Non-memoryless Analog Network Coding in Two-Way Relay Channel

Non-memoryless Analog Net work Coding in T wo-W ay Relay Channel Shengli Zhang 1 , Soung-Chang Liew 2 , Qingfeng Zhou 3 , Lu Lu 2 , Hui Wang 1 1 Department of Comm unicaton Engin eering, Shenzhen Univer sity, China 2 Department of I nformation Engineering, Chinese University of Hong Kong 3 Department of Ele ctronic and Computer E ngineering, Hong K ong University of Science a nd Technology {zsl, wanghsz} @szu .edu.cn, {soung, l l007}@ie.cuhk.edu.hk, enqfzhou @eie.polyu.edu.h k Abstract: Physical-layer Network Coding (PNC) can signif icantly improve the thro ughput of two-way relay chann els. An interestin g variant o f PNC is Analog Net work Cod ing (ANC). A lmost all ANC schemes propo sed to date, however , o perate in a symbol by symb ol manner (memoryless) and cann ot exploi t th e redundant information in channel-coded pack ets to enhance performance. This paper proposes a n on-memoryless ANC scheme. In particular , we design a soft-input soft-output decoder f or the relay node to process the superimposed packets from the two end nodes to yield an estimated MMSE packet for f or warding back to the end nodes. O ur decoder takes into account the correlation among d iff erent symbols in t he packets du e to channel coding, and pro vides significantly i mproved MSE performance. Our analysis shows th at the SNR improvem ent at th e relay n ode is lower bounded by 1/ R ( R is th e code rate) with the simplest LDPC co de (rep eat code). The SNR improvement is also verified by numerical simul ation with LDPC c ode. Our r esults indicate that LDP C codes of different d egrees are p referred in diff erent SNR regions. Generally speaking, smaller d egrees are preferred for lower SNRs. I. Introduction Physical layer network c oding ( PNC) [1] is a promising technique to improve the t hroughput of a two-way rel ay channel (TW RC), in which two end nodes exchange information via a rela y node. In PNC, the t wo e nd nodes send packets simultaneo usly to the relay node. T he relay node then transforms the superi mposed pa ckets to a net work-coded packet for b roadcast back to the end nodes. Each end node then uses their self infor mation to extract the packet o f t he other end nodes fro m the net work-coded p acket. T able 1. C lassificat ion of PNC schemes according to the processing at the relay node Memoryless Re lay Non-memoryl ess Relay Finite Fiel d e.g. PNC [1, 2, 5] e.g. Coded PNC [7, 8]c Infinite Field e.g. AN C [5, 6, 9] ? Beyond the above set-up, there are difference variants of P NC schemes. W e could classif y different P NC sche mes i nto f our categories, as in T able. 1. Fir st, the schemes can be classifie d into memor yless relay a nd non-me moryless rela y , according to whether s ymbol-by-sy mbol (memoryless) o r packet-b y- packet ( non-memoryless) processing is perfor med at the rela y . In our pape r , non-memor yless relay e xploits the c hannel decoding process to enhance the esti mate of the orginal packet; however , it is not necessar y to correctl y decod e the packets as in the trad itional d ecod e-and-forward sc heme [10 ]. Generally spe aking, memoryless rela y sche mes are si mpler to implement. However, for channel-coded packets, memor yless relay schemes do not make use of the correlations among symbols to re move corruption s due to noise. As a result, noi se can acc umulate in a multi-ho p netw ork with multip le re lays. By contrast, the non -memoryless schemes can overcome noise accumulatio n. Second, P NC schemes c an be classified into finite-field P NC (PNCF) and infinite- field PNC (PNCI) [5] according to the field o ver w hich t he net work coding at t he rela y oper ates. Generally