Transactions of Nanjing University of Aeronautics and Astronautics  2018, Vol. 35 Issue (4): 656-663   PDF    
Adaptive Energy Efficient Power Allocation Scheme for DAS with Multiple Receive Antennas
Wang Ying1, Yu Xiangbin1, Wang Hao1, Chu Junya1, Dong Tao2, Qiu Sainan1     
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
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 Lambert W function, 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 with Nt RAs and Nr receive 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 the ith RA is denoted as RAi. The mobile station (MS) is equipped with Nr antennas. For remote transmit antenna i, the corresponding received signals at MS can be expressed as

Fig. 1 A circle-cell DAS structure

$ {\mathit{\boldsymbol{y}}_i} = \sqrt {{p_i}} {\left[ {h_i^1, \cdots ,h_i^{{N_{\rm{r}}}}} \right]^{\rm{T}}}{\kappa _i} + \mathit{\boldsymbol{z}} $ (1)

where the superscript (·)T denotes the transpose, pi the transmit power consumed by the ith RA, hij the composite fading channel coefficient between RAi and the jth antenna of the MS, κi the transmitted symbol from the ith RA with unit energy, and z the complex Gaussian noise vector with zero-mean and variance N0. hij can be modeled as[14]

$ h_i^j = g_i^j{\mathit{\Omega }_i} $ (2)

where gij represents the small-scale fading between RAi and the jth receive antenna of the MS. For Rayleigh fading channel, {gij} are modeled as independent complex Gaussian random variables with zero-mean and unit variance. ${\mathit{\Omega }_i} = \sqrt {{S_i}d_i^{ - {\alpha _i}}} $ denotes the large-scale fading between RAi and the MS, where αi is the path loss exponent and di the distance from RAi to MS. Si is a log-normal shadow fading variable, i.e., 10log10Si is a Gaussian random variable with zero-mean and standard deviation σi.

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

$ R = {\log _2}\left( {1 + \sum\limits_{i = 1}^{{N_{\rm{t}}}} {{\gamma _i}{p_i}} } \right) $ (3)

where ${\gamma _i} = \sum\limits_{j = 1}^{{N_{\rm{r}}}} {|g_i^j{|^2}\mathit{\Omega }_i^2/{N_0}} $ is defined as the channel to noise ratio (CNR) of RAi after employing maximal ratio combining (MRC) at the receiver. It is assumed that per RA power constraint is 0≤piPmax, i, where Pmax, i is the maximum transmit power available at RAi.

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

$ {\eta _{{\rm{EE}}}} = R/\left( {\sum\limits_{i = 1}^{{N_{\rm{t}}}} {{p_i} + {p_{\rm{c}}}} } \right) $ (4)

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

$ {p_{\rm{c}}} = {p_{\rm{s}}} + \xi R $ (5)

where ps is the static circuit consumption term and ξ a constant denoting dynamic power consumption per unit data rate. Obviously, the constant circuit consumption model used in Refs.[9-13] is a special case that ξ in 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 Lambert W function, 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

$ \begin{array}{*{20}{c}} {\mathop {\max }\limits_p {\eta _{{\rm{EE}}}} = \frac{{{{\log }_2}\left( {1 + \sum\limits_{i = 1}^{{N_{\rm{t}}}} {{\gamma _i}{p_i}} } \right)}}{{\sum\limits_{i = 1}^{{N_{\rm{t}}}} {{p_i} + {p_{\rm{s}}}} + \xi {{\log }_2}\left( {1 + \sum\limits_{i = 1}^{{N_{\rm{t}}}} {{\gamma _i}{p_i}} } \right)}}}\\ {{\rm{s}}.\;{\rm{t}}.\;0 \le {p_i} \le {P_{\max ,i}}\;\;\;\forall i \in \left\{ {1, \cdots ,{N_{\rm{t}}}} \right\}} \end{array} $ (6)

where p=[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.

Lemma 1  For the optimization problem

$ \mathop {\max }\limits_{x \ge 0} y\left( x \right) = \mathop {\max }\limits_{x \ge 0} \frac{{\ln \left( {mx + n} \right)}}{{x + s + \xi \ln \left( {mx + n} \right)}} $ (7)

where m>0, n≥1, s>0, the optimal solution x* is obtained as

$ {x^ * } = {\left[ {\hat x} \right]^ + } $ (8)

where

$ \hat x = {m^{ - 1}}\left[ {\exp \left\{ {W\left( {\left( {ms - n} \right){{\rm{e}}^{ - 1}}} \right) + 1} \right\} - n} \right] $ (9)

where [x]+ represents max(x, 0) and W(x) the Lambert W function which is defined as the reverse function of g(x)=xex[16].

