Adaptive Energy Efficient Power Allocation Scheme for DAS with Multiple Receive Antennas

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1. Jiangsu Key Laboratory of Internet of Things and Control Technologies, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R. China 2. Shenzhen Research Institute, Nanjing University, Shenzhen 518057, P.R. China

(Received 11 May 2016; revised 8 March 2017; accepted 21 September 2017)

Abstract:Energy efficiency (EE) of downlink distributed antenna system (DAS) with multiple receive antennas is investigated over composite Rayleigh fading channel that takes the path loss and lognormal shadow fading into account. Our aim is to maximize EE which is defined as the ratio of the transmission rate to the total consumed power under the constraints of the maximum transmit power of each remote antenna. According to the definition of EE, the optimized objective function is formulated with the help of Lagrangian method. By using the Karush-Kuhn-Tucker (KKT) conditions and numerical calculation, considering both the static and dynamic circuit power consumptions, an adaptive energy efficient power allocation (PA) scheme is derived. This scheme is different from the conventional iterative PA schemes based on EE maximization since it can provide closed-form expression of PA coefficients. Moreover, it can obtain the EE performance close to the conventional iterative scheme and exhaustive search method while reducing the computation complexity greatly. Simulation results verify the effectiveness of the proposed scheme.

Key words:distributed antenna system(DAS); energy efficiency(EE); power allocation(PA); composite fading; multiple receive antennas

0 Introduction

Distributed antenna system (DAS) has emerged as a promising technology for future wireless communications, thanks to its ability of enhancing the system capacity, improving the signal quality and reducing the power[1-4]. In DAS, the remote antennas (RAs) are separated geographically and connected to a central control module via dedicated wires, fiber optics, or an exclusive radio frequency link[3]. Traditionally, the spectral efficiency (SE) has been used to measure the efficiency of a communication system[5]. However, it fails to evaluate how the energy is efficiently consumed. Green communication, which pursues high energy efficiency (EE), has drawn increasing attentions these days. Due to the growing energy demand and increasing energy price, pursuing high EE is becoming a mainstream for future mobile systems[6-8].

EE is defined as the sum-rate divided by the total power consumption measured in bit/J/Hz. Based on this, different energy efficient methods have been proposed for DAS[9-13]. In Ref.[9], an approximate power allocation (PA) method through an iterative numerical search was provided for generalized DAS, but the large-scale fading was not considered. An optimal PA algorithm with antenna selection relying on numerical search was proposed for DAS in Ref.[10]. A novel PA algorithm to achieve maximum EE while satisfying SE requirement in downlink multiuser DAS was proposed in Ref.[11]. In Ref.[12], fractional programming theory was adopted to transform the fractional form of non-convex EE optimization into its equivalent subtractive form, leading to an energy efficient PA algorithm in orthogonal frequency division multiple access(OFDMA) system. However, the above algorithms still need iterative calculation. For this, a low-complexity energy efficient PA scheme for DAS was proposed in Ref.[13], but some errors exists in the derivation of Eqs.(7) and (11). Moreover, for analysis convenience, the above studies basically consider single receive antenna and assume the circuit power consumption to be a constant, and thus the derived PA schemes lack generality. Based on this, the EE performance is not studied well, and the corresponding performance improvement and practicability will be limited.

Therefore, a composite fading channel including path loss, log-normal shadowing and Rayleigh fading is presented for DAS considering the practical case. According to this, an energy efficient PA optimization problem for DAS with multiple receive antennas is formulated by means of Lagrange multiplier method. Besides, a more practical circuit power consumption model is considered which includes both static part and dynamic part. By using the Karush-Kuhn-Tucker (KKT) conditions and the LambertWfunction, an adaptive energy efficient scheme is derived and closed-form PA coefficients are obtained. It is shown that this scheme can effectively lower the computation complexity when compared with the conventional scheme with dual loops iteration, and may obtain almost the same EE as the latter.

1 System and Channel Models

A distributed antenna system withNtRAs andNrreceive antennas in a single-cell environment is considered as shown in Fig.1. The RAs are distributed in the cell and linked to the base station (BS, also named as RA1) via dedicated wired connection, and theith RA is denoted as RAi. The mobile station (MS) is equipped withNrantennas. For remote transmit antennai, the corresponding received signals at MS can be expressed as

(1)

(2)

For the DAS, the achievable data transmission rate for the MS can be expressed as

(3)

Energy efficiency is usually defined as the ratio of data transmission rate to the total power consumption, i.e.

(4)

wherepcdenotes the circuit consumption which can be modeled as a linear function of throughput[15]

pc=ps+ξR

(5)

wherepsis the static circuit consumption term andξa constant denoting dynamic power consumption per unit data rate. Obviously, the constant circuit consumption model used in Refs.[9-13] is a special case thatξin Eq.(5) equals 0.

2 Energy Efficient Power Allocation and Algorithm Procedure

In this section, the optimized objective function on PA for maximizing EE is firstly formulated. Then, by using the KKT conditions and the LambertWfunction, a suboptimal closed-form energy efficient PA scheme is developed for DAS, and the corresponding algorithm procedure is presented.

The optimized objective function of the optimal PA can be expressed as

s.t.0≤pi≤Pmax,i∀i∈{1,…,Nt}

(6)

wherep=[p1,…,pNt]T.Since the optimization problem in Eq.(6) is non-convex, it is hard to find the optimal solution directly. For this, the following lemmas and corollaries are introduced.

