Optimal Energy-Efficient Power Allocation Scheme with Low Complexity for Distributed Antenna System

In this paper, by maximizing the energy efficiency (EE), an optimal power allocation scheme is developed for downlink distributed antenna system (DAS). Different from conventional optimal power allocation schemes that need iterative calculation, the developed scheme can provide closed-form power allocation and no iteration is required. Based on the definition of EE, the optimized objective function is firstly formulated, and then a computationally efficient algorithm is proposed to obtain the optimal number of active remote antennas and the corresponding power allocation. Using the optimal number, the multidimensional solution for the optimized function is transformed into searching one-dimensional solution. As a result, closed-form expression of power allocation coefficients is attained. Numerical results verify the effectiveness of the proposed scheme. The scheme can obtain the same EE as the conventional optimal scheme but with lower complexity, and it has more accuracy than the existing low-complexity scheme.


Introduction
Recently, green communication, where energy efficiency (EE) is pursued [1][2][3], has received much attention, since energy demand and prices increase dramatically.As an important technology for green communication, distributed antenna system (DAS) can expand cell coverage and improve the EE [4][5][6].For this reason, different energyefficient power allocation (PA) schemes have been developed for DAS [7][8][9][10][11].In [7], using iterative search and numerical calculation, an approximate PA scheme is proposed for generalized DAS.An adaptive PA scheme for achieving maximum EE while satisfying spectral efficiency requirement in multiuser DAS is developed in [8].In [9], by maximizing the upper bound of average EE, a subopti-mal energy-efficient PA scheme is presented for DAS over composite fading channel.An optimal PA algorithm to maximize EE is proposed in [10], where the optimization problem is solved by using the Karush-Kuhn-Tucker (KKT) conditions and Lambert function.Considering that the above schemes need iterations, a low-complexity PA scheme aiming at maximizing EE is presented in [11], but there exist some errors in the theoretical derivations on PA.Hence, the accuracy of the optimal PA cannot be guaranteed.Besides, the above schemes basically consider single receive antenna for analysis convenience, and thus the corresponding EE performance will be limited.
Motivated by the reasons above, we propose a lowcomplexity energy-efficient power allocation scheme for DAS in composite Rayleigh channel, where path-loss, shadowing, fading, and multiple receive antennas are all considered.Also we present a new computationally efficient algorithm to achieve the optimal number of active remote antennas and the corresponding closed-form power allocation coefficients.It is shown that the proposed power allocation scheme and algorithm are both valid, the power allocation scheme can obtain the same energy efficiency as the conventional optimal one [10], and the algorithm has more accuracy than the existing low-complexity algorithm [11].Furthermore, no iteration is required.
The rest of this paper is organized as follows.Section 2 introduces the system model.In Section 3, a lowcomplexity power allocation scheme is developed, and the comparison between different PA schemes is presented.Simulation results are provided in Sec. 4, and Section 5 concludes the paper.
The notations throughout this paper are as follows.Bold lower case letters denote column vectors.The superscript () T denotes transposition, E{} denotes statistical expectation, W() denotes Lambert W function.

System Model
We consider a downlink DAS with N t remote antennas (RAs) and multiple receive antennas operating in a single cell environment as illustrated in Fig. 1, where N t RAs are distributed in the cell, the i-th RA is denoted as RA i , and Base Station (BS) is referred as RA 1 .All RAs are linked to the central processing unit via dedicated wired connection.The mobile station (MS) is equipped with N t antennas, and the received signal at MS from RA i is given by r T 1 , , where P i , i = 1,..., N t , is the transmit power consumed by the RA i , x i stands for the transmitted symbol from RA i with unit energy, z is the complex Gaussian noise vector with zero mean and variance  n 2 , and h i j = g i j  i , j = 1,..., N r , is the composite channel fading coefficient between RA i and the j-th antenna of MS, where g i j denotes the small-scale Rayleigh fading between RA i and the j-th receive antenna,  i = (S i d i - i ) ½ denotes the large-scale fading between RA i and the MS, where  i is the path loss exponent, d i is the distance from RA i to the MS, and S i denotes the shadowing effect.
In general, EE is defined as the ratio of data transmission rate to the total consumed power.Based on this, the EE of DAS can be written as is defined as the channel to noise ratio (CNR) after maximal ratio combining at the receiver.Per RA power constraint is 0  P i  P max, i , where P max, i is the maximum transmit power available at RA i .P c denotes the circuit power and is a constant.