Proof  By taking the derivative of the objective function y(x) with respect to x yields

$ y'\left( x \right) = \frac{{ - \left( {mx + n} \right)\ln \left( {mx + n} \right) + m\left( {x + s} \right)}}{{\left( {mx + n} \right){{\left[ {x + s + \xi \ln \left( {mx + n} \right)} \right]}^2}}} $ (10)

Equating Eq.(10) to zero gives

$ m\left( {x + s} \right) = \left( {mx + n} \right)\ln \left( {mx + n} \right) $ (11)

Using the Lambert W function[16] and considering the non-negativity of x, the optimal closed-form solution of x can be obtained as Eq.(8).

Corollary 1  y(x) achieves the maximum value at x=x*.

Proof  Defining the numerator of Eq.(10) as

$ g\left( x \right) = - \left( {mx + n} \right)\ln \left( {mx + n} \right) + m\left( {x + s} \right) $ (12)

The derivative of g(x) with respect to x is written as

$ g'\left( x \right) = - m\ln \left( {mx + n} \right) < 0 $ (13)

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

If x*>0, we will easily obtain y′(x)>0(0≤x < x*) and y′(x) < 0(x>x*). Therefore, y(x) will reach the maximum value at x=x*. If x*=0, y′(x) always has a negative value for x>0 and thus y(x) is a strictly decreasing function and obtains the maximum value at x=x*=0.

Corollary 2  As a special case when n=1, x* in Eq.(8) is always positive.

Proof  For x>-1/e, the Lambert W function W(x) is an increasing function and W(-1/e)=-1. When n=1(m>0, s>0), (ms-n)/e>-1/e, and thus W(ms-n)e-1>-1. Therefore, it can be easily seen that x* must be positive from Eq.(8).

Considering that the distances between RAi and the MS are different, γi may be different, and thus they can be sorted in descending order as

$ {\gamma _1} > {\gamma _2} > \cdots > {\gamma _{{N_{\rm{t}}}}} $ (14)

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

$ \begin{array}{*{20}{c}} {J\left( {\left\{ {{p_i},{\lambda _i},{v_i}} \right\}} \right) = \frac{{\ln \left( {1 + \sum\limits_{i = 1}^{{N_{\rm{t}}}} {{\gamma _i}{p_i}} } \right)}}{{\sum\limits_{i = 1}^{{N_{\rm{t}}}} {{p_i} + {p_{\rm{s}}}} + \xi \ln \left( {1 + \sum\limits_{i = 1}^{{N_{\rm{t}}}} {{\gamma _i}{p_i}} } \right)}} + }\\ {\sum\limits_{j = 1}^{{N_{\rm{t}}}} {{\lambda _j}{p_j}} + \sum\limits_{j = 1}^{{N_{\rm{t}}}} {{\nu _j}\left( {{P_{\max ,j}} - {p_j}} \right)} } \end{array} $ (15)

where λi and νi are 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].

Using KKT conditions, the optimal values {pi*, λi*, νi*}(i=1, …, Nt) should satisfy the following equations

$ \frac{{\partial J}}{{\partial {p_i}}} = {f_i}\left( {p_1^ * , \cdots ,p_{{N_{\rm{t}}}}^ * } \right) + \lambda _i^ * - \nu _i^ * = 0 $ (16)
$ \lambda _i^ * p_i^ * = \nu _i^ * \left( {{P_{\max ,i}} - p_i^ * } \right) = 0 $ (17)
$ 0 \le p_i^ * \le {P_{\max ,i}},\lambda _i^ * ,\nu _i^ * \ge 0 $ (18)

where

$ \begin{array}{*{20}{c}} {{f_i}\left( {p_1^ * , \cdots ,p_{{N_{\rm{t}}}}^ * } \right) = }\\ {\frac{{{\gamma _i}\left( {\sum\limits_{j = 1}^{{N_{\rm{t}}}} {p_j^ * + {p_{\rm{s}}}} } \right)}}{{\left( {1 + \mathit{\Psi }} \right){{\left[ {\sum\limits_{j = 1}^{{N_{\rm{t}}}} {p_j^ * + {p_{\rm{s}}}} + \zeta \ln \left( {1 + \mathit{\Psi }} \right)} \right]}^2}}} - }\\ {\frac{{\ln \left( {1 + \mathit{\Psi }} \right)}}{{{{\left[ {\sum\limits_{j = 1}^{{N_{\rm{t}}}} {p_j^ * + {p_{\rm{s}}}} + \xi \ln \left( {1 + \mathit{\Psi }} \right)} \right]}^2}}}} \end{array} $ (19)

where $\mathit{\Psi } = \sum\limits_{j = 1}^{{N_{\rm{t}}}} {{\gamma _j}p_j^*} $. For notation simplicity, we rewrite fi(p1*, …, p*Nt) as fi. Substituting Eq.(14) into Eq.(19) leads to

$ {f_1} > {f_2} > \cdots > {f_{{N_{\rm{t}}}}} $ (20)

Lemma 2  The following conclusions hold for any j (j=1, …, Nt).

If fj < 0, pl*=0 (l >j); if fj>0, pk*=Pmax, k(k < j); If fj=0, pl*= Pmax, l (l < j) and pk*=0(k > j).

