Yanfei Dong，Jincheng Dai，Kai Niu，*，Sen Wang，Yifei Yuan

1 The Key Laboratory of Universal Wireless Communications，Ministry of Education，Beijing University of Posts and Telecommunications，Beijing 100876，China

2 China Mobile Research Institute，Beijing 100876，China

*The corresponding author，email: niukai.bupt.edu.cn

Abstract: In order to provide ultra low-latency and high energy-efficient communication for intelligences，the sixth generation(6G)wireless communication networks need to break out of the dilemma of the depleting gain of the separated optimization paradigm.In this context，this paper provides a comprehensive tutorial that overview how joint source-channel coding (JSCC) can be employed for improving overall system performance.For the purpose，we first introduce the communication requirements and performance metrics for 6G.Then，we provide an overview of the source-channel separation theorem and why it may not hold in practical applications.In addition，we focus on two new JSCC schemes called the double low-density parity-check (LDPC) codes and the double polar codes，respectively，giving their detailed coding and decoding processes and corresponding performance simulations.In a nutshell，this paper constitutes a tutorial on the JSCC scheme tailored to the needs of future 6G communications.

Keywords: 6G; joint source and channel coding;doubel LDPC codes;doubel polar codes

I.INTRODUCTION

The fifth generation (5G) wireless communication networks are being deployed worldwide starting in 2020.The three typical communication scenarios for 5G are enhanced mobile broadband(eMBB)，massive machine type communications (mMTC)，and ultrareliable and low latency communications (uRLLC)[1-4].In addition to the features supporting traditional eMBB scenarios，more capabilities targeted for mMTC and uRLLC scenarios in 5G are being standardized，such as massive connectivity，ultrareliability and guaranteed low latency.With the 5G evolving the land，the oceans and skies，there are huge numbers of connected automated devices，and hundreds of billions of sensors will be spread throughout the natural environment and within living organisms.Meanwhile，various systems based on artificial intelligence (AI) will also be deployed in large numbers in cloud/fog platforms and other edge devices，and create a huge number of new applications.The massive application of AI will build a virtual world in wireless communication networks.The 6G needs to realize the extension from the real world to the virtual world，which includes not only the three core elements of human society，information space，and physical world involved in 5G，but also the fourth dimensional element of the virtual world - thegenie[5].The introduction of the genie has transformed the object of communication into a broader sense of intelligences.Real-time interactions among massive intelligences in 6G application scenarios will place a huge transmission burden on 5G networks.The 6G is expected to offer enhanced spectrum/energy efficiency，lower end-to-end latency and increased intelligence，etc.

To meet the requirements of the future，6G needs new paradigm shift.From the first generation (1G)wireless communication systems to 5G，the separation theorem has been one of the principles followed in the design of communication systems.Guided by the separation theorem，the source(resp.，channel)encoder and decoder do not need to consider the statistics of the channel (resp.，source) [6].However，the separation theorem does not apply when the blocklength is finite，and the channel decoder can capitalize on the residual redundancy left by the source code to reduce the overall error rate.Theoretical analysis of finite-blocklength bounds on the best achievable lossy joint source-channel code rate demonstrates that joint source-channel code design brings considerable performance advantages over separate one in the nonasymptotic regime[7].The paradigm update from separate source-channel design to joint source-channel design will bring considerable overall gains to 6G.

As the intersection of two main branches of Shannon theory，joint source-channel coding(JSCC)has always attracted the interest of researchers.For almostlossless JSCC，there are two main approaches.The first approach is to perform joint source-channel decoding at the receiver when the source compression format (e.g.，JPEG) is given [8-11].Variable-Length Codes (VLCs) is a common entropy coding scheme used in most source coding standards (voice，audio，image and video).The joint decoding for VLCs and a linear channel code need to transform the decoding process of VLCs，such as Huffman codes[12，13]，arithmetic codes[14]and Lempel-Ziv code[15].The second approach assumes a Markov model for the source，which is decoded jointly with a factor graph of the channel code.A hidden Markov source model can be combined with a turbo code for JSC decoding(JSCD)[16，17]and can also be extended to the case of unknown parameters [18].Moreover，a joint lowdensity parity-check(LDPC)encoding structures with a joint belief propagation (BP) decoder，called double LDPC (D-LDPC) codes，is proposed in [19，20].The JSCC scheme with D-LDPC codes is a promising research topic in the last decade.Wang Lin et al.has made a lot of meaningful work by introducing protograph LDPC codes into the D-LDPC codes scheme[21-27].