speaking, P NCF generates less extra neous information at the relay a nd is more efficient for do wnlink transmission; while PNCI can match the t wo uplink c hannels to reduce e stimation errors. Since different sc hemes are preferred in dif ferent scenario s, most of them a re of interest and have be en studied to a cer tain extent. Ho wever , to the best of our knowledge, there have been no propo sals o r investigations on non-me moryless PNCI schemes, such a s Analo g Net work Codin g ( ANC). T he reasons could be that: i) it is not straightfor ward for the relay to decode the received p acket h 1,3 X 1 +h 2,3 X 2 ( X 1 and X 2 are th e packets fro m t he t wo end nod es) since it i s not a val id code word; ii) for m ost chan nels, the ach ievable end-to -end code rates are beyond the capa city of the relay node, which does not have self -information o f t he end nodes. The situati on faced by the relay in a PNC s ystem, therefore, is di ff ere nt from t hat in traditional channel coding, which focuses on code ra tes within the capacity r egion 1 . In this paper , w e propose a novel channel decoding (non-memor yless) sc heme at the relay node to enhance t he performance of ANC. Note that with ANC, we do not ai m to successfully deco de X 1 , X 2 , or 1 2 X X ⊕ , but to find a good estimate for h 1,3 X 1 +h 2,3 X 2 . W e assume the same LDPC code i s used at b oth end no des. Fir st, the joint proba bility of the t wo symbols from the end nodes is estimated from the r eceived superimposed symbol. E xploiting the correla tions a mong the coded symbols, this joint probability is then refined by a novel be lief p ropagation algorithm [12] . B ased on the refined joint probability , an MM SE estimate of h 1,3 X 1 +h 2,3 X 2 is obtained as the network-cod ed signals for bro adcast back to the two e nd nodes. As shown in [5, 1 1], MMSE estimate at the rela y achieve s better power allocation to symbols with varying reliabilities; a nd achieves smaller mean square uncorrelated error ( MSUE) at the end n ode th an any ot her forms of estimate for h 1,3 X 1 +h 2,3 X 2 . W e analyze the performance of our scheme in ter ms of the SNR improve ment for a g iven c hannel cod e rate. An achievab le lo wer bound is given. 1 Recently , there have been sever al works on decoding channel code with beyond-capacity rate for point-to-point chan nel [10, 1 1, 17]. II. System Mode l and Notat ions System: W e consider the t wo-way rela y c hannel as s hown in Fig.1, in which nodes N 1 and N 2 exchange infor mation with t he help of relay node N 3 . W e assume that all nodes are half-duplex, i.e. , a nod e c annot recei ve a nd tra nsmit simultaneousl y . This i s an assumption arisi ng from practical consideratio ns b ecause it is difficult for the wireless nod es to remove the str ong interference of its o wn tra nsmitting signal from the recei ved signal. W e also ass ume t hat there is no direct link bet ween nodes N 1 and N 2 . An example in pra ctice is a satell ite communication s ystem in w hich the t w o end nodes on t he earth can o nly communicate with each other v ia the relay satellite. 1 N 2 N 3 N Fig 1: T wo-way relay channel. In this p aper, S i is a vector denoting the uncod ed source packet of node N i ; X i denotes the p acket after channel coding; A i d enotes the transmitted p ackets a fter B PSK m od ulatio n; and Y i denotes the r eceived base-band packet at node N i . Lowercase le tters, { 0,1 }, i s ∈ { 1 , 1 }, i a ∈ − { 0,1 } i x ∈ , o r , i y ∈  denote a symbol in the correspondin g packet. The complex channel coeff icient fro m n ode N i to n ode N j , h i,j , is assumed to be invariant during one pac ket tran smission, and varies independ ently between d