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The optimization of the number of deployed antennas in largescale CLDAS for energy efficiency
EURASIP Journal on Wireless Communications and Networking volume 2023, Article number: 96 (2023)
Abstract
We consider the deployment of a largescale circular layoutdistributed antenna system. To maximize energy efficiency, we define the optimization problem with the approximated analysis and find the number of deployed antennas as a closedform solution. In simulation results, we verify that the analysis based on approximation is accurate and the closedform solution can achieve nearoptimal energy efficiency without an exhaustive search method.
1 Introduction
A massive multipleinput multipleoutput (MIMO) system, also known as a largescale MIMO system, is considered a key technology to guarantee the requirements of nextgeneration communications [1, 2]. Especially, a centralized massive MIMO system can improve overall performances including spectral efficiency (SE) and link reliability by adopting a large number of antennas at the base station. However, the spatial correlation of colocated antennas at the base station limits the performance of a massive MIMO system. Moreover, a centralized massive MIMO system is not efficient to provide highquality service for celledge users.
To handle these problems, the distributed antenna system (DAS) was introduced [3,4,5,6,7]. In DAS, the antennas are geometrically separated from each other within a cell and each distributed antenna (DA) is controlled by a central unit (CU) via dedicated channels. Therefore, the average distance between DAs and users is shorter than the centralized MIMO system and DAS can achieve enhanced performance by mitigating the effects of spatial correlation and path loss. Adopting a large number of antennas has advantages in both a centralized MIMO system and DAS, especially, in view of the SE [8]. However, as the energy consumption used by the antennas rapidly increases, the energy efficiency (EE), which is defined as the total data rate divided by the overall power consumption, of the massive MIMO system decreases [9].
The downlink capacity of multicell DAS is analyzed in [3] and a comprehensive overview of DAS in highspeed trains is introduced in [4]. The authors in [5] proposed an antenna selection and user clustering method to maximize the EE of largescale DAS. In [6], the power allocation method for circular layoutDAS (CLDAS) to maximize the EE was proposed. The authors in [7] analyzed the achievable data rate of largescale CLDAS and optimized the location of DAs to maximize the average data rate. Note that the existing studies mainly focused on resource allocation with numerical approaches, and largescale DAS and EE were rarely considered at the same time. In [5], methods for maximizing the EE of largescale DAS were presented, but they were heuristic algorithms without analysis. Moreover, even though an antenna deployment of the largescale DAS significantly affects the EE of the system, not enough study on that subject has been done yet.
In this paper, we optimize an antenna deployment of largescale CLDAS by finding the number of deployed antennas. Especially, the optimization problem to maximize the EE of the system is defined and we perform an analysis based on an approximation to express the EE by the system parameters, explicitly. With the explicit expression, we solve the approximated optimization problem and obtain the number of deployed antennas as a closedform solution. Simulation results show the validity of the approximated analysis. We also demonstrate that the closedform solution of the optimization problem can attain almost the same EE compared with the optimal EE which is obtained by an exhaustive search method.
