augment (7), we can detemine the tanmit ignal point x fom the finite et in QPSK contellation. The PDF fo each tem coeponding to the following expeion: y h x j p y x x exp N N () j whee,,,.detail deivation of (8) and (9) ae peented in the Appendix. It i clea fom (8) and (9) that each QPSK ymbol take on a diffeent ymbol eo pobability. The analytical eult peented thu fa in peviou wok have been deived fom tudie which examined the SER poblem auming that the ymbol eo pobability of each QPSK ymbol i equally likely. Thu, thee eult cannot be teated a offeing a complete ML olution. Note that anothe advantage in the popoed ML ove C-MRC i it flexibility of combining diffeent modulated ignal fom diffeent node ince each link can be teated independently (ymbol-wie detection). In C-MRC, if the potential eo cannot be accuately modeled, maximizing the SNR would not eult in the impovement of the cheme. Fo completene, we alo examine the cheme in the abence of the CSI of S-R link at the detination node. The detination ignoe the eo poibility at the elay i.e., 3 in (7). Hence, we obtain xˆ ag max p y x p y x x d d x () One can gain inight about () that it i identical with the conventional MRC. Thi mean the combine at the detination doe not explicitly take into account the uncetainty of the elay deciion when S-R link i eoneou. Next, we peent a pecial cae when BPSK i ued, intead. Fo BPSK, the elay deciion poce can be imply modeled a a binay ymmetic channel (BSC) with the pobability of a binay deciion eo at the elay P x x. Thi i becaue the S-R link i modeled a a Quai-tatic Rayleigh fading channel and each channel i equivalent to a pecific AWGN channel having a complex gain of h. When the elay node R eceive the ignal fom the ouce node, it doe a ha deciion x fo y and fowa it to the detination node. The elay deciion poce can be chaacteized by a andom vaiable defined a : x x : x x () Copyight SciRe.
5 9 A. A. AZIZ ET AL. i the pob- with E b Px x whee b ability of bit eo at the elay. Theefoe, p y x can be hown a p y x Px x p y x x P x x p y x x P p y x x P p y x x whee the tanition PDF of p y x x i epeented by y p y x x exp N (3) h x N (4) In (4), x i the complement of x,. Ineting the conditional PDF into (3), the tanition PDF of thi elayed path fo BPSK cae can be hown a y exp N x p y N y exp h x h x N N (5) In bief, the popoed cheme featue: ) A imple detecto which i capable of mitigating the potential eo at the noiy elay. ) A new eo model which eve a the ide infomation fo the detection at the detination by conideing individual SER of the ignal point in QPSK. Since (7) conide individual SER in it deciion, ou famewok alo uit well fo othe type of modulation Nonethele, ince having complete CSI fo S-R link can be eouce-exhautive, we popoe anothe cheme a in (6) fo QPSK conideing only aveage SER fo each modulation..3. Multiple-elay Scheme (MRS) In thi ection, we genealize (7) to multi elay cheme (MRS) with L elay node denoted a Rl, l,, L. The ytem model fo MRS i hown in Figue whee in the t timelot, the ouce boadcat it infomation to the detination (dahed line) and the elay node. In the following l timelot, elay R l e-tanmit the eceived ignal to the detination in TDMA othogonal channel whee the coeponding eceived ymbol at the detination i y, l h, l x, l n, l. Note that the total timelot ae L. The othogonality i obeved due to the containt in the pactical adio tanceive which cannot tanmit and eceive at the ame time. Finally, at S R R l R L Figue. Multiple elay cheme in paallel channel etup. the detination, all ignal tanmitted fom the elay ae combined by the popoed ML combining technique. All elay ae conideed to employ unifom elay function e.g. DEF. Fom (5), we let the backeted tem denote the deciion citeion fo the eceived ymbol fom a elay a Pxl, xpyl, xl, x pˆ P x x p y x x, l, l, l, l xl, Theefoe, fo MRS, we can ewite (5) compactly a ˆ, d d l x l D R l (6) xˆ ag max p y x p (7) The detination will detect the ymbol fom all elay (plu the diect link) by uing the popoed ML auming that the pefect knowledge of evey S-R link i available at the detination. 3. Simulation Reult and Dicuion We analyze Packet Eo Rate (PER) againt aveage SNR in decibel (db). We fit aume the ouce and elay node tanmit with the ame aveage powe E eulting in the aveage SNR, E N (ymmetical netwok). Fo C-MRC, we alo conide the detination ha a pefect knowledge of S-R link (i.e., intantaneou SNR) and pefect channel etimation i aumed. In thi imulation, we only conide blind coopeative elaying cheme whee elay node alway e-tanmit to the detination whethe the ignal i coectly detected o contain eo. No automatic epeat equet (ARQ) potocol i ued to avoid the eo popagation fom the elay node to the detination. Fit, we check the PER pefomance of the popoed cheme a hown in (7) againt the baeline fo multiple elay node i.e. L, and 3. In Figue 3, a expected, the popoed cheme outpefom C-MRC (3) in all cae with aound. db,.5 db and db gap at PER 3 L Copyight SciRe.