Proof

If fj < 0, based on Eqs.(16) and (20), fl=-λl*+νl* < 0 holds for l>j. Thus λl*≠0, and then from the complementary slackness condition in Eq.(17), it can be derived pl*=0.

If fj>0, based on Eqs.(16) and (20), fl=-λl*+νl*>0 holds for l < j. Thus νl*≠0, and then from the complementary slackness condition in Eq.(17), it can be derived that pl*=Pmax, l.

If fj=0, we have fl>0 and fk < 0 for l < j and k > j from Eq.(20), respectively. Based on the two conclusions above, it can be easily derived that pl*=Pmax, l (l < j) and pk*=0 (k > j).

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

$ \begin{array}{*{20}{c}} {\left( {p_i^ * ,\lambda _i^ * ,\nu _i^ * } \right) = }\\ {\left\{ {\left( {0,\lambda _i^ * ,0} \right)\left( {x_i^ * ,0,0} \right)\left| {_{0 < x_i^ * < {P_{\max ,i}}}} \right.,\left( {{P_{\max ,i}},0,v_i^ * } \right)} \right\}} \end{array} $ (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:


With the algorithm above, the energy efficient PA coefficients {pi*} can be obtained. Substituting these coefficients into Eq.(4) obtains the corresponding energy efficiency.

Considering that the optimization problem in Eq.(6) is non-convex, the KKT conditions are not sufficient but necessary condition only. Thus, the solution for Eq.(6) by solving the above KKT condition possibly does not give the optimal one, and may be suboptimal one. Namely, the obtained {pi*} may be suboptimal. As a result, the proposed adaptive PA scheme will become suboptimal as well.

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.

When the dynamic power consumption is considered, the EE maximizing problem can still be transformed into standard convex optimization by the method in Ref.[12] based on fractional programming theory, and thus the extended scheme from Ref.[12] is optimal. For this scheme, during the process of computing each pi*, the dual loops are performed, and both the inner and outer loop need O(log(1/ε)) iterations to guarantee the error tolerance of ε. On the other hand, in our proposed scheme, each pi* can be directly computed by Eq.(24) at each stage successively, and the computation stops immediately when pi*Pmax, i is fulfilled. As a result, our scheme has lower complexity, which can also be seen from Table 1, where the average number of iterations of two schemes is compared.

Table 1 Complexity comparison

3 Simulation Results and Analyses

In this section, the validity of the proposed scheme will be evaluated via computer simulation. For convenience, assume Pmax, i=Pmax for ∀i in the simulations. The RAs are uniformly distributed over a circle with radius r. As a result, the polar coordinate of the BS/RA1 is (0, 0), and the polar coordinates of other RAs are (r, $\frac{{2\pi \left( {i - 1} \right)}}{{{N_{\rm{t}}} - 1}}$), i=2, …, Nt with $r = \sqrt {3/7} D$, where D is the radius of the cell. Also assuming the MS in the cell is uniformly distributed. Unless otherwise specified, the main parameters used in simulations are listed in Table 2. The simulation results are obtained through Monte Carlo simulations, and are illustrated in Figs. 2-5, where the "conventional scheme" denotes the iterative power allocation scheme in Ref.[12] after some extensions, and the exhaustive search means examining all possible power allocation combinations with a resolution of 0.01 W.

Table 2 Simulation parameters

Fig. 2 EE of DAS with different remote antennas

Fig. 3 EE of DAS with different receive antennas

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

Fig. 5 EE of DAS with different path loss exponents

Fig. 2 shows the EE of DAS with different PA schemes, where the proposed scheme, conventional scheme, and exhaustive search method are compared. As shown in Fig. 2, the proposed scheme has the EE performance very close to the exhaustive search method and the conventional scheme for different numbers of remote antennas, and the EE gradually improves and is finally saturated as Pmax increases. This is due to the fact that ${{\hat x}_i}$ in Eq.(23) will become smaller than Pmax when Pmax increases, and thus pi* in Eq.(24) does not change any more as the latter increases.Moreover, because of greater space diversity, the EE of the system with more remote antennas is higher than that with fewer remote antennas as expected. In Table 1, the average numbers of iterations of the proposed scheme and the conventional scheme are compared. It is found that the proposed scheme requires less iteration than the latter, which accords with the complexity analysis in Section 2. Namely, our scheme has relatively lower complexity.

In Fig. 3, the EE of DAS with different receive antennas are plotted as a function of Pmax. 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 antenna Nr increases. Namely, the EE with Nr=2 performs approximately 8.2% better than that with Nr=1, while an extra 4.2% EE gain can be observed for Nr=3 compared with Nr=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, where ξ=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 factor ξ increases as expected. Because ηEE is a decreasing function with respect to ξ. Thus, it is derived that the system EE under ξ=0.1 is lower than that at ξ=0.05, and the system EE at ξ=0.05 is lower that without dynamic power consumption (ξ=0). The above results show that the proposed scheme is also reasonable.

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 exponent αi decreases for both two schemes, which accords with the existing knowledge. The reason is that the decrease of αi means 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).

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