Wireless Network Coding in Network MIMO: A New Design for 5G and Beyond

1 Physical Layer Network Coding in N etwork MIMO: A Ne w Design for 5G and B eyond T ong Peng, Y i W ang, Alister G. Burr and Mohammad Shikh-Bahaei Abstract —Physical layer network coding (P N C ) has been studied to serve wireless network MIMO systems with much lower backhaul load than approaches such as Cloud Radio Access Network (Cloud-RAN) and c o ordi- nated multipoint (CoMP). In t his paper , we present a design guideline of eng ineering a pplica ble P NC to fulfil the request o f high user densities in 5G wireless RAN infrastructure. Unlike compute-and-forward and PNC de- sign criteria for two-way relay cha nnels, the proposed guideline is designed for uplink of network MIMO (N- MIMO) systems. W e sho w that the proposed d esign criteria guarantee that 1 ) t he whole system operate s over binary system; 2 ) the P NC functions utilised at each acc ess point o vercome all singular fade states; 3 ) the destination can unambiguously recover all source messages while the overall backhaul load r emains at the lowest lev el. W e then develop a two-stag e search a lgorithm t o identif y the optimum PNC ma pping functions which greatly reduces the real-time computational complexity . The impact of estimated cha nnel information and reduced number of singular fade states in different QAM modulat ion schemes is studied in this paper . In addition, a sub-optimal search method based o n lookup t a ble mechanism to achieve further reduced computational complexity with limited performance loss is presented. Numerical results show that the proposed schemes achieve low outag e probability with reduced backhaul load. Index T erms —adapt ive PNC, industrial a pplicable, backhaul load, unambiguous detection. I . I N T R O D U C T I O N Over t he past few years, network multiple-input multiple-out put (N-MIMO) technique [1] has re- cei ved significant attention d u e t o it s flexibility , power and capacity adv antage over the centrali zed architectures i n fifth generation (5G) dense cellular networks. Multiple mobil e t erm i nals (MTs) may T . Peng was with the Department of Electronics, Univ ersit y of Y ork, and now is with the Centre for T elecommunications Re- search, Department of Informatics, King’ s college London , UK (e- mail: tong.peng@york.ac.u k, tong.peng@kcl.ac.uk). Y . W ang and A. G. Burr are with the Department of Electronics, Univ ersit y of Y ork, UK (e-mails: yi.wang@york.ac.uk, alister .burr@yo rk.ac.uk). M. Shikh-Bahaei is with the Centre for T elecommun ications Re- search, Department of Informatics, King’ s college London , UK (e- mail: m.sbahaei@kcl.ac.uk). This research is funded by EPSRC NetCoM project EP/K040006/1 and partially by EP SRC I oSIRE project EP /P022723/1. share t h e same radio resources and be served by the correspondi ng access points (APs) where the inter-ce ll interference can be effecti vely mitigated. This was also applied in the coordinated multipoint (CoMP) approach standardized in L TE-A [2, 3]. T h e Cloud Radio Access Network (C-RAN) concept has been proposed in [3] with similar goals. Howe ver a potential issue in thes e approaches is a sign i ficant increased uplo ad burden on t he backhaul (referred to as fronth aul in C-RAN) network between APs and central processin g unit (CPU) on the upli n k, especially for wireless b ackhaul networks. Sev eral methods hav e been studied in previous work in order to add ress this p ro b l em. In [5], W yner -Ziv compression is utilis ed to reduce the backhaul load. Iterativ e in t erference cancellation and com pressiv e sensing algorith ms are designed in [6]-[8] as alter - nativ e solutions, but the total backhaul load rem ai n s typically sev eral times the total user data rate. In [12, 13 ], n ovel approaches have been designed based on physi cal layer network codi n g (PNC) that keep th e total backhaul load equal to the to t al us er data rate. PNC is a scheme im plemented at APs in which each AP attempts to infer and forward combina- tions of the signals over an algebraic field, where the signals are transmi t ted from multipl e sources simultaneous l y and superimp osed in the recei ved constellation. An import ant property of PNC is that the APs decode the joint messages from mult iple sources t o a linear function over th e algebraic field rather t han decode each source symbol individually . On the other aspect, PNC is a multiple-mess age compressing t echni que that makes network through - put greatly improved and keeps the cardinalit y of relay outputs consid erably reduced. Hence PNC is an appeali n g t echnique t o s erve wireless RAN infrastructure with high user density in 5 G and beyond. Pre vious work on PNC mainly focused on a two-way relay channel (TWRC) s cenario to easily double the network t hroughput without rout ine op- erations [9]-[14]. The original PNC was proposed and designed i n a TWRC based on BPSK [11]. Although onl y BPSK was used, PNC contributes to a big idea and m o tiv ates m any research outcomes thereafter , e.g. compute-and-forward (C & F) whi ch generalizes PNC of TWRC to mult iuser relay net- works by utilizing structured nested latt i ce codes [10], and latti ce n et work codi ng [4]. Howe ver , the lattice based network codi ng i n construction A and D operates ov er a finite field and the coset size of the quotient latt ices is typically not b inary-based [11]. Then lattice codes have disadva ntages for engineering application s as no n-binary codes are required over lar ge prime fields. In contrast to the previous work in PNC, we focus on designin g th e PNC approach with conv entional 2 m -ary digital modulation. When QAM modul ation schemes are used at MTs, PNC has to solve t he so-called si ngular fading problem whi ch is typically unav oidabl e at the m ultiple access phase under some circumstances at each AP . Failure to resolve such problem results in network performance degrada- tion. T oshi aki et. al. [15, 16] proposes a scheme, namely the denoise-and-forward, which employs a non-linear 5 QAM PNC m apping to mitigate all singular fade st at es, and gives goo d p erformance. Other researches on this issue have worked on the design of li near functi ons over the integer finite field or ring, e.g. linear PNC (LPNC) [17] which can only be opt i mised for the q -ary PNC mapping where q is a prim e i n Z + . All these approaches, howe ver , do not operate ove r t he binary systems, and hence cannot be readily appli ed in the current mobile com munication networks. The work in [13] provides a solution for impl em enting PNC in binary systems with low modulation orders only . In this paper , we propose an adaptiv e PNC wi t h reduced backhaul load and unambi guous decoding for QAM modulation s chemes and the main contri- butions are listed as follows 1) A PNC design guidelin e for uplink scenarios is proposed, along with a search algorithm based on