6 A. A. AZIZ ET AL. 93 lowe bound become almot contant if lage netwok i applied (multiple elay). Thi pefomance impovement i achieved by employing only a ha-deciion potocol (DEF) at the elay node. Such a ytem deign i pactical fo wiele netwok which uually cannot compomie on the high enegy conumption and longe time delay at the elay. We alo plot no elay cae uing BPSK modulation fo fai compaion. It i clea that the gain fom the popoed ML cheme i ignificantly lage due to coopeative gain. Thi, we invetigate the popoed ML cheme when the detecto conide the aveage SER of QPSK in the deciion citeion (6) a hown in Figue 5. Inteetingly, Figue 3. PER compaion between the popoed ML cheme and C-MRC (dahed line) uing DEF potocol fo L =, and 3 elay. fo, and 3 elay cae epectively. In paticula, we can alo obeve that all the coopeative cheme achieve full oe diveity a obeved fom the lope of the L cuve i.e., db (diveity oe of L ) but a ignificant deceae in diveity gain fo No CSI cae (). Thi eult demontate that the popoed algoithm ha bette accuacy of ymbol detection due to the ufficient tatitic of the eceived ignal y d and. Fo thi eaon, the conditional pobability y p y x can be computed uing the obevation p y x x. A fo the No CSI cae, the diveity oe of i achieved fo all cae. PER pefomance futhe degade fom L 3, and elay cae. We note that when no CSI of S-R link i available at the detination, the deciion by the detination ae done uing ignificantly eoneou aumption that thee ae no detection eo at the elay node. A a eult, they convey fale eliability meaue to the decode and the pefomance i noticeably affected (moe than 5 db lo at PER 3 compaed to the popoed ML cheme fo all cae). In pactice theefoe, a coopeative cheme with DEF which doe not account fo the S-R CSI may not be vey effective. Second, we imulate the lowe bound coeponding to each cae whee a pefect elay i employed i.e., the ame ignal fom the ouce i tanmitted to the detination (complete MRC). A hown in Figue 4, fo PER = 3 the gap between the lowe bound and the popoed cheme i db fo elay cae and 3 db fo and 3 elay cae each. In pefect elay cae, the elay jut eplicate the ame data a the ouce node; hence the pefomance can be egaed a the lowe bound of the coopeative cheme. We can alo notice that the gap to the Figue 4. PER compaion between the lowe bound (dahed line) and the popoed ML cheme (olid line) in multiple elay cheme. The lowe bound ae imulated with pefect elay. Figue 5. PER pefomance when the popoed ML cheme (olid line) uing only aveage SER, againt the baeline C-MRC (dahed line). Copyight SciRe.
