# Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers +
## 4 Results and Discussion
### Genetic Algorithm Designed Demodulation Using K-Dimensional Trees  
Next, we want to investigate an application of the proposed baseband signal processing approach illustrated in Figure 2 in a data transmission system that employs a non-binary low-density parity-check code over the Galois field \(2^{N}\left(\operatorname{GF}\!\left(2^{N}\right)\right)\) with \(N>1\) for forward error correction, but uses BPSK for signalling over an AWGN channel. A data transmission scheme similar to the one studied here was investigated for a lookup table-based information Bottleneck approach in [33]. For a deep introduction to non-binary low-density parity-check codes we kindly refer the reader to [34]. Pairing a non-binary channel code with BPSK offers a particularly interesting use case of the system illustrated in Figure 2. As it will be explained in the following, in the considered setup \(N\) received samples have to be processed for the demodulation of a \(\operatorname{GF}\!\left(2^{N}\right)\) symbol at the receiving end. Hence, this problem perfectly matches the architecture of the considered system. For completeness, it shall be mentioned that it is also common to pair non-binary channel codes with \(2^{N}\)-ary modulation schemes, for example, \(2^{N}\)-PSK. For such a coding and modulation scheme, the demodulation problem can be conducted using the systems investigated in Section 4.1. To do so, one has to use \(p(x|t)\) after the quantization, as it has already been mentioned in Section 3. The data transmission system that includes a non-binary channel code and BPSK modulation studied in this section is sketched in Figure 9.  
The upper part of the figure shows the considered transmitter and the channel. The lower part illustrates the receiver processing including the demodulator designed with a genetic algorithm. In the transmitter, random data bits are mapped onto \(\operatorname{GF}\!\left(2^{N}\right)\) symbols and then encoded using a non-binary low-density parity-check encoder with code rate \(R\). In order to transmit the data over an AWGN channel using BPSK modulation, each output symbol of the encoder is mapped onto \(N\) consecutive BPSK symbols which are transmitted over the channel. At the receiving end, first a coarse analog-to-digital conversion is performed using a \(q\) bit quantizer. \(N\) outputs \(y_{k}\in\{0,1,\ldots,2^{q}-1\}\), \(k\in\{0,1,\ldots,N-1\}\) from this quantizer correspond to the received samples for the transmitted BPSK symbols for a single \(\operatorname{GF}\!\left(2^{N}\right)\) output symbol of the channel encoder in this setup. The scalar channel output quantizer is designed as explained in [11]. Figure 9: Illustration of the considered data transmission scheme which employs a non-binary low-density parity-check code over \(\operatorname{GF}\!\left(2^{N}\right)\). At the receiver, a demodulator using the nearest neighbor search from Section 3.1.2 in \(K\)-dimensional trees is employed. The next crucial task of the communication receiver is to provide symbol probabilities for \(x\in\mathrm{GF}\big{(}2^{N}\big{)}\) to the channel decoder, such that it can perform soft channel decoding. The applied channel decoder performs the iterative sum-product algorithm, also known as belief-propagation decoding, to decode the non-binary low-density parity-check code with a maximum of \(i_{\mathrm{max}}\) decoding iterations. As a result, an output symbol \(x\in\mathrm{GF}\big{(}2^{N}\big{)}\) of the channel encoder forms the relevant random variable \(\mathsf{X}\) for our proposed demodulator. We use a genetic algorithm optimized demodulator which conducts either approximate or exact nearest neighbor search in a \(K\)-dimensional tree. The demodulator determines the index \(t\) of the nearest neighbor \(\mathbf{\theta}_{t}\) as explained in Section 3.1.2 and delivers the symbol probability \(p(x|t)\) to the channel decoder.