this gu i deline to find the opt imal coef ficient mapping matrices, such that a) the global matri x , formed by th e coeffi cient matrices from each AP , guarantees all source symbols to be decoded at CPU; b) the ma- trices st o red at each AP can resolve all s in- gular fade stat es; c) the number of coeffic ient matrices stored at each AP is m i nimised; d ) the proposed alg orithm is generalised to QAM modulation schemes of dif ferent orders; e) the proposed algorithm can be applied in N- MIMO systems with multip l e MTs and APs. 2) The whole scheme operates over binary sys- tems with multip le MTs and APs. As dis- cussed earlier about adv antages of coping with the singu l ar f ading problem in multipl e access stage, PNC p l ays as a reliable rol e not only in TWRC but also in RAN to s erve multiple MTs . In this paper , we in vestigate t he design criteria of engineering appl icable PNC over binary systems for an uplink scenario of 5G N-MIMO system and discuss how to address t he singul ar fading problem with multiple MTs. 3) A regulated PNC approach w h ich fulfils the low latency demand in practical networks is also p resented in t his paper . The regulated approach is developed based on the origi- nal search algorit hm, and the l o okup t able mechanism i s adopted to achieve low latency . In such approach, all t he optim al coefficient mapping m at ri ces to resolve differe nt singul ar fade state comb inations are stored at the APs and the CPU with a corresponding table with their indexes. Instead of s earch amon g the matrix candi dates, the optimal matrix selec- tion algorit hm is replaced by loo king up the index table. The drawback of this approach is di scussed in this paper and we provide a solution to overcome the problem. 4) The im pact of estimated channel information to optimal matrix selection as well as the ef fect of the reduced number of singul ar fa de states t o performance degradation are s tudied. The proposed PNC mapping selection alg o - rithm depends on t he accuracy of channel information at each AP , thus we tested how the estim ated chann el af fects the propo s ed al- gorithm. Utili sation of less nu m ber of si ngular fade st at es contributes to reduced calculation complexity , but performance d egradation is observed. W e discuss these issues in th is paper and giv e potential resoluti o ns. The rest o f t his paper i s organised as fol lows. The introduction of N-M IM O syst em s and definitions of PNC design crit eria are gi ven in Section II and III, respectiv ely . The proposed binary mat ri x adaptiv e s el ecti o n algorithm is derived in Section IV , followed by t he discussion of m ethods to reduce the computational complexity of the proposed algorithm in Section V . Numerical result s are given in Section VI and finally the conclusions are drawn in Section VII. I I . S Y S T E M M O D E L A t wo-stage uplink model of N-MIMO syst em is illustrated in Fig. 1. W e assume MTs and APs are all equipp ed with a single antenna for simplicit y . At the first stage, u MTs transmit symbols to n APs during the same period, which refers to a multi-access stage. W e ha ve studied the impact of synchronisation errors in [19] so i n this paper we assume t he synchroni sation is p erfect for simplicity . Each AP receive s data from all MTs and then infers and forwards a li near combi nation (which is referred to as the network coded symbo ls (NCS ) in t h is paper) of the ent i re messages over a finit e field or ring. The second stage is called b ackhaul stage where n APs forward the NCSs t o CPU via a lossless but capacity limited ‘bit-pipe’. In t h is paper , the links in multi-access stage are modelled as wireless links in order to fulfil th e request of 5G systems; wh i le t he backhaul links may be wireless or deployed on wireline. The techniques p resented in this paper are in particular su itable for wireless backhaul which is normally more cost-ef fecti ve. Each MT employs a 2 m -ary di g ital modulati o n scheme where m d enotes t he mo dulation o rder . Let M : F 2 m − → Ω denotes a one-to-one mapping function, where Ω is the set of all possible complex constellation poi nts. Hence t h e messages w ℓ ∈ F 2 m at the ℓ th MT can be mapped to the complex symbol s ℓ = M ( w ℓ ) , where w ℓ = [ w (1) ℓ , · · · , w ( m ) ℓ ] is an m -tuple with each element w ( i ) ℓ ∈ F 2 . The link between all MTs and the j th AP forms a multiple access channel (MA C), where t h e j th AP observes the noisy , faded and superimpo sed signals at a certain time slot, mathematically giv en by y j = u X ℓ =1 h j,ℓ s ℓ + z j , (1) where z j denotes the additive complex Gaussian noise with zero m ean and va riance σ 2 , and h j,ℓ represents the channel fading coefficient between the ℓ th MT and the j th AP , which i s a random var iable w i th Rayleigh distribution. MT 1 MT 2 · · · AP 1 AP 2 AP 3 AP n · · · C P U MT u Fig. 1. The uplink system diagram. I I I . D E S I G N C R I T E R I A Before presenting the proposed PNC design cri- teria, we list th ree main cons traints for a PNC t o b e engineering applicable: 1) The PNC decoding must operate over F 2 ; thus, the NCS need to be binary-based. 2) The PNC mapp i ng function must be well designed su ch that all singular fade states can be resolved. 3) The PNC mappi ng functions must ensure t hat CPU can unam biguously recov er all sou rce messages. W e gi ve d etails of the proposed design criteria for PNC in N-MIMO systems in this section, which relaxes all three aforementioned constraints . A. Engineering Appli cable PNC Function W e are primarily concerned with the MA C phase between u MTs and th e j th AP in t he design of the PNC mapping function. Inst ead of using PNC approaches performi ng linear com binations on symbol level, such as [9], we design a m ethod to encode PNC directly at the b i t level which allows the APs to operate over a binary field for i n dustrial application. Definition 1: The bit-level lin ear network codi ng function of the j th AP for u M Ts is defined as N j : ( M j , w ) − → x j , (2) and mathematically expressed as x j = N j ( M j , w ) = M j ⊗ w , (3) where w , [ w 1 , · · · , w u ] T denotes an mu × 1 j oint message set wit h w ∈ F mu × 1 2 , and each w ℓ stands for a 1 × m binary data vector at the ℓ th MT . M j denotes a matrix with F t × mu 2 , where t stands for the size of network coded vector at th e j th AP and t ≥ m , and ⊗ denotes t h e mult iplication over F 2 . x j ∈ F t × 1 2 is call ed the network cod ed vector (NCV) which consists of all t linear network coded bit s x j = [ x (1) j , x (2) j , · · · , x ( t ) j ] T . (4)  It is obviou s that each coded b it x ( i ) j is indeed a linear combination of all source bits over F 2 , thus, x ( i ) j = M ( i, 1) j ⊗ w (1) 1 ⊕ · · · ⊕ M ( i,um ) j ⊗ w ( m ) u , (5) where ⊕ denotes the addition operation over F 2 , and M ( i, 1) j denotes the entry at t he i th row and the 1 st column of M j . Definition 2: W e define t he const ellation set which con t ains all possible superim p o sed symbols at the j th AP over a given channel coefficient vec tor h j , [ h j, 1 , · · · , h j,u ] as s j, △ , [ s (1) j, △ , · · · , s (2 mu ) j, △ ] , where s ( τ ) j, △ = u X ℓ =1 h j,ℓ s ℓ , ∀ s ℓ ∈ Ω , τ = 1 , 2 , · · · , 2 mu . Theor em 1: For the MAC l ink between u MTs and the j th AP , th ere exists a surjectiv e functio n Θ : s j, △ − → x j , (6) when the size of NCV t < mu . Pr oof: Since M is a bijectiv e function, we hav e the following relationship x j N j ⇐ = w ⇐ = = ⇒ M M − 1 s , (7) where ⇐ = and ⇐ = = ⇒ represent surjective and b i jec- tiv e relationships, respectively . s , [ s 1 , · · · , s u ] = [ M ( w 1 ) , · · · , M ( w u )] stands for the set that con- tains th e modul ated symbol s at all MTs. Following (6), for each element in s , there exists a superim- posed constellation point s j, △ at a given channel coef ficient vector h j , and this proves Theorem 1. W e call Θ the PNC mapping function which m aps a s u p erimposed constellation poi nt to an NCV and plays the key role i n PNC encoding, where this PNC encoding performs estimation of the possibl e NCV outcomes x j for the j th AP , based on the received signals y . Let X j denote the vector-based random var iable with its realization x j . The a p o steriori probability of the ev ent X j = x j conditioned on the MA C outputs Y j = y j is Pr( X j = x j | y j , h j ) = Pr( Y j | X j = x j , h j )Pr( X j = x j ) Pr( Y j = y j ) = P ∀ w : N j ( M j , w )= x j Pr( Y j | w , h j )Pr( w ) Pr( Y j = y j ) = P ∀ s :Θ( s j, △ )= x j Pr( Y j | S j, △ = s j, △ )Pr( S = s ) Pr( Y j = y j ) . (8) The condition al probabi lity density fun cti on is given by Pr( Y j | S j, △ = s j, △ ) = 1 √ 2 π σ 2 exp  − | y j − s j, △ | 2 2 σ 2  . (9) The a p o steriori L-value L x j for th e e vent X j = x j is L x j = log    P ∀ s :Θ( s j, △ )= x j Pr( Y j | S j, △ = s j, △ )Pr( S = s ) P ∀ s :Θ( s j, △ )= 0 Pr( Y j | S j, △ = s j, △ )Pr( S = s )    , (10) where 0 is a l ength- t all-zero vector over F t × 1 2 . B. Resolving the Singu l ar F ading W e have s et u p the PNC mappi ng approach in binary s y stems, which establis hes the fun damental PNC sys tem structure av ailable for practical engi- neering application. The next upcom ing prob lem lies in how to resolve the singular fading in t h e multiple access stage. In this section, we demo n - strate that the PNC mapping function Θ proposed above is capable of resolvi ng all singular fade stat es with a simple des i gn approach. W e first define the singular fade states as follows Definition 3: The singular fade state (SFS) at the j th AP is defined as the channel fading coeffic ients h j which makes s ( τ ) j, △ = s ( τ ′ ) j, △ when τ 6 = τ ′ .  In other words, for a given channel coef ficient vector h j , if two o r more element s in the s et s j, △ are th e same, h j is an SFS. No rmally , singular fading is u n a v oidable at MA C stage, and multi u ser detection is in princip l e i nfeasible if the j th AP expects to decode all source messages. PNC is capa- ble to overcome SFS problem when the coincident superimposed constellati o n poi nts are well labelled by the NCV x j , which helps CPU to recover all source messages. Definition 4: If a set of constellation points re- cei ved at APs are mapped to the sam e NCV , we call this set a cluster , denoted as c ( τ ) , [ s ( τ 1 ) j, △ , s ( τ 2 ) j, △ , · · · ] , (11) where s ( τ i ) j, △ denotes th e i th cluster members. In a singular fading, if the values of cluster members are the same then this cluster i s called a clash .  Definition 5: Given two clusters c ( τ ) and c ( τ ′ ) that th e constellatio n point s in wh i ch are mapped to d i f ferent NCVs, then the m inimum inter-cluster distance, also kno wn as the minimu m distance be- tween these two d i f ferent NCVs, is defined as d min = min Θ( s ( τ i ) j, △ ) 6 =Θ( s ( τ ′ k ) j, △ ) | s ( τ i ) j, △ − s ( τ ′ k ) j, △ | 2 , (12) ∀ s ( τ i ) j, △ ∈ c ( τ ) , ∀ s ( τ ′ k ) j, △ ∈ c ( τ ′ ) , i = 1 , 2 , ..., k = 1 , 2 , ... Theor em 2: The PNC m app i ng function Θ cannot resolve singular fading if the minim u m int er -cluster distance d min = 0 . Pr oof: When d min = 0 , the posterior prob abi l- ity o f some outcom es of X j will be very sim ilar (in terms of (8)). This definitely introduces the am- biguities i n estimating th e real NCV x j , especially when a superimposed constellati on point labelled by one NCV is close to another point that is labelled by another NCV . Hence the singular PNC mapping function is in princip le not capable of decoding the NCV reliably . Normally the dimens ion of NCV x j is t × 1 at the j th AP , m ≤ t ≤ mu . When the num ber of source i n creases (a lar ge MA C), the singular fading problem becomes more s e vere and t h e method of SFS values calculation is d i f ferent. H owev er , by simply increasing the dimens ion t of NCV (thus , increasing t he number of rows of M j ), there def- initely exists non -singular PNC fun cti on which is capable of resolving a kind of SFS. Remark 1: W e can obtain non-singular PNC map- ping function Θ j for the j th AP if the cardinali ty t of the PNC encoding out com es are determined in terms of the following criterion t = ar g min m ≤ t 0 is a dis t ance threshol d . Remark 1 rev eals the second design criterion for PNC mappi ng function Θ j over a u -MT and 2 m - ary digit al modulation MA C, wh ich guarantees t he reliable PNC encoding wi th the minimum poss ible cardinality expansion. C. Algebraic W ork for Unambig uous Decodabil ity W e hav e set up two design guidelines of the engineering applicable PNC approach for uplink scenarios. The next criterion is that th e CPU can guarantee all source messages to be unambiguously recov ered. W e need to carefully d esign each M j , j = 1 , 2 , · · · , n , so that M = [ M 1 , · · · , M n ] T includes a number of ro w coefficients which form s the following theorem: Theor em 3: Assume M = M n × n ( R ) , where the coef ficients are from a commutat ive ring R . Source messages are d rawn from a subs et of R and all source m essages can be unambiguously d ecoded at the destination if and only if the determin ant of the transfer matrix is a u nit in R , det( M ) = U ( R ) . (14) Pr oof: W e first prove that (14) giv es the suf fi- cient and necessary condit ions that make a m atrix B inv ertible in N-MIMO n etworks. Suppose B is in vertible, then there exists a matri x C ∈ M n × n ( R ) such t hat BC = CB = I n . This im plies 1 = det( I n ) = det( BC ) = det( B )det( C ) . According to the definition of a unit, we say det( B ) ∈ U ( R ) . W e know B · adj( B ) = adj( B ) · B = det( B ) I n . If det( B ) ∈ U ( R ) , we have B · (det( B ) − 1 adj( B )) = (det( B ) − 1 adj( B )) B = det( B ) − 1 det( B ) = I n . (15) Hence, C = (det( B ) − 1 adj( B )) is t he in verse of B since BC = CB = I n . If B is in vertible, then its inv erse B − 1 is uniquely determined. Assuming B has two in verses, s ay , C and C ′ , then B · C = C · B = I n , (16) B · C ′ = C ′ · B = I n , (17) hence we ha ve C = C · I n = C · B · C ′ = I n · C ′ = C ′ . (18) It proves the uniqueness of the inv ertible m atrix B over R . Assume a 6 = a ′ , B · a = F , B · a ′ = F ′ , and F = F ′ . This means a = B − 1 · F = B − 1 · F ′ = a ′ . (19) This cont radicts a 6 = a ′ . Hence it ensures unam- biguous decodability: B · a 6 = B · a ′ , ∀ a 6 = a ′ . (20) Definition 6: The i deal in R generated by all ν × ν minors of M m × n ( R ) is denoted by I ν ( M m × n ( R )) , where ν = 1 , 2 , · · · , r = min { m, n } .  A ν × ν minor o f M m × n ( R ) i s the determin ant of a ν × ν matrix obtained by deleting m − ν ro w s and n − ν colum ns. Hence there are  m ν  n ν  minors of size ν × ν . I ν ( M m × n ( R )) is the ideal of R generated by all these minors. Design Criterion : T h e destination is abl e to un- ambiguously decode u source messages if: 1) u ≥ max { ν | Ann R ( I ν ( M j )) = h 0 i} , ∀ j = 1 , 2 , · · · , n , 2) M j = arg max M j n I  − → Y ; − → F j o , where h x i denotes the ideal generated by x . Condition 1 can be proved as foll ows. According to Laplace’ s theorem, ever y ( ν + 1) × ( ν +1 ) minor of M m × n ( R ) m ust lie in I ν ( M m × n ( R )) . This suggest s an ascending chain of ideals in R : h 0 i = I r +1 ( M j ) ⊆ I r ( M j ) ⊆ · · · ⊆ I 1 ( M j ) ⊆ I 0 ( M j ) = R . (21) Computing t h e annihilato r o f each ideal in (21) produces another ascending chain of ideals, h 0 i = Ann R ( R ) ⊆ Ann R ( I 1 ( M j )) ⊆ · · · ⊆ Ann R ( I r ( M j )) ⊆ Ann R ( h 0 i ) = R. (22) It is obvious that: Ann R ( I k ( M j )) 6 = h 0 i ⇒ Ann R ( I k ′ ( M j )) 6 = h 0 i , ∀ k ≤ k ′ . (23 ) The maximum value of ν which satisfies Ann R ( I ν ( M j )) = h 0 i guarantees that I k ( M j ) ∈ R , ∀ k < ν . Hence we define the rank of M j as rk( M j ) = max { ν | Ann R ( I ν ( M j )) = h 0 i} . Sup- pose that M k ∈ M m × p ( R ) and M k ′ ∈ M p × n ( R ) , then r k( M k M k ′ ) ≤ min { rk( M k ) , rk( M k ′ ) } , and we can easily prove that 0 ≤ rk( M m × n ( R )) ≤ min { m, n } . Thus, i n order to guarantee there are at least u unambig u ous lin ear equations a v ailable at the CPU, rk( M j ) must be at least u , ∀ j = 1 , 2 , · · · , n . The special case of condition 1 is th at t h e entry of the coeffic ient m atrix M j ∈ M m × n ( F ) is from a finite field F ∈ F . Then condition 1 of the above Design Criterion may be changed to “the maximum nu m ber of lin early independent ro ws (or columns)” since Ann R ( I ν ( M j )) = h 0 i if and only if I ν ( M j ) 6 = 0 . In other words, the largest ν such that the ν × ν minor of M j is a non-zero divisor represents ho w many reliable linear comb inations the j th layer may produce. Hence conditio n 1 is a strict definition which ensures unambiguous decod- ability of the u sources. Condition 2 ensures that the selected coefficient matrix maximi s es the mutual information of the particular layer , giving finally the maximum overa ll throug hput. I V . B I NA RY M A T R I X A DA P T I V E S E L E C T I O N A L G O R I T H M D E S I G N According to the design criteria proposed in the pre vious section, we can summarise th at giv en a QAM modulation scheme, t he opt imal binary PNC mapping function contains the following properties 1) it maximis es the mini mum distance between diffe rent NCVs ; 2) the compos i ted global mapping matrix is in- vertible. In order t o achieve these p roperties and applica- bility in practical N-MIM O system s , we propose a binary matrix adaptiv e s election (BMAS) algorithm based on t he design criteria introd u ced in Section III. T h e BMAS alg orithm is divided into two stages, one is called Off-line search, in which an exhaustiv e search is implement ed among all m × mu b i nary mapping matrices to find a s et of candidate matrices which resolves all SFSs with the above property 1); the other one is called On-line search, in which a selection from the candid at e matrices i n order to obtain the in vertible mapping matri x accord- ing t o property 2) is executed. The computatio n al complexity of the proposed algorithms i s mainly caused by the Off-line s earch, especially in higher order modulation schemes due to the increased number o f SFSs and mat ri ces, and this Off-line search algorithm only needs to be do n e once for each modul at i on scheme. The candidate mapping matrices found in Off-line search are stored at APs and CPU in order t o im plement the On-line search in real-time transmissio n . A. Off-Line Searc h Algorithm Define a set W j oin t , [ w j o 1 , w j o 2 , · · · , w j o N ] which contains all possi ble binary joint message combinations wit h N = 2 um , so that each w j o i in this set stands for a 1 × mu binary joint message ve c- tor from u M T s , for i = 1 , 2 , · · · , N . By applying a modulation scheme M o ver each w j o i in W j oin t , a joint modulation set S j oin t , [ s j o 1 , s j o 2 , · · · , s j o N ] T is obtained, where s j o i = [ s ( j o i ) 1 s ( j o i ) 2 · · · s ( j o i ) u ] stands for the i th combination of u mod ulated symbols and s ( j o i ) ℓ ∈ Ω for ℓ = 1 , · · · , u . The next step is to calculate the NCS s ( q ) n, △ and its corre- sponding NCV x ( q ) i,n under all L SFS circumst ances, mathematically giv en by s ( q ) n, △ = h v ( q ) S F S s T j o n , x ( q ) i,n = M i ⊗ w ( q ) j o n , ( 24) n = 1 , · · · , N , q = 1 , · · · , L, i = 1 , · · · , N 2 , where h v ( q ) S F S denotes the channel coef ficient vector causes SFS. Due to the property of SFS, the same s ( q ) n, △ could be obtained with different s j o n sets in a clash. In that case, th ese joint symbol sets shoul d be encoded to the same NCV according to t he unambiguous decodability theorem. The next st ep i s to s tore the m apping matrices which can resolve one SFS and also con t ains a high possibility t o form an in vertible global mapping matrix when combi ning with o t her selected mapping matrix candidates in other SFSs. Detailed descripti on of the Off-line search i s ill ustrated in Algorithm s 2 and 3 in Ap- pendix A. B. On-Line Searc h Algori thm The proposed Of f-lin e search is implemented before the transmission to reduce the number of mapping mat ri ces utilised in t he On-line s earch. In the real-time transm ission, the propo sed On-li ne search, which contains the same st eps in Algorithm 3 but wit h a much smaller v alue of K , is appl i ed at the CPU to select th e optim al mapping m atrix for each AP . When the optimal mapping matrix is selected, the indexes of th e selected m apping matrices will be sent back to each AP t h rough t he backhaul channel and at each AP , an estimat o r calculates the conditional probabi l ity of each pos sible NCV given the optimal mapping function. The estim ator returns the log-likelihood ratio (LLR) of each bit o f x j which is then applied t o a