## 2 The Information Bottleneck Method and Coarsely Quantized Information Bottleneck Signal Processing
### An Example of Information Bottleneck Receiver Design with Iterative Detection and Decoding
#### 2.3.4 Comparison of Iterative Receiver Performances  
In this section, we investigate the bit error rate performance of the proposed information bottleneck receiver and compare it to the one of several reference receivers. As mentioned above, the proposed information bottleneck receiver from Figure 6 was designed for a manually chosen design \(E_{b}/N_{0}\) and was not matched to the actual \(E_{b}/N_{0}\) on the channel in the results presented next. The lookup tables were constructed offline with the information bottleneck method. The bit error rate performances of the considered receivers are compared in Figure 10 for a number of \(i_{\mathrm{FB}}=0\) and \(i_{\mathrm{FB}}=5\) feedback iterations of the decision feedback channel estimation. The applied LDPC code was a length of \(8000\)\((d_{\mathrm{v}},d_{\mathrm{c}})=(3,6)\) regular LDPC code from [63]. In the studied scenario, we considered a block fading channel that was constant for \(B=35\) symbol durations before a new independent channel coefficient was drawn, which weighted the transmitted symbols according to Equation (9). The first \(P=3\) symbols transmitted for each of these blocks were the known pilot symbols for the initial channel estimation. The quantization bit widths of the information bottleneck receiver were chosen as follows. We used \(q=5\) bits per sample for channel output quantization in the real and imaginary parts of the received signal, respectively, \(q^{\rm ce}=8\) bits for channel estimation and \(q^{\rm det}=5\) bits for the detection lookup table. We also used \(q^{\rm det}=5\) bit messages in the information bottleneck LDPC decoder. The decoder applied \(i_{\rm max}=25\) iterations in each of the outer feedback iterations for the channel estimation. In each outer iteration, the \(N_{\rm FB}=20\) most reliable bit decisions from this decoder in each block were fed to the feedback channel estimator illustrated in Figure 7 to supplement the channel estimation. The red curves in Figure 10 show the bit error rate performance of the proposed information bottleneck receiver with these parameters for \(i_{\rm FB}=0\) and \(i_{\rm FB}=5\) outer iterations of the decision feedback loop for channel estimation. A comparison of the receiver's performance for \(i_{\rm FB}=0\) and \(i_{\rm FB}=5\) outer iterations of the channel estimation feedback loop clearly illustrates the gain resulting from improving the initial channel estimate based on the decision feedback from the channel decoder. The blue curves in Figure 10 correspond to the conventional receiver chain from Figure 6 with the respective numbers of feedback iterations \(i_{\rm FB}=0\) and \(i_{\rm FB}=5\) and also used \(i_{\rm FB}=25\) iterations of the conventional belief propagation LDPC decoder. We want to stress again that this receiver does not suffer from any mentionable quantization loss. Despite that, it can clearly be observed that the performance of the quantized information bottleneck receiver chain that only processed unsigned integers and replaced all conventional arithmetical operations in the signal processing algorithms with simple lookup operations for \(i_{\rm FB}=5\) iterations was tremendously close to the one of the conventional reference receiver. It is also observable that without feedback iterations (i.e., \(i_{\rm FB}=0\)), the loss of the information bottleneck receiver was slightly higher. The reason for this is that all of the lookup tables in the information bottleneck receiver were designed only for a single design \(E_{b}/N_{0}\), which we optimized to yield the best performance for the case  
Figure 10: Bit error rates of different quantized receivers and the conventional receiver from Figure 6, which did not suffer from a quantization loss at all. The quantized information bottleneck receiver with \(q=5\) bit channel output quantization, \(q^{\rm ce}=8\) bit channel estimation and \(q^{\rm det}=5\) bit detection and LDPC decoding met the performance of the double-precision reference receiver for \(i_{\rm FB}=5\) feedback iterations.