## 3 Channel Quality Metric in WSN
\begin{array}{rl}\xi[k]^{2}=&2/(k+1)\{\\ &\underset{j=0}{\overset{k}{\sum}}[\sigma_{n}(-b+\lambda_{ji,R})\varphi(\frac {-b-\lambda_{jk,R}}{\sigma_{n}})+(\lambda_{jk,R}^{2}+\sigma_{n}^{2})Q(\frac{-b -\lambda_{jk,R}}{\sigma_{n}})]\\ &+\underset{j=0}{\overset{k}{\sum}}[\sigma_{n}(b+\lambda_{jk,R})\varphi( \frac{b-\lambda_{jk,R}}{\sigma_{n}})+(\lambda_{jk,R}^{2}+\sigma_{n}^{2})Q( \frac{b-\lambda_{jk,R}}{\sigma_{n}})]\\ &+\underset{i=1}{\overset{k}{\sum}}\underset{j=0}{\overset{k}{\sum}}[\sigma _{n}(-b+\lambda_{ji,R})\varphi(\frac{-b-\lambda_{ji,R}}{\sigma_{n}})+(\lambda_{ ji,R}^{2}+\sigma_{n}^{2})Q(\frac{-b-\lambda_{ji,R}}{\sigma_{n}})]\\ &-\underset{i=1}{\overset{k}{\sum}}\underset{j=0}{\overset{k}{\sum}}[\sigma _{n}(b+\lambda_{ji,R})\varphi(\frac{b-\lambda_{ji,R}}{\sigma_{n}})+(\lambda_{ ji,R}^{2}+\sigma_{n}^{2})Q(\frac{b-\lambda_{ji,R}}{\sigma_{n}})]\}\end{array}  \tag{14}\]  
where \(\alpha\) is the attenuation factor with a PDF of \(f_{\alpha,\eta-\mu}\), and \(Q(\cdot)\) is a complementary cumulative distribution function (CCDF). From Equation (14), the receiver can analyze the NDA-EVM value of different constellation size without requiring the transmitter to explicitly send each modulation order. In other words, if the receiver gets the NDA-EVM value of symbols with current modulation order, the NDA-EVM value for any other modulation order can be derived easily in the same channel condition (e.g., if the NDA-EVM for 16 QAM is confirmed, the NDA-EVM for QAM/64 QAM/256 QAM can be obtained by using Equation (14) without retransmission at the same channel condition). The key of adaptive modulation is to choose the optimum modulation order to match the channel conditions, while achieving the expected SER. To obtain the modulation order for which NDA-EVM can tolerate the expected SER, we need to confirm the relationship between NDA-EVM and SER. The SER as a function of NDA-EVM is derived in Appendix A, which is  
\[P_{S}=1-\{\frac{\xi[k]^{2}-F}{2\sigma_{n}^{2}}+\frac{1}{1+k}\underset{i=1}{ \overset{k}{\sum}}[Q(\frac{-b-\lambda_{ii}}{\sigma_{n}})-Q(\frac{b-\lambda_{ ii}}{\sigma_{n}})]\}^{2}, \tag{15}\]  
where  
\[\begin{array}{  
five states represented, by the letters A, B, C, D and E. A is the worst state that cannot establish communication under target SER (e.g., NoTx, No Transmission). States B, C, D and E denote the states from the former stage to QAM, 16 QAM, 64 QAM and 256 QAM, respectively. The pseudocode for adaptation strategy based on NDA-EVM is shown in Algorithm 1. Note that, if transmitting in particularly bad fading channel states, even the modulated QAM symbols with the minimum constellation size will not satisfy the expected SER (i.e., \(P_{S}(\xi^{n}[k(1)])>S_{TH}\)), and the transmitted sensor node will not send any data (NoTx, No Transmission). In this case, the transmitted sensor node will remain the outage state at time \(n+1\), until the new turn (i.e., time \(n+2\)) of adaptive adjustment. ```
Input:\(k\in K=\{1,3,7,15\}\) (1-QAM,3-16 QAM,7-64 QAM,15-256 QAM), \(j=\{1,2,3,4\}\)
1 initial\(k=1\), \(M^{0}=QAM\)
2 begin
3\(NDA\)-\(EVM_{all}=\{\xi^{n}(k)\,|\,k\in K\}\)
4\(SER_{all}=\{P_{S}[\xi^{n}(k)]\,|\,k\in K\}\);
5 if\(P_{S}(\xi^{n}[k(1)])>S_{TH}\)then
6\(M^{n+1}=NoTx\), the \(n+2\)turn start from \"initial\"
7 if\(P_{S}(\xi^{n}[k(4)])<S_{TH}\)then
8\(M^{n+1}=k(4)\), the next turn start from \"begin\"
9 else
10 while\(k(j)<4\)do
11\(k=arg\,\max\{k\,|\,P_{S}(\xi^{n}[k(j)])\leq S_{TH}\cap P_{S}(\xi^{n}[k(j+1)])>S_ {TH}\}\)
12\(M^{n+1}=k(j)\), the next turn start from \"begin\"
```  
**Algorithm 1**The Pseudocode for NDA-EVM-AM Strategy.