iff erent pac kets. The two-phase trans mission sc heme i n P hysical la yer Network Coding (PNC) con sists of an uplink phase a nd a downlink phase. In the uplink p hase, N 1 and N 2 transmit to N 3 simultaneousl y . Therefore, N 3 rece ives 3 1,3 1 2, 3 2 3 1,3 1 2, 3 2 3 ( 1 2 ) ( 1 2 ) y h a h a w h x h x w ′ ′ = + + ′ = − + − + (1) where 3 w ′ is the noise at N 3 , whi ch is complex Gaussian with variance 2 σ (identical for all the three nodes), W e assume that the tra nsmit power , the phase diff erence betwee n the transmitted si gnal a nd the local signal at the receive node, and the cha nnel fadi ng effect at the received nod e N j are taken in to account b y h i,j , w hich is assumed to be complex Ga ussian variable with a given variance. W ith soft d ecision de modulatio n, the re ceived si gnal at N 3 can be expressed as ( ) 3 3 1,3 2 , 3 1,3 1 2, 3 2 3 2 2 y y h h h x h x w ′ = − − − = + + (2) where t he Ga ussian noise 2 3 3 (0, ) w w CN σ ′ = − ∈ and its vector versio n for the o verall p acket is W 3 . Herea fter , we write the r eceived packet Y 3 as a function of t he transmitted packet h 1,3 X 1 +h 2,3 X 2 . In the do wnlink phase, a memoryless s ystem would j ust receive a superi mposed symb ol 3 y , process it and br oadcast it to both end nodes. For example, the ANC scheme [9 ] simply amplifies 3 y by a fixed factor and broa dcasts 3 3 x y α = , where α is a scaling factor to satisfy the p ower constraint. Indee d, 3 x is a linear MMSE esti mation of h 1,3 x 1 +h 2,3 x 2 . W e can write th e signals rec eived by N 1 a nd N 2 as 1 3,1 3 1 2 3 , 2 3 2 y h x w y h x w = + = + (3) Consider N 1 . It ob tains its target i nformatio n b y subtracti ng the self-infor mation as 1 1 3,1 1,3 1 3,1 2, 3 2 3,1 1 ' y y h h x h h x h w w α α α = − = + + . (4) The above equation is of the same form as that in a point-to-point transmission system: it consists of the ta rget signal p lus noise. Thus, N 1 could decod e S 2 as in the point-to-point s ystem. Examples o f other m emor yless schemes are [5, 6 ]. T his paper is diff ere nt from these previo us works in that it prop oses to use non -memoryless esti mation to e xploit t he cor relations among the symbols in a channel-cod ed packet. In this ca se, the e stimate of each symbo l h 1,3 x 1 +h 2,3 x 2 depends on the whole received p acket, and it c an be expressed as 3 3 ( ) x func Y = . (5) LDPC codes: An integral p art of o ur new signal processing s che me at t he relay is in fact a c hannel d ecoding scheme, except that we do not a im to al ways s uccessfully dec ode the individual p ackets at the relay . The chan nel de coding scheme dep ends on t he channel codes ado pted at the end nodes. For simplicity , we assume the same LDPC cod e [14] is used at the end nodes. LDPC code is attractive in that it is capacit y appr oaching [1 5]. An LDPC code can be characterized by a spar se parity c heck matrix H . Suppose the uncoded pa cket length is N- K and the coded packet length is N . Deno te the ( ) N K N − × parity check matrix by H and the corresponding ( ) N N K × − generator matrix by G . Then we have ( ) 0 i i i i X S GS HX = Γ = = . (6) and the cod e rate is 1 - K / N . III. Channel Decodi ng Algorit hm This section elab orates the proposed channel decodin g scheme at t he relay . Althou gh we foc us on regular LDPC code in this p aper , extensions to o ther channel co des, such as RA code a nd T urb o code, are str aightforward. At the r elay , the target is t o obtain a refined e stimate of 1,3 1 2, 3 2 h X h X + based on the received packet Y 3 and the redundanc y contained i n the c hannel-cod ed packets, X 1 and X 2 . T o do so, we use b elief prop agation to dec ode the joint prob ability de nsity