## 3 Design of Source Codes
### 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  
\[C=\frac{1}{2}\log_{2}(1+2R\cdot E_{s}/N_{0}), \tag{4}\]  
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 \(H(p)\), the Shannon limit \(\frac{E_{s}}{N_{0}}|_{s}\) is determined from the condition \(H(p)R<C\), 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 \(b^{i}(1\leq i\leq m-1)\) 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 \(W=(w_{1},w_{2},\ldots,w_{m-1})\), which is obtained as follows  
\[P_{W}(X)=\frac{1}{2}\prod_{i=1}^{m-1}\frac{1+(-1)^{v^{i}}(1-2p)^{w_{i}}}{2}, \tag{5}\]  
where \(v^{i}\in\{0,1\}\) represents a specific bit of the symbol \(X\) at the bit position \(b^{i}\). Figure 2: Tanner graph for joint source and channel decoding. Given the input probability distribution of the modulated symbols \(P_{W}(X)\), the channel capacity of the ASK-modulated AWGN channel is calculated as  
\[C_{W}=\sum_{X\in\mathcal{A}}P_{W}(X)\int_{-\infty}^{+\infty}P_{\alpha}(Y|X)\log_ {2}\frac{P_{\alpha}(Y|X)}{\sum\limits_{X^{\prime}\in\mathcal{A}}P_{W}(X^{ \prime})P_{\alpha}(Y|X^{\prime})}dY, \tag{6}\]  
where \(\mathbf{X}\) is the modulated symbol sequence, \(\mathbf{Y}\) is the received symbol sequence, \(P_{\alpha}(Y|X)=\frac{1}{\sqrt{\pi N_{0}}}\exp\left\{-\frac{(Y-aX)^{2}}{N_{0} }\right\}\), and the associated SNR is calculated as [8, 18]  
\[\frac{E_{s}}{N_{0}}\Big{|}_{W}=\frac{\sum_{X\in\mathcal{A}}P_{W}(X)|aX|^{2}}{ RN_{0}}. \tag{7}\]  
As the limit of the ASK constellations, \(\frac{E_{s}}{N_{0}}\Big{|}_{W}\) is larger than \(\left.\frac{E_{s}}{N_{0}}\right|_{S}\) 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 \(\left.\frac{E_{s}}{N_{0}}\right|_{S}\). 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 (\(H(p)\leq R_{s}\)). Then, we gradually increase the value of \(p\) for \(p<1\) and locate potential candidates \(W\) for each \(p\) as follows:  
\[\mathcal{W}_{\Delta}(p)=\Big{\{}W\in[1,w_{max}]^{m-1}:H(p)R<C_{W},\;\;E_{s}/N_ {0}|_{W}-E_{s}/N_{0}|_{S}<\Delta\Big{\}}, \tag{8}\]  
where \(w_{max}\) is the maximum row weight value specified for \(\mathbf{B}_{s}\), and \(\Delta\) is the maximum gap in dB.