2 System model
Figure 1 illustrates the architecture of a multiuser downlink largescale CLDAS. The system consists of one CU, M DAs which are perfectly controlled by CU, and \(K\left( { \le M} \right)\) users with one receive antenna each. A CU coordinates the operation of DAs and exchanges limited information between a CU and DAs such as configuration parameters, control parameters, and traffic load information [10]. We assume that DAs are uniformly located on the circle with a radius of r and its center is the same as that of the cell. Moreover, K users are uniformly distributed within the cell coverage radius R and the distance from the center of the cell to the kth user is denoted as \(x_k\). Then, the downlink channel can be expressed as
where \({{{\textbf {G}}}} = {\left[ {{{\textbf {g}}}_1^T \cdots {{\textbf {g}}}_k^T \cdots {{\textbf {g}}}_K^T} \right] ^T}\) denotes a \(K \times M\) channel matrix whose \(\left( k,m\right)\) element \(g_{k,m}\) is a complex channel coefficient associated with the mth DA and the receive antenna of the kth user, \({{{\textbf {g}}}_k}\) is the kth user’s channel vector of \(1 \times M\) size, \({{\textbf {L}}}\) and \({{\textbf {H}}}\) are a largescale fading and a smallscale fading matrix, respectively, and “\(\circ\)” denotes the Hadamard product. Note that \(g_{k,m}=\beta _{k,m}h_{k,m}\) where \(\beta _{k,m}\) and \(h_{k,m}\) are the \(\left( k,m\right)\) elements of \({{\textbf {L}}}\) and \({\textbf {H}}\). We assume that the largescale fading factor \(\beta _{k,m}\) is mainly composed of shadow fading factor \(\zeta\) and path loss factor \(d_{k,m}^\alpha\) where \(\zeta\) is a random variable with lognormal distribution with standard deviation of \(\sigma _{sh}^2\), \({d_{k,m}}\) is the distance between the mth DA and the kth user, and \(\alpha\) is a path loss exponent, respectively. Therefore, \(\beta _{k,m}\) can be modeled as \(\beta _{k,m}=\sqrt{ \frac{\zeta }{d_{k,m}^{\alpha }}}\). In the case of a smallscale fading, \(h_{k,m}\) is zero mean circularly symmetric complex Gaussian random variable [11, 12]. Defining a received signal of kth user as \(y_k\), we can represent a received signal vector of K users, \({{\textbf {y}}} = {\left[ {{y_1} \cdots {y_k} \cdots {y_K}} \right] ^T}\), as
where \({{\textbf {V}}} = \left[ {{{{\textbf {v}}}_1} \cdots {{{\textbf {v}}}_k} \cdots {{{\textbf {v}}}_K}} \right]\) denotes an \({M \times K}\) precoding matrix, \({{{\textbf {v}}}_k}\) is the kth user’s precoding vector of \(M \times 1\) size, \({{\textbf {P}}}\) is a \({K \times K}\) diagonal matrix for power allocation whose kth diagonal term is given by \(\sqrt{P_{k}}\), \({{\textbf {s}}} = {\left[ {{s_1} \cdots {s_k} \cdots {s_K}} \right] ^T}\) is a transmitted signal vector, \(s_{k}\) is the kth user’s transmit symbol, and \({{\textbf {n}}} = {\left[ {{n_1} \cdots {n_k} \cdots {n_K}} \right] ^T}\) is a noise vector. We assume that \(E\left[ {{{\textbf {s}}}{{{\textbf {s}}}^H}} \right] = {{{\textbf {I}}}_K}\) and \(E\left[ {{{\textbf {n}}}{{{\textbf {n}}}^H}} \right] = {N_0}{{{\textbf {I}}}_K}\) where \({{{\textbf {I}}}_K}\) is the identity matrix of size K and \(N_0\) is the noise spectral density. Then, the total data rate of the system, \({R_{tot}}\), is given by
where \(R_{k}\) is the kth user’s data rate and W is the system bandwidth. To define the EE, we assume that the overall power consumption of the system, \(P_{tot}\), is given by
where \({\eta _{PA}}\) is the inefficiency of power amplifier, \({P_T} = \sum \limits _{k = 1}^K {{P_k}}\), \({P_R}\) and \({P_C}\) denote the power consumption of RF chains and circuits, respectively, and \({P_{BH,m}}\) represents the power consumption for the mth backhaul link of the dedicated channels [13, 14]. Then, the EE of the system can be represented as
3 Optimization of the number of deployed antennas for energy efficiency
To maximize the EE, we optimize the antenna deployment of largescale CLDAS. Considering that DAs are uniformly located on the circle, the optimization of the antenna deployment is equivalent to finding the number of DAs. Therefore, the optimization problem can be defined as
where \({M^*}\) denotes the optimal number of deployed antennas. However, taking into account that \({R_{tot}}\) corresponding to the numerator of the EE is not an explicit function of M, we cannot solve the optimization problem in (6), directly. Furthermore, the Hadamard product of \({{{\textbf {G}}}}\) makes it hard to find the closedform solution. Therefore, we will solve the optimization problem with approximation.
