## 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 process of the information bottleneck channel estimator delivers the joint distribution \(p\left(\tilde{h}^{\text{re}},h^{\text{re}}\right)=p\left(h^{\text{re}}|\tilde {h}^{\text{re}}\right)p\left(\tilde{h}^{\text{re}}\right)\), which has to be used further to construct an information bottleneck detection lookup table. Its design is explained in the next subsection. A very similar concept of information bottleneck channel estimation can also be used in the feedback iterations of the iterative information bottleneck receiver structure from Figure 6 to implement a turbo-like channel estimation scheme. After decoding the LDPC code, the receiver can identify the most reliable bit decisions \(\hat{s}_{l}\) and their corresponding received samples \(w_{l}^{\text{re}}\). Let \(w_{l}^{\text{re}}\), \(l\in\{0,1,\ldots,N_{\text{FB}}-1\}\) denote the quantized integer indices \(y_{k}^{\text{re}}\), which correspond to the most reliable bit decisions in the decoded codeword. Moreover, let \(\hat{s}_{l}\) denote the respective hard decision BPSK symbols corresponding to the respective decision bits. The task of an information bottleneck feedback channel estimator is to extract the information on \(h^{\text{re}}\) from \(\hat{s}_{l}\) and \(w_{l}^{\text{re}}\). Please note that the transformation  
\[\tilde{w}_{l}^{\text{re}}= \begin{cases}w_{l}^{\text{re}}&\hat{s}_{l}=+1\\ 2^{q}-w_{l}^{\text{re}}-1&\hat{s}_{l}=-1.\end{cases}  \tag{15}\]undoes the influence of \(s_{l}\) and makes all \(\tilde{w}_{l}^{\rm re}\) look like quantization indices received for the transmitted symbol \(s_{l}=+1\) if the decision \(\hat{s}_{l}\) is correct. The integers \(\tilde{w}_{l}^{\rm re}\) can be used in the feedback channel estimation scheme shown in the bottom part of Figure 7. The design of the concatenated compression mappings is completely equivalent to the one for the forward channel estimator above, with the only difference being that one assumes the transmitted symbol to be \(s_{l}=+1\) due to the transformation from Equation (15). In fact, this transformation is a simple way to make the feedback channel estimation lookup tables independent of the symbol decision and thus keep them minimal in size. However, we want to mention that this method only works for BPSK modulation. Higher-order modulation schemes are not studied in this article. The presented information bottleneck channel estimation could be implemented for higher-order modulation schemes equivalently, but this might also require feeding the symbol decisions \(\hat{s}_{l}\) to the feedback channel estimation lookup tables. The final compression mapping \(p(\hat{h}^{\rm re}|\hat{h}^{\rm re}_{\rm EW},\hat{h}^{\rm re}_{\rm FB})\) combines the relevant information on \(h^{\rm re}\) from the forward and feedback channel estimator lookup tables to finally deliver the integer \(\hat{h}^{\rm re}\) that is highly informative about \(h^{\rm re}\). The obtained \(\hat{h}^{\rm re}\) can be used for detection in the next step.