Lemma1For the optimization problem

(7)

wherem>0,n≥1,s>0, the optimal solutionx*is obtained as

(8)

where

(9)

where [x]+represents max(x,0) andW(x) the LambertWfunction which is defined as the reverse function ofg(x)=xex[16].

ProofBy taking the derivative of the objective functiony(x) with respect toxyields

(10)

Equating Eq.(10) to zero gives

m(x+s)=(mx+n)ln(mx+n)

(11)

Using the LambertWfunction[16]and considering the non-negativity ofx, the optimal closed-form solution ofxcan be obtained as Eq.(8).

Corollary1y(x) achieves the maximum value atx=x*.

ProofDefining the numerator of Eq.(10) as

g(x)=-(mx+n)ln(mx+n)+m(x+s)

(12)

The derivative ofg(x) with respect toxis written as

g′(x)=-mln(mx+n)<0

(13)

Thus,g(x) is a strictly decreasing function.

Ifx*>0, we will easily obtainy′(x)>0(0≤xx*). Therefore,y(x) will reach the maximum value atx=x*. Ifx*=0,y′(x) always has a negative value forx>0 and thusy(x) is a strictly decreasing function and obtains the maximum value atx=x*=0.

Corollary2As a special case whenn=1,x*in Eq.(8) is always positive.

ProofForx>-1/e, the LambertWfunctionW(x) is an increasing function andW(-1/e)=-1. Whenn=1(m>0,s>0), (ms-n)/e>-1/e, and thusW((ms-n)e-1)>-1. Therefore, it can be easily seen thatx*must be positive from Eq.(8).

Considering that the distances between RAiand the MS are different,γimay be different, and thus they can be sorted in descending order as

γ1>γ2>…>γNt

(14)

The Lagrangian duality function of Eq.(6) is constructed as follows

(15)

whereλiandνiare the introduced Lagrange multipliers.

As the constraints of the optimization problem in Eq.(6) are linear, they satisfy linearity constraint qualification[17]. Therefore, the duality gap is zero, which implies that KKT conditions are necessary for optimality[18].

(16)

(17)

(18)

where

(19)

f1>f2>…>fNt

(20)

Lemma2The following conclusions hold for anyj(j=1,…,Nt).

Proof

According to the complementary slackness condition in Eq.(17), a possible set of PA solutions can be divided into three mutually exclusive cases as

(21)

After further derivation of Eq.(21) based on Lemma 2, an adaptive PA scheme is presented, and the corresponding algorithm procedure is summarized as follows:

The conventional iterative PA scheme in Ref.[12] will be used to solve our EE maximizing problem by some extensions since it does not consider the dynamic power consumption and only considers single receive antenna. Then, the complexity comparison of these two schemes is provided.

Table 1 Complexity comparison

3 Simulation Results and Analyses

Table 2 Simulation parameters

Fig.2 EE of DAS with different remote antennas

Fig.3 EE of DAS with different receive antennas

In Fig.3, the EE of DAS with different receive antennas are plotted as a function ofPmax. It can be found that the EE performance of the proposed scheme is almost the same as that of the conventional scheme. Moreover, the EE of the system can increase as the number of receive antennaNrincreases. Namely, the EE withNr=2 performs approximately 8.2% better than that withNr=1, while an extra 4.2% EE gain can be observed forNr=3 compared withNr=2. Because the increase of the receive antennas will bring about more spatial diversity gain. Based on the analysis above, the application of multiple receive antennas does improve the EE performance obviously. These results further indicate that the proposed scheme is valid.

In Fig.4, the EE performances with different schemes and dynamic power consumption factors are compared, whereξ=0, 0.05, 0.1 are considered. From Fig.4, it is found that the proposed scheme exhibits the EE performance very close to the conventional iterative scheme. Besides, for these two schemes, their EEs both decrease as the dynamic circuit power consumption factorξincreases as expected. BecauseηEEis a decreasing function with respect toξ. Thus, it is derived that the system EE underξ=0.1 is lower than that atξ=0.05, and the system EE atξ=0.05 is lower that without dynamic power consumption (ξ=0). The above results show that the proposed scheme is also reasonable.

Fig.4 EE of DAS with different dynamic power consumption factors

Fig.5 EE of DAS with different path loss exponents

Fig.5 shows the EE of DAS with different path loss exponents, where the proposed scheme and the conventional scheme are compared. It is observed that the EE of the system can increase as path loss exponentαidecreases for both two schemes, which accords with the existing knowledge. The reason is that the decrease ofαimeans the decrease of path loss, which reduces the impact on EE. Besides, the proposed scheme can obtain almost the same EE as the conventional scheme.

4 Conclusions

The energy efficiency for DAS with multiple receive antennas in composite Rayleigh fading channel is investigated, and an adaptive energy-efficient PA scheme for downlink DAS is developed. This scheme considers both the static and dynamic parts of circuit power consumptions, and can provides closed-form expression for PA coefficients. Moreover, it has lower complexity than the existing iterative and search schemes due to closed-form calculation and less iteration. Simulation results show that the proposed scheme is valid, and may obtain the energy efficiency close to that of the existing iterative scheme and exhaustive search scheme.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.61571225, 61571224), the Fundamental Research Funds for the Central Universities, the Research Founding of Graduate Innovation Center in NUAA (No.kfjj20160409), the Qing Lan Project of Jiangsu, Shenzhen Strategic Emerging Industry Development Funds (No.JSGG20150331160845693), and the Six Talent Peaks Project in Jiangsu Province(No.DZXX-007).