Constellation Mapping for Physical-Layer Network Coding with M-QAM Modulation

Constellation Mapping for Phys ical-La yer Network Coding with M-QAM Modulation Shiqiang W ang ∗ † , Qingyang Song ∗ , Lei Guo ∗ and Abbas Jamalipour ‡ ∗ School of Informatio n Science and E ngineerin g, Northeastern Un iv ersity , Shenyang 11 0819 , P . R. China † Departmen t of Electrical an d Electronic Engineerin g, Imperial College Lond on, SW7 2AZ, United Kingdom ‡ School of Electrical an d I nform ation Enginee ring, University of Sy dney , NSW , 2 006, Australia Email: sh iqiang.wang1 1@imperial.ac.u k, songqing yang@ise.neu .edu.cn, guolei@ise.neu. edu.cn, a.jamalipo ur@ieee.org Abstract —The denoise-and-forward (DNF) method of physical- layer network codin g (PNC) is a promising approach for wireless relaying networks. In this paper , we consid er DNF-based PNC with M -ary quadrature amplitude modulation ( M -QAM) and propose a mapping scheme that maps the sup erposed M -QAM signal to coded symbols. The mapping scheme supp orts both square and n on-square M -QAM modulations, with various orig- inal constellation mappings (e.g. binary-coded or Gray-coded). Subsequen tly , we ev alu ate the symbol error rate and bit error rate (BER) of M -QAM modulated PNC that u ses the proposed mapping scheme. Afterwa rds, as an application, a rate adaptation scheme for the DNF method of PNC i s proposed. Si mulation results show that th e rate-adaptive PNC i s advantageous in various scenarios. 1 Index T erms —Constellation mappin g, denoise-and-forward (DNF), physical-layer network coding (PNC), quadrature am- plitude modulation (QAM), rate adaptation, wireless networks. I . I N T RO D U C T I O N Physical-layer n etwork cod ing (PNC) [1]–[ 3] has emerged as a new co ding p aradigm th at can significan tly impr ove the throug hput perfo rmance of wireless relaying networks. T he basic id ea is to allow node s to tra nsmit simultaneo usly to th e relay . After receiving a super posed signal, the relay performs a mapping op eration to the superposed signal, and subsequ ently forwards the resulting sign al to the destination n odes. The denoise-an d-for ward ( DNF) metho d of PNC h as b een shown to outp erform th e amplify-an d-for ward (AF) meth od, b ecause it a voids no ise amp lification [4]. When u sing DNF with binary ph ase-shift keying (BPSK) or quadr ature phase-shift keying (QPSK) modu lations, the superpo sed sig nal can be mapped to the co ded signal with the XOR operation [2 ]. Howev er , the XOR m apping m ethod may not be suitable for hig her level modulation s. For instance, for the 4 -ary pulse amp litude mod ulation (4-P AM), th ere exists ambiguity between the “10” and “00” bits wh en using XOR mapping , as shown in Fig. 1. Th is is also the case for M - ary quadratu re amplitu de m odulation ( M -QAM) with M > 4 , 1 c  20 12 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in an y current or future media, includi ng reprinting/ republishi ng this m ateria l for advert ising or promotional purposes, creating ne w colle cti ve works, for resale or redistribut ion to serve rs or lists, or reuse of any copy righted component of this work in other works. 00 01 10 11 00 01 10/ 00 11 01 00 10/ 00 Ambiguity Original bits: Mapped bits: Original constellations: Superposed constellations: Fig. 1. Constell ations and the corresponding bits with XOR mapping for 4-P AM modulati on. which is an extension to M -P AM. Ther efore, we hav e to find alternative map ping m ethods f or high-level mod ulations. Because M -QAM is widely applied to rate-ad aptive co m- munication systems, it is significant th at PNC su pports g eneral M -QAM mod ulations. A clustering-based m apping metho d that suppo rts general M - QAM modulation s f or asynchro nous DNF was pro posed in [5], and a non- unifor m constellation design for M -P AM to r esolve the a mbiguity p roblem was propo sed