function s of the 2-tup le x =( x 1 , x 2 ), denot ed by 1 2 ( , ) P x x , from Y 3 . In or der to perform c hannel decodin g, we regard the 2 N -t uple ( X 1 , X 2 ) as one virtual co de and the corresp onding vector ver sion co nstraint is 1 2 1 2 ( , ) ( , ) (0, 0) HX H X X HX HX = = = . (7) Consider one check node which is connected b y k ed ges in Fig. 2. The abo ve constraint can be expressed in scalar for m as 1 2 1 1 ( [ 1 ], [ 2], [ ]) ( [ ], [ ]) (0, 0 ) k k i i g x x x k x i x i = = = = ∑ ∑  (8 ) From the above equation, the virtual code is eq uivalent to a 4-ary LDPC co de, and it s c orre sponding T a nner Grap h is shown in Fig. 2 . The belie f pro pagation deco ding algorit hm can then be designed ac cordingly . Let P h, t denote the message p assed be tween a c heck node and a variab le node (code nod e). T he message i s associated with the edge fro m node h to node t , where o ne of h or t is a variable node, a nd the other is a c heck node. Le t , [ 1 , ] k P k N ∈ , b e the message from the k -th (o rdered from top to bo ttom as i n Fig. 2) evide nce nod e to the k -th cod e no de, where N is t he length of t he cod ed pac ket. Fig 2: T ann er G raph of the virtual (3, 6) L DPC code Message fo rm: , 0, 0 0,1 1, 0 1,1 ( , , , ) h t P p p p p = is a vector , in which p i,j is t he prob ability that the correspo nding variable nod e ( h or t ) takes on the pair of values o f ( , ) , { 0,1 } i j i j ∈ . 0, 0 0 ,1 1, 0 1, 1 ( , , , ) k P p p p p = is a vector , in which p i,j is the prob ability that the k th coded symbol is ( i, j) gi ven the k- th received s ymbol. Message In itial V alues : All t he messages associa ted with the edge s in Fi g. 2 are initialized to (1 /4, 1/4, 1/4, 1/4) excep t for the messages o n the edges incide nt to the evidence nod es, which conta in information on the received signal. The message fro m an evidence node is computed from the correspondin g received symbol 3 y as follo ws: ( ) 0, 0 0 ,1 1,0 1 ,1 1 2 3 1 2 3 1 2 3 1 2 3 2 2 3 1,3 2 , 3 3 1,3 2 ,3 2 2 2 2 3 1,3 2, 3 3 1,3 2 ,3 2 2 ( , , , ) Pr( 0, 0 | ), Pr( 0, 1 | ), Pr( 1 , 0 | ), Pr( 1 , 1 | ) | | | | 1 exp( ), exp( ), 2 2 | | | | exp( ), exp( ) 2 2 k P p p p p x x y x x y x x y x x y y h h y h h y h h y h h β σ σ σ σ = = = = = = = = = =  − − − − − + =     − + − − + +    (9) where β is a nor malizing factor to make sure that the four prob abilities sum to one. Message Up date Ru les : Parallel to the generic u pdate rules in [1 2], w e also h ave the same mes sage upd ate rules a t o ur check nodes and variable nodes. N ote t hat the messages fro m the evid ence nodes to the code no des remain the sa me without b eing cha nged d uring the iterations of t he decod ing process. Fig 3: message updates for deco d ing the virtual co de in Fig. 2 Update Equa tions fo r Output Message s Goin g Ou t of a V ariab le Node When the node d egree is 2, each output m e ssage is the same as the other i nput message. Fig. 3 (a) illustrates the case in whic h the node degre e is 3 . When the pro bability vector s of the two input messages, 0, 0 0 ,1 1, 0 1, 1 ( , , , ) P p p p p = and 0, 0 0,1 1,0 1, 1 ( , , , ) Q q q q q = (associated with the edge from y to x a nd the ed ge from s' to x, respectivel y), arrive at a cod e node of de gree three (except the lowest cod e node), the prob ability that the co de symbol is (0, 0) is obtained as follows: 0, 0 0 , 0 Pr( (0, 0) | , ) 4 x P Q p q λ = = (10) where = P r( ) Pr( ) / Pr( , ) P Q P Q λ and the two i nput me ssages are assu med to be independen t given the va lue of t he variab le node, i.e., Pr( | , ) Pr( | ) P Q x P x = . Given the l -depth neighborhoo d of the edge is c ycle free (c ycle free condit ion), this assumptio n is true