soft decision d ecoder . Note that the LLR algorithm does not require to detect individual symbols transmitted from each MT but a linear combination of the binary m essages. Finally the NCV at each AP will be forwarded t o the CPU and the original data from all MTs can be recov ered by m u ltiplying the in verse of the gl obal binary PNC mapping m atrix. V . A N A L Y S I S A N D D I S C U S S I O N In this section, we discuss ho w to apply the proposed BMAS algorithm to a general N-MIMO network with multipl e M Ts and APs, inclu d ing util- isation of reduced number of SFSs and d iscussion of resolving the SFS problem with more than 2 MTs. W ith a study of the properties of SFSs, a regulated On-line search algorithm based on l ookup table mechanism with a small performance degradation i s proposed in this s ecti o n in order to fulfil the request of low latency in 5G RANs. A. Image SFSs and Pr incipal SF S s Since an exhaustive search is carried out among all t × mu binary matrices in the prop osed Off- line search alg o rithm, the computati onal com p lexity increases due to a l arge num ber of SFSs as well as an increased v alue of um in higher order modulation schemes wi th large numb er of MTs. For example, i n 2 -MT and 2 -AP case, the number of SFSs need to be resolved at each AP is L = 13 in 4 QAM and L = 389 in 16 QAM. Thus for t he 4 QAM case, at least 13 binary matrices with each si ze of 2 × 4 sh ould be stored at each AP for On-line s earch. When 16 QAM scheme is employed at each MT , at least 389 of 4 × 8 binary matrices need to be stored which results in a huge in creased number of candidates in real-ti m e computation. In the prop o sed BMAS algorithm, we resolve this problem b y keeping the number of useful SFSs m inimum. According to our research of NCV calculation expressed in (24), we found some of diffe rent SFSs g enerate the same clashes whi ch can be resolved by the same bin ary matrices. W e then define such SFSs as image SFSs (iSFSs) and k eep only one in the proposed search algorithm. Howe ver , this probl em still remains when higher order modulati on schemes are employed at MTs. In addition, due to a lar ger constell at i on in a higher or- der QAM scheme, a fe w SFSs cannot be resolved by a binary mappi ng matrix. In order to address these problems, we focused on the occurrence probability of an SFS in higher modulation schemes and noticed that n ot all SFSs occur frequently , so that we can ignore those “nonactive” SFSs with low app earance probabilities to minimise the number of mapping matrices utilised i n the p rop osed On-li ne search. W e define th e SFSs with high appearance probabili ties as principal SFSs (pSFSs) and a trade-off between the performance de gradation and t he number of pSFSs used in Off-line search is i llustrated in the next section. B. Calculat ion of Singul ar F ade S tates W e i l lustrate how to determin e a singular fading for a QAM modulation schem e with a simple net- work first wit h u = 2 MTs, and discuss t h e SFS calculation issue in a network wit h more than 2 MTs l at er . Follo wing Definition 3 , given a QAM modulation scheme, for singular fading we hav e s ( τ ) j, △ = s ( τ ′ ) j, △ when τ 6 = τ ′ in a constellation. Then mathematically , an SFS can be derived as s ( τ ) 1 , △ = s ( τ ′ ) 1 , △ , hs ( τ ) = hs ( τ ′ ) , for s ( τ ) 6 = s ( τ ′ ) , h 1 , 1 s ( τ ) 1 + h 1 , 2 s ( τ ) 2 = h 1 , 1 s ( τ ′ ) 1 + h 1 , 2 s ( τ ′ ) 2 , where h j,ℓ denotes the channel coefficient between the ℓ th MT and the j th AP , and s ( τ ) , [ s ( τ ) 1 s ( τ ) 2 ] refers to a joint sy mbol set wh i ch contains the m od- ulated symbols at both M T s , and s ( τ ) 6 = s ( τ ′ ) means at least one symbol is differ ent in s ( τ ) and s ( τ ′ ) . Then we define v S F S = [ v (1) S F S , v (2) S F S , · · · , v ( L ) S F S ] as the set contains all u nique value of SFSs and calculated by v ( q ) S F S = h 1 , 2 h 1 , 1 = s ( τ ) 1 − s ( τ ′ ) 1 s ( τ ′ ) 2 − s ( τ ) 2 , ∀ s ( τ ) l , s ( τ ′ ) l ∈ Ω . (25) By substituting the QAM modul ated symbo l s with all pos s ible com b inations to (25), we can find al l SFS values for this QAM scheme when u = 2 MTs. In the mult iple-MT ( u > 2 ) case, (25) is no longer suitable for SFS v alues representation due to the increased number of MTs i n MA C stage. In this case, t he relationshi p between the values of SFSs and channel coefficients are no longer able to expressed by a simple ratio between di f ferent channel coeffi cients, e.g. the SFSs form differ ent surfaces with infinite values in 3 -MT case. This is stil l an open issue in the literature for PNC design and we gi ve our potent ial solution here. One solution is to ut ilise clashes in s tead of calculatio n of the values of SFS s in the proposed algorith m . In the multip le-MT case, an SFS still causes clashes and different clashes can be always found according to Definition 5 , then an opt imal bi n ary matrix is found if it maps the superimposed symbols within a clash to the same NCV and kee p the v alue of d min maximised at the s ame time, withou t SFS calculation. Another solution to this issue is to divide the whole networks into multiple 2 -M T s u bnetworks. One way to achiev e this goal is by all o cating diffe rent pairs of MTs to different frequencies or time slo ts. In this case, t he superim posed sym- bol at an AP is al ways from two MTs and then SFSs can be calculated by (25). The only issu e of this appro ach is that m ultiple On-lin e search algorithms for different MT pairs are required to be implemented whi ch may cause extra computational complexity and latency time. An alternative way is to consider two MTs with the s imilar channel strength and transmi t power as the prime MT pair and trade the other receiv ed s ignals as additional noise. According to ou r research in [19 ], when a strong signal with a much higher energy com apred to the rest of received s ignals is receive d at an AP in the mult i ple access st age, i t is difficult to find an optimal matrix achieving unambig ous recovery due to t h e hi gh int erference from t h i s stron g sign al s [19]. Thus the 2 MTs whose recei ved signals are allocated at the similar energy level in the multipl e access st age can be paired to form a subnet work for PNC encoding and by pairing di f ferent MTs and APs, t h e multiple-MT -m u ltiple-AP case is replaced by multiple 2 -MT - 2 -AP cases. C. Re gu lated BMAS Searc h Algorithm In order to fulfil the request of low latency in some scenarios, we present a re gulated BMAS (R- BMAS) approach with a lookup table m echanism i n this subsection. According to the definiti on of clash , the superim p o sed sy mbols in a clash hav e an intra- cluster di stance of ‘ 0 ’ and by the calculation in (25), the clash groups in an SFS are mainl y determi ned by the absolut e value and the angle of the ratio of two channel coef ficients. So following the d esi gn rules in the propose algorithm, we