in [6]. Howe ver , the co mplexity of those methods are relativ ely high. I n [2], a mappin g schem e was p roposed for synchro nous D NF with M -QAM m odulation . Th e scheme assumes that signal additions dir ectly tran slate to sym bol- wise modu lar L ( where L = √ M ) addition s, which ma y not b e suitable fo r Gray- coded sym bols [7] or n on-squ are constellations. Some o ther existing work f ocus o n mo re theo retical as- pects on PNC with h igh-level mo dulation and rate ad aptation. Ref. [ 8] pr oposed a precod ing technique to improve the perfor mance o f PNC in r ate-adaptive systems. A hier archical modulatio n scheme to reso lve the issue of asymmetric node positioning was prop osed in [9]. Also, the in tegration of lattice codes with PNC was discussed in [10], which , howe ver, is difficult to implement in p ractice [6]. In this pape r , we fo cus on a practical solu tion to syn- chrono us PNC with M -QAM mod ulation. W e p ropose a simple but ef fective c onstellation mapping metho d, which supports both square and n on-squa re constellation s, as well as Gray -coded symbols (th at ar e wid ely used in pr actical commun ication sy stems). Follo win g that, we ap ply th e pro- posed mapping scheme to a ra te adapti ve system , which selects the a pprop riate mo dulation level based on the channel R S 1 S 2 D 1 D 2 x 1 x 1 x 2 x 2 x 2 x 1 x c x c 1 st phase (intended signal) 1 st phase (interference signal) 2 nd phase Fig. 2. Netw ork topology . status. Regardin g the sync hronizatio n issues, note that we have a separ ate paper [ 11] which foc uses on phase- le vel synchro nization for PNC. The re mainder of this pap er is organized as follows. Sec- tion II discusses the conditio ns of uniq ue d ecodability and introdu ces the mapping s cheme. Section III ev aluates the perfor mance o f PNC with M -QAM modula tion using the propo sed map ping schem e. The rate adap tation sch eme is de- scribed and analyze d via simulations in Section IV . Section V draws con clusions. I I . C O N S T E L L AT I O N M A P P I N G In this sectio n, we discuss the condition s of un ique dec od- ability for M -P AM and M -QAM modulatio ns and pr opose the cor respond ing mapping schem es b ased on constellation analysis. A. System Mod el W e co nsider a network topo logy as shown in Fig. 2, in which R denotes the re lay , S 1 and S 2 denote the sou rce nod es, D 1 and D 2 denote the co rrespon ding destination nodes, x 1 and x 2 respectively denote th e signals transmitted by the sour ce nodes S 1 and S 2 , and x c denotes the signal tran smitted by the relay R . The packet exchange p rocess using PNC consists of two commun ication phases. In the first com munication phase, the source nodes s imultaneou sly tr ansmit their p ackets t o the relay . After th e r elay r eceiv es the superp osed signal, it m aps the su- perposed signal to a new signal, which carries a co ded version of th e p ackets. Then , in the second comm unication pha se, the relay send s the cod ed packet to the cor respond ing destination nodes. From the cod ed p acket, th e destination can decod e the packet it expects to receiv e, which is possible as lo ng as the oppor tunistic listener D 1 (or D 2 ) has overheard the pac ket sent by the nea rby sou rce n ode S 2 (or S 1 , correspon dingly) . Let s 1 and s 2 respectively d enote th e two ind ependen t symbols that ar e carr ied by the signals x 1 and x 2 , the mapping function s c = C ( s 1 , s 2 ) maps th e super posed signal, which carries the symbo ls s 1 and s 2 , to a cod ed symbol s c , which is then br oadcasted with the signal x c . Note tha t, a t the relay , only the super posed sign al can be extracted. T he relay is not aware of the individual values of s 1 and s 2 . Howe ver, we use s 1 and s 2 as variables of the mapp ing fun ction to simplify o ur subsequen t analysis. In this paper, we consider the case wh ere the two