## 2 System Model  
As shown in Figure 1, a downlink model of two-tier heterogeneous networks is considered in this paper. The first tier includes one MBS and the maximum transmit power of the MBS is \(P_{\text{M}}^{\text{max}}\). The MBS is configured with a sector antenna array and the coverage area is divided into three macrocells that are 120 degrees from each other. In the second tier, there are \(N_{\text{P}}\) PBSs whose maximum transmit power is \(P_{\text{P}}^{\text{max}}\) randomly distributed in the coverage of macrocells. It is assumed that the number of picocells in each macrocell is the same. In order to improve spectral efficiency, the MBS and the PBSs share the same spectral resources. The users of each BS use NOMA or full power mode to communicate, and the users who transmit in NOMA mode recover the signals at the receivers through SIC technology. In this model, a downlink NOMA HetNets which consists of one MBS denoted by \(b=1\) and \(N_{\text{B}}-1\) PBSs denoted by \(b=2,3,\ldots,N_{\text{B}}\) is considered. Let the number of users of BS \(b\) is denoted by \(N_{\text{U}}^{b}\). Moreover, each user can be served by only one BS and perfect channel state information is considered. The total available system bandwidth is denoted by \(B\) and is divided into \(N_{\text{SC}}^{b}\) (\(2N_{\text{SC}}^{b}\geq N_{\text{U}}^{b}\)) orthogonal subchannels with the bandwidth \(B/N_{\text{SC}}^{b}\) by BS \(b\). Let, \(n=1,2,\ldots,N_{\text{SC}}^{b}\) represent the set of subchannels of BS \(b\). We assume that the greatest number of NOMA users on each subchannel is 2, and the user with the worst signal to interference-plus-noise ratio (SINR) will transmit with full power on the subchannel if the number of users of BS is odd. We assumed that the weak user is denoted by \(u=1\) and the strong user is denoted by \(u=2\) on the subchannel \(n\) of BS \(b\). The NOMA superimposed signal transmitted by the subchannel \(n\) of BS \(b\) can be expressed as:  
\[x_{b}^{n}=\sqrt{a_{b}^{n}s_{1,b}^{n}}+\sqrt{(1-a_{b}^{n})}s_{2,b}^{n}, \tag{1}\]  
where \(s_{1,b}^{n}\) and \(s_{2,b}^{n}\) are the signals of weak users and strong users. \(a_{b}^{n}\) is the power allocation factor of the subchannel \(n\) of the BS \(b\) and \(1/2<a_{b}^{n}<1\) since the BS needs to allocate more  
Figure 1: System model. power for weak users. The received signals of two users on the subchannel \(n\) of BS \(b\) are given by:  
\[y_{u,b}^{n}=h_{u,b}^{n}\sqrt{p_{b}^{n}}x_{b}^{n}+\sum\limits_{b^{\prime}=1,b^{ \prime}\
eq b}^{N_{\text{BS}}}h_{u,b^{\prime}}^{n}\sqrt{p_{b^{\prime}}^{n}}x_{b^ {\prime}}^{n}+\omega_{u,b}^{n}, \tag{2}\]  
where \(h_{u,b}^{n}\) represents the channel gain between BS \(b\) and user \(u\) on the subchannel \(n\), including the channel gain of large-scale fading and small-scale Rayleigh fading. \(p_{b}^{n}\) represents the transmit power of BS \(b\) on the subchannel \(n\) and \(\omega_{u,b}^{n}\sim CN\left(0,\sigma^{2}B/N_{\text{SC}}^{b}\right)\) represents the additive white Gaussian noise (AWGN); the mean is zero and the variance is \(\sigma^{2}B/N_{\text{SC}}^{b}\) of BS \(b\) on the subchannel \(n\). \(\sum_{b^{\prime}=1,b^{\prime}\
eq b}^{N_{\text{BS}}}h_{u,b^{\prime}}^{n}\sqrt{ p_{b^{\prime}}^{n}}x_{b^{\prime}}^{n}\) represents the aggregate interference from other BSs, including co-tier interference and cross-tier interference. It is assumed that a strong user can successfully demodulate the signal of a weak user and that means a strong user can cancel the interference from the signal of a weak user in the same NOMA group.