ha ve: Theor em 4: The m apping matrix wh ich resol ves an SFS can al ways resol ve the non-singul ar fade states with the values close to this SFS. Pr oof: When a non-singul ar fade state (nSFS) happens, different superimpo sed symbols receiv ed at an AP will not be coincided which means no clash is observe d. When t his nSFS h o lds a simil ar absolute value and rotation angle to an SFS, the superimposed symb o ls, which form a clash in thi s SFS, will form a clu s ter i n thi s nSFS with a smal l er intra-cluster distances compared to inter-cluster dis - tances to other clusters. In th is case, the mappin g matrices, t hat are capable to resolve the SFS by mapping the coincided superim posed symbols in a clash to the same NCV and kee p different NCVs as far as possi ble, can achiev e th e maxim um d min in this nSFS by mapping the superim posed symbol s in the cluster to the same NCV . An example of thi s theorem i s given in Appendix B. According t o Theorem 4, the propos ed On-line search app roach could be replaced by a lookup table based mechanism for the optimal mapping matrices selection. A table cont ains all SFS combin ations and their corresponding in vertible mu × mu optimal binary mapping matrices could be established in the Of f-line search. Durin g the real-ti m e transmissio n , when the channel coeffi cients are estimated at the j th AP , the value of the fade s tate v F S j is calculated by v F S j = h j, 2 /h j, 1 , (26) and then the clo s est SFS to this nSFS is o b tained by d ( q j ) F S = min | v F S j − v ( q j ) S F S | 2 , (27) for q = 1 , · · · , L, j = 1 , · · · , n. (28) The ind ex q j will be forwarded to the CPU. By checking the table, the CPU send the optimal map- ping m at ri x index back to the APs for PNC encod- ing. The algorithm is summarised in Algorithm 1. As sh own i n Algo ri t hm 1, mos t of the calcula- tions for mapping s election ha ve been done before the transm ission. At the s am e t ime, the latency is reduced by applying the re g u lated On-line search in Algori thm 1 instead of Algorithm 3. Howe ver , a Algorithm 1 Regulated Binary Matrices Adaptive Selection (R-BMAS) Algorithm Off-line Sear ch 1: for i = 1 : L do ⊲ each SFS 2: Apply Algorithm 2 and 3 for M i 3: end for 4: for i 1 = 1 : L do ⊲ all SFS combinations 5: . . . 6: f or i n = 1 : L do ⊲ all n APs 7: G =    M i 1 . . . M i n    8: δ ← det( G ) | F 2 ⊲ determinant over F 2 . 9: if δ = 1 then 10: G l ← G l ∪ G 11: Add l as the opti mal m apping matrix for SFS combination [ i 1 · · · i n ] 12: end if 13: end f or 14: . . . 15: end for On-line Sear ch 16: for j = 1 : n do ⊲ each AP 17: v F S j = h 2 ,j /h 1 ,j , j ∈ [1 , 2 ] ⊲ fade st ate 18: f or i = 1 : L do 19: d ( i j ) S F = | v F S j − v ( i ) S F S | 2 20: end f or 21: [ d ( k j ) , k j ] = min d ( i j ) S F ⊲ SFS index 22: end for 23: Forward [ k 1 · · · k n ] to CPU 24: Look u p the table to obtain the op t imal mapping matrix index l 25: Send the index back to A Ps disadvantage of the R-BMAS algori t hm is the per - formance degradation caused b y sub-optimal global mapping matrices stored for some SFS com b ina- tions to achiev e unambi guous recovery at t h e CPU. In order to overcome this problem , a combination of the two proposed algorithms can be uti lised. During t he Off-line search, a request of BMAS algorithm i mplementation could be stored in the table establ ished i n R-BMAS algorithm for the SFS combi n ations that need to be resolved by a suboptim al global mappi ng matrix. Thus the real- time comp u tational complexity and latency time is reduced and the traf fic in backhaul network is restricted to the total user dat a rate at the same time. D. Computa t ional Complexity an d Backhaul Load W e in vestigate t h e computati onal complexity of the ideal CoMP , non-ideal CoMP and the proposed algorithms in this s u bsection t o illustrate the advan- tage of the proposed algorithms. In ideal CoMP , the bandwid th of backhaul net- work is as s umed unlimited so that the recei ved signal y j in (1) at each AP wil l be forwarded t o the CPU for j o int multiu s er M L detection, given by ˆ s = arg min s ∈ Ω u k y − Hs k , (29) where s denotes the u × 1 sym bol vector at the M T s and ˆ s is th e estimated version, and H stands for the n × u channel matrix. In non-ideal CoMP , the backhaul network is bandwidth-lim ited and each AP empl oys an LLR based m ultiuser det ecti o n algorithm to esti mate the transmitted sym bols from each MT , mathematicall y giv en by χ i,ℓ = P w i,ℓ =0 P ( s ℓ ) p ( y j | s ℓ ) P w i,ℓ =1 P ( s ℓ ) p ( y j | s ℓ ) , i = 1 , 2 , · · · , m, (30) where χ i,ℓ stands for the LLR correspondin g to the i th bit of the bin ary m essage vector w ℓ . A scalar quantizer which quant i zes χ i,ℓ into binary bits is employed after the est imation. The quant i sed bits are sent to t h e CPU via the b ackhaul network. A detailed computation of (30) is giv en in [18]. Con- sider a quantisati o n scheme of 2 bi ts is empl oyed at each AP and 4 QAM is utilis ed at 2 MTs in a simple 5 -node sys tem, then 4 LLRs are calculated by (30) at each AP and a to t al 16 bits are sent via t h e backhaul n et work. In order to achie ve a good performance in terms of error rate and outage probability , a qu antization scheme with a larger number of quantized bits is required and in thi s case, there is a trade-off between the performance and the backhaul load in non-ideal CoMP . In the proposed BMAS algorith m , each AP esti- mates the linear combination of the messages from MTs based on the ML rule rather th an decoding individual s ymbols (such as ML detection in the ideal CoMP), mathematically giv en by (8) - (10). In order to minimis e the com putational complexity , an exhaustiv e search before the real-time t ransmission is implemented which con t ains t h e majori t y of com- putational complexity in th e p rop osed algori t hm, and the proposed On-li n e search is implemented during the transmis sion with a reduced num ber of mapping matrix candidates, e.g. K = 5 m atrices at each AP for 4 QAM modulation s cheme in Algo- rithm 3. In t h e proposed R-BMAS algorithm, calculation s in the mapping selection are replaced by a loo k up table mechanism and the com putation compl exity is reflected in dis tance comparis o n in (2 7). Then an LLR estimation of each bit in the NCV is appl ied which is the same as in the BMAS algorithm. Follo wing the above 5 -nod e example, as illu s trated in (3), a binary mapping matrix with the minimum size o f 2 × 4 is selected at each AP to encode the 4 message bits from both MTs into the NCV , which results i n a total 4 bits b ackhaul load. Note an AP could empl oy a mapp i ng matrix with the maximum size of mu × mu to generate th e NCV and in this case, the other APs wi ll not participate in t h e PNC encoding because they fail t o recei ve any us eful signals and the total backhaul l oad i s stil l equal to mu . V I . N U M E R I