# Low-Resolution Precoding for Multi-Antenna Downlink Channels and OFDM +
## 4 Performance Metrics
### Achievable Rates
\begin{split} x^{(b)}&=(x_{1}^{(b)},\ldots,x_{S}^{(b)} )^{T}\\ y^{(b)}&=(y_{1}^{(b)},\ldots,y_{S}^{(b)})^{T}\end{split} \]  
for \(b=1,\ldots,B\), and then the following average rate will approach \(\mathrm{I}_{q,s}(\mathbf{X};\mathbf{Y})/S\) bpcu as \(B\) grows:  
\[R_{\mathrm{a}}=\frac{1}{B}\sum_{b=1}^{B}R_{\mathrm{a}}^{(b)} \tag{17}\]  
where  
\[R_{\mathrm{a}}^{(b)}=\frac{1}{S}\sum_{i=S_{p}+1}^{S}\log_{2}\!\left(\frac{q_{x ^{(b)},y^{(b)}}\!\left(y_{i}^{(b)}\,\middle|\,x_{i}^{(b)}\right)^{s}}{\sum_{a}P (a)\,q_{x^{(b)},y^{(b)}}\!\left(y_{i}^{(b)}\,\middle|\,a\right)^{s}}\right). \tag{18}\]  
We choose the Gaussian auxiliary density  
\[q_{\mathbf{x},y}(y|x)=\frac{1}{\pi\sigma_{q}^{2}}\exp\!\left(-\frac{\left|y-h \cdot x\right|^{2}}{\sigma_{q}^{2}}\right) \tag{19}\]  
where for pilot-aided transmission (PAT) the receiver computes joint maximum likelihood (ML) estimates with sums of \(S_{p}\) terms:  
\[ \begin{split} h&=\frac{\sum_{i=1}^{S_{p}}y_{i}\cdot x _{i}^{*}}{\sum_{i=1}^{S_{p}}\left|x_{i}^{2}\right|}\\ \sigma_{q}^{2}&=\frac{1}{S_{p}}\sum_{i=1}^{S_{p}} \lvert y_{i}-h\cdot x_{i}\rvert^{2}.\end{split}  \tag{20}\]  
For the data-aided detector we replace \(S_{p}\) with \(S\) in (20). Note that for the Gaussian channel (19) the parameter \(s\) multiplies \(1/\sigma_{q}^{2}\) in (16) or (18), and optimizing \(s\) turns out to be the same as choosing the best parameter \(\sigma_{q}^{2}\) when \(s=1\). Summarizing, we use the following steps to evaluate achievable rates. Suppose the coherence time is \(S/T_{F}\) OFDM symbols where \(S\) is a multiple of \(T_{F}\). We index the channel symbols by the pairs \((\ell,m)\) where \(\ell\) is the OFDM symbol and \(m\) is the subcarrier, \(1\leq\ell\leq S/T_{F}\), \(0\leq m\leq T-1\). We collect the pilot index pairs in the set \(\mathcal{S}_{p}\) that has cardinality \(S_{p}\), and we write the channel inputs and outputs of UE \(k\) for OFDM symbol \(\ell\) and subcarrier \(m\) as \(\hat{u}_{k}[\ell,m]\) and \(\hat{y}_{k}[\ell,m]\), respectively. 1. Repeat the following steps (2)-(4) \(B\) times; index the steps by \(b=1,\ldots,B\);
2. Use Monte Carlo simulation to generate the symbols \(\hat{u}_{k}[\ell,m]\) and \(\hat{y}_{k}[\ell,m]\) for \(k=1,\ldots,K\), \(\ell=1,\ldots,S/T_{F}\), and \(m=0,\ldots,T-1\);
3. Each UE estimates its own channel \(h_{k}\) and \(\sigma_{q,k}^{2}\), i.e., the channel estimate (20) of UE \(k\) is  
\[ \begin{split} h_{k}&=\frac{\sum_{(\ell,m)\in \mathcal{S}_{p}}\hat{y}_{k}[\ell,m]\cdot\hat{u}_{k}[\ell,m]^{*}}{\sum_{(\ell,m) \in\mathcal{S}_{p}}\left|\hat{u}_{k}[\ell,m]\right|^{2}}\\ \sigma_{q,k}^{2}&=\frac{1}{S_{p}}\sum_{(\ell,m)\in \mathcal{S}_{p}}\lvert\hat{y}_{k}[\ell,m]-h_{k}\cdot\hat{u}_{k}[\ell,m]\rvert ^{2}.\end{split}  \tag{21}\]  
For the data-aided detector, in (21) we replace \(\mathcal{S}_{p}\) with the set of all index pairs \((\ell,m)\), and we replace \(S_{p}\) with \(S\);4. Compute \(R_{\text{a}}^{(b)}\) in (18) for each UE \(k\) by averaging, i.e., the rate for UE \(k\) is \[R_{\text{a},k}^{(b)}=\frac{1}{S}\sum_{(\ell,m)\in\mathcal{S}_{p}}\log_{2}\!\left( \frac{q_{\hat{\mathbf{u}}_{k}}\hat{\mathbf{g}}_{k}(\hat{\mathbf{y}}_{k}[\ell,m]\mid\hat\mathbf{{u}}_{k}[\ell,m])^{s}}{\sum_{a}P(a)\;q_{\hat{\mathbf{u}}_{k},\hat{\mathbf{y}}_{k}}(\hat {\mathbf{y}}_{k}[\ell,m]\mid a)^{s}}\right)\] (22) where \(\hat{\mathbf{u}}_{k}\) and \(\hat{\mathbf{y}}_{k}\) are vectors collecting the \(\hat{\mathbf{u}}_{k}[\ell,m]\) and \(\hat{\mathbf{y}}_{k}[\ell,m]\), respectively, for all pairs \((\ell,m)\). For the data-aided detector we set \(\mathcal{S}_{p}=\mathcal{O}\) in (22);
5. Compute \(R_{\text{a}}\) in (17) for each UE, i.e., the average rate of UE \(k\) is \(R_{\text{a},k}=\frac{1}{B}\sum_{b=1}^{B}R_{\text{a},k}^{(b)}\);
6. Compute the average UE rate \(\overline{R}_{\text{a}}=\frac{1}{K}\sum_{k=1}^{K}R_{\text{a},k}\). Our simulations showed that optimizing over \(s\geq 0\) gives \(s\approx 1\) if the channel parameters are chosen using (21).