## 2 System Model  
The downlink of OFDMA-based wireless sensor network considered in this paper is shown in Figure 1. In the OFDMA systems for the sensor networks, each sensor is treated as one user. At the transmitter, the signals with \(N\) orthogonal subcarriers for \(K\) users are performed by IFFT and are transmitted through the fading channel. Since the knowledge of channel state information (CSI) is needed for the adaptive resource allocation of OFDMA system, the channel information is obtained by means of channel estimation adopted in the system and is sent to the adaptive resource allocation algorithm through feedback channels. The transmitter allocates different users with different numbers of subcarriers and different sizes of power according to the adaptive resource allocation algorithm. In this paper, it is assumed that a total number of \(K\) users in the OFDMA system with \(N\) subcarriers share the transmit power constraint \(P_{tot}\). The total bandwidth \(B\) is equally divided into \(N\) subbands and the bandwidth of the subcarriers can be denoted as \(B_{n}=B/N\). Therefore, the additive white Gaussian noise (AWGN) variance can be expressed as \(\sigma^{2}=N_{0}B/N\), where \(N_{0}\) is the noise power spectral density. Since each user has a different fading characteristic in the subchannels, it is assumed that \(g_{k,n}\) is the channel gain for user \(k\) in the subchannel \(n\), and the \(k\)th user's received signal-to-noise ratio (SNR) on subcarrier \(n\) can be described as \(\gamma_{k,n}=p_{k,n}h_{k,n}\), where \(p_{k,n}\) is the power allocated to user \(k\) on subcarrier \(n\), \(h_{k,n}\) is the subchannel SNR which is expressed as \(h_{k,n}=g^{2}_{k,n}/\sigma^{2}\). In consideration of the bit error rate (BER) constraint, it is assumed that QAM modulation is performed as in [23]. Then the BER for user \(k\) on the \(n\)th subcarrier signal is defined as  
\[\mathrm{BER}_{{}_{\mathrm{MQAM}}}\left(\gamma_{k,n}\right)=0.2\exp\!\left[ \frac{-1.6\gamma_{k,n}}{2^{\gamma_{k,n}}-1}\right] \tag{1}\]  
where \(r_{k,n}\) is the number of bits in each data symbol. \(r_{k,n}\) can be solved by  
\[\gamma_{k,n}=\log_{2}\!\left(1\!+\!\frac{\gamma_{k,n}}{\Gamma}\right) \tag{2}\]  
In Equation (2), \(\Gamma\) is a constant SNR gap [24], which is the function of the BER. \(\Gamma\) is defined as  
\[\Gamma=-\ln\left(5\!\times\!\mathrm{BER}\right)/1.6 \tag{3}\]  
The objective of the additive resource allocation algorithm is to optimize the subcarrier and power allocation, which, in another word, means the maximization of the total system sum rate under proportional fairness and total power constraints. The benefit of proportional fairness in the system is that we have the ability to set the rate ratios among users [25, 26]. Therefore, each user can receive its target data rate, and avoid the situation that the users with low channel gains are unable to receive any data rate. Figure 1: The downlink of orthogonal frequency division multiple access (OFDMA) system for wireless sensor network. Thus, the mathematical model of the resource allocation of OFDMA system is formulated as  
\[\max_{c_{k,n},\rho_{k,n}}\frac{B}{N}\sum_{k=1}^{K}\sum_{n=1}^{N}c_{k,n}\log_{2} \left(1+\frac{p_{k,n}g_{k,n}^{2}}{\sigma^{2}\Gamma}\right) \tag{4}\]  
subject to: C1: \(c_{k,n}\in\left\{0,1\right\}\)\(\forall k,n\)  
* \(p_{k,n}\geq 0\)\(\forall k,n\)
* \(\sum_{k=1}^{K}c_{k,n}=1\)\(\forall n\)
* \(\sum_{k=1}^{K}\sum_{n=1}^{N}c_{k,n}P_{k,n}\leq P_{tot}\)
* \(R_{i}:R_{j}=\varphi_{i}:\varphi_{j}\)\(\forall i,j\in\left\{1,\cdots,K\right\},i\
eq j\)  
where \(c_{k,n}\) can only be either 1 or 0, representing whether subcarrier \(n\) is used by user \(k\) or not. \(p_{k,n}\) is the power allocated to user \(k\) in the subcarrier \(n\).