## 2 The Information Bottleneck Method and Coarsely Quantized Information Bottleneck Signal Processing
### An Example of Information Bottleneck Receiver Design with Iterative Detection and Decoding  
In this section, we illustrate the idea of information bottleneck signal processing by considering data transmission with binary phase shift keying (BPSK) modulation over a frequency flat block fading channel. The presented study continues and extends the investigation from our conference publication [60]. We consider data transmission over a frequency-flat block fading channel. The complex-valued continuous received signal at time instance \(k\) in the symbol clock is given by  
\[\hat{y}_{k}=\tilde{h}s_{k}+\tilde{n}_{k}=\hat{y}_{k}^{\text{re}}+j\hat{y}_{k}^{ \text{im}}, \tag{9}\]  
where \(\tilde{h}\) is a complex channel coefficient from a zero-mean circularly complex Gaussian process that is modelled as a constant for \(B\) symbol durations and \(s_{k}\in\{-1,+1\}\) is the transmitted BPSK symbol. The noise sample \(\tilde{n}_{k}\) is also a realization from a complex zero-mean white Gaussian process with variance \(\sigma_{\tilde{n}}^{2}\). Please note that in our notation, we use a tilde to indicate that a signal has not been quantized and hence is continuous, and the superscripts re and im distinguish real and imaginary parts, respectively. For clarity, Figure 5 provides an overview of the considered transmitter that is explained in the following. The transmitter transmits codewords from a regular LDPC code with a codeword length \(N^{\text{LDPC}}>>B\) over the channel using BPSK modulation. In order to simplify the receiver-side detection and decoding, the transmitter multiplexes \(P\) known pilot symbols into a transmitted codeword with a distance of \(B-P\) symbols such that each block of \(B\) symbols weighted with the same channel coefficient \(\tilde{h}\) starts with a few known pilot symbols. The resulting burst structure is also sketched at the bottom of Figure 5. These pilots shall serve as a reference signal to enable a simple receiver-side channel estimation. Figure 6 shows a conventional receiver structure for the considered transmitter that uses classical signal processing operations and does not employ any quantization beyond the numerical limits of double-precision at the top. This receiver shall serve as a reference receiver that illustrates the performance of the considered data transmission scheme if no quantization is applied at all. It initially employs minimum mean squared error (MMSE) channel estimation based on the inserted pilot symbols. Based on the obtained channel estimate, it then performs a matched soft demodulation. The term matched reflects that the demodulator takes into account the uncertainty of the estimated channel coefficient. The demodulator then delivers log-likelihood ratios (LLRs) and provides them to a belief propagation decoder for the decoding of the LDPC code. This decoder performs \(i_{\text{max}}\) decoding iterations. After one such decoding round, the receiver has estimates of the transmitted codeword bits. The reliability of the different codeword bits can be judged by analyzing the absolute magnitudes of the decision LLRs from the decoder. In order to further improve the detection and decoding quality, the receiver then iteratively employs decision feedback (FB) of the \(N_{\text{FB}}\) most reliable decision bits within a block to virtually  
Figure 5: Overview of the considered transmitter. \(P\) pilot bits are multiplexed into a length \(N^{\text{LDPC}}>>B\) LDPC codeword periodically with a distance of \(B-P\) bits, and the resulting data stream is modulated using BPSK modulation. enlarge the number of available pilot symbols used for channel estimation. The considered receiver implements a hard-decision turbo channel estimation scheme. Its decision feedback loop is repeated \(i_{\text{FB}}\) times, and then the final decision bits are provided. Underneath the conventional receiver chain in Figure 6, the proposed information bottleneck receiver chain is depicted. At first glance, both receiver chains look similar.