## 3 Information Mapping of Encoded Radio Transmission under Time-Varying Disturbance
### Information Mapping Based on MI-ESM  
Assuming that an information stream is encoded by some FEC codes into blocks of length \(Q\) symbols. Given the finite-alphabet transmitted constellation \(\mathbb{Q}=\{q_{i}\}\) (\(i=1,2,\ldots,M\)), the symbols of \(\mathbb{Q}\) are chosen according to a uniform probability distribution. To evaluate the interference effect on a single code block, the mutual information-based quality model is shown in Figure 4.  
For RSTT, each symbol of the received signal can be expressed as  
\[y=hx+n \tag{2}\]  
where \(h\) is the channel gain, \(x\) refers to the transmitted signal, and \(n\) refers to the stochastic disturbance in the HSR environment. The symbol mutual information (\(SI\)) of each symbol is defined as  
\[SI=I(X;Y)=\sum_{i}\int p\Big{(}y\big{|}q_{i}\Big{)}P(q_{i})\log_{2}\frac{p \Big{(}y\big{|}q_{i}\Big{)}}{p(y)}dy \tag{3}\]  
where \(q_{i}\) represents the \(i\)-th (\(i=1,2,\ldots,M\)) symbol in the transmitted constellation \(\mathbb{Q}\).  
To calculate the \(SI\) of a transmitted symbol, the corresponding \(SINR\) of this symbol should be determined in the first place. For modern FEC codes combined with interleaving technology, like turbo codes and low-density-parity-check (LDPC) codes, it is typically reasonable to assume a uniform distribution of the errors within a code block [28]. In order to deal with the problem of interleaved codes in a non-Gaussian channel, the block-fading channel model [29, 30] is adopted. The block-fading channel represents a memoryless channel with random channel gain or random interference that is constant for a block of symbols, and then changes to a new independent value from a block to another [31]. That means, the \(SINR\) of each symbol in this block remains the same despite the burst noise. Therefore, the effective symbol \(SINR\) of the received symbol \(q_{i}\) is:  
\[\gamma_{i}=\frac{\big{|}q_{i}\big{|}^{2}}{\sigma_{w}^{2}+\sigma_{A}^{2}} \tag{4}\]  
Figure 4: Illustration of an MI-based quality model. SI: symbol mutual information.  
where \(\sigma_{w}\) and \(\sigma_{A}\) refer to the standard deviation of AWGN and the pulsed interference, respectively. Since the background AWGN is relatively weak compared to the pulses, the effective noise power of this symbol can be estimated by the average power of noise during one code block:  
\[\sigma_{e}^{2}=\sigma_{w}^{2}+\sigma_{A}^{2}\approx\int\limits_{0}^{\infty}n^{ 2}\cdot dAPD_{n}(n) \tag{5}\]  
where \(n\) is limited to the noise within a single code block length. To simplify the calculation, the signal information when transmitting symbol \(q_{i}\) is approximate as [32]  
\[SI_{i}(\sigma_{e},m)=-\log_{2}\frac{1}{M}\sum\limits_{j=1}^{M}\exp(-\frac{d_{ ij}^{2}/\sigma_{e}{}^{2}}{3-\exp(-d_{ij}^{2}/(4\sigma_{e}{}^{2}))}) \tag{6}\]  
where \(m\) is the number of information bits that compose the \(i\)-th symbol, and \(d_{ij}\) denotes the Euclidean measure between the \(i\)-th and \(j\)-th signal points in the constellation \(\mathbb{Q}\).  
Furthermore, \(RBIR\) is computed from the \(SI\) value of symbols comprising a code block [33], which is obtained from the \(SINR\)\(\gamma_{k}\) during one block. Thus, \(RBIR\) is given as [34]  
\[RBIR=\frac{\sum\limits_{k=1}^{Q}SI(\gamma_{k})}{\sum\limits_{k=1}^{Q}m_{k}}= \frac{E[S_{k}]}{E[m_{k}]} \tag{7}\]  
where \(SI_{k}\) is the mutual information of the \(k\)-th (\(k\) = 1,2,..., \(Q\)) transmitted symbol in the block.