for iterations up to l in the decoding algorithm. As in the pro of for the LDPC codes in [ 14], the prob ability t hat t he cycle free conditio n is tr ue for our coder in Fi g. 2 should also go to 1 as the le ngth of t he co de goes to infinity . That is, l beco mes larger and larger . In a similar way , we can obtai n that 0,1 0,1 Pr( (0,1 ) | , ) 4 x P Q p q λ = = , 1,0 1,0 Pr( ( 1 , 0) | , ) 4 x P Q p q λ = = , and 1, 1 1, 1 Pr( ( 1 , 1 ) | , ) 4 x P Q p q λ = = . T hus, the outp ut mes sage at the variable node is 0, 0 0 , 0 0 ,1 0,1 1 , 0 1,0 1,1 1, 1 ( , ) 4 ( , , , ) VAR P Q p q p q p q p q λ = (11) Since the s ummation o f the thre e probab ilities should b e 1, we require 0, 0 0 , 0 0,1 0 ,1 1,0 1,0 1, 1 1, 1 1 / ( ) / 4 p q p q p q p q λ = + + + for normalizatio n. When the node degre e is k , a nd the k -1 input messages a re 1 1 , k P P −  . T hen, w e can obtain the o utput message in an induction way as 1 1 1 1 1 1 0, 0 0,1 1 ,0 1 ,1 1 0 0 0 ( , ) ( , , , ) k k k k k i i i i i i i i VAR P P p p p p λ − − − − − = = = = = ∏ ∏ ∏ ∏  (12 ) where λ is the nor malization fa ctor . Update Eq uations for Ou tput Messa ges Going O ut of Check Nodes: When the node d egree is 2, each output m e ssage is the same as the other i nput message. Fig. 3 (b) illustrates the c ase in which the node degree is 3. Based on the f de fined in ( 8), a nd using s imilar computatio n as in (8), the p robability t hat the var iable node symbol x is (0,0) given the t wo inp ut m essages 0, 0 0 ,1 1, 0 1, 1 ( , , , ) P p p p p = and 0, 0 0 ,1 1,0 1 ,1 ( , , , ) Q q q q q = (associated w ith the edg e fro m x' to s and the ed ge fro m x ' to c, resp ectively) is 0, 0 0, 0 0 ,1 0 , 1 1,0 1,0 1 ,1 1,1 0 ,0 Pr( (0, 0) , ) = ( , ) x P Q p q p q p q p q g P Q = = + + + (13) In a similar way , we ca n o btain that Pr( (0,1 ) | , ) x P Q = , Pr( ( 1 , 0 ) | , ) x P Q = and Pr( ( 1 , 1 ) | , ) x P Q = , which ar e denoted by 0,1 (P,Q) g , 1,0 (P,Q) g , 1,1 (P,Q) g respectively . As a result, t he output message a t the check node is 0, 0 0 ,1 1,0 1, 1 ( , ) ( ( , ), ( , ), ( , ), ( , )) ( , ) CHK P Q g P Q g P Q g P Q g P Q g P Q = = (14 ) When t he node degree is k an d the k -1 i nput messa ges to the variable node are 1 1 , k P P −  . Then, we can o btain the o utput message in a recursive manner a s 1 1 1 2 3 1 ( , , ) ( ( ( , ), ) ) k k CHK P P g g g P P P P − − =    (1 5) S top R ules : The messages ar e upda ted in an itera tive wa y and the iteration stops when the follo wing rules are gi ven. W e first c heck whether the decod ing of X 1 is succe ssful. W e make a hard decision on x 1 by takin g the marginal pro bability: 0, 0 0 ,1 1,0 1, 1 1 0 if 1 otherwise p p p p x + ≥ +  =    (16) If 1 0 HX =  , then the decod e of X 1 is succe ssful and we stop its ite ration by setting the mes sages associated with ea ch ed ge as 0, 0 1,0 0 ,1 1, 1 1 0, 0 1,0 0 ,1 1, 1 1 ˆ ( , , 0, 0) if 0 ˆ ( 0, 0, , ) if 1 p p p p x P p p p p x + + =  =  + + =  . (17) It can be ver ified that the val ue of 1 X  will not change a n y more with the message update r ules under the setting in (1 7). This is equivalent to subtr acting the interfer ence o f X 1 fro m Y and deco ding X 2 alone. Similarl y , we can make a hard decision on X 2 and check whether it has b een cor rectly deco ded. If both X 1 and X 2 have been correctly decoded , the iteration stops. H owever, in general, it may not always be p ossible to decod e the pac kets succe ssfull y . So we need to set a maximum nu mber of iter ations. The Rela y Outpu t : When the deco ding p rocess s tops, the rela y will ge nerate the output symbols based on the output of the decoder . If the number o f iterations is les s than the maximum value, th en both pa