Low-complexity Power Allocation Scheme
In this section, we will develop an optimal power allocation scheme with low complexity, and present a computationally efficient algorithm to obtain the optimal number of active remote antennas and the corresponding PA coefficients.
Subject to the power constraint, we firstly formulate the objective function of the optimal PA as t where P = [P 1 , …, P N t ] T .Without loss of generality, we assume that  1 >  2 > …>  N t .It has been proved in [10] that the optimal PA solution will have the following general form: According to this, the optimization of PA in (3) will become to search the optimal number of active RAs (i.e., N 0 ) and the corresponding PA.The optimized problem in (3) can be converted into finding the optimal solution of the following segmented function: where Obviously, ( 5) is a continuous function.Taking the derivative of ( ) where Taking the derivative of A k (P k ) with respect to P k gives It means that A k (P k ) is decreasing in any segment.With (7), we have:   It means that A k (P k ) is always greater than A k + 1 (P k + 1 ).According to the analysis above, the optimal PA, 0 * N P will be achieved.Specifically: is decreasing function for Similarly, .
A P P P P is changed to Equating 0 0 ( ) With (14), we have: Using (15) and the Lambert function [12], we can obtain the solution of 1 exp 1 where W() denotes the Lambert function.Equation ( 16) provides a closed-form expression of P * N 0 in (4).Based on the analysis above, a low-complexity optimal PA algorithm is proposed, and correspondingly, the algorithm procedure is summarized as follows: Algorithm 1 Efficient PA calculation method 1: Sort the i  with descending order.2: Calculating the V k (0), for k = 1,…, N t , and finding K that makes V K (0) be the largest among {V k (0)}.Using the above algorithm, the optimal number of active RAs is determined correctly, and the multi-dimensional root finding for the optimized objective function becomes one-dimensional root finding.As a result, closedform power allocation is computed and no iteration is required.For this algorithm, the number of Lambert W function is calculated by one or zero.Note that in [11], a lowcomplexity PA scheme is also proposed for maximizing the energy efficiency in DAS, but the scheme cannot always guarantee the optimal values since there are errors in [11, (9)] and [11, (11)], i.e., the derived derivative V´k (P k ) and optimal PA P * N 0 are not accurate so that the obtained power value may exceed the given maximum transmit power.For example, using the CNRs as { i } i = 1, …, 7 = {6.3608e+05,4.8028e+05, 1.6016e+05, 1.3177e+05, 6.6877e+04, 6.4877e+03, 1.2921e+03}, we plot the corresponding numerical results of the energy efficiency in Fig. 2.
As shown in Fig. 2, the proposed scheme can be in good agreement with the conventional optimal scheme [10], but the existing low-complexity scheme [11] has obvious difference because the derived ( 9) and (11) in [11] have errors, which result in inaccurate power allocation.Taking the example of Fig. 2, let P max,i = P max = 0.5, then V´2(0) = -0.1460< 0, so N 0 = 1 according to the algorithm in [11] (this algorithm is based on the [( 9),11]), but in fact, V´2(0) = 0.0417 > 0, and correspondingly, N 0 = 2 according to our algorithm.Besides, using [( , but according to our algorithm, ) Obviously, the result from [(11), 11] is larger than P max, N 0 , whereas our result can satisfy the maximum power constraint.
The above results indicate that the proposed scheme is valid.Besides, the complexity analysis measured by the number of Lambert function calculation and comparison among three schemes are listed in Tab. 1. From Tab. 1, it is shown that the proposed scheme has lower calculation complexity than the other two and is more accurate than the existing scheme [11].

Simulation Results
In this section, we will assess the validity of the proposed scheme via computer simulation.In simulation, it is assumed that P max , i = P max (i = 0, …, N t ) for analysis convenience.The BS (RA 1 ) is in the center of the cell and the  RAs are evenly and symmetrically placed in the cell.The main parameters used in simulations are listed in Tab. 2.
The simulation results are shown in Fig. 3 and Fig. 4.
Figure 3 illustrates the EE performance of DAS with different receive antennas.Considering that the optimal scheme in [10] is based on single receive antenna, we extend the scheme in [10] to multiple receive antennas case after some modifications.From Fig. 3, it is observed that the EE of the proposed scheme is the same as that of the conventional optimal scheme [10].Moreover, the EE increases with the growth of the number of receive antennas.Namely, the EE of DAS with three receive antennas (N r = 3) is higher than that with two receive antennas (N r = 2) because of more spatial diversity gain, and due to the same reason, the EE of DAS with two receive antennas (N r = 2) is higher than that with single receive antenna (N r = 1).The results above indicate that the developed scheme is effective and reasonable, and the application of multiple receive antennas does improve the EE performance effectively.
In Fig. 4 we plot the EE performance of DAS with different path loss exponents, where two receive antennas are considered.It is found that the EE of the proposed scheme is still consistent with that of conventional optimal 0 0. scheme [10].Besides, the EE decreases with the growth of the path loss exponent  as expected.Namely, the EE of DAS with  = 4 is lower than that with  = 3.5, and the EE of DAS with  = 3.5 is lower than that with  = 3.This is because the system with higher path loss exponent will experience greater path loss, which leads to a decline in EE performance.

Conclusions
We have developed an optimal energy-efficient power allocation scheme for DAS using multiple receive antennas in composite Rayleigh fading channel, and proposed a computationally efficient algorithm to obtain the optimal number of active RAs.Based on this, optimal power allocation is derived, and the corresponding closedform expression is attained.The results indicate that our scheme has lower complexity than the existing schemes.Moreover, it can obtain the same EE performance as the existing optimal scheme, and provide the optimal active antenna number and power allocation accurately.Benben WEN was born in Henan, China.He is currently working towards the M.Sc.degree at Nanjing University of Aeronautics and Astronautics.
served as a technical program committee of International Conference on Communications (ICC 2016, 2015), Globecom'2006, International Conference on Communications Systems (ICCS 2008, 2010), International Conference on Communications and Networking in China (Chinacom 2010, 2014) and International Conference on Wireless Communications and Signal Processing 2011.He has been a member of IEEE ComSoc Radio Communications Committee (RCC) since May 2007.His research interests include MIMO, distributed antenna systems, energy efficiency, adaptive modulation and cross-layer design.Weiye XU was born in Neimengu, China.She received her M.S. degree in Communication and Information Systems from Hohai University.She is an associate professor at the School of Communication Engineering at Nanjing Institute of Technology.Her research interests include digital communication, MIMO technique, and distributed antenna systems.
on this, considering that A k (P k ) is decreasing function in a segment, the A k (P k ) in different segments has the following relation: