Rate-Compatible Protograph LDPC Codes for Source Coding in Joint Source—Channel Coded Modulation Systems

Abstract

Using the source residual redundancy in a joint source and channel coded modulation (JSCCM) system to achieve a shaping gain has proven to be a good solution for probabilistic amplitude shaping. In a JSCCM system, rate-compatible source codes are essential for adapting to dynamically changing source probabilities. However, conventional rate-compatible source codes are not appropriate for JSCCM systems, since they may yield a loss in the shaping gain. Moreover, a code design that depends solely on the source decoding threshold appears to be ill suited, thus complicating the search for good source codes. In this paper, we propose a rate-compatible family of source codes for JSCCM systems based on an achievable system rate analysis and the source protograph extrinsic information transfer (PEXIT) algorithm. The proposed codes not only have good source decoding thresholds but also obtain shaping gains over the whole considered range of source probabilities. Numerical results show that the proposed rate-compatible family of source codes can significantly improve the bit-error-rate (BER) performance of the JSCCM system.

1. Introduction

Shannon stated that the capacity of an additive white Gaussian noise (AWGN) channel is achieved when the transmitted signal follows a Gaussian distribution. However, many communication systems utilize uniformly distributed symbols, resulting in a shaping loss of up to 1.53 dB [1,2]. To overcome this loss, geometric shaping (GS) and probabilistic shaping (PS) have been proposed by imitating the capacity that achieves the symbol distribution. GS uses equiprobable symbols on a non-equispaced constellation, which was extensively studied in [3,4,5,6,7]. PS approaches, which assign equispaced constellation points with different probabilities, have been demonstrated to be effective means of achieving shaping gains [8,9,10,11,12]. To integrate PS with bit-interleaved coded modulation (BICM) systems [13], a probabilistic amplitude shaping (PAS) scheme with a symbol-level distribution matcher (DM) [14] was introduced in [8]. Subsequently, a bit-level DM was proposed to improve the throughput of distribution matching [9]. The complexity of a binary DM was reduced by utilizing simplified sign-bit shaping for high-spectral-efficiency coding [10]. The design of protograph low-density parity-check (LDPC) and non-binary codes for PAS systems can be found in [11,12].

In general, the input source bit sequence of a PAS scheme is assumed to be uniformly distributed, and a DM is utilized to generate non-uniformly distributed symbols from uniformly distributed input data bits. However, redundancy often resides in natural sources, such as a non-uniform distribution of source symbols and source memory. Moreover, redundancy is also found in the source encoder’s outputs due to the suboptimality of the compression scheme. Various solutions have been proposed for non-uniform sources, including constellation mappings [15] and optimized protograph low-density parity-check (LDPC) codes with unequal power allocation [16]. For the residual source redundancy after source coding, in [17], a novel PAS strategy was proposed for a joint source and channel coded modulation (JSCCM) system. By using source encoders to transform the non-uniform source bit sequence directly into bitstreams with the desired distribution, the PAS strategy for JSCCM systems avoids the rate loss and the high latency of traditional PAS schemes. Additionally, it is important to match source codes with source probabilities in JSCCM systems to effectively use the residual redundancy. Considering some communication applications with dynamically changing source statistics, rate-compatible source codes in JSCCM systems are essential for adaptive coding.

Substantial efforts have been invested in the source code design for JSCC. In [18], a source protograph extrinsic information transfer (PEXIT) algorithm was proposed, which utilizes the source decoding threshold to evaluate the error-floor level. Further, a rate-compatible family of source codes with good source decoding thresholds was proposed for JSCC systems in [19]. Moreover, several optimized source-linking matrices have been suggested to further improve the error-floor performance [20]. However, the source code designs above only focus on improving the source decoding threshold for the JSCC system, which would yield a loss in the shaping gap for the JSCCM system due to the mismatch between the source probabilities and the source codes. In [17], the source codes in JSCCM systems were optimized for a specific source probability, but they may fail to ensure the PAS performance for source probabilities with slight variations. Consequently, these source codes may not be suitable for adaptive coding in JSCCM systems with dynamically changing source probabilities.

In this paper, we present a study on the construction of a family of rate-compatible source codes for JSCCM systems. The main contributions of this paper are as follows:

• In contrast to the design of source code parameters for a specific source probability, a searching algorithm for the source code parameters is proposed to obtain high shaping gains across the whole considered range of source probabilities for a given source coding rate.
• Combining the achievable system rate analysis and the source decoding thresholds, some design principles for source codes are proposed to simultaneously improve the error-floor quality and guarantee the PAS performance of JSCCM systems for a range of source probabilities.
• A family of source codes with rates ranging from 1/3 to 3/4 is proposed for a JSCCM system by code lengthening; these codes have good source decoding thresholds and obtain shaping gains for different source probabilities, which are attractive for adaptive coding in systems with changing source statistics.

The remainder of the paper is organized as follows. Section 2 presents the JSCCM system model, as well as the encoding and decoding algorithms. Section 3 proposes the design method of a rate-compatible family of source codes. The simulation results and comparisons for the system with the proposed rate-compatible source codes are discussed in Section 4. Finally, Section 5 concludes the paper.

2. System Model

In the JSCCM system, we employ protograph low-density parity-check (LDPC) codes as both source and channel codes. Specifically, let us denote the base matrix of source protograph LDPC codes by ${\mathbf{B}}_s$ and the base matrix of channel protograph LDPC codes by ${\mathbf{B}}_c$. Then, we utilize the progressive edge-growth (PEG) algorithm to generate the sparse binary parity-check matrices ${\mathbf{H}}_s$ and ${\mathbf{H}}_c$ [21]. Later, ${\mathbf{H}}_c$ is converted into a parity-check matrix with a systematic form of ${\mathbf{H}}_c=\left[\mathbf{I}\right|{\mathbf{P}}^{\mathbf{T}}]$, where $\mathbf{I}$ is the identity matrix, and $\mathbf{P}$ is a matrix obtained by transforming ${\mathbf{H}}_c$. After that, a systematic generator matrix of the channel protograph LDPC code ${\mathbf{G}}_c$ can be constructed from ${\mathbf{H}}_c$, which is represented by ${\mathbf{G}}_c=\left[\mathbf{P}\mid \mathbf{I}\right]$.

The block diagram of the JSCCM system is illustrated in Figure 1. The source we consider in this paper is a binary independent and identically distributed Bernoulli source $\mathbf{s}$, where the probability of “1” is represented as p ($p>0.5$). The entropy of $\mathbf{s}$ can be obtained by

$$H\left(p\right)=-p{log}_{2}p-(1-p){log}_2(1-p)<1.$$

(1)

On the transmitter side, a rate-$R_s$ source protograph LDPC encoder and a rate-$R_c$ channel protograph LDPC encoder are serially concatenated. The source information bit sequence $\mathbf{s}$ is fed to the source encoder to calculate the syndrome ${\mathbf{b}}^T={\mathbf{H}}_s{\mathbf{s}}^T$, which is also the compressed sequence.

Then, the channel protograph LDPC is utilized to protect $\mathbf{b}$ for transmission through the AWGN channel, which generates a vector of the parity bit sequence $\mathbf{c}=\mathbf{b}\mathbf{P}$. Assuming systematic channel encoding, the channel codeword $\mathbf{x}$ can be composed of a parity bit sequence $\mathbf{c}$ and a compressed bit sequence $\mathbf{b}$ as $\mathbf{x}=\left[\mathbf{c}\mathbf{b}\right]$.

For one-dimensional $2^m$-ary amplitude-shift keying (ASK) modulation, every m consecutive coded bits are converted into an ASK symbol $X\in \{\pm 1,\pm 3,\dots ,\pm (2^m-1)\}$. Then, we set the channel code rate $R_c$ to be $(m-1)/m$, contributing to the overall system rate as

$$R=\frac{m-1}{R_s}.$$

(2)

By employing an interleaver, the channel codeword $x=\left[\mathbf{c}\mathbf{b}\right]$ can be organized as

$$\begin{gathered} \begin{array}{cc}\hfill {\mathbf{M}}_{\mathbf{x}}^{\left(1\right)}& =\left[\begin{array}{ccccc}\hfill c_1& b_1^1& b_1^2\hfill & \cdots & b_1^{m-1}\\ \hfill c_2& b_2^1& b_2^2\hfill & \cdots & b_2^{m-1}\\ \hfill ⋮& ⋮& ⋮\hfill & ⋮& ⋮\\ \hfill \underbrace{c_N}& \underbrace{b_N^1}& \underbrace{b_N^2}\hfill & \cdots & \underbrace{b_N^{m-1}}\end{array}\right], \\ \begin{array}{cccccc}\hfill & \hfill b^0& \hfill b^1& b^2\hfill & \hfill \cdots & b^{m-1}\hfill \end{array}\hfill \end{array} \end{gathered}$$

(3)

where N is the lifting factor, and each column forms a bit level $b^i$ for $0\le i\le m-1$. By row-wise mapping with binary-reflected Gray code (BRGC) labeling, the ASK symbol sequence $\mathbf{X}=\left(X_1,X_2,\cdots ,X_N\right)$ for transmission is generated.

On the receiver side, after demodulation and de-interleaving, we employ joint source and channel (JSC) decoding to reconstruct the source bit sequence. The concatenated Tanner graph corresponding to the JSC decoder is shown in Figure 2, where circles denote variable nodes (VNs), and check nodes (CNs) are denoted by squares. The numbers of CNs and VNs are determined by the dimensions of source–channel protomatrices (${\mathbf{B}}_s$, ${\mathbf{B}}_c$), denoted as $m_s$, $m_c$ and $n_s$, $n_c$, respectively. Since the output of the source encoder is the input of the channel encoder, each VN in the systematic part of ${\mathbf{B}}_c$, denoted by an empty circle, is connected to a CN in ${\mathbf{B}}_s$. Thus, in each JSC iteration, extrinsic information is exchanged not only between the VNs and CNs in ${\mathbf{B}}_s$ or ${\mathbf{B}}_c$ but also between ${\mathbf{B}}_s$ and ${\mathbf{B}}_c$ through those connections. In this way, the residual source redundancy after source coding can be utilized in JSC decoding to improve the performance of the channel decoder.

3. Design of Source Codes

In this section, we propose the design of a rate-compatible family of source codes for the JSCCM system with the objective of jointly optimizing the error-floor performance and the PAS performance for a range of source probabilities.

3.1. Achievable System Rate Analysis

As shown in [17], the row weight distribution of the source code plays a critical role in controlling the residual redundancy and optimizing the PAS performance for the JSCCM system. Unlike the approach in [17], this study explores the row weight distribution design for a range of source probabilities, rather than focusing solely on a specific source probability.

For an AWGN channel, the mutual information-maximizing distribution under an average power constraint is a zero-mean Gaussian input X with unit variance, and it yields the capacity C expression

$$C=\frac{1}{2}{log}_2(1+2R·E_s/N_0),$$

(4)

where R is the system rate, $E_s$ is the average energy per source bit, and $N_0$ is the one-sided noise power spectral density.

For an asymmetric source with entropy $H\left(p\right)$, the Shannon limit $\frac{E_s}{N_0}{|}_{\mathrm{s}}$ is determined from the condition $H\left(p\right)R<C$, where R is the target system rate defined in source bits per channel symbol.

For an ASK-modulated AWGN channel, the channel capacity is subject to the input probability distribution of the modulated symbols, which can be determined by the source probability p and the row weight distribution of the source code in the JSCCM system.

In (3), the bit level $b^0$ composed of the parity bits is approximately uniformly distributed. Meanwhile, the bit level in block $b^i(1\le i\le m-1$) composed of the compressed bits can be computed as the modulo-2 sum of the source bits.

Assuming that the bits at the input of the modulator are independent, the probabilities of the ASK symbols $X\in \mathcal{A}$ can be calculated as a function of the row weight distribution $W=(w_1,w_2,\dots ,w_{m-1})$, which is obtained as follows

$$\begin{array}{c}\hfill P_W\left(X\right)=\frac{1}{2}\prod _{i=1}^{m-1}\frac{1+{(-1)}^{v^i}{(1-2p)}^{w_i}}{2},\end{array}$$

(5)

where $v^i\in \{0,1\}$ represents a specific bit of the symbol X at the bit position $b^i$.

Given the input probability distribution of the modulated symbols $P_W\left(X\right)$, the channel capacity of the ASK-modulated AWGN channel is calculated as

$$C_W=\sum _{X\in \mathcal{A}}{P}_W\left(X\right){\int }_{-\infty }^{+\infty }{P}_{\alpha }\left(Y\right|X){log}_2\frac{P_{\alpha }\left(Y\right|X)}{{\displaystyle \sum _{X^{\prime }\in \mathcal{A}}}{P}_W\left(X^{\prime }\right)P_{\alpha }\left(Y\right|X^{\prime })}dY,$$

(6)

where $\mathbf{X}$ is the modulated symbol sequence, $\mathbf{Y}$ is the received symbol sequence, $P_{\alpha }\left(Y\right|X)=\frac{1}{\sqrt{\pi N_0}}exp\left\{-\frac{{(Y-\alpha X)}^2}{N_0}\right\}$, and the associated SNR is calculated as [8,18]

$${\left.\frac{E_s}{N_0}\right|}_W=\frac{{\sum }_{X\in \mathcal{A}}{P}_W\left(X\right){\left|\alpha X\right|}^2}{R{N}_0}.$$

(7)

As the limit of the ASK constellations, ${\left.\frac{E_s}{N_0}\right|}_W$ is larger than ${\left.\frac{E_s}{N_0}\right|}_{\mathrm{s}}$ for a given p and R. Thus, the goal of this work is to design the row weight distribution W that achieves reliable transmission close to the capacity–power function (4); i.e., we want to reliably transmit data over the AWGN channel at an SNR ${\left.\frac{E_s}{N_0}\right|}_W$ that is close to ${\left.\frac{E_s}{N_0}\right|}_{\mathrm{s}}$.

Instead of designing the row weight distribution W for one specific source probability p as in [17], W with high shaping gains across the whole range of source probabilities is sought using Algorithm 1 for a given source coding rate. To implement this search, we start by identifying the smallest p for a given source coding rate $R_s$ according to the lossless source coding theorem ($H\left(p\right)\le R_s$). Then, we gradually increase the value of p for $p<1$ and locate potential candidates W for each p as follows:

$${\mathcal{W}}_{\Delta }\left(p\right)=\left\{W\in {[1,w_{max}]}^{m-1}:H\left(p\right)R<C_W,{\left.E_s/N_0\right|}_W-{\left.E_s/N_0\right|}_{\mathrm{S}}<\Delta \right\},$$

(8)

where $w_{max}$ is the maximum row weight value specified for ${\mathbf{B}}_s$, and $\Delta$ is the maximum gap in dB. Finally, candidates satisfying (8) over the whole considered range of p for a given $R_s$ are chosen, which can maintain a satisfactory PAS performance for varying p.

$$\begin{array}{ccc}\hfill I_s^{\mathrm{ev}}\left(i,j\right)& =& \mathrm{\Phi }\left(x_{i,j}^s\right)J_{\mathrm{BSC}}\left(\frac{1}{2}{\sum }_{k\neg \ne i}\left(x_{k,j}^s{\left[J^{-1}\left(I_s^{\mathrm{av}}\left(k,j\right)\right)\right]}^2\right)+\frac{1}{2}\left(x_{i,j}^s-1\right){\left[J^{-1}\left(I_s^{\mathrm{av}}\left(i,j\right)\right)\right]}^2,p\right)\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill I_s^{\mathrm{ec}}\left(i,j\right)& =& \mathrm{\Phi }\left(x_{i,j}^s\right)\left[1-J\left(\sqrt{{\sum }_{k\neg \ne j}\left(x_{i,k}^s{\left[J^{-1}\left(1-I_s^{\mathrm{ac}}\left(i,k\right)\right)\right]}^2\right)+\left(x_{i,j}^s-1\right){\left[J^{-1}\left(1-I_s^{\mathrm{ac}}\left(i,j\right)\right)\right]}^2}\right)\right]\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill I_{s,\mathrm{APM}}\left(j\right)& =& J_{\mathrm{BSC}}\left(\frac{1}{2}{\sum }_k{x}_{k,j}^s{\left[J^{-1}\left(I_s^{\mathrm{av}}\left(k,j\right)\right)\right]}^2,p\right)\hfill \end{array}$$

(11)

Note:

• The choice of $\Delta$ influences the system design. On the one hand, a small value of $\Delta$, which means a small gap to the Shannon limit, leads to a small number of candidates and hence limits the search for a source code with good source decoding thresholds. On the other hand, increasing $\Delta$ can enlarge the search space for source codes with a good error-floor performance at the expense of the PAS performance. As a consequence, the proposed Algorithm 1 can strike a balance between the PAS performance and the error-floor performance by adjusting the value of $\Delta$.
• The value of $\delta$ is related to the step size of p in Algorithm 1. We use $\delta =0.02$ in our search process based on trial and error.

Algorithm 1 Search for the row weight distribution of the source code

Require: $p_{ini}=0.5,\delta ,R,m_s,n_s,e_s,\mathcal{A},\Delta$   1: while $H\left(p_{ini}\right)\ge R_s$ do   2:    $p_{ini}\leftarrow p_{ini}+\delta$;           \∗find the lowest p for $R_s$∗\   3: end while   4: compute R using (2);           \∗target system rate∗\   5: for $p=p_{ini};p<1$ do   6:   ${\left.\frac{E_s}{N_0}\right|}_{max}\leftarrow \left(2^{2H\left(p\right)R}-1\right)/\left(2R\right)·{10}^{\Delta /10}$;   7:    ${\mathcal{W}}_{\Delta }\left(p\right)\leftarrow \varnothing$;   8:    for every $W\in {[1,n_s{e}_s]}^{m_s-1}$ do   9:      compute $P_W\left(X\right)$ using (5) for all $X\in \mathcal{A}$;   10:      ${\displaystyle \alpha \leftarrow {\left(\frac{R{N}_0·{\left.\frac{E_s}{N_0}\right|}_{max}}{{\sum }_{X\in \mathcal{A}}{P}_W\left(X\right)|X{|}^2}\right)}^{1/2}}$;      \∗scaling amplitudes∗\   11:      compute $C_W$ using (6) with $P_W\left(X\right)$ and $P_{\alpha }\left(Y\right|X)$;   12:      if $C_W\ge C$ then   13:       ${\mathcal{W}}_{\Delta }\left(p\right)\leftarrow {\mathcal{W}}_{\Delta }\left(p\right)\cup \left\{W\right\}$.   14:      end if   15:    end for   16:   $p\leftarrow p+\delta$;   17: end for   18: Selecting the common elements ${\mathcal{W}}_{\Delta }$ in all ${\mathcal{W}}_{\Delta }\left(p\right)$   19: return ${\mathcal{W}}_{\Delta }$.

3.2. Source PEXIT Chart Analysis

As is shown in [18], the source PEXIT analysis can be exploited to evaluate the source decoding threshold. In contrast to the channel decoding threshold, the source decoding threshold represents the maximum compressible entropy as the source code length approaches infinity. Equivalently, the source decoding threshold can also be interpreted as the minimum source distribution $p^{th}$ acceptable for the source code when $p>0.5$.

Let $c_i$ and $v_j$ represent the ith CN and the jth VN in ${\mathbf{B}}_s$ (size $m_s\times n_s$), respectively. Five types of mutual information (MI) are first defined, including a priori mutual information (AMI), extrinsic mutual information (EMI), and a posteriori mutual information (APM), as follows:

•   $I_s^{\mathrm{av}}(i,j)$ denotes the AMI from $c_i$ to $v_j$;
•   $I_s^{\mathrm{ev}}(i,j)$ denotes the EMI from $v_j$ to $c_i$;
•   $I_s^{\mathrm{ac}}(i,j)$ denotes the AMI from $v_j$ to $c_i$;
•   $I_s^{\mathrm{ec}}(i,j)$ denotes the EMI from $c_i$ to $v_j$;
•   $I_{s,\mathrm{APM}}\left(j\right)$ denotes the APM for $v_j$.

In [22], the function $J\left(\sigma \right)$ calculating the MI between a binary bit and the corresponding LLR value following the Gaussian distribution $\mathcal{N}({\sigma }^2/2,{\sigma }^2)$ is given as

$$\begin{array}{cc}J\left(\sigma \right)\hfill & =1-\frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\infty }^{\infty }{e}^{-\frac{{(\xi -{\sigma }^2/2)}^2}{2{\sigma }^2}}{log}_2\left[1-e^{-\xi }\right]d\xi .\hfill \end{array}$$

(12)

The function $J_{\mathrm{BSC}}\left(\mu ,p\right)$ is defined as follows:

$$\begin{array}{cc}{J}_{\mathrm{BSC}}\left(\mu ,p\right)=\hfill & \left(1-p\right)\times \left(1-\mathbb{E}\left[{log}_2\left(1+e^{-sign\left(V\right){\chi }^{(1-p)}}\right)\right]\right)\hfill \\ \hfill & +p\times \left(1-\mathbb{E}\left[{log}_2\left(1+e^{-sign\left(V\right){\chi }^{\left(p\right)}}\right)\right]\right),\hfill \end{array}$$

where V represents the source VN in ${\mathbf{B}}_s$, $\mu$ represents the average LLR value, and ${\chi }^{(1-p)}\sim \mathcal{N}(\mu +ln((1-p)/p),2\mu ),{\chi }^{\left(p\right)}\sim \mathcal{N}(\mu -ln((1-p)/p),2\mu )$.

In addition, an indicator function is defined as

$$\mathrm{\Phi }\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\mathrm{x}=0\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(13)

The algorithm for calculating the source decoding threshold is specified in Algorithm 2. We use $n_{max}=100$ as the maximum iteration number and $\delta =0.02$ as the step size. The source decoding threshold $p_{th}$ is the lowest value that makes $I_{\mathrm{APM}}\left(j\right)=1$ hold for all $j\in [1,n_s]$.

Algorithm 2 Source decoding threshold

Require: ${\mathbf{B}}_s,p=0.5,\delta ,n_{max}$   1: for $j\in [1,n_s]$do   2:    $I_{\mathrm{APM}}\left(j\right)=0.0$;   3: end for   4: while ${\sum }_{j=1}^{n_s}{I}_{\mathrm{APM}}\left(j\right)\ne n_s$ do   5:    $p\leftarrow p+\delta$;   6:    $n_{itr}=0$;   7:    while $n_{itr}<n_{max}$ do   8:      for $(i,j)\in [1,m_s]\times [1,n_s]$ do   9:         Compute $I_s^{\mathrm{ev}}(i,j)$ using (9);   10:         Set $I_s^{\mathrm{ac}}(i,j)=I_s^{\mathrm{ev}}(i,j)$;   11:      end for   12:      for $(i,j)\in [1,m_s]\times [1,n_s]$ do   13:         Compute $I_s^{\mathrm{ec}}(i,j)$ using (10);   14:         Set $I_s^{\mathrm{av}}(i,j)=I_s^{\mathrm{ec}}(i,j)$;   15:      end for   16:      for $j\in [1,n_s]$ do   17:         Compute $I_{\mathrm{APM}}\left(j\right)$ using (11);   18:      end for   19:      $n_{itr}=n_{itr}+1$;   20:    end while   21: end while   22: $p^{th}=p$;   23: return $p^{th}$.

3.3. Design of Rate-Compatible Source Codes for JSCCM systems

The base matrix of the source protograph LDPC code can be defined as ${\mathbf{B}}_s=\left({\mathcal{B}}_{i,j}\right)$, where all ${\mathcal{B}}_{i,j}$ variables represent edges connecting $c_i$ and $v_j$. A good source protograph LDPC code for the JSCCM system should have two properties:

(1) Asmentioned in [18], the source decoding threshold indicates the performance of the error-floor level, which is the lowest BER when $E_s/N_0$ approaches infinity. The protograph LDPC codes with low source decoding thresholds $p^{th}$ for $p>0.5$ are preferred for source coding. In [19], it is shown that some degree-2 VNs in protograph LDPC codes can improve the source decoding threshold and thus lower the error floor. Additionally, it is worth noting that the degree-2 nodes cannot form a cycle among themselves.

(2) According to the analysis (see Section 3.1), the source code with an appropriate row weight distribution can help to close the shaping gap to the Shannon limit. Thus, the row weight distribution candidates identified by Algorithm 1 are utilized as constraints in the design of source codes to ensure the PAS performance.

With all of that in mind, consider the following example for the design of a rate-3/4 base matrix, which has 4 VNs and 3 CNs, i.e., size $3\times 4$. Firstly, we institute one degree-2 variable node (column of weight 2) for a good source decoding threshold. Therefore, the base matrix is:

$${\mathbf{B}}_s=\left[\begin{array}{cccc}{\mathcal{B}}_{1,1}& {\mathcal{B}}_{1,2}& {\mathcal{B}}_{1,3}& 0\\ {\mathcal{B}}_{2,1}& {\mathcal{B}}_{2,2}& {\mathcal{B}}_{2,3}& 1\\ {\mathcal{B}}_{3,1}& {\mathcal{B}}_{3,2}& {\mathcal{B}}_{3,3}& 1\end{array}\right],$$

(14)

The variables ${\mathcal{B}}_{i,j}$ designate the remainder of the protograph to be designed. Then, candidates for the row weight distribution W are sought via Algorithm 1. The imposed constraints on the elements are

$$\left\{\begin{array}{c}{\mathcal{B}}_{1,1}+{\mathcal{B}}_{1,2}+{\mathcal{B}}_{1,3}=w_1\hfill \\ {\mathcal{B}}_{2,1}+{\mathcal{B}}_{2,2}+{\mathcal{B}}_{2,3}=w_2-1\hfill \\ {\mathcal{B}}_{3,1}+{\mathcal{B}}_{3,2}+{\mathcal{B}}_{3,3}=w_3-1\hfill \\ {\mathcal{B}}_{1,j},{\mathcal{B}}_{2,j},{\mathcal{B}}_{3,j}]\ne [0,1,1]\hfill & \mathrm{for}\mathrm{j}\in \{1,2,3\}\hfill \end{array}\right.$$

(15)

where $W=[w_1,w_2,w_3]$ is one of the candidates for the row weight distribution. For $R_s$ = 3/4, m = 4, $\Delta =0.5$ dB, and $\delta =0.02$, the optimized row weight distribution is $(3,3,3)$. According to the constraints in (15), we search for the source code with the lowest $p^{th}$ by using the PEXIT analysis in Section 3.2, and we find:

$${\mathbf{B}}_s^{\prime }=\left[\begin{array}{cccc}1& 1& 1& 0\\ 1& 0& 1& 1\\ 1& 1& 0& 1\end{array}\right].$$

(16)

This code has a source decoding threshold of $p^{th}=0.829$, and thus, $H^{th}=0.66$ with a gap of $R_s-H^{th}=0.09$ to the source coding rate.

It is instructive to note that if we increase the value of $\Delta$, such as $\Delta$ = 0.6 dB, the number of the row weight distribution candidates increases, and we have $(3,3,3)$, $(3,3,5)$, $(3,5,3)$, and $(5,3,3)$. Then, we can find the base matrix with a better source decoding threshold ($p^{th}=0.812$, $H^{th}=0.697$) as

$${\mathbf{B}}_s^{3/4}=\left[\begin{array}{cccc}2& 0& 1& 0\\ 0& 2& 0& 1\\ 0& 3& 1& 1\end{array}\right].$$

(17)

In Figure 3, the BER performance of JSCCM systems with those two source codes is provided. Even though the source code ${\mathbf{B}}_s^{3/4}$ yields a slight loss in the shaping gain, it achieves a competitive advantage over ${\mathbf{B}}_s^{\prime }$ in the high-SNR region when the value of p decreases. For instance, the code ${\mathbf{B}}_s^{\prime }$ suffers from an error floor close to a BER of $4\times {10}^{-3}$ for $p=0.84$, while ${\mathbf{B}}_s^{3/4}$ does not show any error floors down to a BER of ${10}^{-5}$. Thus, ${\mathbf{B}}_s^{3/4}$ is more attractive for dynamically changing p.

A low-rate source code can be constructed from the proposed rate-3/4 protograph found in (17) by code lengthening. Specifically, a base matrix with rate-$R_{s2}$ is constructed by lengthening the base matrix of a rate-$R_{s1}$ ($R_{s1}>R_{s2}$) source code (a base code) in the form of

$${\mathbf{B}}_s^{R_{s2}}=\left[\begin{array}{ccc}\multicolumn{1}{c}{{\mathbf{B}}_s^{R_{s1}}}& |& \multicolumn{1}{c}{{\mathbf{B}}_e}\end{array}\right],$$

(18)

where ${\mathbf{B}}_s^{R_{s1}}$ is the base matrix of the rate-$R_{s1}$ source code, and ${\mathbf{B}}_e$ is an extension matrix.

Similarly, row weight distribution candidates for different source coding rates are sought via Algorithm 1 with m = 4, $\Delta =0.6$ dB, and $\delta =0.02$, which are presented in

Table 1. Row weight distribution candidates for different source coding rates ($\Delta$ = 0.6 dB).

Source Coding Rate	Target System Rate (Bits/Symbol)	Size of ${{B}}_{\mathit{s}}$	${\mathit{W}}_{\mathit{\Delta }}$
$R_s=3/4$	$R=4$	$3\times 4$	$(3,3,3),(3,3,5),(3,5,3),(5,3,3)$
$R_s=3/5$	$R=5$	$3\times 5$	$(5,3,5),(3,3,3),(5,5,5)$
			$(3,3,5),(3,5,3),(5,3,3)$
$R_s=1/2$	$R=6$	$3\times 6$	$(5,5,5),(5,5,7),(7,5,5),(5,7,5)$
			$(7,7,7),(7,7,9),(9,7,7),(7,9,7)$
$R_s=1/3$	$R=9$	$3\times 9$	$w_i\in \{9,11,13\}$, for $i=1,2,3$

. With the same setting of m, the target system rates R are dominated by the source coding rates according to (2). Then, we impose the row weight constraints on the elements of the extension matrix and search for the base matrix with the lowest source decoding threshold for each source coding rate. Finally, we obtain a rate-compatible family of source codes with rates from 1/3 to 3/4.

$${\mathbf{B}}_s^{3/5}=\left[\begin{array}{ccc}& & 2\\ \multicolumn{1}{c}{{\mathbf{B}}_s^{3/4}}& & 2\\ & & 0\end{array}\right].$$

(19)

$${\mathbf{B}}_s^{1/2}=\left[\begin{array}{ccc}& & 2\\ \multicolumn{1}{c}{{\mathbf{B}}_s^{3/5}}& & 2\\ & & 2\end{array}\right].$$

(20)

$${\mathbf{B}}_s^{1/3}=\left[\begin{array}{ccccc}& & 3& 0& 1\\ \multicolumn{1}{c}{{\mathbf{B}}_s^{1/2}}& & 3& 0& 1\\ & & 3& 2& 1\end{array}\right].$$

(21)

Table 2. Source decoding thresholds for the proposed source codes.

Source Codes	W	Source Decoding Threshold	Gap ${\mathit{R}}_{\mathit{s}}-{\mathit{H}}^{\mathit{th}}$
${\mathbf{B}}_s^{3/4}$	$(3,3,5)$	$p^{th}=0.812$, $H^{th}=0.6973$	0.0527
${\mathbf{B}}_s^{3/5}$	$(5,5,5)$	$p^{th}=0.877$, $H^{th}=0.5379$	0.0621
${\mathbf{B}}_s^{1/2}$	$(7,7,7)$	$p^{th}=0.912$, $H^{th}=0.4298$	0.0702
${\mathbf{B}}_s^{1/3}$	$(11,11,13)$	$p^{th}=0.951$, $H^{th}=0.2821$	0.0512

shows the source decoding thresholds of the rate-compatible codes. It is found that the proposed codes exhibit threshold gaps between 0.0512 and 0.0702 to the corresponding source coding rates.

4. Experimental Results

In this section, an analysis of the achievable system rates and the BER results are presented to illustrate the advantage of the proposed codes.

Figure 4, Figure 5, Figure 6 and Figure 7 plot the achievable system rates for the proposed codes with different source probabilities. For instance, in Figure 6, the achievable system rates $R=C_W/H\left(p\right)$ versus $E_s/N_0$ with four typical source probabilities p (0.9, 0.92, 0.94, 0.96) for the proposed code $B_s^{1/2}$ are provided. In addition, the curves for $B_s^{0.94}$, which is proposed for $p=0.94$ in [17], are also provided for comparison. Note that $p=0.9$$\left(H\right(p)=0.469)$ is the smallest probability that satisfies the lossless source coding theorem ($H\left(p\right)\le R_s$) for $R_s=0.5$ when $\delta =0.02$. It can be found that $B_s^{1/2}$ and $B_s^{0.94}$ have similar shaping gaps to the Shannon limits for $p=0.94$ and $p=0.96$ at the target system rate $R=6$. However, as the source probability decreases, the proposed $B_s^{1/2}$ can achieve an advantage over $B_s^{0.94}$. Specifically, for $p=0.9$, the JSCCM system with the proposed $B_s^{1/2}$ can obtain a 0.19 dB shaping gain over $B_s^{0.94}$, reducing the gap to the Shannon limit. Thus, for the whole considered range of source probabilities, the proposed source code can obtain a better PAS performance than $B_s^{0.94}$. Furthermore, in Figure 4, Figure 5, Figure 6 and Figure 7, observe that in a range of source probabilities, the gaps to the Shannon limits for the proposed codes are all within 0.6 dB at the target system rates, which can ensure the PAS performance for varying source probabilities.

