# Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers +
## 4 Results and Discussion
### Genetic Algorithm Designed Demodulation Using K-Dimensional Trees
In the considered case for \(|\mathcal{T}|=1024\) using approximate nearest neighbor search, typically \(n_{\text{dist}}=10\) distances have to be calculated to achieve performance enormously close to the soft demodulation reference system. Please note that for a number of \(\theta_{t}\) vectors which is a power of 2 there is the possibility that \(\log_{2}(|\mathcal{T}|)+1\) distance calculations are needed. This, however only affects one of all possible paths in the tree and happens very rarely, especially if the tree is large. Therefore, the single additional distance calculation may be neglected. Anyway, we will mention it as the worst-case to be precise in the following. For the GF(8) code used here, according to Equation (11) the soft demodulator has to determine \(n_{\text{dist}}=8\) distances to obtain the probabilities \(p(x|\hat{\mathbf{y}})\;\forall x\in\text{GF}(8)\). However, it is important to note that the soft symbol demodulator reference system has a significantly higher complexity anyway. Most importantly, the non-quantized soft demodulator uses 64 bit double floating-point values from the channel. The proposed demodulator circumvents the need of representing and processing the received samples from the channel with high precision, as it directly works on the \(q=4\) bit output integers \(y_{k}\in\{0,1,\ldots,15\}\) from the quantizer. There is no need to represent the quantized received values using real numbers as representation values, as the genetic algorithm learns to directly process the \(q\) bit quantization indices. This alone yields a significant complexity reduction of the receiver because the resolution used for the analogue-to-digital conversion of the receiver can be reduced significantly. In addition, the soft demodulator reference system requires divisions by \(2\,\sigma_{n}^{2}\) and \(2^{N}=2^{3}=8\) evaluations of the very costly exponential function in the considered case of a GF(8) code. Once the right hand side of Equation (11) has been evaluated for all \(x\in\text{GF}(8)\), one needs a normalization step to obtain a valid probability distribution \(p(x|\hat{\mathbf{y}})\) which needs seven summation and eight division operations for the used GF(8) code. All these add on top of the required eight distance calculations. The proposed system with \(|\mathcal{T}|=1024\) and approximate nearest neighbor search trades the required high precision of the analog-to-digital conversion and the numerous mentioned costly operations for typically two (worst case: three) additional distance calculations and very simple thresholding operations during the search in the \(K\)-dimensional tree. Despite this, it achieves almost identical performance as the optimum non-quantized reference scheme. Finally, the curves for \(|\mathcal{T}|=128\) and \(|\mathcal{T}|=32\) for approximate nearest neighbor search in Figure 10 (\(\times\)-markers, \(\bullet\)-markers) reveal that even the demodulators with fewer distance calculations than the optimum soft demodulator can already realize very significant soft processing gains in comparison to the hard decision demodulator. At a bit error rate of \(10^{-4}\), the demodulator with \(|\mathcal{T}|=32\), i.e., typically just five (worst case: six) distance calculations achieves more than 2 dB soft processing gain over \(E_{b}/N_{0}\) in comparison to the hard decision demodulator. The one for \(|\mathcal{T}|=128\) with typically seven (worst case: eight) distance calculations achieves 2.5 dB and has a remaining gap of approximately 0.5 dB to the non-quantized reference scheme. This illustrates that the proposed method allows to flexibly tune the trade-off between complexity and performance.