signals that are sup erposed at the relay have synchro nous phase and equal power . Henc e, igno ring the cha nnel attenu ation and th e noise, the superp osed signal is x 1 + x 2 . Phase synchr onization can be acco mplished with the method described in [ 11]. The equal power chara cteristic can b e achiev ed b y reducing the transmission power of the strong transmitter . As we will see in Section I V , this power reduc tion does not br ing severe perfor mance degradation for rate-ada ptiv e systems, wh ile it makes the map ping scheme design feasible. B. Map ping S cheme for M -P AM Mod ulation W e first discuss the mapping schem e for M -P AM modu la- tion. The mapping schem e h as to b e d esigned to ensure that the packets can b e successfully decoded at the destinations. Based on the Exclusive L aw [5], th e necessary and sufficient condition for u nique decod ability at the destinatio ns is as follows: 1) For any s ′ 1 6 = s 1 , we have C ( s ′ 1 , s 2 ) 6 = C ( s 1 , s 2 ) . 2) For any s ′ 2 6 = s 2 , we have C ( s 1 , s ′ 2 ) 6 = C ( s 1 , s 2 ) . Thus, we hav e the following pr oposition. Pr opo sition 1: For M -P AM mo dulation, the n ecessary and sufficient cond ition for u nique deco dability is that any M consecutive constellatio n points in the c onstellation of the superpo sed signal are m apped to different symbols. Pr oof: Th e co nstellation o f the superp osed signal is a superpo sition of the constellation s of the original sign als. For two ind ependen t symbols s 1 and s 2 , giv en s 1 , the superposed signal x 1 + x 2 has M possible values. Th ese M possible values correspo nd to M c onsecutive constellation po ints in the supe rposed constellation. T he case is similar w hen s 2 is giv en. Hence, the statement that any M consecu tiv e constel- lation po ints a re mapp ed to different sym bols is equiv alen t to condition s 1) and 2) in the E xclusive Law . According to the Exclusive Law , the ne cessity and su fficiency are proved. Note that the co nstellation size of the superposed signal is 2 M − 1 . L et s o,i ( i = 0 , 1 , . . . , M − 1) denote th e orig inal symbol correspo nding to the i th constellation point in th e original con stellation, and s c,j ( j = 0 , 1 , . . . , 2( M − 1)) denote th e co ded symbo l that the j th c onstellation p oint in the super posed constellation is m apped to. Accord ing to Proposition 1, the value of the cod ed symbols can be evaluated by s c,j = s o,j m o d M . (1) Eq. (1) only shows o ne possible m ethod of generating the cod ed symbols. In principle, any genera tion method that satisfies Prop osition 1 can be applied . By g enerating the co ded symbols accor ding to (1), we essentially m ap the superpo sed signal to sy mbols from the o riginal sym bol set. Suppose the k nown symbol is s 1 = s o,i and th e r eceiv ed coded s ymbol is s c = s o,j m o d M . F o r any i , i ′ ∈ 0 , 1 , . . . , M − 1 , i 6 = i ′ , we have s o,i 6 = s o,i ′ . Henc e, whe n s 1 is k nown, we can obtain the value of i . Sim ilarly , at the d estination, 00 01 11 10 00 01 11 10 01 11 00 Original bits: Coded bits: Original constellations: Superposed constellations: Known symbol Possible constellations 00 01 11 10 Original bits: Original constellations: Decoded symbols s o, 0 s o, 1 s o, 2 s o, 3 s o, 0 s o, 1 s o, 2 s o, 3 s c, 0 s c, 1 s c, 2 s c, 3 s c, 5 s c, 6 s c, 4 Fig. 3. Constell ations, corresponding bits, and the decoding process of Gray- coded 4-P AM modulation with the proposed mapping method. Assume the kno w n symbol is s o, 2 , then the size of the possible superposed constella tion is reduced to 4, and there is a one-to-one m apping between the coded symbol and the symbols that the destin ation expe cts to recei ve. the value of j mod M can be obtaine d fro m the received coded sym bol s c . T he values of i an d j can be regarded as scalars of the kn own signal x 1 and the super posed sig nal x 1 + x 2 , respectively . Th erefore, the scalar of the signal x 2 is j − i . When the cod ed symbols are gener ated ac cording to ( 1), th e sup erposed sign al is mapped to sy mbols fr om the original sym bol set b y ap plying a mo d M oper ation to its scalar . Hence, the symbo l s 2 , which the destination intends to receive, can be ev aluated with a similar appro ach: s 2 = s o, ( j − i ) m o d M = s o, ( j mo d M − i ) mo d M (2) Eq. (2) is ap plicable because the destination kn ows j mo d M and i . Fig . 