3.1 Approximated optimization problem
As already explained, we cannot easily analyze the channel matrix \({\textbf {G}}\) of largescale CLDAS because it is defined by the Hadamard product. To handle this problem, we approximate \({\textbf {G}}={{\textbf {L}}} \circ {\textbf {H}}\) as \({\varvec{{\bar{G}}}} = {\varvec{{\bar{L}}}}{{\textbf {H}}}\) which is based on matrix multiplication. With sufficiently large M, we can assume that an approximated largescale fading matrix \({\varvec{{\bar{L}}}}\) is a diagonal matrix and it can be expressed as [12]
where
r is the radius of largescale CLDAS, \(x_k\) is the distance from the center of the cell to the kth user, and \(\alpha\) is a path loss exponent. Note that \({}_2{F_1}\left( {\mathrm{{ }} \cdot \mathrm{{ }},\mathrm{{ }} \cdot \mathrm{{ }};\mathrm{{ }} \cdot \mathrm{{ }};\mathrm{{ }} \cdot \mathrm{{ }}} \right)\) in (8) is the Euler type of the hypergeometric function and defined as [15]
where \(B\left( {\mathrm{{ }} \cdot ,\mathrm{{ }} \cdot \mathrm{{ }}} \right)\) indicates the beta function. Therefore, assuming \({\varvec{{\bar{G}}}} = {\varvec{{\bar{L}}}}{{\textbf {H}}}\), we can approximate \(R_{tot}\) as
where \({{{{\textbf {h}}}_k}}\) is the kth row vector of \({{\textbf {H}}}\). Compared with (3), \({{{\tilde{R}}}_{tot}}\) is easy to analyze because \({{{{\textbf {g}}}_k}}\) based on the Hadamard product is replaced by \({{{{\textbf {h}}}_k}}\). Utilizing the statistical characteristics of \({{{{\textbf {h}}}_k}}\), we can express (10) in terms of M according to the type of precoder and power allocation.
From now on, to focus on the effect of the antenna deployment, we assume a zeroforcing beamforming (ZFBF) with equal power allocation. Therefore, the precoding matrix \({{\textbf {V}}}\) is given by \({{\textbf {V}}} = {{{\textbf {H}}}^H}{\left( {{{\textbf {H}}}{{{\textbf {H}}}^H}} \right) ^{  1}}{{\textbf {D}}}\) and \(P_k=P_T / K\). Note that the power normalization matrix \({\textbf {D}}\) is a diagonal matrix whose kth diagonal element \(D_k = \frac{1}{{\sqrt{{{\left[ {{{\left( {{{\textbf {H}}}{{{\textbf {H}}}^H}} \right) }^{  1}}} \right] }_{k,k}}} }}\) [16]. Since ZFBF eliminates interuser interference, we only need to calculate the average desired channel gain of ZFBF, \(E\left[ {{{\left {{{{\textbf {h}}}_k}{{{\textbf {v}}}_k}} \right }^2}} \right]\), in (10). Considering that \({\left {{{{\textbf {h}}}_k}{{{\textbf {v}}}_k}} \right ^2} = D_k^2\) is a Chisquare random variable whose distribution is defined by \(f\left( x \right) = \frac{{{e^{  x}}{x^{M  K}}}}{{\left( {M  K} \right) !}}\) [17], by applying the integration by parts, the average desired channel gain of ZFBF, \(E\left[ {{{\left {{{{\textbf {h}}}_k}{{{\textbf {v}}}_k}} \right }^2}} \right]\), can be calculated as
Therefore, we can rewrite \({{{\tilde{R}}}_{tot}}\) as
Finally, approximating \(\beta _{k,avg}^2 \approx \beta _{avg}^2 = \left( {\sum \limits _{k = 1}^K {\beta _{k,avg}^2} } \right) /K\), we can obtain an approximated optimization problem with explicit expression of M as
where
3.2 Closedform solution
In this subsection, we prove that the objective function, \(\mathrm{{E}}{\mathrm{{E}}^{approx}}\), of (13) is concave to M and find the solution as a closedform by using the LambertW function.