C A L R E S U L T S In thi s section, we illustrate the out age probability performances of the proposed BMAS algorithm, R- BMAS algorithm and CoMP in a 5 -node system which includes 2 MTs, 2 APs and 1 CPU. As mentioned in pre vious section s, the 5 -node network is the smallest network to apply t h e proposed algo- rithms so that we use this network as a baseline t o illustrate the adv antage of the propos ed algorithms . The proposed algorithm s can be adapted to an N- MIMO network with more nodes and we hav e d is- cussed the respective potential issues and solutions in Section V . In the simulations, we assume the m ulti-access links are wi reless and the backhaul is wired which allows only binary data to be transmitted. Each no de contains 1 ant enna for transmissio n and recei ving, and 4 QAM/ 16 QAM modulation s chemes is em- ployed at bot h MT s . W e employ the con volutional code as an example and m o re po w erful channel code can b e utilis ed in order to enhance the reliabil i ty , such as LDPC [20]. In the sim ulation of ideal CoMP , we assume the backhaul capacity is lim i tless and the channel coef ficients are exchanged in order to i m plement a joint ML d etection algo ri t hm. In the non-ideal CoMP scenario, quant i zer with di fferent quantization bits ( 2 bits and 4 bits) are empl oyed at each A P to in vestigate the out age p rob ability performance. Fig. 2 il lustrates t he outage probabiliti es of the proposed BMAS algorithm in 4 QAM and 16 QAM schemes. As we can see from the figure, the outage probability curves achie ve the same diversity order . The ideal CoMP achie ves the optimal p erformance in both 4 QAM and 16 QAM cases due to the unlim - ited backhaul capacity and joint detection. When the backhaul load is capacity limited, non-ideal CoM P with 2 -bit and 4 -bit quantizer , which results in a total of 8 bits and 16 bit s backhaul load respectiv ely , are implem ented in the simulation. Compared t o th e ideal CoMP , a 8 dB and 13 dB performance degrada- tion in the non-ideal CoMP can b e observed; whilst the degradation is limit ed to only approxim at el y 3 dB by the p roposed algori t hm. In the proposed BMAS algorithm, the b ackhaul load is equal to the total number of bits which is 4 bits for 4 QAM, i.e. it is smaller than that in both non-ideal CoMP approaches. Instead of obtaining all SFSs-resolv able mapping matrix candidates for 16 QAM schemes, we consider on ly 4 , 12 and 50 pSFSs in Off -line search algorithm with comput ational complexity reduct i on. Note that the iSFS s are removed before selecti n g these 4 , 12 and 50 pSFSs in the sim u lation. As illus- trated in the figure, approximately 10 dB degradation in outage performance is seen when usin g onl y 4 pSFSs in the proposed On-line search algori t hm. When 12 and 50 pSFSs are used, the degradation is reduced t o 7 dB and 5 dB respectively and the gap will reduce when more pSFSs are consi dered in th e proposed BMAS algorithm. In Fig. 3, out age probability comparisons bet w een the proposed BMAS algorithm and R-BMAS algo- rithm are shown. When 4 QAM is used at both MTs, outage probability performance of BMAS algorithm is 1 dB better than t h at of R-BMAS algorithm due to random suboptimal mapping matrices are stored in table used in the R-BMAS algorithm. In term of reduced computati onal com plexity in the R-BMAS algorithm, an index searching among 25 binary 4 m apping matrices are implemented to resolve all p o s sible SFS combinations ( 5 pSFSs at each AP). When 16 QAM is employed at both MTs, the gap between the BMAS algori t hm and the R- BMAS algorit hm depends on t h e number of pSFSs utilised in the alg o rithm. When only 4 pSFSs are used for optimal mappin g m atrix selection in both 0 5 10 15 20 25 Eb/N0 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Outage Prob. Proposed BMAS Algorithm, 16QAM, 4 pSFSs Proposed BMAS Algorithm, 16QAM, 12 pSFSs Proposed BMAS Algorithm, 16QAM, 50 pSFSs Ideal CoMP, 16QAM Non-Ideal CoM, 4QAM, 8 bits backhaul Non-Ideal CoM, 4QAM, 16 bits backhaul Proposed BMAS Algorithm, 4QAM, 4 bits backhaul Ideal CoMP, 4QAM Fig. 2. Outag e Probab ility of the Pro posed BMAS Algorithm in 4 QAM and 16 QAM. algorithms, BMAS achie ves about 5 dB gains in outage performance compared to R-BMAS. The big gap in 4 pSFSs case is caused by inefficient m ap- ping matrix candidates in the R-BMAS algorit h m. W ith the nu m ber of pSFSs increasing to 50 , more pSFS candidates are used in the R-BMAS algorithm which im p rov es the o u tage performance about 7 d B and reduces the gap to 1 dB only . 0 5 10 15 20 25 Eb/N0 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Outage Prob. Proposed Regulated On-line Search Algorithm, 16QAM, 4 pSFSs Proposed PNC Algorithm, 16QAM, 4 pSFSs Proposed Regulated On-line Search Algorithm, 16QAM, 50 pSFSs Proposed PNC Algorithm, 16QAM, 50 pSFSs Proposed Regulated On-line Search Algorithm, 4QAM Proposed BMAS Algorithm, 4QAM Fig. 3. Outag e Probab ility of the Pro posed BMAS Algorithm vs Regulated On-line Search Algor ithm. W e have in vestigated ho w estimated channel state information (CSI) af fects the network performance, and illustrated the result comparisons in Fig.s 4 and 5. The values o f FSs are calculated by t he first equation in (25) so that during the t ransm ission, the accuracy of the CSI in access link is i m portant 0 5 10 15 20 25 Eb/N0 0% 10% 20% 30% 40% 50% 60% 70% 80% Mis-mapping Perc. pilot length = 1 pilot length = 2 pilot length = 5 pilot length = 10 pilot length = 20 Fig. 4. Miss-Mapping Probab ilities with Different Pilot Lengths. because it determines if the opti mal m apping matrix can be selected. In Fig. 4, we illustrate the im p act of estimated CSI to t he optimal mapping selection. The term “mis-mapping ” means the optimal mapping matrix is not selected. As we can see from the figure, the m is-mapping percentage decreases in any pilot lengt h circumstances with increase of E b / N 0 . By using a short-leng th pil ot sequence, the mis- mapping percentages are quite high which is caused by the fact that inaccurate fade states are calculated by using t he estimated CS I. The lo w m i s-mapping percentage s h o wn in Fig. 4 l eads to bett er out age probability performance in Fig. 5. For example, at 15 d B, t he outage probabili t y usi n g only 1 pilot symbol is 1 0 − 2 which refers to a mis-mapping percentage of 22% ; while using 10 pi lot symbo ls, the outage prob abi lity reduces t o 3 × 10 − 3 and a mis-mapping percentage of only 8% is achiev ed. By comparing the outage performances in Fig. 5 with perfect and estim ated CSI, we can conclude that pilot sequence with length of 10 is good enough for the proposed BMAS algorithm. V I I . F U T U R E W O R K A N D C O N C L U S I O N In thi s paper we present a design guideli ne of en- gineering applicable phys ical layer network coding in the uplink of N-MIMO networks. The proposed design criteria guarantee unambigu ous reco