Interference mitigation for non-orthogonal multiple access in heterogeneous networks

Abstract

Non-orthogonal multiple access (NOMA) technique is a promising alternative for enhancing network sum-rate while increasing the interference experienced by the users. In heterogeneous networks, the signal of a NOMA user is affected by three types of interference; intra-cell, inter-cell, and inter-tier. In an environment with high interference, using orthogonal multiple access (OMA) along with NOMA is beneficial to improve the edge user performance. This work proposes a hybrid model, where NOMA and OMA are deployed together in the same spectrum band. The performance of hybrid model is compared to NOMA and OMA for heterogeneous networks by stochastic geometry analysis. The analysis includes pico- and femto-cell alternatives to elaborate varying interference conditions. Using the capacity analysis formulation of NOMA, a power level optimization framework is presented for maximum sum-rate and fairness objectives. Besides, two pairing methods, near-to-near and near-to-far pairing, are included from the literature to reveal pairing effect on user-specific and system-level performance. The numerical results indicate NOMA trade-offs for increasing edge user capacity by power optimization and resistance of hybrid model to interference under different conditions.

Appendices

Appendix 1: Optimal power levels for maximum fairness

The problem definition in Eq. (32) is firstly converted to

$$\begin{aligned} \begin{array}{rl} \max _{a,\xi } &{} \quad \xi , \\ \text {subject to} &{} \quad {\mathcal {C}}_{i,j}^{\text{near}}\ge \xi , \\ &{} \quad {\mathcal {C}}_{i,j}^{\text{far}}\ge \xi \end{array} \end{aligned}$$

(36)

to be able to model with the Karush–Kuhn–Tucker (KKT) conditions [43]. The Lagrangian function for the problem becomes

$$\begin{aligned} L = -\xi + \lambda ^{\text{near}}(\xi -{\mathcal {C}}_{i,j}^{\text{near}}) +\lambda ^{\text{far}}(\xi -{\mathcal {C}}_{i,j}^{\text{far}}), \end{aligned}$$

(37)

where $\lambda ^{\text{near}}$ and $\lambda ^{\text{far}}$ are respective Lagrangian multipliers for near and far user capacity lower bounds. Therefore, KKT conditions for the problem are given as

$$\begin{aligned} -\lambda ^{\text{near}}\frac{\partial {\mathcal {C}}_{i,j}^{\text{near}}}{\partial a} -\lambda ^{\text{far}} \frac{\partial {\mathcal {C}}_{i,j}^{\text{far}}}{\partial a}&= 0, \end{aligned}$$

(38)

$$\begin{aligned} -1 + \lambda ^{\text{near}} + \lambda ^{\text{far}}&= 0, \end{aligned}$$

(39)

$$\begin{aligned} \lambda ^{\text{near}}( \xi - {\mathcal {C}}_{i,j}^{\text{near}})&= 0, \end{aligned}$$

(40)

$$\begin{aligned} \lambda ^{\text{far}}( \xi - {\mathcal {C}}_{i,j}^{\text{far}}) &= 0, \end{aligned}$$

(41)

where Eqs. (38) and (39) are derived from derivative of Eq. (37) with respect to a and $\xi$. By definition, the sign of $\frac{\partial {\mathcal {C}}_{i,j}^{\text{near}}}{\partial a}$ and $\frac{\partial {\mathcal {C}}_{i,j}^{\text{far}}}{\partial a}$ are positive and negative, respectively. Considering the signs of these derivatives, Eqs. (38) and (39) provides that $\lambda ^{\text{near}}$ and $\lambda ^{\text{far}}$ have non-zero values and their sum equals to 1. Using this outcome on Eqs. (40) and (41), the solution is achieved when $\xi = {\mathcal {C}}_{i,j}^{\text{near}}= {\mathcal {C}}_{i,j}^{\text{near}}$.

Appendix 2: Optimal power levels for maximum sum-rate

The Lagrangian function for the problem in Eq. (34) is given as

$$\begin{aligned} L = -{\mathcal {C}}_{i,j}^{\text{near}}- {\mathcal {C}}_{i,j}^{\text{far}}+ \lambda ^{\text{near}} (C_{\text{min}} - {\mathcal {C}}_{i,j}^{\text{near}}) + \lambda ^{\text{far}} (C_{\text{min}} - {\mathcal {C}}_{i,j}^{\text{far}}), \end{aligned}$$

(42)

where $\lambda ^{\text{near}}$ and $\lambda ^{\text{far}}$ are the respective Lagrangian multipliers for the near and far user capacity lower bounds. To solve the problem, the KKT conditions [43] are given as

$$\begin{aligned} - \frac{\partial {\mathcal {C}}_{i,j}^{\text{far}}}{\partial a}(1 + \lambda ^{\text{far}}) -\frac{\partial {\mathcal {C}}_{i,j}^{\text{near}}}{\partial a} (1 + \lambda ^{\text{near}})= & {} 0, \end{aligned}$$

(43)

$$\begin{aligned} \lambda ^{\text{near}}(C_{\text{min}} - {\mathcal {C}}_{i,j}^{\text{near}})= & {} 0, \end{aligned}$$

