## 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
with \(i_{\text{FB}}=5\) feedback iterations. It is, however, possible to also tune the design \(E_{b}/N_{0}\) to minimize the performance gap with respect to the non-quantized receiver without feedback iterations to achieve a negligible loss over \(E_{b}/N_{0}\) for \(i_{\text{FB}}=0\)[61]. The numbers of bits \((q,q^{\text{ce}},q^{\text{det}})=(5,8,5)\) used in the information bottleneck signal processing units were determined by carefully analyzing which settings resulted in the optimum performance. While for the detection and decoding stage we found \(q^{\text{det}}=5\) bit processing sufficient, during our investigations, we noted that the channel estimation appeared to require a larger bit width of \(q^{\text{ce}}=8\) bits to perform that close to the non-quantized reference system. A possible intuitive explanation is that other than for all the detection and decoding parts of the receiver, the relevant variable of the channel estimator \(\text{H}^{\text{re}}=f_{\text{Q}}(\tilde{\text{H}}^{\text{re}})\) is not binary. Instead, with a \(q\)-bit channel output quantizer \(f_{\text{Q}}(. )\) in place, it can take \(2^{q}\) different values. This intuitively suggests that in order to achieve close-to-optimum performance, the channel estimation stage requires a larger bit width to preserve the relevant information. As another reference, the bit error rate of an 8-bit fixed-point implementation of the conventional receiver is included in Figure 10 for \(i_{\text{FB}}=5\) feedback iterations in black. This receiver used the \(Q5.3\) fixed-point format in the detection and channel estimation stages and the \(Q4.4\) fixed-point format for the messages in the belief propagation decoder. In this notation, \(Qm.n\) denotes using \(m\) bits for the integer part and \(n\) bits for the fractional part of the processed fixed-point numbers. In order to not disadvantage the conventional receiver chain, all other possible combinations of 8-bit fixed-point formats were also tested, but the chosen selection offered the best performance. The 8-bit fixed-point conventional receiver was clearly outperformed by the information bottleneck receiver. This is particularly interesting because the information bottleneck receiver only used 8-bit integers in its channel estimation stage and shorter 5-bit integers in all other detection and decoding stages. So far, Figure 10 shows that the performance of the proposed receiver was very close to that of the conventional receiver with double-precision if \(i_{\text{FB}}=5\) feedback iterations were performed. It is, however, still unclear whether or not the same information bottleneck receiver suffers from performance degradation if fewer feedback iterations are performed. As a result, Figure 11 shows the bit error rates of both receivers for all \(i_{\text{FB}}\in\{0,1,2,3,4,5\}\). The figure illustrates that the performance of the information bottleneck receiver can compete with the double-precision conventional receiver for all investigated numbers of feedback iterations \(i_{\text{FB}}\). The figure also indicates that the gains of the feedback loop were very significant in the first three feedback iterations for both receivers and then became less significant. It was found that using more than five feedback iterations hardly improved the bit error rate performance of any investigated receiver. As the most important result so far, we can summarize that the information bottleneck design of the proposed receiver yielded a quantized information bottleneck receiver with very simple and homogenous operations that could deliver performance practically identical to that of a conventional receiver with double-precision arithmetic. This receiver employs advanced signal processing concepts, such as decision feedback for quantized channel estimation and quantized LDPC decoding.