As another coding scheme in addition to LDPC codes in 5G，polar codes can also be applied to the JSCC field.Polarization phenomenon exists not only in channel coding [28] but also in source coding[29].For the lossy source coding，polar codes can achieve the rate-distortion bound for a binary symmetric source [30].For the lossless source coding，polar codes can achieve the optimal compression rate asymptotically [31].Furthermore，polar codes is a good candidate to exploit the residual redundancy leaved by the source encoder in the JSCD scheme[32].By judging the validity of decoded words，a joint polar list decoder and language decoder is proposed to exploits the redundancy of language-based sources[33].The systematic polar codes [34] can be utilized in a JSCD scheme for correlated sources[35].Considering joint source-channel polarization，a JSCC scheme using a quasi-uniform systematic polar code (Qu-SPC)has been proposed [36].In our previous work，inspired by D-LDPC codes，we proposed double polar(D-Polar)codes combing the source polar coding and the channel polar code[37].

The main contribution of this paper is，thus，to provide a comprehensive tutorial on the topic of LDPC/Polar-based JSCC for future 6G communications.The overarching goal is to give a tutorial on the emerging research contributions that address the major opportunities and challenges in developing JSCC frameworks for enhanced spectrum/energy efficiency and lower end-to-end latency.To the best of our knowledge，this is the first tutorial that gathers the state-of-the-art and emerging research contributions related to the use of JSCC to improve the overall performance of 6G.Our main contributions include:

· We describe the performance metrics of 6G in terms of future communication needs and illustrate the gains that JSCC can bring to 6G.Also，a basic framework of JSCC for 6G is given.

· After providing an introduction to the basics framework of JSCC for 6G，we elaborate on the JSCC scheme using D-LPDC codes.The coding and decoding process of the D-LDPC JSCC system is described in detail.

· Then，we introduce another new JSCC scheme based on polar codes.This scheme，called DPolar codes，consists of a source polar code and a channel polar code.Further，based on the belief propagation(BP)decoders of the polar codes，we introduce a JSCD algorithm for D-Polar codes，namely the turbo-like BP(TL-BP)decoder.

The rest of this tutorial is organized as follows.In Section II，we describe the requirements for 6G and why JSCC should be applied.Section III presents the JSCC scheme using D-LDPC codes，as well as its various variants and optimization methods.In Section IV，we present the encoder/decoder architecture of DPolar codes and the potential for future development.Finally，conclusions are drawn in Section V.

II.PERFORMANCE METRICS

With the proposal of the genie，6G networks must meet the communication needs arising from the convergence of the virtual and real worlds.The 5G networks are limited to real-world application scenarios with key performance metrics including 20 Gbps peak data rate，0.1 Gbps user experienced data rate，1ms endto-end latency，3 times spectrum efficiency，and 100 times energy efficiency compared to the fourth generation(4G)wireless communication systems.However，as applications expand，5G networks will not be able to meet future requirements in 2030+.

In March 2019，Bell Laboratory suggested several critical performance metrics of 6G networks.The peak data rate needs to be greater than 100 Gbps.The data rate experienced by the user ranges between 1 and 10 Gbps.The density of connections is 107devices/km2.The delay should be less than 0.3 ms.Energy efficiency is projected to be 10 times that of 5G networks.The capacity is predicted to be 10，000 times more than that of 5G networks.The indoor positioning accuracy is 10 cm level，and outdoor positioning accuracy is 1 m level.The reliability need to reach six“9” reliability，which indicates that the bit error rate(BER) or block error rate needs to reach 10-6level.In[38]，the following key performance metrics of 6G networks was proposed to compare to 5G networks:≥1 Tbps peak data rate，1 Gbps user experienced data rate，10-100 μs latency，1000 km/h mobility，107 devices/km2connection density，1 Gbps/m2area traffic capacity，and 10-100 times energy efficiency and 5-10 times spectrum efficiency.The authors of [39]have a similar perspective on the major performance metrics of 6G networks.In[40]，6G is expected to provide Tbps-level peak rates，Gbps-level user experience rates，3-5 times higher spectral efficiency，10 times lower latency and 100 times higher connection density compared to 5G networks.In[41]，some essential performance metrics，including as 99%coverage percent，-130dBm receiver sensitivity and 99.999% reliability，was proposed.The increase in data rate will bring an increase in energy consumption，and to compensate for this，the energy efficiency must increase by more than 100 times compared to 5G networks.

The achievement of high spectral efficiency and energy efficiency requires new transmission technologies.In the future 6G networks，independent consideration of channel coding can hardly bring considerable gains.Separate design guidelines derived from Shannon’s source and channel separation theorem need to be reconsidered.If the system design is severely constrained in complexity or delay，the source and channel encoders may be largely suboptimal.In such cases，JSCC can improve the overall system performance.The basic framework of the 6G-oriented JSCC system is shown in Figure 1.The JSCC is executed in the transmitter and the corresponding JSCD is performed in the receiver.In this JSCC system，the residual redundancy in the source code can be utilized to reduce the overall error rate.The reduction in overall system error rate will result in end-to-end latency，spectral efficiency and energy efficiency improvements across the board.The LDPC codes and the polar codes are adopted by 5G networks standards for their excellent performance.For 6G networks，the extension of LDPC and Polar codes adopted in 5G networks to the source compression field to form a JSCC scheme is to be considered.In the following，we present two JSCC systems，called D-LDPC codes and D-Polar codes，respectively.

Figure 1. The block diagram of the JSCC system of 6G.

III.DOUBLE LDPC CODES

The D-LDPC codes were introduced by Fresia [19]，which uses two concatenated LDPC codes in the transmitter.In the 5G new radio (NR)，the LDPC codes have replaced turbo codes for data channels.The standardization of 5G does not involve source compression.A prominent feature of the D-LDPC codes is the use of LDPC codes in source compression.

3.1 Structure of D-LDPC Codes

The block diagram of the D-LDPC JSCC system is shown in Figure 2.The transmitter of this system is a serial connection of two LDPC codes，i.e.，both source compression and channel coding are in LDPC codes.First，one LDPC code is employed to compress the source s into b，and then another LDPC code is used to encode b into c to counteract channel noise.The joint BP algorithm is adopted at the receiver for JSCD to reconstruct the source.

In the following we will depict the D-LDPC JSCC system from a graphical model perspective.As shown in Figure 3，the D-LDPC codes can be represented by a Tanner graph，in which the left side indicates the source code and the right side indicates the channel code.The squares represent the check nodes (CNs)and the circles indicate the variable nodes(VNs).The whole Tanner graph of the D-LDPC codes can be represented by a single joint parity-check matrix HJ.This matrix is given by

Figure 2. The block diagram of the D-LDPC JSCC system.

Figure 3. The Tanner graph of the D-LDPC JSCC system.

where Hs，Hcare the source LDPC code parity-check matrix and the channel LDPC code parity-check matrix respectively.The matrix HL1specifies connections between the CNs of source LDPC code and the VNs of channel LDPC code，and the matrix HL2specifies connections between the CNs of channel LDPC code and the VNs of source LDPC code.

For regular and irregular LDPC codes，the Hsand Hccan be obtain by PEG algorithm with a given degree distribution.For protograph LDPC codes，the matrix HJis obtained by performing “copy-andpermute” operation on the base matrix BJusing the PEG algorithm.Corresponds to the joint parity-check matrix HJ，the joint base matrix BJcan be given by

where Bsis a base matrix of source code with sizems×ns，Bcis a base matrix of channel code with sizemc×ncand two linking matrix BL1with sizems×ncand BL2with sizemc ×nsrepresent the connectivity from CNs of source code to VNs of channel code and from VNs of source code to CNs of channel code，respectively.In the following，we will elaborate the encoding and decoding process of the D-LDPC JSCC system.

3.1.1 Encoder

An independent and identically distributed (i.i.d.)Bernoulli source with success probabilityp ＜1/2 is given，which generates the source sequence s.The coding steps of the D-LDPC JSCC system are as follows:

(1)The source sequence s is first compressed using the source LDPC code parity-check matrix

where the size of HsisMs×Ns，s is a column vector ofNsrows representing theNs-source bits，and the column vector b is a compressed sequence ofMsbits.

(2) Let spdenotes the part of s connected by the CNs of the channel code.Then we cascade spand b vertically into[sp;b].

(3) Let GL2crepresents an (Ns+Nc - Mc)×(Ns+Nc) generator matrix whose parity-check matrix is given by the horizontal cascade of the matrices HL2and Hc.Then the codeword is obtain by

(4)After puncturing the sp，transmit c.If the channel code employs a punctured protogaph LDPC code，the corresponding parity bits are punctured before the codeword is transmitted.

3.1.2 Decoder

As shown in Figure 3，a joint sparse bipartite graphs form a BP decoder for JSCD of D-LDPC codes.The BP decoder operates in a circular iterative fashion.In one iteration，first，the variable nodes inform the check nodes of their log-likelihood ratios (LLRs)，and then the check nodes respond with their LLR constraints on each variable node.The BP decoder terminates when the preset maximum number of iterations is reached or when the stopping criterion is satisfied.We describe this decoder with the following notation:

·LsvandLcvare，respectively，the initial LLRs of the source code VNs and the channel code VNs.

·Ltv，c(Ltc，v)are the LLRs passed from thev-th VN(c-th CN) to thec-th CN (v-th VN) in thet-th iteration.

The initialization ofLsvandLcvis based on the source and channel statistical characteristics，respectively.For a given i.i.d.binary Bernoulli source with Pr(1)=p，theLsvcan be given by

To make the codeword transmitted over a binary input additive white Gaussian noise (BI-AWGN) channel，Lcvis calculated as follows.

whereyi= (1-2ci)+nσ，ciis the(i=v-Ns)-th bit of the codeword c，andnσis the Gaussian channel noise with varianceσ2.

After initialization is complete，the BP decoder updatesLtv，candLtc，vin every iteration.Let us consider the message update in thet-th iteration.The update of the LLRs passed form the VNs to the CNs is given by

And the update of the LLRs passed form the CNs to the VNs:

wherec=1，...，Ms+Mc.

After the termination of the iteration，the decoder performs hard-decision on the posterior LLR of the source LDPC code VNs to reconstruct the source bits，i.e.，the source bits are estimated by

whereL(sv)is the posterior LLR the source bitsvforv= 1，...，Ns.Supposet=Twhen the iteration terminates.TheL(sv)can be computed by

3.2 Joint PEXIT Algorithm of Double LDPC Codes

The extrinsic information transfer (EXIT) chart that tracks the mutual information (MI) at each iteration gives an excellent visual representation of the message-passing decoder，providing the performance prediction of LDPC codes[42].It can be extended to protograph LDPC codes as protograph EXIT(PEXIT)chart [43].The PEXIT algorithm was employed to predict the convergence of the source code of D-LDPC codes in[22].Then，the joint PEXIT(JPEXIT)analysis was proposed to calculate the decoding threshold of the joint base matrix BJfrom a global perspective，which can provide the error performance prediction of the D-LDPC codes[23].

The following notation is used in the rest of the section:

Ive(i，j): the extrinsic MI fromj-th VN toi-th CN

Ice(i，j): the extrinsic MI fromi-th CN toj-th VNIva(i，j): the prior MI fromj-th VN toi-th CNIca(i，j): the prior MI fromi-th CN toj-th VN

Iapp(j): the MI between a posterior LLR evaluated byj-th VN and the corresponding source bitsj

The MI between a binary bit and its corresponding LLR valueL ～(σ2/2，σ2)can be represented byJ(σ)，which is given by

Letbi，jdenote the edge number connecting a VNvjto a CNci.Assuming the channel is AWGN，the proposed JPEXIT algorithm for D-LDPC is described as follows.

1)The MI Update From VNs to CNs:Ifbi，j≠0，

forj= 1，··· ，nsandi= 1，··· ，ms+mc.The functionJBSCis defined as

whereI(V;χ) is the MI between the VN of the source andχ(1-p)～ N(μ+Lsv，2μ) andχp ～N(μ-Lsv，2μ) withLsv= ln((1-p)/p).Forj=ns+ 1，··· ，ns+ncandi= 1，··· ，ms+mc，ifbi，j≠0，

Moreover，we haveIca(i，j) =Ive(i，j) forj=1，··· ，ns+ncandi=1，··· ，ms+mc.

2) The MI Updates From CNs to VNs:Forj=1，··· ，ns+ncandi=1，··· ，ms+mc，ifbi，j≠0，

and setIva(i，j)=Ice(i，j).

3)The APP-LLR MI Evaluation:Forj=1，··· ，nsandi=1，··· ，ms，

3.3 Optimization of Double LDPC Codes

The optimal channel LDPC code in a separate system is not optimal in the D-LDPC JSCC system [23，44]，which indicates that the LDPC code employed in the D-LDPC JSCC system needs to be designed specifically.The improvement of the water-fall region and error-floor region is two main optimization topic of this JSCC system.

3.3.1 Optimization of the error-floor region

The high error-floor limits the application of the DLDPC codes in the communication system.In the joint base matrix BJ，the complete-source-variablelinking (CSVL) protomatrix，i.e.[] determines the error-floor of the D-LDPC codes.The generalized source protograph EXIT (GSP-EXIT) algorithm was proposed to evaluate the source threshold，considering the CSVL protomatrix [45].The GSP-EXIT curves consist of the inner-code curve and the outercode curve.The inner-code curve can be obtained by applying the P-EXIT analysis to the CSVL protomatrix.By assuming perfect LLR messages are passed from the VNs and CNs in BL2，a series inner-code curve can be plotted for differentp.Then，the P-EXIT analysis is applied to the matrix[BsBL1].The outercode curve is depicted by assuming perfect LLR messages are passed from the VNs and CNs in BL1.Aspincreases，the decoding tunnel between the inner-code curve and the outer-code curve becomes narrower.The source decoding thresholdpthis the valuepwhen the tunnel is nearly closed.

In[44]，the differential evolution[46]algorithm was used to optimize the source decoding threshold of Bsfor BL2=0.And the differential evolution is extented to search optimal Bsfor BL2≠ 0，where the source decoding thresholdpthfound by the GSP-EXIT algorithm is used as the cost function，i.e.，

The Φ(Bs，BL2) returns the source decoding thresholdpth，and Ω(Bs，BL2) = 1 when all the required conditions are satisfied.The required conditions in the differential evolution searching algorithm can be concluded as follows，

· The minimum column weight of Bsis 3 (except the fixed columns)，and the minimum row weight of Bsis 2;

· Except for the fixed entries，the other entries inBshave a maximum value of.

where theis preset.

3.3.2 Optimization of the water-fall region

The introduction of the link matrix BL2effectively lowered the error-floor for D-LDPC codes.However，the increasing of the size of the link matrix BL2results in the loss of the water-fall performance.The number of non-zero column in BL2is not necessarilyns，i.e.，

where the size of B′L2isms×nadd.The performance of the water-fall region will degrade asnaddincreases.For high-entropy sources，the matrix B′L2with multiedges has better performance in water-fall region than that with single-edges[47].Several design principles guiding the optimization of the water-fall region can be concluded as follows，

· The kind of multi-edge (2) is suggested inB′L2;

· The number of row weight and column weight no lessw(w ≥2)is no less than min(ms，nadd);

· The row and column rank is no less than min(ms，nadd).

The search forB′L2can utilize the differential evolution algorithm，where the decoding threshold obtained by the JPEXIT algorithm is used as the cost function，i.e.，

The Φ(BJ) returns the decoding threshold，and Ω(B′L2) = 1 when all the proposed design principles are satisfied.

In addition，the overall optimization of D-LDPC can improve the performance of the water-fall region and the error-floor region，such as extending the curve-fitting algorithm to JSCC systems[24]，searching for the optimal BJwith differential evolution algorithm[25]，and the multi-objective differential evolution(MODE)algorithm[27].

3.4 Performance of Double LDPC Codes

In this section，we show the performance evaluation results of the D-LDPC codes.Given an i.i.d Bernoulli source withp= 0.04.The rate of the source code and the channel code are set to 1/2，i.e.，the overall system code rate is 1.As shown in Figure 4，the best performance of the D-LDPC JSCC system，represents by，is obtained after the optimization of the base matrix using a joint design and optimation algorithm based on the idea of MODE algorithm [27].Andrepresents the separation-based counterpart for the.It can be seen thathave 2.25dB performance gains compared with.The performance of the D-LDPC JSCC system optimized based on the joint protograph extrinsic information transfer(JPEXIT)algorithm is represented by BJ.Optimizing the distribution of degree-2 variable nodes in the base matrix also enhances the performance of the system，which is represented byBJ3.The performance of the D-LDPC JSCC system with regular LDPC codes is represented by (Regular LDPC，Regular LDPC).And(R4JA，AR4JA)indicates the use of repeat by 4 jagged accumulate(R4JA)code in source coding and Accumulate repeat by 4 jagged accumulate (AR4JA) code in channel coding.From the Figure 4，it can be seen that the D-LDPC JSCC system can gain significantly with respect to the separationbased counterpart.The D-LDPC system performance using the optimized protograph-LDPC code is better than that of the regular LDPC code.In addition，we note that the D-LDPC system with direct channel protograph-LDPC codes，i.e.，(R4JA，AR4JA)，has inferior performance to the regular LDPC codes.

