## 2 System Model of NDA-EVM-AM in an NLOS Scenario  
For the case of a single-input single-output (SISO) system, we consider two sensor nodes adaptively adjust modulation order to transmit the information in an NLOS scenario of WSN.  
Since a variable-power scheme increases the peak level of co-channel interference [21] imposed on other sensor nodes of WSN, the transmitter keeps constant-power in adaptive modulation. The information sequence \(d[n]\) is mapped into complex MQAM signals of modulation order \(M\) (e.g., \(x[n]\)); then, \(M\in\{4,16,64,256\}\). Considering the \(\eta-\mu\) fading channel as a channel model in NLOS condition, MQAM signals received can be written as:  
\[y[n]=ge^{-j\theta}x[n]+w \tag{1}\]  
where \(w\) is an independent and identically distributed (i.i.d) complex Gaussian noise with a zero mean and variance \(\sigma_{n}^{2}\); and \(g\) and \(\theta\) are the channel gain and phase shift, respectively, which are both introduced by the fading channel. To achieve the best performance, the optimum receiver for the signal received is consists of matched filter, whose output is multiplied by the corresponding complex-valued (conjugate) channel gain \(ge^{j\theta}\). The effect of this multiplication is to compensate for the phase shift in the channel. Assuming after matched filtering and sampling with perfect symbol timing, the phase shift \(\theta\) can be compensated without error, the baseband discrete-time complex-valued signal is obtained as [22; 23]:  
\[y[n]=\alpha x[n]+w \tag{2}\]  
In Equation (2), \(\alpha\) is attenuation factor (i.e., \(0<\alpha<1\)), which can be obtained by normalize channel gain. Due to the probability density function (PDF) of the fading power \(z=\alpha^{2}\) can be expressed as [24; 25; 26; 27],  
\[f_{Z,\eta-\mu}(z)=\frac{z^{2\mu-1}}{\theta_{1}^{s}\theta_{2}^{s}\Gamma(2\mu)} \Phi_{2}^{(2)}(\mu,\mu;2\mu;\frac{-z}{\theta_{1}},\frac{-z}{\theta_{2}}) \tag{3}\]  
where \(\Phi_{2}^{(2)}(\cdot)\) is the confluent Lauricella function [28], \(\theta_{1}=\frac{1}{2\mu(h+H)}\) and \(\theta_{2}=\frac{1}{2\mu(h-H)}\), where \(h=\frac{2+\eta^{-1}+\eta}{4}\), \(H=\frac{\eta^{-1}-\eta}{4}\) in Format 1 and \(h=\frac{1}{1-\eta^{2}}\), \(H=\frac{\eta}{1-\eta^{2}}\) in Format 2; and \(\mu=\frac{E^{2}\left\{a^{2}\right\}}{2\mathrm{var}\left\{a^{2}\right\}}[1+( \frac{H}{h})^{2}]\). The PDF of \(\alpha\) can be derived as  
\[f_{\alpha,\eta-\mu}(\alpha)=\frac{2\alpha^{4\mu-1}}{\theta_{1}^{\mu}\theta_{2 }^{\mu}\Gamma(2\mu)}\Phi_{2}^{(2)}(\mu,\mu;2\mu;\frac{-\alpha^{2}}{\theta_{1} },\frac{-\alpha^{2}}{\theta_{2}}) \tag{4}\]  
The system model of adaptive transmission between two sensor nodes is illustrated in Figure 1. We estimate the NDA-EVM at time \(n\) and adapt the modulation order, the adaptation takes place at a multiple of the symbol duration. After MQAM signals are transmitted through \(\eta-\mu\) channels, the receiver estimates the NDA-EVM to evaluate the quality of the channel according to the received MQAM signals. The availability of this channel quality metric at the transmitter allows it to adapt its transmission scheme relative to the channel variation. The adaptation strategy is modeled as NDA-EVM state machine where each state is represented by a couple formed by a modulation order. This issue will be discussed in more detail in Section 4. We neglect the delay of feedback between two sensor nodes and assume that the feedback path does not introduce any errors.  
Figure 1: The system model of nondate aided error vector magnitude based adaptive modulation (NDA-EVM-AM) between the sensor nodes.