## 3 Information Mapping of Encoded Radio Transmission under Time-Varying Disturbance
### Long-Term Coded RBIR under Time-Varying Disturbance Based on Nonlinear Optimization  
Assuming the number of code blocks affected during the interference is \(N_{BL}\) and \(RBIR\)-to-\(BLER\) mapping in AWGN channel is \(f_{\text{R2B}}(RBIR)\), the average \(BLER\) of these \(N_{BL}\) blocks is denoted as \(BLER_{ave}=\frac{\sum_{i=1}^{N_{BL}}f_{\text{R2B}}(RBIR_{i})}{N_{BL}}\). To guarantee the validity of dynamic equivalence, \(RBIR_{L}\) is defined as the inverse mapping of \(BLER_{ave}\) in the AWGN channel, expressed as  
\[RBIR_{L}\equiv f_{\text{R2B}}{}^{-1}(BLER_{ave}) \tag{8}\]  
Given the \(RBIR\) series of each code blocks under the disturbance, i.e., \(\{RBIR_{i}\}\) (\(i\) = 1,2,..., \(N_{BL}\)), the \(RBIR_{L}\) of the radio receiver should be obtained by  
\[RBIR_{L}=\frac{1}{N_{BL}}\sum\limits_{i=1}^{N_{BL}}\beta_{i}RBIR_{i} \tag{9}\]  
where \(\{\beta_{i}\}\) (\(i\) = 1,2,..., \(N_{BL}\)) is the weighting coefficient of the \(i\)-th \(RBIR\). Therefore, the major problem is to find a proper parameter \(\{\beta_{i}\}\) that could map the RBIR of successive blocks to a long-term equivalent metric \(\text{RBIR}_{L}\). As stated in (8), the \(RBIR_{L}\) is closely related to \(f_{R2B}(RBIR)\). Generally, it can be approximated as a parametric function related to the encoding parameters such as the block size and coding rate (BCR) [32], expressed as  
\[BLER=f_{\text{R2B}}(RBIR)=0.5 erfc\bigg{(}\frac{RBIR-b_{BCR}}{\sqrt{2}c_{\text{ odd}}c_{BCR}}\bigg{)} \tag{10}\]where _erfc_(_x_) stands for the Gaussian complementary error function. The parameters \(b_{BCR}\) and \(c_{BCR}\) mainly depend on the code rate and block size, respectively, while the adjusting factor \(c_{cod}\) denotes the practical coding loss from the Shannon limit determined by the type of channel codes [27]. Due to the nonlinearity of _erfc_(_x_), it is appropriate to apply the optimally nonlinear weighting method [35] so as to achieve the coefficients \(\{\beta_{i}\}\). Intuitively, code blocks with different RBIR make a different contribution to \(BLER_{ave}\), so \(\beta_{i}\) cannot have a linear relation with \(BLER\). Therefore, to specify the relation between \(RBIR\) and \(\beta_{i}\), a parametric function \(g(f_{R2B}(RBIR)\),\(t)\) is brought in to describe the contribution of the \(i\)-th block, where the argument \(t\) is the key factor that determines the degree of nonlinear. In this problem, function \(g\) is defined as  
\[g=\left(f_{R2B}(RBIR),t\right)^{t}=f_{R2B}^{t},\ t\in[0,\infty) \tag{11}\]  
The function \(g\) is illustrated in Figure 5. As is shown, in the case of \(t=1\), \(\beta\) is in proportion to \(f_{R2B}(RBIR)\), just referring to the linear relation. As for \(t>1\), the data block with the worse error performance has a higher weight, and vice versa. In fact, \(t\) is mainly dependent on the function \(f_{R2B}(RBIR)\) other than the value of _RBIR_[35]. Therefore, the weighting coefficient \(\beta_{i}\) is in the form of:  
\[\beta_{i}=\frac{g_{i}}{\sum\limits_{k=1}^{N_{BL}}g_{k}}=\frac{f_{R2B}^{t}(RBIR _{i})}{\sum\limits_{k=1}^{N_{BL}}f_{R2B}^{t}(RBIR_{k})} \tag{12}\]  
The error between the estimated \(BLER\) from the RBIR\({}_{\text{L}}\) and the exact \(BLER_{ave}\) is expressed as \(e=BLER_{ave}-f_{R2B}(RBIR_{L})\). Based on MSE (minimum squared error) principle, \(\{\beta_{i}\}\) is determine by  
\[\beta_{i}=\arg\min_{t\in[0,\infty)}e^{2}=\arg\min_{t\in[0,\infty)}\left[\sum \limits_{i=1}^{N_{BL}}\frac{f_{R2B}(RBIR_{i})}{N_{BL}}-f_{R2B}\left(\sum \limits_{i=1}^{N_{BL}}\frac{f_{R2B}^{t}(RBIR_{i})RBIR_{i}}{N_{BL}\sum\limits_ {k=1}^{N_{BL}}f_{R2B}^{t}(RBIR_{k})}\right)\right]^{2} \tag{13}\]  
(13) reaches the optimal solution when \(\text{d}e^{2}/\text{d}t=0\). It is very difficult to get the accurate analytical solution of \(t\) in consideration of _erfc_(_x_), so we turned to approximate the numeric results instead. Assuming that \(f_{R2B}(x)=0.5\textit{erfc}(x)\), Figure 6 shows the value of the \(BLER\) MSE as the variation of \(t\), when \(\{RBIR_{i}\}\) obey different distributions.