Figure 4. BER simulation results of the JSCC system based on the D-LDPC codes.

IV.DOUBLE POLAR CODES

The D-Polar codes is one of our latest proposed JSCC systems [37]，which is a joint application of source polarization [29] and channel polarization [28].The D-Polar codes JSCC system directly introduces the source polar code in the source compression to form a double coding structure，which is different from other JSCC schemes based on polar codes [33，36].In the D-Polar JSCC system，the source polarization is first performed to compress the source sequence，and another channel polar code is used to protect the compressed sequence.

Source polarization refers to the polarization of the entropy.The basic idea of source polarization is to transform a pair of identical binary sources into two distinct sources of different entropies: one higher and one lower than the original binary sources.By repeating such a pairwise polarizing operation on a set ofNs= 2nsi.i.d binary sources，a set ofNspolarized sources of varying entropies can be obtained.Whennsis large，some of these polarized sources are almost non-redundant (i.e.，entropy tends to 1)，while the rest are almost fully redundant(i.e.，entropy tends to 0).Therefore，the polarizedNssources can be divided into two parts，thehigh-entropysources and thelow-entropysources，where the high-entropy sources concentrates almost all the entropy of the originalNssources，and the low-entropy sources can be almost completely determined by the high-entropyNsand the statistical characteristics of the original sources.The high-entropy sources is the result of source compression using polarization codes.In this way，it is sufficient for the channel encoder to ensure that the highentropy sources can be reliably transmitted through the channel.The low-entropy sources can be recovered in the receiver by source decoding.

Channel polarization is a method for constructing coding sequences that reach the symmetric capacity of any given binary-input discrete memoryless channel (B-DMC)，and based on this method Ar?kan proposed polar codes.The channel polarization operation consists of two steps: the channel combining operation and the channel splitting operation.The channel combining operation combinesNcindependent copies of a given B-DMCWin a recursive manner to produce a vector channelWNc，whereNc=2nc，nc ≥0.Then the channel splitting operation is to split the vector channelWNcback into a set ofNcbinary-input polarized channelsW(i)Nc.Different from the underlying raw channelW，the polarized channelsW(i)Nctakes both the channel received samples and the previous decoded bits as the channel output.AsNcbecomes large，in the set ofNcpolarized channels，some of these polarized channels are nearly perfect while the rest of them are totally noisy.The basic idea of polar codes is to use those perfectly polarized channels to carry the bits that need to be transmitted.Combining the source polarization and channel polarization，we can use those nearly perfect polarized channels to carry the high-entropy sources obtained by source polarization.

4.1 The Joint Source and Channel Coding Using Double Polar Codes

The block diagram of the D-Polar JSCC system is shown in Figure 5.The transmitter is composed of two concatenated encoders: the source polar encoder and the channel polar encoder.As in Section III，the sources are all set to a binary Bernoulli source，where the probability of‘1’isp ＜1/2.In the D-Polar JSCC system，the source sequence of lengthNsis first compressed intoKsbits using a source polar code，and then encoded into a codeword of lengthNcby a channel polar code.In the following，we describe the coding process of the D-Polar codes in detail.

The first step is to perform source compression using the polar code specified by the parameter vector(Ns，Ks，H)，whereNs=2nsis the length of original source sequence，Ksis the length of the compressed source sequence and specifies the size ofH.The ratioKs/Nsis called the compression rate.ForHan arbitrary subset of{1，...，Ns}，we will refer to it as the high-entropy set.Let the row vector s represent the source sequence of sizeNs.The source polarization operation is accomplished by multiplying s with the generation matrix G?ns.

where b represents the compressed source sequence，and G?nsis thens-order Kronecker power of F =.The polarized source indexed byHis extracted from b as the output of the source encoder，denoted by bH.

The second step is to perform channel coding using the polar code specified by the parameter vector(Nc，Kc，A，uAc)，whereNc= 2ncis the codelength，Kcis the code dimension and specifies the size ofA.The ratioKc/Ncis called the code rate.ForAan arbitrary subset of{1，...，Nc}，we will refer to it as the information set.Let u denote the sequence to be encoded，which can be divided into two parts: information bits indexed byAand frozen bits indexed byAc.The frozen bits carriesuAcknown to the transmitter and the receiver.The polar codes can be adopted in the channel encoder including non-systematic polar codes and systematic polarization codes.

