# Low-Resolution Precoding for Multi-Antenna Downlink Channels and OFDM +
## 4 Performance Metrics  
### Achievable Rates  
We use generalized mutual information (GMI) to compute achievable rates [46; 47], (Ex. 5.22) which is a standard tool to compare coded systems. Consider a generic input distribution \(P(\mathbf{x})\) and a generic channel density \(p(\mathbf{y}|\mathbf{x})\) where \(\mathbf{x}=(x_{1},\ldots,x_{S})^{T}\) and \(\mathbf{y}=(y_{1},\ldots,y_{S})^{T}\) each have \(S\) symbols. A lower bound to the mutual information  
\[\mathrm{I}(\mathbf{X};\mathbf{Y})=\sum_{\mathbf{x},\mathbf{y}}P(\mathbf{x})p(\mathbf{y}|\mathbf{x})\log_{2 }\biggl{(}\frac{p(\mathbf{y}|\mathbf{x})}{\sum_{a}P(\mathbf{a})\;p(\mathbf{y}|\mathbf{a})}\biggr{)} \tag{12}\]  
is the GMI  
\[\mathrm{I}_{q,s}(\mathbf{X};\mathbf{Y})=\sum_{\mathbf{x},\mathbf{y}}P(\mathbf{x})p(\mathbf{y}|\mathbf{x}) \log_{2}\biggl{(}\frac{q(\mathbf{y}|\mathbf{x})^{s}}{\sum_{a}P(\mathbf{a})\;q(\mathbf{y}|\mathbf{a })^{s}}\biggr{)} \tag{13}\]  
where \(q(\mathbf{y}|\mathbf{x})\) is any auxiliary density and \(s\geq 0\). In other words, the choices \(q(\mathbf{y}|\mathbf{x})=p(\mathbf{y}|\mathbf{x})\) for all \(\mathbf{x},\mathbf{y}\) and \(s=1\) maximize the GMI. However, the idea is that \(p(\mathbf{y}|\mathbf{x})\) may be unknown or difficult to compute and so one chooses a simple \(q(\mathbf{y}|\mathbf{x})\). The reason why \(p(\mathbf{y}|\mathbf{x})\) is difficult to compute here is because we will measure the GMI across the end-to-end channels from the \(\dot{u}_{k}[m]\) to the \(\dot{y}_{k}[m]\) and the quantized precoding introduces non-linearities in these channels. The final step in evaluating the GMI is maximizing over \(s\geq 0\). Alternatively, one might wish to simply focus on \(s=1\), e.g., see [48]. We study the GMI of two non-coherent systems: classic PAT and data-aided channel estimation. For both systems, we apply memoryless signaling with the product distribution  
\[P(\mathbf{x})=\prod_{i=1}^{S_{p}}1(x_{i}=x_{p,i})\cdot\prod_{i=S_{p}+1}^{S}P(x_{i}) \tag{14}\]  
where the \(x_{p,i}\) are pilot symbols, \(1(a=b)\) is the indicator function that takes on the value \(1\) if its argument is true and \(0\) otherwise, and \(P(x)\) is a uniform distribution. Joint data and channel estimation has \(S_{p}=0\) so that we have only the second product in (14). At the receiver we use the auxiliary channel  
\[q(\mathbf{y}|\mathbf{x})=\prod_{i=1}^{S}q_{\mathbf{x},\mathbf{y}}(y_{i}\,|\,x_{i}) \tag{15}\]  
where the symbol channel \(q_{\mathbf{x},\mathbf{y}}(. )\) is a function of \(\mathbf{x}\) and \(\mathbf{y}\). Observe that \(q_{\mathbf{x},\mathbf{y}}(. )\) is invariant for \(S\) symbols and the channel can be considered to have memory since every symbol \(x_{\ell}\) or \(y_{\ell}\), \(\ell=1,\ldots,S\), influences the channel for all \"times\" \(i=1,\ldots,S\). The GMI rate (13) simplifies to  
\[\sum_{\mathbf{x},\mathbf{y}}P(\mathbf{x})p(\mathbf{y}|\mathbf{x})\sum_{i=S_{p}+1}^{S}\log_{2} \biggl{(}\frac{q_{\mathbf{x},\mathbf{y}}(y_{i}\,|\,x_{i})^{s}}{\sum_{a}P(a)\;q_{\mathbf{x },\mathbf{y}}(y_{i}\,|\,a)^{s}}\biggr{)}. \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**System** & \(\mathbf{N}\) & \(\mathbf{K}\) & \(\mathbf{T}\) = \(\mathbf{T_{F}+T_{c}}\) & \(\mathbf{L}\) & **Constellation** & \(\mathbf{b}\) & **Fading Statistics** \\ \hline A & 128 & 16 & 270 & \(=\) 256 + 14 & 15 & \([16,64]\)-QAM & 2,3 & Flat and Rayleigh uniform PDP \\ \hline B & 64 & 8 & 35 & \(=\) 32 + 3 & 4 & \(\{\)4-32\(\}\)-PSK & 2 & Rayleigh uniform PDP \\ \hline C & 80 & 8 & 277 & \(=\) 256 + 21 & 22 & 16-QAM & 2 & Rayleigh uniform PDP \\  & & 286 & \(=\) 256 + 30 & varies & & & Winner2 NLOS C2 urban \\ \hline D & 128 & 16 & 410 & \(=\) 396 + 14 & 15 & 64-QAM & 2 & Rayleigh uniform PDP \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System parameters for the simulations. One may approximate (16) by applying the law of large numbers for stationary signals and channels. The idea is to independently generate the \(B\) pairs of vectors  
\[