(44)

$$\begin{aligned} \lambda ^{\text{far}}(C_{\text{min}} - {\mathcal {C}}_{i,j}^{\text{far}})= & {} 0. \end{aligned}$$

(45)

The optimal values for $\lambda ^{\text{near}}$, $\lambda ^{\text{far}}$, and a can only be obtained by finding a relationship between $\frac{\partial {\mathcal {C}}_{i,j}^{\text{far}}}{\partial a}$ and $\frac{\partial {\mathcal {C}}_{i,j}^{\text{near}}}{\partial a}$. The derivative of the near user capacity can be obtained as

$$\begin{aligned} \frac{\partial {\mathcal {C}}_{i,j}^{\text{near}}}{\partial a}= & {} \int \limits _0^\infty \frac{1}{\tau } \frac{\partial }{\partial a} \Pr \left( \Psi _i > \frac{\tau }{a} \right) d\tau \end{aligned}$$

(46)

$$\begin{aligned}= & {} \int \limits _0^\infty \frac{\tau }{(\tau +1)a^2} f_{\Psi _i}\left( \frac{\tau }{a} \right) d\tau \end{aligned}$$

(47)

$$\begin{aligned}= & {} \int \limits _0^\infty \frac{u}{1+au} f_{\Psi _i}\left( u\right) du \end{aligned}$$

(48)

$$\begin{aligned}= & {} \int \limits _0^{1/a} \Pr \left( \Psi _i > \frac{v}{1-av}\right) dv, \end{aligned}$$

(49)

where $f_{\Psi _i}(\cdot )$ presents PDF of the i-th user SINR. Through derivations, Eq. (46) is obtained by moving the derivative inside the integral. Equation (47) is obtained by the derivative of the CDF of $\Psi _i$ is its PDF. Equation (48) is obtained by $\frac{\tau }{a} = u$ transformation in integral. Lastly, Eq. (49) is obtained by $\int _0^\infty \Pr (\Psi >g^{-1}(t))dt = \int g(t)f_{\Psi }(t)dt$ transformation between integrals of tail probability and PDF [42] as $\Psi$ is always positive.

Similarly, the derivative of the far user capacity can be obtained as

$$\begin{aligned} \frac{\partial {\mathcal {C}}_{i,j}^{\text{far}}}{\partial a}= & {} \frac{\partial }{\partial a} \int \limits _0^{\frac{1-a}{a}} \frac{1}{\tau +1} \Pr \left( \Psi _j>\frac{\tau }{1-a-a\tau }\right) d\tau \end{aligned}$$

(50)

$$\begin{aligned}= & {} \int \limits _0^{\frac{1-a}{a}} \frac{1}{\tau +1} \frac{\partial }{\partial a} \left( \Pr \left( \Psi _j> \frac{\tau }{1-a-a\tau }\right) \right) d\tau \nonumber \\{} & {} \quad - \frac{1}{a^2}\Pr \left( \Psi _j > \frac{\frac{1-a}{a}}{1 - a - a \frac{1-a}{a}}\right) \end{aligned}$$

(51)

$$\begin{aligned}= & {} -\int \limits _0^{\frac{1-a}{a}}\frac{\tau }{(1-a-a\tau )^2} f_{\Psi _j} \left( \frac{\tau }{1-a-a\tau } \right) d\tau \end{aligned}$$

(52)

$$\begin{aligned}= & {} -\int \limits _0^\infty \frac{u}{1+au} f_{\Psi _j}(u) du \end{aligned}$$

(53)

$$\begin{aligned}= & {} -\int \limits _0^{1/a} \Pr \left( \Psi _j > \frac{v}{1-av} \right) dv. \end{aligned}$$

(54)

In Eq. (51), the derivative operator moved inside the integral by the Leibniz rule [44]. The second term in Eq. (51) equals to 0. Equation (52) is obtained by the derivative of the CDF of $\Psi _i$ is its PDF. Equation (53) is obtained by integral transformation $u = \frac{\tau }{1-a-a\tau }$. Lastly, Eq. (54) is obtained by the transformation between integrals of tail probability and PDF [42].

As the users are ordered according to their SINRs, $\Psi _i > \Psi _j$ condition is always true for $j>i$. By $\Pr (\Psi _i>x)>\Pr (\Psi _j>x)$ condition and Eqs. (49) and (54), we can conclude that

$$\begin{aligned} \frac{\partial {\mathcal {C}}_{i,j}^{\text{near}}}{\partial a} > - \frac{\partial {\mathcal {C}}_{i,j}^{\text{far}}}{\partial a} \end{aligned}$$

(55)

condition is always true. Adding Eq. (55) condition to the KKT conditions in Eqs. (43), (44), and (45), the optimal solution is achieved by selecting the Lagrangian multipliers as $\lambda ^{\text{near}}=0$ and $\lambda ^{\text{far}}>0$. Therefore, the optimal solution of a can be calculated by finding the root of ${\mathcal {C}}_{i,j}^{\text{far}}= C_{\text{min}}$ equation.