· For the non-systematic polar code，the information bitsuAcarries the compressed sequence bH.The codeword x can be given by

where G?nsis thenc-order Kronecker power of

· For the systematic polar code，the codeword x can be divided two parts: systematic bits indexed byBand parity-check bits indexed byBc.We can setB=A[34]and let systematic bitsxBcarries bH.The parity-check bits xBcis calculated as follows.

where uAis given by

4.2 The Joint Source and Channel Decoding of Double Polar Codes

The BP decoder is a common soft-in and soft-out(SISO) algorithm for polar codes.In this section，we will introduce a JSCD scheme based on BP algorithm.

4.2.1 The turbo-like BP decoding

Figure 6 depicts the structure of the turbo-like BP(TLBP) decoder，which is presented in [37].The TL-BP decoder consists of a channel BP decoder and a source BP decoder.In the TL-BP decoding process，the channel BP decoder can provide the source BP decoder with a priori message associated with the high-entropy bits，while the source decoder can provide the channel BP decoder with a priori information associated with the information bits or parity-check bits.The information interaction between the channel BP decoder and the source BP decoder will lead to an overall performance improvement.

Figure 5. The block diagram of the D-Polar JSCC system.

The channel decoder is activated first after the channel samples are is obtained.The polar codes can be represented in the form of a factor graph.Figure 7 illustrates the factor graph for an(8，4)polar code.This factor graph consists of 3 stages，where each stage contains 4 processing elements (PEs).Each PE element connects 4 adjacent nodes.Thus the channel polar codes used in the D-Polar JSCC system can be represented by annc-stage factor graph.The channel BP decoder is iteratively applied over this factor graph containing(nc+1)Ncnodes.Each(i，j)-index node of the factor graph is associated with left-to-right and right-to-left messages.The soft messages of every node are updated and propagated among adjacent nodes during iteration process.LetLti，jandRti，jdenote the right-to-left and left-to-right messages associated with the(i，j)-index node in thet-th iteration，respectively.Before the iteration starts，Lti，jandRti，jis initialized.TheRti，1associated with the leftmost column node is initialized based on the information setA.Fori ∈Ac，theRti，1is initialized to+∞.AndLti，nc+1associated with the rightmost column node are initialized to the channel received LLRs.In addition，the initialization of the channel BP decoder needs to take into account the a priori information from the source BP decoder.LetLcadenote the a priori information received from the source BP decoder.Note that when the channel BP decoder is first activated，Lcaequals 0.Lcais used to initializeRti，1fori ∈Aif the channel encoding employs a non-systematic polar code，orLcais add toLti，nc+1fori ∈Bif the channel encoding uses a systematic polar code.In one iteration，the decoder first updatesRti，jserially in the order ofj= 1，2，...，nc+1 and then updatesLti，jserially in the reverse order ofj=nc+1，nc，...，1.The channel BP decoder iteration process is terminated when the number of iterations reaches the preset maximum value.The output of the channel BP decoder is divided into two cases.For non-systematic polar codes，the channel BP decoder outputs the messagesLti，1indexed by the information seti ∈A，and for systematic polar codes，the channel BP decoder outputs the messagesRti，nc+1indexed byi ∈B.The messages output by the channel BP decoder is sent to the source BP decoder as a priori information of high-entropy bits.

Figure 6. The turbo-like BP JSCD for D-Polar codes.

Figure 7.The factor graph of a polar codes with codelength of 8.

Figure 8. BER Performance Comparison of D-Polar Codes JSCC Systems.

Figure 9. BER simulation results of the D-Polar JSCC system with various compression rate and source distribution.

Similar to the channel polar code，the source polar code employed in the D-Polar JSCC system can be represent by anns-stage factor graph.For the sake of clarity，in the source BP decoder，we uselti，jandrti，jto denote the right-to-left and left-to-right messages associated with the (i，j)-index node in thet-th iteration.The initialization of the source BP decoder is based on the source statistics and the a priori information received from the channel BP decoder associated with the high-entropy bits.The initialization ofrti，1associated with the leftmost column node is same as Eq.(5).And thelti，ns+1associated with the rightmost column node is initialized based on the high-entropy setH.LetLscdenote the a priori information received from the channel BP decoder.Fori ∈H，thelti，ns+1is initialized toLsc.In one iteration，the decoder first updatesrti，jserially in the order ofj= 1，2，...，ns+1 and then updateslti，jserially in the reverse order ofj=ns+1，ns，...，1.The source BP decoder iteration process is terminated when the number of iterations reaches the preset maximum value.After the termination of the iteration，the source BP decoder outputrti，ns+1indexed byi ∈His sent to the channel BP decoder.Then the channel BP decoder will be activated again.

