the same time-frequency resource,e.g..the same subcarriers=[..].These codewords of channel codes within the same orthogonal frequency-division multiplexing are then mapped into m baseband symbols via modulation, (OFDM)symbols in 802.1lax.For the j-th time-frequency denoted by s=[sv.0,..,sv.m-1],and sent out. resource unit (e.g.,1 subcarrier within 1 OFDM symbol),the uplink MU-MIMO channel can be presented as follows Quantized update vector A Yuoj Svo.j Wuo.j Slice =H, a. Quantized update yector chunk Quantized undate vecto chank yuwv1-1.j Svvl-13] WuU1-1.j Quantized update vector chunk a. Encod where wj is i.i.d.random variable following circularly nor- (source codes,checksum Bit-sequence CRC mal distribution C(0,202),and denotes the additive noise of antenna u for the j-th time-frequency resource unit.C Parity bits denotes the observation of u,sv.jES(where SCC)denotes Codeword x Modalate,interpolate,up-convert the baseband symbol of v obtained via modulation,and the channel gain matrix H[hh where vector RF signal =h以o…,h2,uw- Fig.3.Encoding process of each client V.OVERVIEW OF PHYARITH Here we provide an overview of PhyArith based on the A.Source Codes model of federated learning and uplink MU-MIMO channel Regarding the source codes,in the quantized update vector presented above.PhyArith consists of two phases,i.e.,the client-side encoding phase and the server-side decoding phase. chunk av=[.,dv.j,avj+1,..].each its element avj is a signed integer,and is encoded into c bits bvjo,bv.j At the client-side,each quantized update vector A?is sliced into chunks,which are encoded into aligned digital sequence within b =[6v.0,...,b.k1]using two's-complement rep- resentation [15].according to the following rule through the source codes and channel codes encoder by ap- pending checksums and parity bits,and simultaneously trans- (Lav./2--1]mod 2, av,j0, mitted through the uplink MU-MIMO channel.The source (2c+avj)/2-1-1]mod 2,o.w. (3) codes taken here are based on two's-complement representa- tion [15],and the channel codes are based on systematic low- where I=0,...,c-1,bv.jo is the most significant bit and density parity check(LDPC)codes [6],such that the additionis the least significant bit.Here we should ensure and subtraction operations during aggregation can be unified, 2e-1≥max{lau,llv∈V,j∈N}.Compared to other systems and error correction capability can be provided.The appended for representing signed numbers (e.g.,one's complement),the checksum is the cyclic-redundancy-checksum (CRC)[17]. advantage of two's-complement is that the fundamental arith- such that its linear property can be exploited to verify the metic operations of addition,subtraction,and multiplication correctness of the directly recovered summation.At the server- are identical to those for unsigned binary numbers,such that it side,from the superimposed RF signal uo,, 1 would be much convenient for our PhyArith to directly decode the summations of a large number of updates chunks are the summation of update vectors. directly recovered through a customized decoder derived from B.Checksum the sum-product algorithm.After verifying their correctness based on the linear property of CRC,these summations are The checksum appended to each bit-sequence of source averaged and assembled into (BA+1)/IVI,which codes is cyclic-redundancy-checksum (CRC)[17].Together is then taken to update the global model to train. with the bit-sequence encoded from a.they form the source- word b..We use CRC in PhyArith because it is a linear VI.ENCODING PROCESS OF PHYARITH function with the following property In this section,we demonstrate the encoding process of PhyArith.As shown in Fig.3,the quantized update CRC(b⊕b)=CRC(b)⊕CRC(b）)=0⊕0=0，(4) vector A of client v is sliced into fixed size chunks where denotes bitwise-XOR,and CRC()returns the check- first,i.e,Ag=L…,aw,…].For each chunk a= sum of input in the form of sequence of bits.This property ,+1,],vZ.it is encoded into a bit-can help us verify the correctness of the decoded summation sequence of the source codes.The obtained bit-sequence of of update vectors.We present this verification process in detail source codes is then appended with its checksum (we use in Section VII-C. cyclic-redundancy-checksum,i.e.,CRC,in PhyArith),and forms thek bits sourceword of the channel codes,denoted C.Channel Codes by vector b=[6.0,...,bv.k-].Each such sourceword is In order to provide error correction capability in PhyArith, further encoded into the n bits codeword of the channel codes the k bits sourceword be,consisting of the bit-sequence by appending a sequence of parity bits,denoted by vector encoded from quantized update vector chunk a and its CRC,the same time-frequency resource, e.g., the same subcarriers within the same orthogonal frequency-division multiplexing (OFDM) symbols in 802.11ax. For the j-th time-frequency resource unit (e.g., 1 subcarrier within 1 OFDM