3 shows an examp le of the mapp ing and decodin g p rocess, when using Gray- coded 4-P AM mod ulation. As shown in Fig. 3, Pro position 1 ensures that, given a known symbol, th ere exists a one-to -one mappin g between th e c oded bits and the expected bits. It can a lso be verified that the coded bits in Fig. 3 are genera ted according to (1) and th e expected bits can b e decoded by applying (2). C. Map ping Scheme for M -QAM Mo dulation M -QAM modulatio n is a two-dimension al extension to M - P AM modulatio n. Hence, its conditio ns f or uniqu e deco d- ability can be easily ob tained from th e Exclusive L aw and Proposition 1. Cor ollary 1: For M -QAM modula tion with square constel- lations (i.e. the co nstellation size is M = L × L , with L being the n umber of co nstellation po ints on each of th e in-p hase and quadrature axes), the necessary and sufficient cond ition for unique decod ability is that the constellation points in any L by L sq uare in the co nstellation o f the sup erposed sign al are m apped to different symbols. Pr oof: Accord ing to Pro position 1, when only con sid- ering the in-p hase componen t or the q uadratur e com ponen t of the superposed signal, the unique deco dability condition is equiv alent to: a ny L consecutive constellation poin ts ( in terms of either the in- phase branch or the quad rature b ranch) are mapped to d ifferent symbols. It follows that, when jo intly con - sidering the in-ph ase and qu adrature co mponen ts, the unique 1 j s o ,0 s o ,1 s o ,2 s o ,3 s o ,4 s o ,5 s o ,6 s o ,7 1 j s c ,0 s c ,1 s c ,2 s c ,5 s c ,6 s c ,7 s c ,10 s c ,11 s c ,12 s c ,3 s c ,4 s c ,8 s c ,9 s c ,13 s c ,14 s c ,15 s c ,16 s c ,17 s c ,20 s c ,21 s c ,22 s c ,19 s c ,23 s c ,24 Known symbol s c ,18 Possible constellations (a) (b) Fig. 4. Constel lation s of 8-QAM: (a) original signal, (b) superposed signal. Assume the known symbol is s o, 7 , the possible superposed constel latio n points are those shown in the gray re gion. Because the superposed con- stella tion point s in any 3 by 3 square are m apped to dif ferent sym bols, the destina tion can decode the expected symbol based on the code d symbol. decodab ility condition is equ i valent to: the co nstellation points in any L by L square are map ped to d ifferent symbols. The gen eration method of coded symbols an d the decod- ing pr ocess are similar w ith (1) and (2), except that two indexes (one fo r th e in-p hase com ponen t an d one for th e quadra ture compon ent) sh ould be u sed. L et s o,k,k ′ ( k , k ′ = 0 , 1 , . . . , L − 1 ) den ote the origin al symbol co rrespon ding to the ( k , k ′ ) th constellatio n point in the original constellation , and s c,l,l ′ ( l, l ′ = 0 , 1 , . . . , 2( L − 1)) denote th e cod ed symbo l correspo nding to the ( l , l ′ ) th constellation point in the sup er- posed constellation . A possible r elationship between the coded and original sym bols is s c,l,l ′ = s o,l m o d L ,l ′ mo d L . (3) Suppose the known symbol is s 1 = s o,k,k ′ and th e rece i ved coded sym bol is s c = s o,l mo d L,l ′ mo d L , then the expected symbol s 2 can be dec oded b y s 2 = s o, ( l − k ) m od L , ( l ′ − k ′ ) mo d L = s o, ( l mo d L − k ) mo d L, ( l ′ mo d L − k ′ ) mo d L . (4) Cor ollary 2: For M -QAM modu lation with no n-squar e constellations (b ut the vertical an d horizontal spacing s between constellation po