## 3 Channel Quality Metric in WSN  
NDA-EVM is defined as the root-mean-squared (RMS) value of the received symbols \(y\) and the estimated transmitted symbols \(\hat{x}\), the expression can be expressed as  
\[\xi[M]=\sqrt{\sum\limits_{n=1}^{N}{\left|y[n]-\hat{x}[n]\right|^{2}}/{NP_{0}}} \tag{5}\]  
where \(y[n]\) is the \(n\)th received symbols and \(\hat{x}[n]\) can be obtained from the received symbols by using the ML estimation. The symbol power is normalized, \(P_{0}=1\). For simplicity, the index \(n\) is dropped for the remainder of this discussion. Considering \(x\) is a MQAM signals with modulation order \(M\), then \(x\in\{S_{1},\ldots,S_{i},\ldots,S_{M}\}\), \(S_{i}\) is the \(i\)th constellation point in the constellation set, which can be expressed as  
\[ \begin{array}{l}S_{i}=S_{i,R}+jS_{i,I}=(2i-k)b\ +\ j(2m-k)b,\\ i,m=0,1,\cdots k;b\ =\sqrt{3/2(M-1)}\ \&\ k=\sqrt{M}-1\end{array}  \tag{6}\]  
where \(b\) is the normalized symbol amplitude. Because of the independence and symmetry of the real and imaginary parts of the MQAM symbols, only the real part of the signal is considered. Thus, we can rewrite Equation (5) as  
\[ \begin{array}{l}\xi[k]^{2}=2E\{(y_{R}-\hat{x}_{R})^{2}\}\\ =2\sum\limits_{i=0}^{k}{P(\hat{x}_{R}=S_{i,R})}\int_{-\infty}^{+\infty}{(y_{R}- S_{i,R})^{2}f(y_{R}|x_{R}=S_{i,R})dy_{R}}\end{array} . \tag{7}\]  
By using Equation (2), the conditional PDF of the received symbol \(y_{R}\) is  
\[f(y_{R}|x_{R}=S_{i,R})=\frac{1}{\sigma_{n}}\varphi(\frac{y_{R}-\alpha S_{i,R}} {\sigma_{n}}), \tag{8}\]  
where \(\varphi(\cdot)\) is the PDF of a standard normal distribution. By the total probability theorem:  
\[P(\hat{x}_{R}=S_{i,R})=\sum\limits_{j=0}^{k}{P(x_{R}=S_{j,R})}\int\limits_{D_ {i,R}}{f(y_{R}|x_{R}=S_{j,R})dy_{R}}. \tag{9}\]  
where \(D_{i,R}\) is the decision region for symbol \(S_{i,R}\), which is confirmed by the ML Criterion,  
\[D_{i,R}=\left\{ \begin{array}{ccc}-\infty<y_{R}\leq S_{0,R}+b,&if&i=0;\\ S_{i,R}-b<y_{R}\leq S_{i,R}+b&if&1\leq i\leq k-1;\\ S_{k,R}-b<y_{R}<\infty&if&i=k.\end{array} \right. \tag{10}\]  
All constellation points are transmitted with equal probability such that \(P(x_{R}=S_{j,R})=1/(1+\text{k}),\forall j\). From the definition of conditional probability, \(f(y_{R}|\hat{x}_{R}=S_{i,R})\) is evaluated as follow  
\[f(y_{R}|\hat{x}_{R}=S_{i,R})=\frac{\sum\limits_{j=0}^{k}{f(y_{R}|x_{R}=S_{j,R} )}}{\sum\limits_{j=0}^{k}{\int\limits_{D_{i,R}}{f(y_{R}|x_{R}=S_{j,R})dy_{R}}}},\ y_{R}\in D_{i,R} \tag{11}\]  
Using Equations (9) and (11), the expression in Equation (7) is reduced to  
\[ \begin{array}{l}\xi[M]^{2}&=2E\{(y_{R}-\hat{x}_{R})^{2}\}\\ &=\frac{2}{k+1}\sum\limits_{i=0}^{k}{\sum\limits_{j=0}^{k}{\int\limits_{D_{i,R }}{(y_{R}-S_{i,R})^{2}\frac{1}{\sigma_{n}}\varphi(\frac{y_{R}-\alpha S_{i,R}} {\sigma_{n}})dy_{R}}}}\end{array}  \tag{12}\]Letting \(v=y_{R}-S_{i,R}\), so \(\widetilde{D}_{i,R}=D_{i,R}-S_{i,R}\), and letting \(\lambda_{ji,R}=-S_{i,R}+\alpha S_{j,R}\), Equation (12) can be reduce to  
\[\xi[M]^{2}=\frac{2}{k+1}\underset{i=0}{\overset{k}{\sum}}\underset{j=0}{ \overset{k}{\sum}}\underset{D_{i,R}}{\overset{k}{\sum}}\underset{\sigma_{n}}{ \overset{\sigma}{\sigma}}\varphi(\frac{v-\lambda_{ji,R}}{\sigma_{n}})dv \tag{13}\]  
and after some manipulation, Equation (13) is reduced to  
\[