7 94 A. A. AZIZ ET AL. thei pefomance i almot the ame in all SNR egion a C-MRC. By uing aveage SER in the algoithm, the obevation in p y x x i equally weighted fo all QPSK ymbol. A a eult, thi ha caued ome degadation in PER pefomance compaed to the popoed pue ML cae (7) a hown in Figue 3. In fact, the convegence of the two cheme i alo expected due to the appoximation at high SNR of the baeline cheme [6]. Note that keeping pefect CSI of S-R link at the detination can be enegy conuming and involve highe computation i.e., and equied fo pefect CSI, but only i equied fo the aveage cae. Thu, thi popoed ML tategy poe a pactical olution which contitute a good tadeoff between the pefect CSI equiement and eo ate pefomance Finally, in Figue 6, we imulate the popoed cheme and C-MRC when only one elay node i ued but we vay the aveage SNR fo R-D link and we keep the aveage SNR fo S-D link and S-R link the ame, d. We imulate the cheme at thee diffeent cenaio of R-D link quality: + 5 db (+5 db), 5 db ( 5 db) and (equal). In view of thi eult, we infe that the popoed cheme can outpefom C-MRC when R-D link ha ufficiently high SNR quality (+5 db) with db gap at PER = -3 and.5 db gap at PER = - fo low SNR quality ( 5 db). When R-D link ha highe SNR compaed to S-D o S-R link, the combined ignal at the detination i dominated by the eoneou ignal fom the elayed link. Thu, the PER pefomance i degaded futhe compaed to the cae of equal SNR. Thi ignificant pefomance demontated by the popoed ML cheme ende it moe uitable when the elayed link ha poo SNR quality. In C-MRC, one can alo check fom (3) that the uboptimality of C-MRC become inheently evident when the weighted ignal fom the elayed link become lage than that of the diect link. C-MRC effectivene i lagely conditioned on the link quality of R-D link ove S-D link (diect path). Supiingly, it can be eaily een that the popoed ML cheme achieve the advantage of the coopeative gain by uing only imple DEF potocol with inceaing impovement in multiple elay cheme. In paticula, although the eceived ignal at the elay ae noiy and only DEF i ued at the elay, the popoed cheme impove achievable PER pefomance. Futhemoe, the popoed cheme alo even povide bette eo ate pefomance at low and high SNR of R-D link which become an added advantage compaed to C-MRC. 4. Concluion In thi pape, we developed a new ignal combining tategy baed on ML citeion fo coopeative elay cheme which account the potential eo at elay. The eo ae expeed a the Gauian Q-function fo each SER in QPSK ymbol. By applying thee expeion in the detection at the detination, we can accuately model the tanition pobabilitie fo the eoneou tanmiion fom noiy elay. The popoed ML cheme i upeio to the conventional C-MRC in PER pefomance in all cae unde the ame CSI equiement. The popoed ML cheme ha moe flexibility in implementation compaed to C-MRC depending on CSI at the detination. In fact, unlike C-MRC, the popoed cheme emain eilient to popagation eo even if R-D link ha diffeent SNR quality. In addition, with DEF potocol, thi popoed ML cheme ha hifted the poceing complexity to the detination node whilt the elay node can coneve the enegy and imple data poceing. Thu, thi make thi popoed tategy amenable to implementation epecially fo eouce-containt envionment uch a wiele eno netwok. 5. Acknowledgement Thi tudy ha been uppoted by the Scientific Reeach Gant-in-aid of Japan No , the Telecommunication Advancement Foundation (TAF) and the A-STEP by JST (Japan Science and Technology Agency). 6. Refeence Figue 6. PER compaion between the popoed ML cheme (olid line) and C-MRC (dahed line) when the aveage SNR of R-D link, γ vaie at L = elay cae. [] J. Laneman, D. Te and G. Wonell, Coopeative Diveity in Wiele Netwok: Efficient Potocol and Outage Behaviou, IEEE Tanaction on Infomation Theoy, Vol. 5, No., Decembe 4, pp doi:. Copyight SciRe.
