Energy efficiency optimization for uplink traffic offloading in the integrated satellite-terrestrial network

Abstract

Offloading traffic from terrestrial areas to low-orbit satellite networks, the ultra-long-distance and ultra-large-range connectivity can be achieved by on-board routing and transmission. This paper investigates the relay-assisted NOMA-enabled uplink traffic offloading in the integrated satellite-terrestrial network. The joint energy efficiency maximum problem is formulated and then decomposed into two subproblems, i.e., the decoding sequence and power allocation (DS-PA) problem, and the users-satellites-subchannels association and transmit power allocation (USSA-TPA) problem. For DS-PA problem, the decoding sequence constraint is removed, and then the successive convex approximation method is utilized to solve it. To tackle the USSA-TPA issue, we reformulate it into two subproblems and resort to the alternate iteration method to deal with them iteratively. The numerical results show that the proposed joint optimization of energy efficiency for uplink traffic offloading cooperated with NOMA scheme outperforms other methods and their combinations. The performance of proposed USSA scheme and power allocation method is also discussed.

Appendices

Appendix 1

Proof of Lemma 1.

Given the power allocation scheme of users, the original problem can be transformed into a maximum sum rate problem, which is formulated as follows

$$\mathop {\max }\limits_{\Phi } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{n = 1}^{N} {\log_{2} } (1 + \frac{{p_{\Phi (n)} |h_{\Phi (n)} |^{2} }}{{\sum\nolimits_{n^{\prime} = \Phi (n) + 1}^{N} {p_{n^{\prime}} |h_{n^{\prime}} |^{2} + \sigma^{2} } }})$$

(42)

$${\kern 1pt} s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} C_{1} :{\kern 1pt} {\kern 1pt} {\kern 1pt} p_{\Phi (n)} |h_{\Phi (n)} |^{2} \ge (2^{{r_{n} }} - 1)(\sum\nolimits_{n^{\prime} = \Phi (n) + 1}^{N} {p_{n^{\prime}} |h_{n^{\prime}} |^{2} + \sigma^{2} } )$$

(43)

$$C_{2} :{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} r_{N} \le r_{N - 1} \le \cdots \le r_{1}$$

(44)

According to Ref [34], Appendix A], the objective function is confirmed to be concave with respect to each $p_{n}$. The minimum required rate in TS is presented in C₁, where the order of $r_{n}$ is shown in C₂. By introducing the Lagrangian multiplexer $\upsilon_{n}$, the Lagrangian-formed function can be expressed as

$$\begin{gathered} L(\Phi ,\upsilon ) = \sum\limits_{n = 1}^{N} {\log_{2} (1 + \frac{{p_{\Phi (n)} |h_{\Phi (n)} |^{2} }}{{\sum\nolimits_{n^{\prime} = \Phi (n) + 1}^{N} {p_{n^{\prime}} |h_{n^{\prime}} |^{2} + \sigma^{2} } }})} - \hfill \\ \sum\limits_{n = 1}^{N} {\upsilon_{n} } ((2^{{r_{n} }} - 1)(\sum\limits_{n^{\prime} = \Phi (n) + 1}^{N} {p_{n^{\prime}} |h_{n^{\prime}} |^{2} } + \sigma^{2} ) - p_{\Phi (n)} |h_{\Phi (n)} |^{2} ) \hfill \\ \end{gathered}$$

(45)

where the decoding sequence variable $\Phi$ can be calculated by the dual optimum $D(\upsilon ) = \mathop {\sup }\limits_{\Phi } L(\Phi ,\upsilon )$. Adopting KKT conditions, we can easily obtain $\upsilon_{n} = 1 + \sum\nolimits_{j = 1}^{n} {\upsilon_{j} (2^{{r_{n} }} - 1)(\sum\nolimits_{n^{\prime} = \Phi (n) + 1}^{N} {p_{n^{\prime}} |h_{n^{\prime}} |^{2} + \sigma^{2} } )}$. It can be observed that $\upsilon_{n} > 0$ for all $n = 1,2, \cdots ,N$. Thus, we redefine the minimum required rate as

$$r_{n}^{*} = \log_{2} \left( {1 + \frac{{p_{n} |h_{n} |^{2} }}{{\sum\nolimits_{{n^{\prime } = n + 1}}^{N} {p_{{n^{\prime } }} |h_{{n^{\prime } }} |^{2} + \sigma^{2} } }}} \right) - \psi_{n} ((2^{{r_{n} }} - 1)\left( {\sum\limits_{{n^{\prime } = \Phi (n) + 1}}^{N} {p_{{n^{\prime } }} |h_{{n^{\prime } }} |^{2} } + \sigma^{2} ) - p_{\Phi (n)} |h_{\Phi (n)} |^{2} } \right)$$

(46)

Accordingly, the order of the minimum required rate can be rewritten as $r_{N}^{*} (\psi_{N} ) \le r_{N - 1}^{*} (\psi_{N - 1} ) \le \cdots \le r_{1}^{*} (\psi_{1} )$. And the Lagrangian-related variable $\psi_{n}$ is calculated as

$$\psi_{n} = \upsilon_{n} + \frac{{\partial r_{n}^{*} }}{{\partial p_{n} }}(2^{{r_{n} }} - 1)\left( {\sum\limits_{n^{\prime} = \Phi (n) + 1}^{N} {p_{{n^{\prime } }} |h_{{n^{\prime } }} |^{2} } + \sigma^{2} } \right)$$

