## 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
Therefore, the integer \(h^{\rm re}\) is the index of the quantization region the continuous \(\tilde{h}^{\rm re}\) falls into; that is, \(h^{\rm re}=f_{\rm Q}(\tilde{h}^{\rm re})\). From an information theoretical perspective, the task of a pilot-based channel estimator is to map the unsigned integers \(v_{0}^{\rm re},v_{1}^{\rm re},\ldots,v_{P-1}^{\rm re}\) onto \(\tilde{h}_{\rm FW}^{\rm re}\) such that the mutual information \(\mathrm{I}(\tilde{h}_{\rm FW}^{\rm re};H^{\rm re})\) is maximized. Please note that we add the index forward (FW) to distinguish the channel estimation from the one conducted in the feedback (FB) iterations, which will be discussed later. An intuitive approach to designing an information bottleneck equivalent to the pilot-based channel estimation in the forward path of the conventional receiver is to design an information bottleneck compression mapping \(p\left(\hat{h}_{\rm FW}^{\rm re},|v_{0}^{\rm re},v_{1}^{\rm re},\ldots,v_{P-1 }^{\rm re}\right)\) which aims to maximize \(\mathrm{I}(\tilde{h}_{\rm FW}^{\rm re};H^{\rm re})\). However, designing and implementing this compression mapping as a lookup table would typically result in prohibitive complexity, since the cardinality of the observation random variable \((\mathrm{V}_{0}^{\rm re},\mathrm{V}_{1}^{\rm re},\ldots,\mathrm{V}_{P-1}^{\rm re })\) is \(2^{q\cdot P}\). For example, with \(q=5\) for bit quantization and \(P=6\) pilot symbols, this would result in a lookup table with more than \(10^{9}\) entries. This cardinality exceeds the affordable runtime and space complexities of all available information bottleneck algorithms, as these algorithms typically have to handle a dense matrix representation of \(p(x,y)\) that consists of \(|\mathcal{X}||\mathcal{Y}|\) real numbers. Moreover, the resulting lookup table would be prohibitively large. In order to cope with the resulting complexity, a particularly useful feature of information bottleneck compression mappings is that they can be concatenated to reduce the complexity. Such a concatenation is shown in an information bottleneck graph in the upper part of Figure 7 for the considered channel estimation scheme for the pilot-basedforward channel estimation. As shown, the task of processing \(P\) quantized samples is split into a series of \(P-1\) concatenated compression mappings, each of which processes two inputs. This way, the space complexity of the component compression mappings that we implement as lookup tables is drastically reduced to a concatenation of lookup tables with \(2^{2q}+(P-2)2^{q^{\text{ce}}+q}\) entries, where \(q^{\text{ce}}\) denotes the number of bits needed to represent the intermediate results \(\hat{h}_{m}^{\text{re}}\) from Figure 7. Please note that this number of bits is a design choice, and it can be adjusted by the choice of the cardinality of the compression variable of the information bottleneck algorithm that is used. For simplicity, all compression mappings shall use the same output bit width \(q^{\text{ce}}\) such that all intermediate results \(\hat{h}_{m}^{\text{re}}\) and the final output \(\hat{h}_{\text{FW}}^{\text{re}}\) are from the same set \(\left\{0,1,\dots,2^{q^{\text{ce}}}-1\right\}\). With \(q^{\text{ce}}=5\) and \(P=6\) pilot symbols, as considered before, the overall size of the lookup tables to implement the concatenated scheme from Figure 7 is reduced from roughly \(10^{9}\) to \(5120\). All compression mappings appearing inside the forward channel estimator from Figure 7 preserve information on the same variable \(\text{H}^{\text{re}}\).