One information interaction between the channel BP decoder and the source BP decoder is called one external iteration.The TL-BP decoder stops when the number of external iterations reaches a preset maximum.Finally，the TL-BP decoder performs a harddecision on theL(si) =lti，1+rti，1to reconstruct the source sequence.

4.3 Performance and Complexity of Double Polar Codes

In this section，we give the results of the performance evaluation of the D-Polar codes.Given a i.i.d binary Bernoulli source withp= 0.07.The length of the source sequence isNs= 512，the codelength of the channel coding isNc，and the overall code rate is 1/2.The codeword is modulated by BPSK and sent through the BI-AWGN channel.

As shown in Figure 8，the D-Polar code JSCC system with channel systematic polar code has almost same performance compared with the non-systematic polar code version at-2.5dB.Then，as the Eb/N0increases，the systematic version of D-Polar JSCC system will exhibit performance advantages，such as a 0.46dB gain at BER = 10-3.The D-Polar JSCC system achieves more than 3dB gain compared to the scheme where the source sequence is directly executed without compression for channel coding and then transmission.The D-Polar JSCC system can obtain more than 2dB gain compared to the Qu-SPC proposed in [36].The introduction of source polar code helps the D-Polar JSCC system over the Qu-SPC scheme，although excessive compression of the source polar code results in the emergence of an error floor.The absence of an error floor is an advantage of the Qu-SPC scheme.Compared with the DLDPC scheme[20]，the D-Polar JSCC system can obtain 0.46dB gain at BER=10-3.The Sepa-BP represents a version of TL-BP that does not perform external iterations.The gain obtained by the D-Polar codes JSCC system compared to Sepa-BP indicates that the external iteration is efficient.Note that Ebrefers to energy per source bit rather than nonredundant information bit.One problem with D-Polar codes is the presence of error floor.Since systematic polar codes can obtain better performance，we use systematic polar codes in the following simulations.

The error floor of the D-Polar codes is due to incomplete source polarization caused by the limited code length.When the overall code rate is given，the DPolar codes JSCC system can reduce the error floor by changing the compression rate.As shown in Figure 9，when 512 source bits are compressed to 307 bits，the error level is below 10-3.When 512 bits compressed to 358 bits，the error level is below 10-4.In addition，when 512 bits are compressed to 384 bits，the error level is close to 10-5.Moreover，the statistical characteristics of the source have a significant impact on the JSCC system.When 512 source bits are compressed to 307 bits，the gains that can be obtained forp=0.04 andp= 0.02 compared top= 0.07 are 0.69dB and 1.04dB at BER = 10-3，respectively.Meanwhile，there is a significant decrease in error floor.

Table 1 compares the average iteration number between the TL-BP decoder with D-Polar codes[37]and the BP decoder with D-LDPC codes with[20].In the simulation，we use an early iteration stopping criterion[48]to avoid redundant iterations.The specific parameter of the decoder used can be found in[37].As can be seen in Table 1，both D-Polar codes and D-LDPC codes，the average iteration number decreases rapidly as Eb/N0increases from-1.5dB to 0dB.The average number of iterations of the TL-BP decoder with DPolar codes is lower than that of the BP decoder with D-LDPC codes.However，when Eb/N0is greater than 0dB，the decrease in the average iteration number becomes slow due to the error floor.

Table 1.Average iteration number of the D-Polar codes and D-LDPC.

V.CONCLUSION

In this paper，we first describe the communication requirements and performance metrics of 6G，giving a basic framework for a JSCC system.Secondly，we focus on the coding and decoding processes of two JSCC systems，called D-LDPC codes and D-polar codes，respectively.Finally，the performance evaluation illustrates that these two JSCC schemes can achieve significant gains under long and short codes，respectively.

ACKNOWLEDGEMENT

This work was supported by National Natural Science Foundation of China(No.92067202，No.62001049，&No.62071058)，and Beijing Natural Science Foundation under Grant 4222012，and Beijing University of Posts and Telecommunications-China Mobile Research Institute Joint Innovation Center.