## 2 The Information Bottleneck Method and Coarsely Quantized Information Bottleneck Signal Processing
### An Example of Information Bottleneck Receiver Design with Iterative Detection and Decoding
#### 2.3.2 Information Bottleneck Detection  
In a conventional receiver with high precision, the obtained MMSE channel estimate \(\hat{h}\) from Equation (10) can be used for soft demodulation. The LLRs of the soft demodulator are given by  
\[L^{\rm ch}(s_{k})=\frac{1}{1+\frac{1}{p_{+}\sigma_{h}^{2}/\sigma_{h}^{2}}} \cdot\frac{4\,{\rm Re}\Big{\{}\hat{h}^{*}\hat{r}_{k}\Big{\}}}{\sigma_{h}^{2}}, \tag{16}\]  
where \({\rm Re}\{.\}\) denotes the real part of a complex number and \(\cdot^{*}\) denotes the conjugate complex. Using these channel LLRs, it is easy to apply the LLR-based belief propagation decoding algorithm in the conventional receiver at the top of Figure 6.  
When the information bottleneck channel estimator from the preceding section is applied, it delivers the pair \(\left(\hat{h}^{\rm re},\hat{h}^{\rm im}\right)\) of unsigned integers. Moreover, the quantizer delivers a pair \(\left(y_{k}^{\rm re},y_{k}^{\rm im}\right)\) for each symbol \(s_{k}\). The task of an information bottleneck demodulator equivalent for the considered channel is extracting relevant information on \(s_{k}\) from \(\left(\hat{h}^{\rm re},\hat{h}^{\rm im}\right)\) and \(\left(y_{k}^{\rm re},y_{k}^{\rm im}\right)\). Figure 8 shows the information bottleneck graph performing this task. In this graph, one first successively combines \(\hat{h}^{\rm re},y_{k}^{\rm re}\) and \(\hat{h}^{\rm im},y_{k}^{\rm im}\) to obtain \(t_{k}^{\rm re}\) and \(t_{k}^{\rm im}\). The compression mappings \(p\left(t_{k}^{\rm re}|\hat{h}^{\rm re},y_{k}^{\rm re}\right)\) and \(p\left(\hat{i}^{\rm im}_{k}|\hat{i}^{\rm im},y_{k}^{\rm im}\right)\) are identical, as their inputs are identically distributed. To design \(p\left(t_{k}^{\rm re}|\hat{h}^{\rm re},y_{k}^{\rm re}\right)\) with an information bottleneck algorithm, one needs the joint probability distribution  
\[p\left(s_{k},\hat{h}^{\rm re},y_{k}^{\rm re}\right)=\sum_{h^{\rm re}=0}^{2^{ 2}-1}p(h^{\rm re},y_{k}^{\rm re}|s_{k})\underbrace{\frac{p\left(h^{\rm re}, \hat{h}^{\rm re}\right)}{p\left(\hat{h}^{\rm re}\right)}}_{p\left(\hat{h}^{\rm re }|h^{\rm re}\right)}p(s_{k}). \tag{17}\]  
In Equation (17), the distributions \(p\left(h^{\rm re},\hat{h}^{\rm re}\right)\) and \(p\left(h^{\rm re}\right)\) are known from the design of the channel estimation lookup table. The distribution \(p\left(h^{\rm re},y_{k}^{\rm re}|s_{k}\right)\) is again an integral of a multivariate Gaussian distribution, where  
\[p(h^{\rm re},y_{k}^{\rm re}|s_{k})=\int\limits_{-\infty}^{+\infty}\int\limits _{-\infty}^{+\infty}p\left(\hat{h}^{\rm re},\hat{y}_{k}^{\rm re}|s_{k}\right) p\left(h^{\rm re}|\hat{h}^{\rm re}\right)p(y_{k}^{\rm re}|\hat{y}_{k}^{\rm re })\,{\rm d}\,\hat{h}^{\rm re}\,{\rm d}\,\hat{y}_{k}^{\rm re}. \tag{18}\]Again, the integral can be solved using the method from [62]. The required covariance matrix is obtained analogously with Equation (14). An important note is that the design of the final concatenated information bottleneck compression mapping \(p(t_{k}|t_{k}^{\text{re}},t_{k}^{\text{im}})\) inherently delivers the distributions \(p(s_{k}|t_{k})\) and \(p(t_{k})\) and, therefore, \(p(s_{k},t_{k})\). Reviewing Figure 4, the concatenation of the transmitter, channel, quantizer and the proposed concatenation of information bottleneck compression mappings therefore corresponds to a discrete input, discrete output transmission scheme. This discrete input, discrete output transmission scheme delivers \(t_{k}\in\{0,1,\ldots,2^{d^{\text{det}}}-1\}\) for each transmitted modulation symbol \(s_{k}\) at the receiving end, where \(2^{d^{\text{det}}}\) is the output cardinality of the information bottleneck algorithm used to design \(p(t_{k}|t_{k}^{\text{re}},t_{k}^{\text{im}})\). The statistical properties of \((s_{k},t_{k})\) and the preserved relevant mutual information \(\text{I}(\text{S}_{k};\text{T}_{k})\) are completely determined by \(p(s_{k},t_{k})\), which describes a discrete source and channel model for the considered data transmission scheme.