(PMF)between adjacent nodes,the marginal probability of Different from these MUD approaches like LMMSE-IC [22] random variables can be obtained.Generally,suppose client where all random variables sV are approximated by an subset Vi=).For Vi.let the j-th baseband independent circularly symmetric normal distribution,which symbol vector and codeword lost the information of the relationship between these random bit vector.Based on the variables.In order to efficiently utilize the superposition of factorization(6),we build the factor-graph g for decoding RF signals to aggregate updates,here we treat each random the sourceword-sum in PhyArith.In g,as shown in Fig.5,Yj variable pair (svj sv),v,vrE Vi is mutual dependent, denotes the j-th observation check node.And for client subset and use the following function derived from the probability Vi,we let C denote its j-th LDPC codes check node.X density function of multivariate normal distribution to approx- its LDPC codes variable node(corresponding to variable xi.), imate above PMF Mii its modulation check node,Si;its modulation variable Py,→S.（(si,i）=A·exp node(corresponding to variable si.j).Zij its sourceword-sum (-Real(ij -Hsij)TIReal(i.j-Hjsij)/2), (8) variable node (corresponding to variable zi.j),and i.j its sourceword-sum check node. whereis the channel gain matrix of client subset Vi.Covariance matrix Tij is obtained via (22) in Appendix-D.Vector ui.i is given by h=y-∑hE(s), vEV/V where E(sv.j)is the mean of random variable s.j,v Vi,ob- tained based on PMF Ps(sj).And operation Real(): Clw1→R2 is defined as Real(lao,…,aul-i]T)e [Re(ao),Im(co),..,Re(awl-1),Im(awI-1)]T. Algorithm 1 Decoding process of the sourceword-sum LDPC codes variable node Modulation variable node Sourceword-sum variable node nput:g,h,yi where v∈V,j∈{0，…，m-l} LDPC codes check node Modulation check node Sourceword-sum check node Observation check node 0 utput::where i∈{0，…，川-1}，j∈{0，…，k-1} 1:for iterator =0 to MaxIterationNum-1 do Fig.5.Example factor-graph for the sum-product algorithm derived decoder 2: for all variable nodes Si.j,Xi.j,Zi.jg do in PhyArith whereV=2 3: propagate their PMFs given by (11)(12)(13)(14) (15)(16)to their adjacent check nodes B.Decoding of the Sourceword-Sum 4: end for The decoding process of the sourceword-sum on factor- 5: for all check nodes Yi.j,M,i,Ci,j,∑i,i∈gdo graph G is illustrated in Algorithm 1.Given factor-graph g, propagate their PMFs given by (8)(17)(18)(19)(20) the channel gain vector hi,and obersvation yi,where vV, (21)to their adjacent variable nodes j∈{0，.·，m-l},Algorithm1 returns the decoding result 7: end for 2与where i∈{0，·，以-1}，j∈{0，…，k-1}.Algorithm 8:end for 1 has MaxIterationNum iterations.In each iteration,on factor- 9:fori=0:川-1，j=0:k-1do graph g,it first parallelly propagates PMFs from all variable 1o:take()⑦to obtain2 nodes to their adjacent check nodes,and then propagates PMFs 11:end for from check nodes to their adjacent variable nodes.And after these iterations,it takes the following formula to obtain the final decoding result C.Verification of the Sourceword-Sum Here we show how to verify the correctness of the obtained 2=arg max P,→Z（2i,j (7) sourceword-sum zi.Note that for each correct zi -Bb+ 2. where P()denotes the PMF of random variable …+月evI-ib4vI-i,we have zi.j,propagated from check node Si.j to variable node Zi.j, Zi mod 2 given by (21)in Appendix-C. The most challenging part in Algorithm 1 is how to effi- =(mod2)beo⊕…⊕（月e411-.mod2be4wI- ciently and accurately obtain the PMF of si.j propagated from =bo⊕…⊕bwI-a' Yi to Si.j,given by Py→5，(sij) where z;mod2≌[2毫0mod2,·,zk-1mod2].According to (4),we have =A·∑[Pyls0,…,sv1-1)ΠPs,→y(s小 5,3 CRC(zmod2)=CRC(bo)⊕…⊕CRC(b4v-)=0.(PMF) between adjacent nodes, the marginal probability of random variables can be obtained. Generally, suppose client subset Vi = {vi0 , · · · , vi|Vi |−1 }. For Vi , let the j-th baseband symbol vector si,j = [svi0 ,j , · · · , svi |Vi |−1 ,j ] T , and codeword bit vector xi,j = [xvi0 ,j , · · · , xvi |Vi |−1 ,j ] T . Based on the factorization (6), we build the factor-graph G for decoding the sourceword-sum in PhyArith. In G, as shown in Fig. 5, Yj denotes the j-th observation check node. And for client subset Vi , we let Ci,j denote its j-th LDPC codes check node, Xi,j its LDPC codes variable node (corresponding to variable xi,j ), Mi,j its modulation check node, Si,j its modulation variable node (corresponding to variable si,j ), Zi,j its sourceword-sum variable node (corresponding to variable zi,j ), and Σi,j its sourceword-sum check node. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · LDPC codes variable node LDPC codes check node Observation check node Modulation variable node Modulation check node Sourceword-sum variable node Sourceword-sum check node C0,0 C0,n−k−2 C0,n−k−1 Z0,0 Z0,k C1,0 Z1,0 Z1,k X0,n−1 X0,0 X1,n−1 X1,0 Σ0,0 Σ0,k Σ1,0 Σ1,k Y1 Ym−1 M0,0 M0,m−1 M1,0 S0,0 M1,m−1 S1,0 S0,m−1 S1,m−1 Fig. 5. Example factor-graph for the sum-product algorithm derived decoder in PhyArith where |V| = 2 B. Decoding of the Sourceword-Sum The decoding process of the sourceword-sum on factor￾graph G is illustrated in Algorithm 1. Given factor-graph G, the channel gain vector h j v , and obersvation yj , where v ∈ V , j ∈ {0, · · · , m − 1}, Algorithm 1 returns the decoding result z ∗ i,j where i ∈ {0, · · · , |V|−1}, j ∈ {0, · · · , k−1}. Algorithm 1 has MaxIterationNum iterations. In each iteration, on factor￾graph G, it first parallelly propagates PMFs from all variable nodes to their adjacent check nodes, and then propagates PMFs from check nodes to their adjacent variable nodes. And after these iterations, it takes the following formula to obtain the final decoding result z ∗ i,j z ∗ i,j = arg max zi,j PΣi,j→Zi,j (zi,j ), (7) where PΣi,j→Zi,j (zi,j ) denotes the PMF of random variable zi,j , propagated from check node Σi,j to variable node Zi,j , given by (21) in Appendix-C. The most challenging part in Algorithm 1 is how to effi- ciently and accurately obtain the PMF of si,j propagated from Yj to Si,j , given by PYj→Si,j (si,j ) =A · X ∼si,j P(yj |s0,j , · · · , s|V |−1,j ) Y i6=i ′ PSi ′,j→Yj (si ′ ,j ) . Different from these MUD approaches like LMMSE-IC [22] where all random variables sv,j , v ∈ V are approximated by an independent circularly symmetric normal distribution, which lost the information of the relationship between these random variables. In order to efficiently utilize the superposition of RF signals to aggregate updates, here we treat each random variable pair (svl,j , svl ′ ,j ), vl , vl ′ ∈ Vi is mutual dependent, and use the following function derived from the probability density function of multivariate normal distribution to approx￾imate above PMF PYj→Si,j (si,j ) = A · exp ￾ − Real(µi,j − Hi j si,j ) T Γ −1 i,j Real(µi,j − Hi j si,j )/2
, (8) where Hi j = [h j vi0 , · · · , h j vi |Vi |−1 ] is the channel gain matrix of client subset Vi . Covariance matrix Γi,j is obtained via (22) in Appendix-D. Vector µi,j is given by µi,j = yj − X v∈V /Vi h j vE(sv,j ), where E(sv,j ) is the mean of random variable sv,j , v ∈ Vi ′ , ob￾tained based on PMF PSi ′,j→Yj (si ′ ,j ). And operation Real(·) : C |U| → R 2|U| is defined as Real([α0, · · · , α|U|−1] T ) , [Re(α0),Im(α0), · · · , Re(α|U|−1),Im(α|U|−1)]T . Algorithm 1 Decoding process of the sourceword-sum Input: G, h j v , yj where v ∈ V , j ∈ {0, · · · , m − 1} Output: z ∗ i,j where i ∈ {0, · · · , |V| − 1}, j ∈ {0, · · · , k − 1} 1: for iterator = 0 to MaxIterationNum − 1 do 2: for all variable nodes Si,j , Xi,j , Zi,j ∈ G do 3: propagate their PMFs given by (11) (12) (13) (14) (15) (16) to their adjacent check nodes 4: end for 5: for all check nodes Yi,j , Mi,j , Ci,j , Σi,j ∈ G do 6: propagate their PMFs given by (8) (17) (18) (19) (20) (21) to their adjacent variable nodes 7: end for 8: end for 9: for i = 0 : |V| − 1, j = 0 : k − 1 do 10: take (7) to obtain z ∗ i,j 11: end for C. Verification of the Sourceword-Sum Here we show how to verify the correctness of the obtained sourceword-sum z ∗ i . Note that for each correct zi = βˆ vi0 bvi0 + · · · + βˆ vi |Vi |−1 bvi |Vi |−1 , we have zi mod 2 =(βˆ vi0 mod 2)bvi0 ⊕ · · · ⊕ (βˆ vi |Vi |−1 mod 2)bvi |Vi |−1 =bvi0 ⊕ · · · ⊕ bvi |Vi |−1 , where zi mod 2 , [z ∗ i,0 mod 2, · · · , z∗ i,k−1 mod 2]. According to (4), we have CRC(zi mod 2) = CRC(bvi0 ) ⊕ · · · ⊕ CRC(bvi |Vi |−1 ) = 0.