Abstract
In this paper, the performance of uplink distributed antenna system (DAS) with DevicetoDevice (D2D) communication is investigated over composite Rayleigh fading channels, and an energyefficient power allocation (PA) scheme is developed for D2D communication underlaying DAS. Firstly, we establish the uplink DAS model with D2D communication. Then, the optimization problem for energy efficiency (EE) maximization subject to maximum total power constraint and the minimal rate constraints of cellular user and D2D user is formulated. Based on the pseudoconcave of objective function in optimization problem, we propose an optimal PA scheme with the bisection method to obtain the optimal solution of the optimization problem. The simulation results demonstrate the effectiveness of our proposed scheme. The proposed optimal PA scheme can achieve better EE performance than the conventional equal PA scheme, and the same EE as the PA scheme based on twodimensional search method but with lower complexity.
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Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments which improve the quality of this paper greatly. This work is supported by National Natural Science Foundation of China (61571225), Natural Science Foundation of Jiangsu Province in China (BK20181289), and Open Research Fund of National Mobile Communications Research Laboratory of Southeast University (2017D03).
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Appendices
Appendix I
Considering the minimum rate constraints, we can obtain
From (13), it is easily obtained the lower bound and the upper bound of α
where \( {r}_1={2}^{R_{c,\min }}1,{r}_2={2}^{R_{d,\min }}1 \).
Let \( {\alpha}_1=\frac{r_1+{m}_2{r}_1P}{m_1P+{m}_2{r}_1P},{\alpha}_2=\frac{m_3P{r}_2}{m_3P+{m}_4{r}_2P} \), we can get α ∈ [α_{1}, α_{2}].
Appendix II
For the given P, let \( {\left.\frac{\partial {\eta}_{EE}\left(\alpha \right)}{\partial \alpha}\right}_{P={P}_{\mathrm{min}}}=0 \), we can get the quadratic equation
where
Let f(α) = n_{1}α^{2} + n_{2}α + n_{3}, we discuss the candidate solutions for f(α) = 0 below.
Set α_{3}, α_{4} as solutions for f(α) = 0, next we judge the symbol with upper and lower bounds of [α_{1}, α_{2}].

(1)
If f(α_{1})f(α_{2}) ≤ 0, there must be the only solution in [α_{1}, α_{2}]. We set a unique value α_{3} ∈ [α_{1}, α_{2}].

(2)
If f(α_{1})f(α_{2}) > 0, there are three cases of solutions.

Case 1:
f(α) has no solution in [α_{1}, α_{2}].

Case 2:
f(α) has one solution in [α_{1}, α_{2}], then the unique value is extreme point of f(α).

Case 3:
f(α) has two solutions in [α_{1}, α_{2}].
Due to α_{3}, α_{4} ∈ (0, 1), hence
Then we can get the following formula by \( \varDelta ={n}_2^24{n}_1{n}_3>0 \)
However, the symbol of n_{1} is uncertain so we have to discuss it under the following two cases.
From (17), we obtain
And then we further get n_{1} > 0, n_{2} < 0, 2n_{1} + n_{2} > 0 and n_{3} > 0. Moreover from (16), 2n_{1} + n_{2} = 2(1 + k_{4}).
(k_{2}k_{3} + k_{1}(−k_{3} + k_{4} + k_{2}k_{4})) ≈ k_{2}k_{3} + k_{1}(−k_{3} + k_{4} + k_{2}k_{4}) < k_{2}(k_{3} − k_{1}). Due to 2n_{1} + n_{2} > 0, there must be k_{3} > k_{1}.
Furthermore, according to \( \sqrt{n_2^24{n}_1{n}_3}<2{n}_1+{n}_2 \), we derive
And from (16),
n_{1} + n_{2} + n_{3} = (1 + k_{4})(−k_{3} + k_{1}(1 + k_{2} − k_{3} + k_{4} + k_{2}k_{4})) ≈ − k_{3} + k_{1}(1 + k_{2} − k_{3} + k_{4} + k_{2}k_{4}) < k_{1} − k_{3} < 0, which is in contradiction to (20). Thus, the case (i) does not exist.
Similarly from (17), we get
And then we further get n_{1} < 0, n_{2} > 0, 2n_{1} + n_{2} < 0, n_{3} < 0. By (18), we have
Combined with (16) and (22), we can derive n_{3} ≈ k_{1}(1 + k_{3}) − (1 + k_{2})k_{3}(1 + k_{4}) > k_{1} − k_{3}. Because of n_{3} < 0, then k_{1} < k_{3}.
Meanwhile, n_{2} ≈ k_{2}k_{3}(1 + k_{4}) + k_{1}(k_{4} − k_{3}) < k_{4}(k_{1} − k_{3}) < 0, but we have derived n_{2} > 0. This result is conflictive. Thus, the case (ii) does not exist. From the above, the case 3 does not exist.
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Wang, G., Yu, X. & Teng, T. EnergyEfficient Power Allocation Scheme for Uplink Distributed Antenna System with D2D Communication. Mobile Netw Appl 26, 1225–1232 (2021). https://doi.org/10.1007/s11036019013432
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Keywords
 DevicetoDevice Communication
 Distributed Antenna System
 Energy Efficiency
 Power Allocation