# Improved ISR for IEEE 802.11 ah Nonlinearity Compensation via Adjustable Constellation Borders
## 2 System Model  
The system structure of the OFDM transceiver with the ISR technique used in this study is fully consistent with that used in [19] for a fair performance comparison, whose block diagram is shown in Figure 1. At the transmitter, the information bits are modulated by \(\mathcal{M}\)-ary quadrature amplitude modulation (QAM) with average power normalized to one. Subsequently, different types of subcarriers are mapped to generate time-domain OFDM signals \(\mathbf{x}=\left[x_{0},\ \dots,\ x_{N-1}\right]^{T}\) using an inverse discrete Fourier transform (IDFT), yielding  
\[\mathbf{x}=\mathbf{\mathcal{F}}_{\mathbf{N}}^{H}\mathbf{X}, \tag{1}\]  
where \(\mathbf{X}=\left[X_{0},\ \dots,\ X_{N-1}\right]^{T}\) is the transmitted subcarrier vector based on the 802.11 ah channel bandwidth 1 (CBW1) specification [1]; \(\mathbf{\mathcal{F}}_{\mathbf{N}}\) is the \(N\times N\) discrete Fourier transform (DFT) matrix, the entry of which is given by \(\left[\mathbf{\mathcal{F}}_{N}\right]_{(n,k)}\triangleq\frac{1}{\sqrt{N}}e^{-j( \frac{2\pi}{N})nk}\), \(0\leq n,\ k\leq N-1\); and \([\cdot]^{H}\) stands for Hermitian transpose. After padding a length-\(P\) cyclic prefix (CP), i.e., \(\mathbf{x}=\left[x_{N-P},\ \dots,\ x_{N-1},x_{0},\ \dots,\ x_{N-1}\right]^{T}\), the baseband output signal is distorted by [PERSON]'s PA nonlinearity model [20], whose amplitude-to-amplitude (AM/AM) conversion is featured as  
\[G(x_{n})=g\cdot x_{n}\left[1+\left(\frac{g\cdot|x_{n}|}{A_{sat}}\right)^{2p} \right]^{-\frac{1}{2p}}, \tag{2}\]  
where \(A_{sat}\) is the saturation level specified by the value of input back-off (IBO, in dB) from \(A_{sat}=x_{RMS}\cdot 10^{IBO/20}\); \(x_{RMS}=\sqrt{\sum_{n=0}^{N-1}{\left|x_{n}\right|^{2}}/N}\); \(p\) controls the smoothness of conversion; \(g\) specifies the voltage gain of the amplifier. In this study, \(g=1\) is used for generality, and \(p=2\) is used for simulating the low-complexity PA of IoT devices.  
At the receiver, after CP removal and performing DFT, the equivalent frequency-domain received signal \(\mathbf{Y}=\left[Y_{0},\ \dots,\ Y_{N-1}\right]^{T}\) is given by  
\[\mathbf{Y}=\mathbf{H}\mathbf{\mathcal{F}}_{N}\cdot G\left(\mathbf{\mathcal{F}}_{\mathbf{N}}^{H}\bm {X}\right)+\mathbf{W}, \tag{3}\]  
where \(\mathbf{W}=\left[W_{0},\ \dots,\ W_{N-1}\right]^{T}\), \(W_{k}\in\mathcal{CN}(0,\ N_{0,\ F})\) is the frequency-domain noise vector; \(\mathcal{CN}(0,\ \sigma^{2})\) denotes a zero-mean complex Gaussian random variable with variance \(\sigma^{2}\); and \(\mathbf{H}=\left[H_{0},\ \dots,\ H_{N-1}\right]^{T}=\mathbf{\mathcal{F}}_{\mathbf{N}}\mathbf{h}\) is the frequency-domain fading channel vector, where \(\mathbf{h}=\left[h_{0},\ \dots,\ h_{L-1},\ 0,\ \dots,\ 0\right]^{T}\) is the \(L\)-tap channel impulse response (\(L\leq P\)). Finally, the recovered subcarrier vector \(\widehat{Q}^{(i)}=\left[\widehat{Q}_{0}^{(i)},\ \dots,\ \widehat{Q}_{N-1}^{(i)}\right]^{T}\) is gained by performing \(i\)-iterations of ISR on the equalized subcarrier vector \(\mathbf{Q}\), where  
\[\mathbf{Q}=\mathbf{Y}\mathbf{H}^{-\mathbf{1}}=\mathbf{H}\mathbf{\mathcal{F}}_{\mathbf{N}}\cdot G\left(\mathbf{ \mathcal{F}}_{\mathbf{N}}^{H}\mathbf{X}\right)\mathbf{H}^{-\mathbf{1}}+\mathbf{W}\mathbf{H}^{-\mathbf{1}}. \tag{4}\]