## 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  
In the considered conventional receiver chain, the task of the channel estimation is to obtain a reliable estimate \(\hat{h}\) of \(\hat{h}\) from the knowledge of \(P\) pilot symbols and their respective received samples. We denote these received samples by \(\hat{v}_{0},\hat{v}_{1},\ldots,\hat{v}_{P-1}\). The MMSE channel estimate is then given by  
\[\hat{h}=\frac{1}{P+\sigma_{h}^{2}/\sigma_{h}^{2}}\sum_{k=0}^{P-1}\hat{v}_{k} \hat{v}_{k}^{\text{P}}, \tag{10}\]  
Figure 6: Conventional and information bottleneck receiver chains for LDPC-encoded data transmission over a frequency-flat fading channel. The conventional receiver uses quasi-continuous received samples \(\hat{y}_{k}\) with double-precision and processes them in state-of-the-art detection and decoding algorithms. The information bottleneck receiver works on quantization indices and implements all signal processing using information bottleneck lookup tables. where \(s_{k}^{\rm P}\in\{-1,+1\}\) are the known pilot symbols. It is obvious, that the evaluation of Equation (10) requires high-precision arithmetic. In a digital receiver, a high quantization resolution is required such that the quantized received samples adequately approximate the continuous \(\tilde{v}_{k}\) to obtain a good estimate with high accuracy. As mentioned above, we assume that the conventional reference receiver is implemented using double floating-point precision and therefore does not suffer from any mentionable quantization loss. In the information bottleneck receiver, however, the aim is to directly process the \(q\)-bit integer indices \(v_{k}^{\rm re}=f_{\rm Q}(\tilde{v}_{k}^{\rm re})\) and \(v_{k}^{\rm im}=f_{\rm Q}(\tilde{v}_{k}^{\rm im})\). The function \(f_{\rm Q}(. )\) describes a channel output quantizer that is also designed to preserve the maximum relevant information with the information bottleneck method as explained in [35]. Please note that the design of the channel output quantizer \(f_{\rm Q}(. )\) is not discussed in detail in this article. However, we want to mention that it requires handling continuous observed random variables with the information bottleneck method, which has not been discussed in this article so far. A simple method to deal with continuous observations for the quantizer design problem is approximating the continuous variables as very finely quantized discrete variables. More details on the quantizer design with a desired maximum preservation of relevant information can be found in [35; 61]. The quantization function \(f_{\rm Q}(. )\) conducts threshold decisions on the real and imaginary parts of the received signal \(\tilde{y}_{k}\). It delivers quantization indices from the set \(\{0,1,\ldots,2^{q}-1\}\). This reflects an analog-to-digital conversion with \(q\) bits per sample in the real and imaginary parts, respectively. The obtained unsigned integers neither approximate the continuous received samples nor can they be used to obtain a channel estimate with an arithmetic rule such as that in Equation (10). Essentially, the task of channel estimation is to extract information on the unknown channel coefficient \(\tilde{h}\) from \(P\) pairs of integers \(\left(v_{k}^{\rm re},v_{k}^{\rm im}\right)\) that correspond to the coarsely quantized received samples for the pilot symbols. Please note that due to the considered channel model and the fact that the pilot symbols are BPSK symbols, it is possible to handle \(v_{k}^{\rm re}\) and \(v_{k}^{\rm im}\) independently. In the following, the signal processing of \(v_{k}^{\rm im}\) will not be discussed, as it is equivalent to that of \(v_{k}^{\rm re}\). Note that the quantizer \(f_{\rm Q}(. )\) leads to the fact that up to the distortion caused by the receiver noise and the influence of \(s_{k}\in\{-1,+1\}\), the real part of the continuous channel coefficient \(\tilde{h}\) is also observed in a quantized manner.