## 3 The Processing Method for Integrated Radar-Communication Signals  
### Improved Normalized Minimum-Sum Algorithm in LDPC  
Some scholars have simulated communication systems with integrated LDPC in Rayleigh fading channels, and the results indicate that the use of the sum-product algorithm (SPA) significantly enhances system performance [27]. The SPA in traditional LDPC involves a large number of multiplications, resulting in high computational complexity during implementation [28]. Therefore, the probability information is represented using likelihood ratios, leading to the log-likelihood ratio belief propagation (LLR BP) algorithm [23]. In this case, the initial messages for the channels are as follows [23]:  
\[L(p_{i})=\frac{\ln p_{i}(0)}{\ln p_{i}(1)}=\ln\frac{P(v_{i}=0|y_{i})}{P(v_{i}= 1|y_{i})} \tag{7}\]  
Here, \(v_{i}\) represents the \(i\)-\(th\) variable node, and \(p_{i}(b)\) represents the posterior probability of codeword \(c_{i}=b\) when \(y_{i}\) is received at the receiving end. The external information passed by the calibration node to the variable node is as follows:  
\[r_{ji}=\ln\frac{r_{ji}(0)}{r_{ji}(1)}=\ln\frac{1+\prod\limits_{\vec{r}\in V(j)/ i}\Bigl{(}1-2q_{\vec{r}j}(1)\Bigr{)}}{1-\prod\limits_{\vec{r}\in V(j)/i} \Bigl{(}1-2q_{\vec{r}j}(1)\Bigr{)}} \tag{8}\]  
in which, \(r_{ij}\) represents the external information passed from check node \(j\) to variable node \(i\), and \(V(j)/i\) denotes the set of other variable nodes connected to \(j\)-\(th\) check node, except for the \(i\)-\(th\) variable node. The simplification of external information yields the following with the introduction of \(\tanh(x)=(e^{x}-e^{-x})/(e^{x}+e^{-x})\):  
\[r_{ji}=2\tanh^{-1}(\prod\limits_{\vec{r}\in V(j)/i}\tanh\frac{q_{\vec{r}j}}{2}) \tag{9}\]  
Even with the introduction of \(\tanh\) in the logarithmic domain, it still consumes a significant amount of hardware resources. In this paper, the minimum-sum belief propagation (min-sum BP) algorithm is introduced to convert complex function operations into symbol value operations and numerical comparison operations. Here, the minimum value is used to replace the multiplication results:  
\[\prod\limits_{\vec{r}\in V(j)/i}\tanh\left|\frac{q_{\vec{r}j}}{2}\right|\approx \min\limits_{\vec{r}\in V(j)/i}(2\tanh\left|\frac{q_{\vec{r}j}}{2}\right|) \tag{10}\]  
The check node update value of the min-sum BP algorithm is greater than that obtained by the LLR BP algorithm, i.e.,  
\[2\tanh^{-1}(\prod\limits_{\vec{r}\in V(j)/i}\tanh\frac{q_{\vec{r}j}}{2})<\min \limits_{\vec{r}\in V(j)/i}(\tanh\left|q_{\vec{r}j}\right|) \tag{11}\]  
Therefore, in this paper, we multiply this outer information by the multiplicative scaling factor \(\alpha\) to obtain the LDPC-improved normalized min-sum algorithm (NMSA). Hence, the external information transmitted from check nodes to variable nodes can be simplified as follows:  
\[r_{ji}=\alpha\cdot\prod\limits_{\vec{r}\in V(j)/i}\operatorname{sgn}(q_{\vec {r}j})\cdot\min\limits_{\vec{r}\in V(j)/i}(\tanh\left|q_{\vec{r}j}\right|) \tag{12}\]  
As depicted in Figure 4, it is discernible that the convergence velocity is maximized when \(\alpha=0.8\). Consequently, in the subsequent simulation experiments, we intend to configure the corresponding multiplicative diminishing factor to \(0.8\). During the decodingdecision process, after iterations, the posterior probability of variable nodes is given by the following:  
\[q_{ij}=\ln\frac{p_{i}(0)\prod\limits_{f^{\prime}\in\text{C}(i)}r_{ji}(0)}{p_{i}(1 )\prod\limits_{f^{\prime}\in\text{C}(i)}r_{ji^{\prime}}(1)}=L(p_{i})+\sum \limits_{f^{\prime}\in\text{C}(i)}r_{ji} \tag{13}\]  
If the value is greater than 0, i.e., \(P(v_{i}=0)>P(v_{i}=1)\), \(v_{i}\) can be adjudicated as 0.