ckets have b een corr ectly decod ed, a nd we make a hard dec ision on the p robab ility tuple. Fina lly , t he k -th s ymbol to be b roadcast is generate d by 3 , , {0 ,1 } [ ] [ ] [( 1 2 ) ( 1 2 ) ] i j i j x k p k i a j b ∈ = × − + − ∑ . (18) In the abo ve eq uation, we si mply set ( a , b ) to ( h 1,3 , h 2,3 ) to match t he uplin k channel 2 . T hen, we ca n regar d the broad cast symbol as { } 3 1,3 1 2 , 3 2 3 | x E h x h x Y = + . (19) In other word s, x 3 is a n MMSE estimate based o n the observation of the whole pac ket. T his co ntrasts with t he memoryless PNCI MMSE estimate gi ven b y { } 3 1 ,3 1 2 ,3 2 3 | x E h x h x y = + . IV. Perfo rmance Analysis MMSE relay is a good rela y scheme [5, 11] which c an minimize the MSUE and achieve smaller BER at the e nd nodes. M MSE is closel y relate d to uncoded BER and mutual information [16 ]. In this sect ion, we a nalyze the p erforman ce of the pr oposed non-me moryless A NC scheme in ter ms o f minimum m ean s quare error (MMSE ) at th e r elay nod e. The MSE of the pr oposed scheme is defi ned as { } 2 3 1,3 1 2 ,3 2 | | mse E x h x h x = − − . (20) 2 Besides pure r eal-field network coding descri bed above, a hybrid PNC is also possible : if the two end packets can be deco d ed, the broadcas t packet could be 3 1 2 X X X = ⊕ ; otherw ise, it i s 3 X as given by (18) (an indica tor in the packet header can be used to indicate w hich has been sent). Eve n for our real-fie ld network coding form as in (1 8), there are more possible values for ( a , b ). T his consideration w ill be addressed in future w ork. For comparison, w e also pr esent the MM SE o f the “conventional” memoryless (un-deco ded) ANC given in [5]. The MM SE estimate o f the received signal is [5 ] 3 , 1,3 2, 3 , ' ( ( 1 2 ) ( 1 2 )) i j i j x p h i h j = − + − ∑ (21) where p i,j takes o n the values in (9). T he corr esponding MSE is { } 2 3 1,3 1 2, 3 2 _ | ' | mse con E x h x h x = − − ( 22) When the chan nel c oefficients are fi xed, mse_con is a function of the Gau ssian noise variance 2 σ , and we denote it by 2 1 _ ( ) mse con f σ = (23 ) W e no w i nvestigate the SNR i mprovement, which is defi ned as the extra SNR neede d by the memoryles s scheme to achieve t he same MSE as our non-me moryless sche me. Specifically , the SNR impr ovement is 1 1 1 1 ( ) _ 1 0 log ( _ ) f mse SNR snr c on snr f mse con − − ∆ = − = . (24) Lower bound: W e first present a lower bound of the SNR improve ment. Consider the naïve rep eat cha nnel code in which each symb ol is rep eated q ti mes, i.e. , t he (1 , q ) LDP C cod e. T he r elay node may co mbine the same symbols with maximu m ratio combinatio n (MRC) 3 and the resultin g SNR i s raised by a factor of q . Therefore , the SNR improve ment i n (24) is also 10 l o g( ) q , w it h respect to the me moryless ANC scheme. This result can be easil y exte nded to the case of non-integer repeat factor q by r epeating some bits q     times and other bits q     +1 ti mes. Since repeat code is o ne specific LDPC code, we have Propo sition 1 as follo ws: Pr opositio n 1: For a channe l code with rate R , the SNR improvement i s lower b ounded by 10 log( ) SNR R ∆ ≥ − . (25) The simple repe at c ode works well i n lo w S NR. W hen the channel is go od, the mo re so phisticated LDPC cod es per for m better and the lower bou nd 10 l og( ) R − is not ti ght in general. V. Numerical Si mulation This section gives numerical simulation results to show the performance i mprovement o f the p ropo sed non-memoryle ss ANC sche me. In our simulation, the parit y check matrix H of the regular LDPC co de is rando mly gener ated according to the Gallager ’ s method. T he length of the code d pa cket N =180 0, while three 3 The