# Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers +
## 4 Results and Discussion
### Genetic Algorithm Designed Demodulation Using K-Dimensional Trees
Please note that the distribution \(p(x|t)\) is obtained as a side product of the genetic algorithm optimization, as it is inherently determined to compute \(\mathrm{I}(\mathsf{X};\mathsf{T})\) (cf. Equations (7) and (8)). After decoding, the decoded information symbols are transformed into the decision bits by reversing the transmitter-sided bit-to-symbol mapping. For brevity, we provide the parameters that characterize the data transmission scheme used in this section further in Table 2.  
We compare the bit error rate performances of the considered data transmission scheme including the proposed demodulation technique with state-of-the-art methods in a bit error rate simulation. Due to the fact that the optimum parameters \(\mathbf{\theta}\) depend on the channel \(E_{b}/N_{0}\), we have designed the proposed tree-based demodulators for different \(E_{b}/N_{0}\) offline, stored them together with the corresponding distributions \(p(x|t)\) and used them in the simulation. The space complexity of storing the \(K\)-dimensional tree is linear in \(|\mathcal{T}|\), i.e., \(\mathcal{O}(|\mathcal{T}|)\). Therefore, storing the obtained demodulators for the different \(E_{b}/N_{0}\) is technically not challenging and only needs a few kilobytes of memory. As a result, the construction costs of the tree were one-time costs that only affected the genetic algorithm optimization, but not the demodulator implementation. We have used the same optimization settings for the genetic algorithm as in Sections 4.1.2 and 4.1.3 (cf. Table 1). As a result, the design of the demodulators could be conducted offline, such that no on-the-fly generation was required. Conducting the genetic algorithm optimization only needed a few minutes on a standard computer. As the toughest reference system, we consider a demodulator which has access to the continuous received samples \(\mathbf{\hat{y}}=[\hat{y}_{0},\hat{y}_{1},\ldots,\hat{y}_{N-1}]\) in double floating point precision, i.e., no quantizer is involved. In this case, the a posteriori distribution \(p(x|\mathbf{\hat{y}})\) is determined for each symbol in the transmitted codeword and delivered to the channel decoder for decoding. Assuming equally likely symbols \(x\in\mathrm{GF}\big{(}2^{N}\big{)}\), it is given by  
\[p(x|\mathbf{\hat{y}})\propto\prod_{k=0}^{N-1}\exp\left(-\frac{|\hat{y}_{k}-[ \mathsf{x}]_{k}|^{2}}{2\,\sigma_{n}^{2}}\right)=\exp\left(-\frac{\sum\limits_{ k=0}^{N-1}|\hat{y}_{k}-[\mathsf{x}]_{k}|^{2}}{2\,\sigma_{n}^{2}}\right)\quad \forall x\in\mathrm{GF}\Big{(}2^{N}\big{)}, \tag{11}\] \begin{table}
\begin{tabular}{c l c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(q\) & bit width for channel output quantization & 4 \\ \(i_{\mathrm{max}}\) & number of belief propagation decoding iterations & 20 \\ \(2^{N}\) & field order of non-binary low-density parity-check code & 8 \\ \(K_{\mathrm{cw}}\) & uncoded information per codeword (in \(\mathrm{GF}(8)\) symbols) & 130 \\ \(N_{\mathrm{cw}}\) & codeword length of non-binary code (in \(\mathrm{GF}(8)\) symbols) & 260 \\ \(R\) & code rate of non-binary low-density parity-check code & 0.5 \\ \(d_{\mathrm{c}}\) & check node degree of applied regular code & 4 \\ \(d_{\mathrm{v}}\) & variable node degree of applied regular code & 2 \\ \(d(\mathbf{y},\mathbf{\theta}_{t})\) & distance type used in \(K\)-dimensional tree demodulator & Euclidean \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overview of parameters of data transmission system. where \(\mathbf{x}\) is a vector with the BPSK symbols transmitted over the channel for symbol \(x\in\mathrm{GF}(2^{N})\) and \([\mathbf{x}]_{k}\) denotes the \(k\)-th element of this vector. Please note that this demodulator also requires calculating \(2^{N}\) squared Euclidean distances in the argument of the exponential (one for each Galois field symbol). In addition, it needs several divisions and the evaluation of the exponential function. Especially the latter is costly in digital hardware.