8 A. A. AZIZ ET AL. 95 9/TIT [] A. Sendonai, E. Ekip and B. Aazhang, Ue Coopeation Diveity Pat I and Pat II, IEEE Tanaction on Communication, Vol. 5, No., Novembe 3, pp doi:.9/tcomm [3] A. Noatinia and T. E. Hunte, Diveity though Coded Coopeation, IEEE Tanaction on Wiele Communication, Vol. 5, No., Febuay 6, pp doi:. 9/TWC.6.65 [4] D. Chen and J. N. Laneman, Modulation and Demodulation fo Coopeative Diveity in Wiele Sytem, IEEE Tanaction on Wiele Communication, Vol. 5, No. 7, July 6, pp doi:.9/twc [5] A. A. Aziz, Y. Iwanami and E. Okamoto, On the Impovement of Maximum Likelihood in Multiple Relay Sytem, Poceeding of IEEE Wiele Communication & Netwoking Confeence, Sydney, 8- Apil, pp. -6. doi:.9/wcnc [6] T. Wang, A. Cano, G. Giannaki and N. Laneman, High Pefomance Coopeative Demodulation with Decode-and- Fowa Relay, IEEE Tanaction on Wiele Communication, Vol. 55, No. 7, July 7, pp [7] B. Djeumou, S. Laaulce and A. G. Klein Combining Decoded and Fowaed Signal in Gauian Coopeative Channel, Poceeding of IEEE Intenational Sympoium on Signal Poceing and Infomation Technology, Vancouve, 7-3 Augut 6, pp [8] M. R. Souyal and H. You, Diveity Pefomance of a Pactical Non-Coheent Detect-and-Fowa Receive, Po- ceeding of IEEE Global Telecommunication Confeence, New Olean, 3 Novembe-4 Decembe 8, pp. -6. [9] M. Benjillali and L. Szczecinki, A Simple Detect-and- Fowa Scheme in Fading Channel, IEEE Communication Lette, Vol. 3, No. 5, May 9, pp doi:.9/lcomm [] M. N. Khomuji and E. G. Laon, Receive Deign fo Wiele Relay Channel with Regeneative Relay, Poceeding of IEEE Intenational Confeence on Communication, Glagow, 4-8 June 7, pp [] J. P. K. Chu, R. S. Adve and A. W. Eckfo, Uing the Bhattachayya Paamete fo Deign and Analyi of Coopeative Wiele Sytem, IEEE Tanaction on Wiele Communication, Vol. 8, No. 3, Mach 9, pp doi:.9/twc.8.87 [] K. Lee and L. Hanzo, MIMO-Aited Ha veu Soft Decoding-and-Fowaing fo Netwok Coding Aided Relaying Sytem, IEEE Tanaction on Wiele Communication, Vol. 8, No., Januay 9, pp doi:.9/t-wc [3] X. Hu, M. Benjillali and L. Szczecinki. Bit Eo Rate fo Rectangula QAM with Abitay Contellation Mapping in Nakagami-m Channel, Wiley Jounal on Wiele Communication and Mobile Computing, Vol. 8, No., 8, pp doi:./wcm.437 [4] M. S. Alouini and A. J. Goldmith, A Unified Appoach fo Calculating Eo Rate of Linealy Modulated Signal ove Genealized Fading Channel, IEEE Tanaction on Communication, Vol. 47, No. 9, Septembe 999, pp doi:.9/ Copyight SciRe.
9 96 Appendix A. A. AZIZ ET AL. integal a follow Deivation of Symbol Eo Rate (SER) of QPSK Signal in Gay Mapping Employing the two-dimenional (-D) Gauian Q- function epeentation, we peent cloed-fom expeion fo the individual SER of each QPSK ignal. Figue 7 illutate the ignal point fo QPSK when Gay mapping i ued. Let u denote I and Q a the in-phae and quadatue component epectively. Since each complex ymbol of QPSK coepond to two binay bit, a peented in Figue 7 we aign the epective ymbol (,), (,), 3 (,),4 (,) accoingly.. Deivation of ε In thi ub-ection, we deive the ymbol eo pobability of 3. Figue 7 depict the diffeence of Euclidean ditance between and othe ymbol. Hee we aume 3. Fit, it i convenient to define eveal aumption ued in the analyi. Conide the i-th eceived ignal vecto i ( Xi, Yi), i {,,3,4} of QPSK tanmitted ove an AWGN channel with the channel gain h. Hence, ( X i, Y i) in i i given by the following in-phae and quadatue component X, Y co t n, in t n (8) i i X Y whee n and n ae jointly Gauian with zeo mean X and equal vaiance Y N uch that the expected value i X Y E n E n E N (9) The ymbol eo pobability when i ent and 3 i detected can be hown by the -D Gauian pobability X Y exp N N dxdy () () can be analytically calculated and we can e-wite it a efc 4 N Q E N E whee efc() i the complementay eo function.. Deivation of ε () In thi ub-ection, we deive the tanition pobability of a ignal point that fall into an adjacent quadant e.g., Similaly, fom () X Y exp N N dxdy () Then, () can be analytically calculated and we obtain E E efc efc N 4 N E E Q Q N N (3) Poof: Deivation of uing, and 3 which i equivalent to the tanda cloed-fom expeion of the aveage pobability of eo fo QPSK in AWGN channel [4]. In thi ub-ection, we povide an altenative deivation fo the aveage SER of QPSK,. We pove that the um of all individual SER of QPSK ymbol amount to the aveage SER of QPSK. Since 3, Figue 7. Signal QPSK ymbol and ymbol eo pobabilitie. E E 3 efc efc N 4 N (4) Copyight SciRe.
10 A. A. AZIZ ET AL. 97 Theefoe, the aveage SER fo QPSK i imply 3 E E efc efc N 4 N E E Q Q N N (5) Fo compaion, we alo invetigate the effect of having only aveage SER in the popoed ML algoithm. Fom (7), by etting 3, we have the following citeion p y x x 3 j p y x x j xˆ ag max pd yd x p y x x x j p y x x (6) (6) can be viewed a uboptimal ince it only conide aveage SER fo all ymbol in QPSK modulation. Copyight SciRe.
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