## 2 The Information Bottleneck Method and Coarsely Quantized Information Bottleneck Signal Processing
### An Example of Information Bottleneck Receiver Design with Iterative Detection and Decoding
#### 2.3.1 Information Bottleneck Channel Estimation
The design of the compression mappings from Figure 7 requires feeding the joint distributions  
\[p\left(h^{\text{re}},v_{0}^{\text{re}},v_{1}^{\text{re}}|s_{0}^{\text{p}},s_{1} ^{\text{p}}\right)=p\left(h^{\text{re}},v_{0}^{\text{re}}|s_{0}^{\text{p}} \right)p\left(h^{\text{re}},v_{1}^{\text{re}}|s_{1}^{\text{p}}\right)\frac{1}{ p(h^{\text{re}})} \tag{11}\]  
and  
\[p\left(h^{\text{re}},\hat{h}_{m-1}^{\text{re}},v_{m+1}^{\text{re}}|s_ {0}^{\text{p}},s_{1}^{\text{p}},\ldots,s_{m+1}^{\text{p}}\right)=\\ p\left(h^{\text{re}},\hat{h}_{m-1}^{\text{re}}|s_{0}^{\text{p}},s_{1}^ {\text{p}},\ldots,s_{m}^{\text{p}}\right)p\left(h^{\text{re}},v_{m+1}^{\text{re} }|s_{m+1}^{\text{p}}\right)\frac{1}{p(h^{\text{re}})}\text{ for }m\geq 1. \tag{12}\]  
to an information bottleneck algorithm. One question that is still open is how to obtain the joint distributions \(p\left(h^{\text{re}},v_{m}^{\text{re}}|s_{m}^{\text{p}}\right)\) needed to evaluate Equations (11) and (12). Recall that \(h^{\text{re}}\) and \(v_{m}^{\text{re}}\) are quantization indices of \(\hat{h}^{\text{re}}\) and \(\hat{v}_{m}^{\text{re}}\), respectively. Hence, \(h^{\text{re}}=f_{\text{Q}}\left(\hat{h}^{\text{re}}\right)\) and \(v_{m}^{\text{re}}=f_{\text{Q}}\left(\hat{v}_{m}^{\text{re}}\right)\), where \(f_{\text{Q}}(. )\) characterizes the threshold decisions of the channel output quantizer. The quantizer \(f_{\text{Q}}(. )\) for \(\tilde{h}^{\text{re}}\) and \(\tilde{v}^{\text{re}}\) can also be described by \(p\left(h^{\text{re}}|\tilde{h}^{\text{re}}\right)\) and \(p(v_{m}^{\text{re}}|\tilde{v}_{m}^{\text{re}})\). This allows expressing the joint distribution \(p\left(h^{\text{re}},v_{m}^{\text{re}}|s_{m}^{\text{p}}\right)\) as  
\[p(h^{\text{re}},v_{m}^{\text{re}}|s_{m}^{\text{p}})=\int\limits_{-\infty}^{+ \infty}\int\limits_{-\infty}^{+\infty}p\left(\tilde{h}^{\text{re}},\tilde{v} _{m}^{\text{re}}|s_{m}^{\text{p}}\right)p\left(h^{\text{re}}|\tilde{h}^{\text {re}}\right)p\left(v_{m}^{\text{re}}|\tilde{v}_{m}^{\text{re}}\right)\text{ d}\,\tilde{h}^{\text{re}}\text{ d}\,\tilde{v}_{m}^{\text{re}}. \tag{13}\]  
In this integral, the factors \(p\left(h^{\text{re}}|\tilde{h}^{\text{re}}\right)\) and \(p(v_{m}^{\text{re}}|\tilde{v}_{m}^{\text{re}})\) gather all the probability masses of the continuous \(\tilde{h}^{\text{re}}\) and \(\tilde{v}_{m}^{\text{re}}\), which are mapped onto the same pair \((h^{\text{re}},v_{m}^{\text{re}})\) by application of the channel output quantizer on these variables. Therefore, they determine the integration area for a particular pair \((h^{\text{re}},v_{m}^{\text{re}})\). The distribution \(p\left(\tilde{h}^{\text{re}},\tilde{v}_{m}^{\text{re}}|s_{m}^{\text{p}}\right)\) for the considered channel model is a multivariate Gaussian distribution. Both components have a zero mean. As a result, the multivariate Gaussian distribution \(p\left(\tilde{h}^{\text{re}},\tilde{v}_{m}^{\text{re}},|s_{m}^{\text{p}}\right)\) is fully characterized by the covariance matrix  
\[\mathbf{C}_{\tilde{h}^{\text{re}},\tilde{v}_{m}|\tilde{s}_{m}^{\text{p}}}= \frac{\sigma_{\tilde{h}}^{2}}{2} \begin{bmatrix}1&s_{m}^{\text{p}}\\ s_{m}^{\text{p}}&1\end{bmatrix} +\frac{\sigma_{\tilde{h}}^{2}}{2} \begin{bmatrix} 0&0\\ 0&1\end{bmatrix} . \tag{14}\]  
The integral from Equation (13) with the considered Gaussian distributions and covariance matrices characterized by Equation (14) is easily solved numerically using the method described in [62]. This allows one to obtain the joint distributions \(p(h^{\text{re}},v_{m}^{\text{re}}|s_{m}^{\text{p}})\) and hence construct the concatenated structure of information bottleneck compression mappings from the top of Figure 7 using Equations (11) and (12). We apply the KL-means algorithm from [57] to construct the proposed information bottleneck channel estimator, as this algorithm can be highly parallelized and is very fast.