ints still need to be equal o r mu ltiples of th e minimum spacing ), suppose the ma ximum numb er of constel- lation points seen on e ither the in-phase ax is or the q uadratu re axis is L ′ , a sufficient conditio n for u nique decod ability is that the constellation points in any L ′ by L ′ square in th e constellation of th e superp osed signal are mapp ed to different symbols. Pr oof: For two sym bols s 1 and s 2 , giv en s 1 (or s 2 ), the possible constellation points of the superp osed signal x 1 + x 2 do not exceed a n L ′ by L ′ square. Hence, whe n mappin g the co nstellation points in any L ′ by L ′ square to d ifferent symbols, we h av e un ique dec odability . Note that Coro llary 2 only provides a sufficient con dition, because the number of the possible sup erposed constellation points is smaller than L ′ × L ′ , when either s 1 or s 2 is g iv en. This fact implies that there may exist map ping schemes that ensure unique de codability , in which some o f th e c onstellation points in an L ′ by L ′ square are mapped to the same symbo l. Howe ver, Cor ollary 2 provides a simple way o f designing the mapping scheme fo r non-squ are M -QAM mo dulation , such as 8-QAM ( as shown in Fig . 4), 32-QAM etc. For n on-squ are M -QAM, the cod ed sym bol set is generally larger than th e or iginal symbol set. For instance, with 8- QAM, the cod ed sym bol set h as 3 × 3 = 9 symb ols; and with 32 - QAM, the coded symbo l s et has 6 × 6 = 36 symbo ls. It follows that more bits ar e n ormally neede d to tran smit the coded symbol. Howev er , considerin g th at the th eoretical thro ughpu t gain of PNC over conventional network co ding (CNC) is 1.5 [2], PNC is th eoretically still b eneficial as long as the number of the additional bits is not larger than 50 % of the numb er of bits used to tr ansmit the o riginal symbo l. I t should also be noted that the ad ditional bits ne ed to be rearran ged to match with the size of th e o riginal con stellation. The cod ed symbols fo r non -square M -QAM can be g ener- ated by extendin g the or iginal constellation to a squar e con- stellation with L ′ × L ′ symbols. Then, the extended superposed constellation h as (2 L ′ − 1) × (2 L ′ − 1) symbols. Based on the extended constellation s, the generation of cod ed symbo ls and the decoding process for non-squar e M - QAM is the same with square M - QAM. The extended con stellations are only u sed for calcula tion pur poses, and th e original co nstellations are still used for tran smission. Consequently , some points in the extended constellation s m ay not exist in actu al transmissions. D. Complexity of Gen erating Coded Symbols For M -P AM o r M -QAM, (1) or (3), respectiv ely , must be ev aluated for every superpo sed constellation po int, to obtain all the relatio nships between the supe rposed constellation points and th e cod ed sym bols. The size of the superpo sed constellation is 2 M − 1 for M -P AM and (2 L − 1) 2 = 4 M − 4 M 1 / 2 + 1 for squa re M -QAM. For non- square M - QAM, note that the maximum possible value of L ′ does not exceed M , hen ce the la rgest p ossible superposed constellation size is (2 M − 1) 2 = 4 M 2 − 4 M + 1 . Theref ore, the com plexity of the proposed co ded symbol g eneration process fo r M -P AM or squ are M - QAM is O ( M ) ; and fo r no n-squar e M -QAM , it is O ( M 2 ) . This complexity is lower than that of t he clustering- based method [5] (which is O ( M 4 ) due to its necessity o f calculating and comp aring the distances between constellation points) and th e non -unifo rm constellation d esign method [6 ] (which is O ( L ! × L !) becau se it ne eds to en umerate all the possible mappings). I I I . P E R F O R M A N C E W I T H M - Q A M M O D U L AT I O N In th is section, we evaluate the symbo l error r ate (SER) and bit er ror rate ( BER) perfo rmance of PNC with M - QAM modulatio n using the pr oposed m apping