To further verify the merits of the proposed source codes, we present some bit-error-rate (BER) results of JSCCM systems with 16ASK for different source probabilities. For all simulations, an AWGN channel was assumed, and the information block length was fixed at 3600 bits. Additionally, the maximum number of iterations was set to 100. For source codes with different coding rates, the same protograph LDPC code (3/4-rate-${\mathbf{B}}_c^U$ [23]) was employed as the channel code. The existing source code with a good PAS performance for JSCCM systems (1/2-rate-${\mathbf{B}}_s^{0.94}$ [17]), the code with a good source decoding threshold (1/3-rate-${\mathbf{B}}_s^{LFloor}$ [19]), and 3/5-rate regular LDPC were utilized as benchmark codes. The base matrices for these benchmark codes are provided as follows:

$${\mathbf{B}}_s^{0.94}=\left(\begin{array}{cccccc}1& 1& 3& 3& 1& 0\\ 1& 0& 5& 2& 0& 1\\ 1& 2& 3& 3& 1& 1\end{array}\right).$$

(22)

$${\mathbf{B}}_s^{LFloor}=\left(\begin{array}{cccccccccccc}3& 2& 1& 1& 0& 1& 0& 0& 2& 0& 2& 0\\ 2& 3& 1& 0& 1& 0& 1& 0& 3& 2& 0& 0\\ 3& 3& 0& 0& 0& 0& 0& 1& 3& 0& 0& 1\\ 3& 0& 1& 2& 2& 1& 1& 1& 2& 2& 1& 2\end{array}\right).$$

(23)

$${\mathbf{B}}_s^{regular}=\left(\begin{array}{ccccc}2& 1& 1& 1& 0\\ 1& 0& 1& 2& 1\\ 0& 2& 1& 0& 2\end{array}\right).$$

(24)

The BER curves of JSCCM systems with the proposed source codes for different source probabilities are depicted in Figure 3 and Figure 8, Figure 9 and Figure 10.

Figure 8 provides the BER comparison of ${\mathbf{B}}_s^{3/5}$ and ${\mathbf{B}}_s^{regular}$ with different source probabilities. Despite having the same row weight distribution ($W=[5,5,5]$), the proposed ${\mathbf{B}}_s^{3/5}$ performs better than ${\mathbf{B}}_s^{regular}$ for different source probabilities. For instance, ${\mathbf{B}}_s^{3/5}$ achieves coding gains in all five different source statistics. Furthermore, it is obvious that ${\mathbf{B}}_s^{regular}$ has a higher error floor than the proposed code for $p=0.9$ and $p=0.91$.

In Figure 9, even though ${\mathbf{B}}_s^{0.94}$ has a slightly better performance in the waterfall region than the proposed code for $p=0.95$ and $p=0.96$, the proposed ${\mathbf{B}}_s^{1/2}$ achieves a significant advantage at high $E_s/N_0$. Specifically, ${\mathbf{B}}_s^{1/2}$ obtains 2 dB, 0.59 dB, 0.42 dB, and 0.23 dB gains over ${\mathbf{B}}_s^{0.94}$ at a BER of ${10}^{-4}$ for $p=0.93$, $p=0.94$, $p=0.95$, and $p=0.96$, respectively. For $p=0.92$, our proposed code achieves a lower error floor ($8\times {10}^{-4}$) than ${\mathbf{B}}_s^{0.94}$ ($7\times {10}^{-3}$).

In Figure 10, it can be found that the source code ${\mathbf{B}}_s^{1/3}$ outperforms ${\mathbf{B}}_s^{LFloor}$ by 1.5 dB and 2.4 dB at a BER of ${10}^{-4}$ for $p=0.97$ and $p=0.98$, respectively.

The above observations imply that the proposed rate-compatible source codes are expected to provide a better system performance for JSCCM systems with varying source probabilities than conventional source codes.

5. Conclusions

In this paper, a rate-compatible family of source codes is proposed based on an achievable system rate analysis and the PEXIT algorithm. Simulation results demonstrate that the proposed source codes exhibit good source decoding thresholds as well as high shaping gains for the JSCCM system across a wide range of source probabilities, making them well suited for practical applications with changing source probabilities.