symbol), the uplink MU-MIMO channel can be presented as follows   yu0,j · · · yu|U|−1,j   = Hj   sv0,j · · · sv|V |−1,j   +   wu0,j · · · wu|U|−1,j   , where wu,j is i.i.d. random variable following circularly nor￾mal distribution CN (0, 2σ 2 ), and denotes the additive noise of antenna u for the j-th time-frequency resource unit. yu,j ∈ C denotes the observation of u, sv,j ∈ S (where S ⊂ C) denotes the baseband symbol of v obtained via modulation, and the channel gain matrix Hj = [h j v0 , · · · , h j v|V |−1 ], where vector h j v = [h j v,u0 , · · · , hj v,u|U|−1 ] T . V. OVERVIEW OF PHYARITH Here we provide an overview of PhyArith based on the model of federated learning and uplink MU-MIMO channel presented above. PhyArith consists of two phases, i.e., the client-side encoding phase and the server-side decoding phase. At the client-side, each quantized update vector Av t is sliced into chunks, which are encoded into aligned digital sequence through the source codes and channel codes encoder by ap￾pending checksums and parity bits, and simultaneously trans￾mitted through the uplink MU-MIMO channel. The source codes taken here are based on two’s-complement representa￾tion [15], and the channel codes are based on systematic low￾density parity check (LDPC) codes [16], such that the addition and subtraction operations during aggregation can be unified, and error correction capability can be provided. The appended checksum is the cyclic-redundancy-checksum (CRC) [17], such that its linear property can be exploited to verify the correctness of the directly recovered summation. At the server￾side, from the superimposed RF signal [yu0,j , · · · , yu|U|−1,j ] T , the summations of a large number of updates chunks are directly recovered through a customized decoder derived from the sum-product algorithm. After verifying their correctness based on the linear property of CRC, these summations are averaged and assembled into P v∈V (βvAv t +γv1)/|V |, which is then taken to update the global model to train. VI. ENCODING PROCESS OF PHYARITH In this section, we demonstrate the encoding process of PhyArith. As shown in Fig. 3, the quantized update vector Av t of client v is sliced into fixed size chunks first, i.e., Av t = [· · · , av, · · · ]. For each chunk av = [· · · , av,j , av,j+1, · · · ], av,j ∈ Z, it is encoded into a bit￾sequence of the source codes. The obtained bit-sequence of source codes is then appended with its checksum (we use cyclic-redundancy-checksum, i.e., CRC, in PhyArith), and forms the k bits sourceword of the channel codes, denoted by vector bv = [bv,0, · · · , bv,k−1]. Each such sourceword is further encoded into the n bits codeword of the channel codes by appending a sequence of parity bits, denoted by vector xv = [xv,0, · · · , xv,n−1]. These codewords of channel codes are then mapped into m baseband symbols via modulation, denoted by sv = [sv,0, · · · , sv,m−1], and sent out. Quantized update vector Av t Slice Quantized update vector chunk Quantized update vector chunk · · · Quantized update vector chunk av Encode (source codes, checksum) Bit-sequence CRC Encode (channel codes) Sourceword bv Parity bits Codeword xv Modulate, interpolate, up-convert RF signal Fig. 3. Encoding process of each client A. Source Codes Regarding the source codes, in the quantized update vector chunk av = [· · · , av,j , av,j+1, · · · ], each its element av,j is a signed integer, and is encoded into c bits {bv,j0 , · · · , bv,jc−1 } within bv = [bv,0, · · · , bv,k−1] using two’s-complement rep￾resentation [15], according to the following rule bv,jl = ( ⌊av,j/2 c−l−1 ⌋ mod 2, av,j ≥ 0, ⌊(2c + av,j )/2 c−l−1 ⌋ mod 2, o.w., (3) where l = 0, · · · , c − 1, bv,j0 is the most significant bit and bv,jc−1 is the least significant bit. Here we should ensure 2 c−1 ≥ max{|av,j ||v ∈ V, j ∈ N}. Compared to other systems for representing signed numbers (e.g., one’s complement), the advantage of two’s-complement is that the fundamental arith￾metic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers, such that it would be much convenient for our PhyArith to directly decode the summation of update vectors. B. Checksum The checksum appended to each bit-sequence of source codes is cyclic-redundancy-checksum (CRC) [17]. Together with the bit-sequence encoded from av, they form the source￾word bv. We use CRC in PhyArith because it is a linear function with the following property CRC(bv ⊕bv′ ) = CRC(bv)⊕CRC(bv′ ) = 0⊕0 = 0, (4) where ⊕ denotes bitwise-XOR, and CRC(·) returns the check￾sum of input in the form of sequence of bits. This property can help us verify the correctness of the decoded summation of update vectors. We present this verification process in detail in Section VII-C. C. Channel Codes In order to provide error correction capability in PhyArith, the k bits sourceword bv, consisting of the bit-sequence encoded from quantized update vector chunk av and its CRC