scheme, at d iffer - ent commun ication phases. Additive white Gaussian noise (A WGN) ch annel is assum ed. T he n etwork top ology und er consideratio n is shown in Fig. 2 . W e assum e that the receiver decodes the signals ba sed on the minimu m distance d ecision rule [12]. A. Sec ond Commu nication Ph ase W e start our analysis with th e second comm unication phase, where the symbo ls are sent w ith general M -QAM modula- tions. W e summar ize related th eories regar ding M -QAM in this subsection, to sup port o ur further d iscussions. Let the distance between neighb oring constellation p oints at the receiver be 2 d ; as shown in [ 7], th e SER is constrain ed by p s ≤ 4 Q   s 2 d 2 N 0   , (5) where Q ( · ) is the tail pro bability of the standard nor mal distribution and N 0 denotes the power spectr al d ensity of noise. For M -QAM with squar e constellations, acco rding to [7], we can evaluate the exact SER by p ′ s = 1 −   1 − 2( L − 1 ) L Q   s 2 d 2 N 0     2 , (6) where L = √ M . Assume that the baseband sign al is shaped with a raised cosine pu lse with roll-off factor β = 1 , we can also write the SER with r espect to th e av erage signa l-to-noise ratio (SNR) γ at the re ceiv er [7]: p ′ s = 1 − 1 − 2( L − 1 ) L Q r 3 γ M − 1 !! 2 . (7) Because when an erroneo us symbo l is rece i ved, at least on e bit is wro ng and at most all th e bits th at the sy mbol carries are wr ong, the BER p b is constrained by p s log 2 M ≤ p b ≤ p s . (8) B. Rela y in the F irst Communica tion Phase In the first co mmunica tion ph ase, th e re lay R recei ves a superp osed signal. For synchr onous PNC with ide ntical power , the superpo sed signal is still a n M - QAM sign al; an d accordin g to the discussions in Section I I, adjacen t superpo sed constellation p oints are map ped to different (coded ) symbols. Hence, the SER is still con strained by (5). For squ are M - QAM, the num ber of superp osed con stel- lation points in either the in-ph ase bran ch or the qua drature branch is 2 L − 1 . Ignoring the fact that ther e is repe tition of coded sym bols (i.e. different su perposed c onstellation po ints can be map ped to the same symb ol), the SER o f the super- posed signal c an b e obtained by su bstituting 2 L − 1 for L in (7), y ielding p ′ s - sup ≈ 1 − 1 − 4( L − 1 ) 2 L − 1 Q r 3 γ M − 1 !! 2 . (9) The BER con straint has th e same f orm as ( 8). C. Op portunistic Listeners in the Fir st Communica tion Phase In the first communication p hase, the d estination D 1 (or D 2 ) overhears the symbols sent by its nearby sou rce node S 2 (or, correspo ndingly , S 1 ). During this proc ess, the sign al sent by S 1 (or, co rrespon dingly , S 2 ) c an be regarded as interfere nce. As a result, the distance between neigh boring con stellation points c an b e reduced . Let the n eighbo ring distance in the constellation o f the r eceived inte rference signal be 2 d ′ , the maximum distance reduction is 2( L − 1) d ′ , because the inter- ference signal is in the s ame modulatio n as the intend ed s ignal. The maximum value is achieved wh en the two signals have a phase difference of nπ / 2 (where n is an arbitrary integer) and the interfere nce sign al c arries a symb ol wh ich lies on the outer bound ary of the c onstellation. According to th e aforem entioned discussions, we ca n obtain an u pper boun d of the SER: p s - opp ≤ 4 Q   s 2(max { 0 , d − ( L − 1) d ′ } ) 2 N 0   = 4 Q  max { 0 , d − ( L − 1) d ′ } r 2 N 0  = 4 Q   max    0 , s 2 d 2 N 0 − ( L − 1) s 2 d ′ 2 N 0      . (10 ) For squar e M -Q AM mo dulations, let f ( α ) = ma x    0 , r 3 γ M − 1 − α ( L − 1 ) s 3 γ ′ M − 1    , (11 ) and g ( α ) = 1 −  1 − 2( L − 1 ) L Q ( f ( α ))  2 , (12) where γ ′ is th e average SNR of the interferenc e signal at the receiver . Then, th e SER f or square M -QAM is bou nded by g (0) ≤ p ′ s - opp ≤ g (1) . (13) The lower b ound in (13) ha s the same form with (7). It correspo nds to the ca se where n o in terferen ce exists. Considering that the sym bols from the sou rce no des are equally probable, we can app roximate the SER by taking an av erage value of the in terference term in (11), yielding p ′ s - opp ≈ g (1 / 2) . (14) At r elativ ely hig h SNRs, it is mo re likely that only o ne bit error exists in an erron eous sym bol. Hence, the BER fo r square M -QAM can b e app roximated b y p ′ b - opp ≈ g (1 / 2 ) log 2 M . (15) The BER is fu rther evaluated with simula tions. W e conside r Gray-cod ed square M -QAM modulation s with different val- ues of M , and each sour ce no de sends 10,00 0 independ ent symbols. It is assumed th at th e phase difference b etween 0 10 20 3 0 40 10 -4 10 -3 10 -2 10 -1 10 0 QPS K, S NR=9.8 dB T heor etic al appro xim ated v alue T heor etic al l owe r bou nd 0 10 20 3 0 40 10 -4 10 -3 10 -2 10 -1 10 0 16- QA M, S NR=16.5 dB 0 10 20 3 0 40 10 -4 10 -3 10 -2 10 -1 10 0 64- QA M, S NR=22.6 dB 0 10 20 3 0 40 10 -4 10 -3 10 -2 10 -1 10 0 256 -QAM, S NR = 28.5 dB Power rat i o of th e in ten ded signal to th e in terf eren ce sig n al ( d B) BE R Sim ulated T heor etic al upper bound Fig. 5. BER at the opportu nistic listeners in the first communic ation phase vs. power ratio of the intended signal to the interfere nce s ignal, with various modulati ons. the inten ded signal and the inte rference signal is random ly distributed within [0 , 2 π ) , because the two signals are ind epen- dent with each o ther . The SNR of the intend ed sign al ( without interferen ce) is set to a value ensuring that the lower bou nd o f the BER is 10 − 3 . W e evaluate th e BER und er different power ratios of the intended signal to the interferen ce sign al. The results are shown in Fig. 5 . W e ca n ob serve in Fig . 5 that the simulated values appro ach the app roximate values, par ticularly when the p ower r atio of the intended signal to the interference sign al is high. Meanwhile, both of these values ar e constraine d by the u pper and lower bound s. When the SNR is low , the appr oximated values are con stant, particularly when M is large. Th is is because the approx imated SER, which is e valuated from (1 4), has reached its max imum value, causin g the ap proxima ted BER also to r each its max imum value. I V . R AT E - A DA P T I V E P N C By introd ucing M -QAM suppor t for the DNF method of PNC, the data rate can be adjusted . In th is sectio n, we propo se a simp le rate a daptation scheme, in which the data is sent at the m aximum p ossible d ata rate, under a specific channel conditio n an d max imum BER req uirement. The data rate is adjusted by using different mo dulations, includ ing BPSK, QPSK, 1 6-QAM, 32- QAM (no n-squar e), 64-QAM, 128-QAM (n on-squar e), and 256 -QAM. The rate ad aptation scheme selects the highest le vel of modulation that satis fies the BER constraint. Th e upp er bo unds of the BER, as d iscussed in Sectio n III, ar e used in th is calculation. The p erforma nce of the rate-adaptive PNC is e valuated with simulations. Its throu ghput is co mpared with other relayin g methods, including CNC, conventional 4- phase rela ying, and direct transmission without relay ing. W e a lso enable rate ad ap- tation in th e relayin g method s that are used for com parison. The simu lation settings are very similar with those in [13]. 