# Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers +
## 4 Results and Discussion
### Genetic Algorithm Designed Demodulation Using K-Dimensional Trees
Our aim is to approach the performance of this non-quantized reference system with the proposed demodulation techniques as closely as possible while circumventing most of the costly signal processing operations. For reference, we also consider a very simple demodulator. This demodulator performs a hard decision on the BPSK symbols on the channel and maps this hard decision onto the corresponding \(\mathrm{GF}(2^{N})\) symbol directly. The decoder then is fed with a distribution that mimics \(p(x|\tilde{\mathbf{y}})\) with probability 1 for the hard decision symbol and 0 for all others. This system, of course, cannot profit from soft information from the demodulation process. We use it to illustrate the gains of using soft demodulation in the data transmission system. Figure 10 shows bit error rate performances of the considered data transmission system over \(E_{b}/N_{0}\) for the different applied demodulation techniques. Of course, the non-quantized soft-decision reference system (\(\diamond\)-markers) has the best possible performance, as it suffers from no quantization loss at all. Comparing it to the system with hard demodulation (\(\otimes\)-markers) shows that at an exemplary bit error rate of \(10^{-4}\) a soft demodulation gain of more than 3 dB over \(E_{b}/N_{0}\) exists for this data transmission system with a non-binary low-density parity-check code over \(\mathrm{GF}(8)\). Interestingly, for the proposed \(K\)-dimensional tree demodulators with different cardinalities \(|\mathcal{T}|\) the shown results indicate that with proposed genetic algorithm optimization of the vectors \(\theta_{t}\), one can learn very powerful demodulators which can approach the performance of the optimum considered reference scheme up to a very slight loss over \(E_{b}/N_{0}\). For the demodulator with exact nearest neighbor search and \(|\mathcal{T}|=1024\) (\(\circ\)-markers), almost the full soft processing gain, i.e., more than 3 dB over \(E_{b}/N_{0}\) can be realized, even though a coarse \(q=4\) bit channel output quantizer is in place. The remaining gap to the non-quantized reference demodulator is just 0.2 dB at a bit error rate of \(10^{-4}\). At the same time, most of the signal processing operations inside the demodulator degenerate to simple  
Figure 10: Bit error rate results for non-binary \(\mathrm{GF}(8)\) low-density parity-check encoded data transmission over an AWGN channel with the parameters mentioned in Table 2. The proposed \(K\)-dimensional tree demodulators can achieve performance very close to the considered optimum reference system. threshold decisions along the axis of the vectors in the processing of the search in the \(K\)-dimensional tree described in Section 3.1.2. Even if the absolute number of vectors \(\theta_{t}\) is very large, only very few distance calculations need to be performed. This goes back to the logarithmic average search complexity in the \(K\)-dimensional tree described in Section 3.1.2. As a result, using large output cardinalities like \(|\mathcal{T}|=1024\) which are required to achieve performance so close to the optimum reference scheme is easily possible here. With the simple nearest neighbor search approach from Section 3.1.1, in contrast, using such a large cardinality \(|\mathcal{T}|\) is practically infeasible. Another very interesting observation from Figure 10 is that the bit error rates obtained with exact and approximate nearest-neighbor search for the same output cardinalities \(|\mathcal{T}|\) superimpose (cf. \((\circ,+)\)-markers \((\square,\times)\)-markers, \((\circ,\bullet)\)-markers). This finding is in fact very important because it highlights that the genetic algorithm automatically learns the different mapping rule applied inside the demodulator and tunes the parameters \(\theta\) accordingly. As it has been explained in Section 3.1.3 switching to approximate nearest neighbor search yields a fixed search complexity \(\mathcal{O}(\log_{2}(|\mathcal{T}|))\).