# Improved ISR for IEEE 802.11 ah Nonlinearity Compensation via Adjustable Constellation Borders
## 3 Improved ISR
### Observation for Standard Decision  
In the conventional ISR [19], the step of subcarrier regularization in (7) utilizes a typical decision region as the standard \(\mathcal{M}\)-ary QAM to reset the symbols of data subcarriers to their closest constellation points. Such action may introduce a serious decision errors, even if a symbol only slightly exceeds the constellation borders (in Figure 2a, a received symbol belonging to the alphabet \(S_{A}\) is wrongly demodulated to \(S_{B}\), where \(S_{A},S_{B}\in\{S_{0},\ \ldots,\ S_{M-1}\}\)). However, these symbols are likely to be correctly demodulated within the next few iterations if they are reserved as reconstructed samples, similar to saturated signals in the time domain. Once a symbol can be successfully demodulated, it considerably contributes to the restoration process.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Vector Element** & **Value of \(K_{T}^{\text{ISR}}\)** & **Labeling Index** & **Carrier** & **Carrier** \\  & & **(\(\mathcal{L}_{T}^{\text{ISR}}\))** & **Index** & **Type** \\ \hline \(K_{T}^{ISR}(0)\) & 0 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(0,0)}=1\) & 0 & DC \\ \(K_{T}^{ISR}(1)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1,1}\end{bmatrix}_{(1,1)}=0\) & 1 & data \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(K_{T}^{ISR}(6)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{6,6}\end{bmatrix}=0\) & 6 & data \\ \(K_{T}^{ISR}(7)\) & 1 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{7}\end{bmatrix}_{(7,7)}=1\) & 7 & pilot 1 \\ \(K_{T}^{ISR}(8)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{8,8}\end{bmatrix}=0\) & 8 & data \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(K_{T}^{ISR}(13)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(13,13)}=0\) & 13 & data \\ \(K_{T}^{ISR}(14)\) & 0 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(14,14)}=1\) & 14 & guard \\ \(K_{T}^{ISR}(15)\) & 0 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(15,15)}=1\) & 15 & guard \\ \(K_{T}^{ISR}(16)\) & 0 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(16,16)}=1\) & \(-16\) & guard \\ \(K_{T}^{ISR}(17)\) & 0 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(17,17)}=1\) & \(-15\) & guard \\ \(K_{T}^{ISR}(18)\) & 0 & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(18,18)}=1\) & \(-14\) & guard \\ \(K_{T}^{ISR}(19)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{1}\end{bmatrix}_{(19,19)}=0\) & \(-13\) & data \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(K_{T}^{ISR}(24)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{2}\end{bmatrix}_{(24,24)}=0\) & \(-8\) & data \\ \(K_{T}^{ISR}(25)\) & \(-1\) & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{2}\end{bmatrix}_{(25,25)}=1\) & \(-7\) & pilot 1 \\ \(K_{T}^{ISR}(26)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{2}\end{bmatrix}_{(26,26)}=0\) & \(-6\) & data \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(K_{T}^{ISR}(31)\) & \(\mathcal{M}\) kinds & \(\begin{bmatrix}\mathcal{I}_{T}^{ISR}\\ \mathcal{I}_{3}\end{bmatrix}_{(31,31)}=0\) & \(-1\) & data \\ \hline \hline \end{tabular}  
* The position of pilot subcarriers may change in each OFDM slot for performing “traveling pilots”.  
\end{table}
Table 1: 802.11 ah (CBW1)-based frequency-domain prior knowledge (\(N=32\)).