0 50 100 150 200 250 1 2 3 4 5 6 Source−to−opportunistic−listener distance (m) Throughput (bps/Hz) PNC CNC 4−phase relaying No relaying Fig. 6. Throughput vs. the distance between the source nodes and their correspond ing opportunistic listeners, with rate- adapti ve transmission. The source no des S 1 and S 2 , and the r elay R ar e located on the same line, with the re lay in the mid dle and the source nodes on opposite sides. Th e distance between each source node and the relay is uniformly d istributed in [125 , 2 50] m. The destination no de D 1 (or D 2 ) is placed at a specific distance from S 2 (or S 1 ). W e co nsider Rician flat-fading chan nels with Rician factor K = 5 dB an d an average power g ain of 1 /r 4 , where r is the distance between two nodes. In the simulations, the maximu m tran smission power is 10 dBm, the noise p ower density is – 174 dBm/Hz, the noise figure is 6 dB, and the rece i ver ban dwidth is 1 MHz. The ma ximum BER constraint is set to 10 − 3 . Each simulation was run with 1 0,000 different random seeds to obtain the overall perform ance. W e ev aluate th e perf ormance for d ifferent distanc es between the source nod es and their cor respond ing oppo rtunistic listeners. The results are shown in Fig . 6 . It can be observed in Fig. 6 that, when the distance is smaller than 140 m, PNC outp erforms the other relayin g meth ods. Specifically , when the distance is zer o, which c orrespon ds to the case wher e the sou rce an d destinatio n nodes overlap, th e throug hput ga in of PNC to CNC is 1 .38, which is slightly lower tha n the theoretical value 1.5. The difference between the theoretical an d simulated thro ughpu t gain s is mainly d ue to the r eduction of th e transmission power of the stro ng trans- mitter when u sing PNC. Howe ver, as men tioned in Section I I, this power red uction makes it feasible to m ap the superpo sed signal to c oded sy mbols. The throu ghputs of both the PNC an d CNC scheme s decrease with the distance, b ecause when the distance is large, the o pportu nistic listeners are less likely to overhear the symb ols sent b y the nearby sourc e nod e. The thr oughp ut of PNC d ecreases faster , due to the presen ce of inter ference at the oppor tunistic listeners. The thro ughpu t of transmitting witho ut relayin g increases with the distance , simply b ecause when the distance is large, the source and destination nodes are closer to each other at certain random instances. V . C O N C L U S I O N In this p aper, we have been focusing on syn chron ous PNC supportin g M -QAM. W e have intro duced a mapping scheme for sup erposed M - P AM and M -QAM signals, which h as n o restrictions on the origin al constellation mapp ings (it can be, for in stance, eith er b inary-c oded or Gr ay-cod ed). Also, bo th square and non-squ are M -QAM mo dulations are considered . W e h av e subseq uently ev aluated the perfo rmance of PNC with M -QAM mod ulation, both analytically and throug h simula- tions. Th e simu lation r esults show that the theoretical upper and lower boun ds, as well as the app roximatio n values, are in accord with the simu lated values. Following tha t, we have introdu ced a rate adap tation scheme for the DNF method of PNC. The rate adaptation scheme consider s B PSK, QPSK, and M -QAM mod ulations. The appropr iate m odulation (an d for M -QAM, the approp riate v a lue o f M ) is selected ac cording to the chan nel status and ma ximum BER con straint. Simulation results show that the rate-adaptive PNC ou tperfor ms other con- ventional r ate-adaptive relaying schemes in various scenarios. A further analysis o n theoretical aspects and optimization of rate ad aptive DNF will be consider ed in our future work. A C K N O W L E D G M E N T W e would like to thank Y ang Huang fo r his suggestions and comments o n this work . This work was supp orted in par t b y the Na tional Natu- ral Science Foundatio n of Chin a (61 1720 51, 6 10711 24), the Fok Y ing T ung Education Found ation (1 21065 ), the Pro- gram for New Century Excellent T a lents in Univ ersity (0 8- 0095, NCET -11 -0075 ), the Specialized Research Fund fo r the Doctora l Pr ogram of High er Education (2011 00421 1002 3, 20110 0421 20035), an d the Fundamen tal Research Funds for the Central Un iv ersities (N100 40400 8, N1102 04001 , N11080 4003 ). R E F E R E N C E S [1] M. Dankberg, M. Miller , , and M. 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