(47)

Based on decoding sequence and KKT conditions transformation, we have

$$\log_{2} \left( {1 + \frac{{p_{\Phi (n)} |h_{\Phi (n)} |^{2} }}{{\sum\nolimits_{{n^{\prime } = \Phi (n) + 1}}^{N} {p_{{n^{\prime } }} |h_{{n^{\prime } }} |^{2} + \sigma^{2} } }}} \right) = r_{n} + \psi_{n} \left( {(2^{{r_{n} }} - 1)\left( {\sum\limits_{{n^{\prime } = \Phi (n) + 1}}^{N} {p_{{n^{\prime } }} |h_{{n^{\prime } }} |^{2} } + \sigma^{2} } \right) - p_{\Phi (n)} |h_{\Phi (n)} |^{2} } \right)$$

(48)

Combining $r_{n}^{*} (\psi_{n} ) = r_{n}$, (46) and (48), the proof is completed.

Appendix 2

Proof of the equivalence between the new transformed problem and the original problem.

The parameter $\varphi_{g,m,k}^{*}$ is achieved only the following condition is satisfied.

$$\begin{gathered} \mathop {\max }\limits_{{(q_{g,m,k}^{T} )^{*} }} \left( {\log_{2} (1 + SINR_{g,m,k}^{PH2} ) - \varphi_{g,m,k}^{*} \frac{{q_{g,m,k}^{T} }}{{\theta_{g} + \eta_{g} q_{g,m,k}^{T} }}} \right) \hfill \\ {\kern 1pt} \quad = \log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} - \varphi_{g,m,k}^{*} \frac{{(q_{g,m,k}^{T} )^{*} }}{{\theta_{g} + \eta_{g} (q_{g,m,k}^{T} )^{*} }} = 0 \hfill \\ \end{gathered}$$

(49)

At first, the adequacy of the condition is proofed. We define the maximum value of the USSA-TPA function as $\varphi_{g,m,k}^{*}$, which can be expressed as

$$\begin{gathered} \varphi_{g,m,k}^{*} = \frac{{\log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} }}{{(q_{g,m,k}^{T} )^{*} }}(\theta_{g} + \eta_{g} (q_{g,m,k}^{T} )^{*} ) \hfill \\ \quad > \frac{{\log_{2} (1 + SINR_{g,m,k}^{PH2} )}}{{q_{g,m,k}^{T} }}(\theta_{g} + \eta_{g} q_{g,m,k}^{T} ) \hfill \\ \end{gathered}$$

(50)

where $(q_{g,m,k}^{T} )^{*}$ is a feasible solution to ${\mathcal{W}}{(}{\mathbf{x,q}}^{{\mathbf{T}}} )$. Then, according to (50), the following formulas can be obtained.

$$\left\{ \begin{gathered} \log_{2} (1 + SINR_{g,m,k}^{PH2} ) - \varphi_{g,m,k} \frac{{q_{g,m,k}^{T} }}{{\theta_{g} + \eta_{g} q_{g,m,k}^{T} }} < 0 \hfill \\ \log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} - \varphi_{g,m,k} \frac{{(q_{g,m,k}^{T} )^{*} }}{{\theta_{g} + \eta_{g} (q_{g,m,k}^{T} )^{*} }} = 0 \hfill \\ \end{gathered} \right.$$

(51)

Consequently, we can summarize that $\log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} - \varphi_{g,m,k} (q_{g,m,k}^{T} )^{*} /(\theta_{g} + \eta_{g} (q_{g,m,k}^{T} )^{*} ) = 0$ is always held under given optimal solution $(q_{g,m,k}^{T} )^{*}$, and the adequacy is confirmed.

And then, we prove the necessity of the equivalence. We assume that $p_{g,m,k}^{*}$ is the optimal solution of the transformed function, and the condition $\log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} - \varphi_{g,m,k}^{*} p_{g,m,k}^{*} /(\theta_{g} + \eta_{g} p_{g,m,k}^{*} ) = 0$ is satisfied. For any available solution $p_{g,m,k}$, they can be expressed as

$$\log_{2} (1 + SINR_{g,m,k}^{PH2} ) - \frac{{\varphi_{g,m,k}^{*} p_{g,m,k} }}{{\theta_{g} + \eta_{g} p_{g,m,k} }}{\kern 1pt} \le \log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} - \frac{{\varphi_{g,m,k}^{*} p_{g,m,k}^{*} }}{{\theta_{g} + \eta_{g} p_{g,m,k}^{*} }} = 0$$

(52)

The above inequality can be transformed as

$$\left\{ \begin{gathered} \log_{2} (1 + SINR_{g,m,k}^{PH2} ) - \varphi_{g,m,k} \frac{{p_{g,m,k} }}{{\theta_{g} + \eta_{g} p_{g,m,k} }} < 0 \hfill \\ \log_{2} (1 + SINR_{g,m,k}^{PH2} )^{*} - \varphi_{g,m,k} \frac{{(p_{g,m,k} )^{*} }}{{\theta_{g} + \eta_{g} (p_{g,m,k} )^{*} }} = 0 \hfill \\ \end{gathered} \right.$$

(53)

Therefore, the optimal solution to the transformed function is also the optimal solution for the original problem, and the proof is completed.