BP algor ithm is equivalent to MRC for the (1, 2) LDP C code. different colu mn degree and row degree pairs, (3, 6 ), (2, 4 ) and (1, 2), are explored to inv esti gate the perfor mance of non-memor yless A NC u nder different c hannel cod es (fixed rate 0.5). T he (1 , 2) case in fact correspo nds to a naïve rep eat code. T he max imu m number o f decoding iterations is set to 20. As in (9), the cha nnel d ecoding only d epends on t he distance between di f ferent constellatio n points of 1 , 3 1 2, 3 2 h x h x + . Here we si mulate the p erformance improvement of non-me moryless ANC under different distances among the co nstellatio n p oints and simpl y set h 1,3 and h 2,3 to real values.. The SNR in our s imulation is defi ned as 2 1 / σ , where σ is the noise variance at the relay nod e. In Fig. 4, the MSE at the rela y node is an average o ver 100 packets and the p arity c heck matri x H is r egenerated for each packet i n a ra ndom way for non-me moryless and memoryless ANC with the chan nel co eff icie nts o f 1 , 3 2 , 3 1 h h = = . As shown in the figure , the ( 1, 2) LDP C code p erforms best and the (3, 6) LDPC c ode performs worst when the SNR i s less then 2. 5 dB . When the S NR is more t han 2.5 dB, the (3, 6) LDPC code b ecomes t he b est while t he ( 1, 2) LDPC code is the worst. For the SNR re gion be ing simulate d, the BER at the rela y node is al ways abo ut 0. 2, since the dec oder can no t differentiate the two t uples (1, -1) and (-1, 1 ) wh en 0 is received. However, this is no t i mportant because we need at the rela y is t he esti mate for 1 , 3 1 2, 3 2 1 2 h x h x x x + = + , not 1 2 ( , ) x x . Fig 4: MMSE perfo rmance when h 1,3 =1 and h 2,3 =1 In Fig. 5 , we simula te the MMSE performance for r andom channel co efficients. W e simulate 1 000 packets to o btain the average pe rfor mance and the channel coe fficients h 1,3 and h 2,3 are randoml y generated for eac h packet with Raylei gh distribution. W e can see that the p erformance of all the thr ee coded schemes degrade with Rayleigh distrib uted cha nnel coefficients. However, the relative per formance is the same as in the previo us simulatio n. All the si mulation res ults sho w that the complex LDP C code s are good a t d istinguishing the constellation p oints spread far apart, wh ile t hey are bad at distinguis hing compa ct constellatio n p oints. The simple repeat co de w or ks in the oppo site way . T he int uition is as follo ws. In lo w SNR regio n, there is a hi gh prob ability that in a co mplex code, a check node is co nnected to t wo or more ver y p oorl y r eceived symbols, and t hat there is one or more bad symbols participa ting in the computatio n of (13) for a variable node. The uncertai nty in variable nodes prop agates to other variab le nodes in the B P inferenci ng process under a complex L DP C code. In co ntrast, the simpler LDPC codes perform well because the variable nodes are not as intertwined to gether . In high SNR region, there is a small probabilit y t hat the check node co nnects to t w o ver y bad symbols. In thi s case, the certainty in variable nodes with go od symbols propagates to other variab les nodes. Fig 5: MMSE perfo rmance when h 1,3 , h 2,3 are Ray leigh distributed random variables with unit vari ance VI. Conclusion In this p aper, we prop osed a memory a mplify-a nd-forward network codin g relay scheme for t wo way r elay channels. Specifically , we pr opose a new sof t input so ft o utput deco ding algorithm to refine the estimate of superi mposed signal at the relay n ode. Our analysis showed that the ne w sche me ca n improve the SN R at the rela y b y 1/R at lea st with rep eat co de. 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