# Genetic Algorithms to Maximize the Relevant Mutual Information in Communication Receivers +
## 4 Results and Discussion
### Quantization of the Channel Output with Minimum Loss of Relevant Information
#### 4.1.1 Information Bottleneck Quantizer Design with the KL-Means Algorithm  
An intuitive application of the information bottleneck method in communications is the design of a channel output quantizer that maximizes the relevant information on the transmitted modulation symbols \(\mathsf{X}\). As already discussed and shown in Figure 3, in this context, \(\mathsf{Y}\) corresponds to the received channel output. If \(\mathsf{Y}\) is continuous, for example, for an AWGN channel, it has to be very finely discretized to \(|\mathcal{Y}|\) uniformly spaced samples on some interval of interest. \(\mathsf{T}\) is the quantized output variable of the quantizer. A \(q\) bit quantizer designed with the Information Bottleneck method maps realizations \(y\) onto quantization indices \(t\in\mathcal{T}=\{0,1,\ldots,2^{q}-1\}\), such that \(|\mathcal{T}|=2^{q}\) and \(\mathrm{I}(\mathsf{X};\mathsf{T})\to\max\). \(\mathrm{I}(\mathsf{X};\mathsf{T})\) is independent of the elements in \(\mathcal{T}\). We consider integer quantization indices that need \(q\) bits in the hardware. As in [26], we consider complex AWGN channels and complex modulation alphabets, such that the continuous received sample at a certain time instance is  
\[y=y^{\mathrm{re}}+j\mathrm{y}^{\mathrm{im}}=(x^{\mathrm{re}}+n^{\mathrm{re}})+ j(x^{\mathrm{im}}+n^{\mathrm{im}}), \tag{10}\]  
where \(n^{\mathrm{re}}+jn^{\mathrm{im}}\) is a realization of a complex valued, circularly symmetric Gaussian process with variance \(\sigma_{n}^{2}\) and mean \(0\) and \(x=x^{\mathrm{re}}+j\mathrm{x}^{\mathrm{im}}\) is a complex modulation symbol. For a simple notation, we assume that \(y\) is already finely discretized using a large number of \(|\mathcal{Y}|\) uniformly spaced samples in a grid on the complex plane with \(\sqrt{|\mathcal{Y}|}\) points for \(y^{\mathrm{re}}\) and \(y^{\mathrm{im}}\), respectively. In addition, we define the vector representation \(\mathbf{y}=[y^{\mathrm{re}},y^{\mathrm{im}}]\) of the received sample. In this situation, we want to quantize \(\mathbf{y}\) to a number of \(|\mathcal{T}|<<|\mathcal{Y}|\) quantization regions. The considered quantizers are particularly useful for phase-shift keying (PSK) signals [26]. An example for 8-PSK under AWGN with noise variance \(\sigma_{n}^{2}=0.5\) is provided in Figure 6. This figure shows the quantization regions obtained with the KL-means algorithm in the complex plane. For this example, \(y^{\mathrm{re}}\) and \(y^{\mathrm{im}}\) were both finely discretized into \(\sqrt{|\mathcal{Y}|}=256\) uniformly spaced samples on the interval \([-1.5,+1.5]\) with properly paying attention to clipping effects. Like this, one obtains a grid with cardinality \(|\mathcal{Y}|=256^{2}=65,536\) in the complex plane. This grid was quantized to \(|\mathcal{T}|=16\) different quantization regions. This implies strong compression. A typical application of the designed quantizer could be in a radio, where the analog-to-digital converter has a resolution of 8 bits for the real and the in-phase component of the received signal, but the signal shall be quantized to be processed further using just 4 bits per sample with minimum relevant information loss. In this example, \(\mathrm{I}(\mathsf{X};\mathsf{Y})\approx 1.49533\) bit and \(\mathrm{I}(\mathsf{X};\mathsf{T})\approx 1.388887\) bit. This indicates that despite the very coarse quantization a significant amount of relevant information on the transmitted modulation symbols (that is, around 92.8%) is preserved. Hence, it illustrates that the KL-means algorithm preserves relevant information. Note that, due to the very complicated shape of the optimized quantization regions obtained using the KL-means algorithm from Figure 6, this quantizer cannot be characterized by simple thresholds for \(y^{\mathrm{re}}\) and \(y^{\mathrm{im}}\).