## 2 The Information Bottleneck Method and Coarsely Quantized Information Bottleneck Signal Processing
### General View on Information Bottleneck Signal Processing for Receiver Design
An interesting fact in this context is that in information theoretical terms, all signal processing of Y can only preserve or lower the information on X as a direct result of the data processing inequality [59] (i.e., \(\text{I}(\text{X};\text{T})\leq\text{I}(\text{X};\text{Y})\)). Due to the typically very advanced channel coding and modulation steps conducted at the transmitter, obtaining an estimate \(\hat{x}\) for \(x\), for example, based on the straightforward application of the maximum likelihood criterion via the equation  
\[\hat{x}=\arg\max_{x}p(y|x) \tag{8}\]  
is often prohibitively complex. The reason for this is that the transmitter spreads information on the user data \(x\) over huge blocks of transmitted symbols to protect it against transmission errors, as already mentioned above. In addition, it might include control data in the transmitted data stream to simplify the receiver-side detection process, for example, for channel estimation. As a result, the digital communication receiver has to aggregate the information on the user data \(\mathsf{X}\) by undoing the transmitter-side encoding and modulation process using suitable algorithms composed in a signal processing chain to process the received \(y\). In order to overcome the complexity of a straightforward maximum likelihood sequence detection, it is common practice to separate and distribute receiver-side signal processing tasks such as synchronization, channel estimation, detection and channel decoding over concatenated signal processing blocks that exchange messages or signals. Unfortunately, the performance of such signal processing algorithms still relies on sufficient precision. The number of bits per sample has to be large enough to adequately represent the received signal in the digital hardware without a degrading quantization distortion. In addition, the required signal processing operations are often computationally demanding. It is well known that a coarse quantization (i.e., a low resolution of the analog-to-digital converter) can help to reduce the implementation efforts of communication receivers, as it directly reduces the number of bits needed to process the signal in the digital hardware. In addition, simplified arithmetical operations can reduce the implementation efforts of optimum or close-to-optimum signal processing algorithms. However, both typically result in performance degradations. Therefore, the fundamental idea of information bottleneck signal processing is to build suboptimum receiver components that, despite being strongly quantized, are designed to preserve the maximum possible amount of relevant information and only use simple arithmetical operations. This design idea inherently minimizes the inevitable performance degradation resulting from the quantization and from using potentially suboptimal signal processing operations. Information bottleneck signal processing has already been applied to various receiver-side signal processing tasks in the past, such as quantized detection and channel estimation [35; 36; 37; 38]. In the following, we will analyze a receiver chain that applies information bottleneck signal processing extensively. In this receiver, we consider the mutual information preserving input/output relations of \(p(t|y)\) to be implemented as lookup tables that replace the conventional signal processing operations.