very of all m essages and th e traffi c i n the backhaul net- work is reduced to the le vel of tot al user data rate at the same t ime. W e then propose an optimal 0 5 10 15 20 25 Eb/N0 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Outage Prob. channel esti (pilot length = 1) channel esti (pilot length = 2) channel esti (pilot length = 5) channel esti (pilot length = 10) Proposed BMAS Algorithm, 4QAM Ideal CoMP, 4QAM Fig. 5. Outage Probability with Different Pilot Lengths. mapping matrix selection algorith m based on the design criteria. In order to reduce the real-time computational complexity , t he proposed algori thm is d ivided into Off-line and On-line parts. An ex- tension study of appl ying the proposed PNC design in binary systems with full -duplex (FD) APs [28]- [29] has been started. Practical PNC design wi th cross layer optim isation in [21]-[24] provide another research d i rection, and the research in [25] focuses on s p ectrum efficienc y solutions and can be ex- tended t o PNC-implemented systems. Moreover , the PNC application with optimal resource allocation [26] and [27] is crit i cal in order t o serve the 5G systems and achie ve massive data transmission s with high accurac y and low latency . Th e proposed algorithm is not only designed for a sim ple 5 - node network but for a general N-MIMO network serves multip l e MTs. In addition, a regulated On- line search algo rithm based on l ookup table m echa- nism is also presented i n order t o further reduce the computational complexity and latency wi t hout much performance degradation. W ith reduced backhaul load, the proposed algorithm s achie ve higher outage probability performance compared to th e p ractical non-ideal CoMP approaches. A P P E N D I X A W e il l ustrate a detail ed Of f-line search algorithm in Al g orithms 2 and 3. Part I indicates how to calculate SFSs and ho w to remove t h e i mage SFS in order to reduce the computatio nal compl exity , while Part II focuses on opt imal mappin g matrix selection. The st eps in the proposed On-line search is the same as that in the Of f-li n e search but with less number of matrix candidates so we will not show the repeated work here. Q d in Algorithm 2 is d efined as a vector contains all th e d min between dif ferent NCSs for e very binary mappi n g matrices in each SFS. Algorithm 2 SFS Calculation and Image SFS Re- move (Of f-line Search Algorithm. Par t I) 1: for i = 1 : L do ⊲ each sin gular fade state 2: h = S ( i ) ⊲ h is a 1 × m vector . 3: f or j = 1 : K do ⊲ each binary matrix 4: [ ξ , T ξ ] = N ( M ( j ) ) 5: ξ f ← F ( T ξ , h ) ⊲ F ( · ) produces all faded NCSs. 6: d min ← D ( ξ f ) ⊲ D ( · ) calculates the minimum distance of all NCSs. 7: Q d ← Q d ∪ d min ⊲ store all d min in Q d . 8: end f or 9: [ β ( i ) , α ( i )] ← C ( Q d ) ⊲ C ( · ) sorts Q d in descendin g order stored in β ( i ) and outputs the rearranged index vector α ( i ) . 10: end for 11: S ′ ← I ( S , α ) ⊲ delete all image sin g ular fade states and S ′ has L ′ singular st ates, L ′ < L . 12: α ← α \ α ( β = 0) ⊲ delete the index element of β = 0 . 13: α ′ ← α ( i |S ′ ) ⊲ α ′ corresponds to o nly S ′ . A P P E N D I X B W e ill u s trate an example of Theorem 4 here. In Fig. 6, a recei ved constell at i on of an SFS with v S F S 1 = i is i llustrated. The number o f MTs is u = 2 and 4 QAM modulation is employed. W e can observe the clashes clearly from the figure and their values can be calculated according to (25), e.g. 4 constellation points are superimposed at (0 , 0) and 2 are at (0 , 2 ) . Then the optim al b inary mapping matrix will encode the superimp osed constellation points in a clash to t h e s am e NCV and maxim ise the dist ance between different NCVs at the same time according to the design criteria. Fig. 7 il l ustrates the receiv ed constellation of all possible s u perimposed symbols of another SFS wi t h v S F S 2 = 1 / 2 + 1 / 2 i . In this case, 2 constel l ation points are superimpos ed at (0 , 1) , (0 , − 1) , (1 , 0) and ( − 1 , 0) , respectiv ely . The optimal m apping matrices Algorithm 3 Binary Matri x candidates Selection for Each AP (Of f-line Search Alg o rithm. Part II) 1: for l L ′ = 1 : L ′ do 2: S † L ′ − 1 ← S ′ \ S ′ ( l L ′ ) 3: θ L ′ − 1 ← E ( S † L ′ − 1 ) ⊲ Index set of S ′ excluding the l th element. 4: f or l L ′ − 1 = θ L ′ − 1 do 5: S † L ′ − 2 ← S † L ′ − 1 \ S † L ′ − 1 ( l L ′ − 1 ) 6: θ L ′ − 2 ← E ( S † L ′ − 2 ) 7: . . . 8: f or l L ′ − n +1 = θ L ′ − n +1 do 9: for i 1 = 1 : K do 10: . . . 11: f or i n = 1 : K do 12: M =    M [ α ( l L ′ , i 1 )] . . . M [ α ( l L ′ − n +1 , i n )]    13: δ ← det ( M ) | F 2 ⊲ determinant over F 2 . 14: if δ = 1 the n 15: R ← R ∪ ( l L ′ · · · l L ′ − n +1 ; i 1 · · · i n ) 16: G ← G ∪ M ⊲ M ↔ G A ( k ) in G has unique address A ( k ) = ( l ( k ) L ′ · · · l ( k ) L ′ − n +1 ; i ( k ) 1 · · · i ( k ) n ) , k = 1 , · · · L ′ ! ( L ′ − n )! . 17: r etur n (21) 18: end if 19: end f or 20: end f or 21: end f or 22: end f or 23: end for 24: [ G A ( k 1 ) · · · G A ( k n ) ] ← X ( G ) ⊲ find n M from G s at i sfying bij ection relations ( l ( k e ) L ′ · · · l ( k e ) L ′ − n +1 ) ⇔ S ′ for k e = k 1 · · · k n . 25: for i = 1 : n do 26: Q i ← [ G i A ( k 1 ) · · · G i A ( k n ) ] ⊲ G i A ( k i ) = M [ α ( l ( k i ) L ′ − i +1 , i ( k i ) i )] 27: end for 28: Output: n stacks Q i with each includi ng L ′ binary matrices. for S F S 1 and S F S 2 are differ ent due to the differ - ences between the cl ash ed constellation points. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Fig. 6. Constellation of the Received Signals at AP , v S F S 1 = i . -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Fig. 7. Constellation of the Received Signals at AP , v S F S 1 = 1 / 2 + 1 / 2 i . Then we consider a recei ved constellati on with a non-singular fading with v nS F S = 7 / 10 + 7 / 10 i illustrated in Fig. 8 . Clearly , no ne of the constel- lation points are sup erim posed but we can easily indicate different clust ers by the distances between constellation po ints. Al so we can find that the dis- tance, which refers th e absolut e value and rotati on angle in (13), between v S F S 1 and v nS F S is smaller than t h at between v S F S 2 and v nS F S . Then according to the unambiguo u s detection theorem, the 4 points around (0 , 0) in Fig. 8 should be mapped to the same NCV to maximise the inter-cluster dist ance. The same criteria s hould be sati s fied by the 8 po ints near (0 , 2) , ( − 2 , 0 ) , (0 , − 2) and ( 2 , 0) . Then th e initi al cluster groups in Fig. 6 and Fig. 8 are the same which leads to the same optimal mapping matrices could be used i n both circumstances. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Fig. 8. Constellatio n o f the Receiv ed Signals at AP , v nS F S = 7 / 10 + 7 / 10 i . R E F E R E N C E S [1] M. V . Clark, T . M. III Willis, L. J. Greenstein, A. J. Rustako, V . 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