First, we define \(\mathrm{{E}}{\mathrm{{E}}^{approx}}\) as a function \(\mathbb \mathcal{{L}}\) of M:
Because the numerator of \(\mathbb \mathcal{{L}}\) is a logarithmic function to M and the denominator of \(\mathbb \mathcal{{L}}\) is a linear function to M, (15) is concave to M. Since the function \(\mathbb \mathcal{{L}}\) of M is concave, we can find a unique global optimal point \(M^*\) of an approximated optimization problem by solving the following equation:
Calculating (16), we can obtain the closedform solution for the number of deployed antenna \(M^*\) as
where \(\kappa = {e^{\lambda + 1}}\), \(\lambda = \mathbb \mathcal{{W}}\left( {\frac{\mu }{e}} \right)\) and \(\mu = \frac{\beta _{avg}^2}{WN_0}\frac{{{P_T}}}{K}\left( {\frac{{{\eta _{PA}}{P_T} + {P_C} +\sum \limits _{m = 1}^M {{P_{BH,m}}}}}{{{P_R}}} + K  1} \right)  1\). Here, \(\mathbb \mathcal{{W}}\) denotes the LambertW function, which is defined as \(\mathbb \mathcal{{W}}\left( z \right) {e^{\mathbb \mathcal{{W}}\left( z \right) }} = z\) where z is any complex number [18]. Considering that the closedform solution (17) is explicitly expressed by K, W, \(P_T\), \(P_R\), \(P_C\), \({P_{BH,m}}\) and other system parameters, we can easily analyze the effect of each system parameter on the number of deployed antennas. Moreover, by substituting (17) into (14), we can obtain the closedform expression of EE. Note that since the closedform solution is optimal for an approximated optimization problem, it cannot guarantee the optimality of the antenna deployment. Therefore, we will verify the accuracy of the closedform solution in the simulation section.
4 Simulation results and discussion
From the simulation results, we verify the validity of the approximation for the analysis and accuracy of the closedform solution. K users are uniformly distributed within the cell coverage and there is the minimum separation distance between a user and a DA, guard distance. Other system parameters used in the simulations are described in Table 1. Note that all the simulation results are obtained by averaging over multiple snapshots.
Figure 2 shows true EE value and approximated EE value based on (5) and (14), respectively, according to the number of deployed antennas, M. We performed simulations for various K, from 5 to 40, and the optimization of the number of deployed antennas is not applied. Comparing true EE value and approximated EE value in the simulations, we can verify that the analysis based on approximation is valid even though the number of deployed antennas is finite. Moreover, we can confirm that (15), a function \(\mathbb \mathcal{{L}}\) of M, is concave as we explained, regardless of the number of users K.
Figure 3 represents the number of deployed antennas over the various number of users and it was obtained by the closedform solution (17) and an exhaustive search method. For the simulation results of an exhaustive search method, we calculated EE for all possible candidates and chose the optimal number of deployed antennas maximizing EE for every channel realization. Compared to an exhaustive search method, the number of deployed antennas based on the analysis differs by a maximum of three, which is only about a 5% difference. This gap between the exhaustive search method and closedform solution comes from the largescale fading approximation and conversion process of a closedform solution (17) to an integer. However, considering the high computational complexity of an exhaustive search method, we can conclude that the closedform solution based on approximation is very effective for the realtime operations of practical systems.
Figure 4 describes the optimal EE by an exhaustive search method and achievable EE by the closedform solution over the various number of users, K. As shown in Fig. 4, utilizing the closedform solution based on the analysis with approximation, we can achieve nearly optimal EE without an exhaustive search method.
5 Conclusion
In this paper, we approximated the function of EE in the largescale CLDAS and optimized an antenna deployment to maximize the EE. Especially, by solving the optimization problem, we found the number of deployed antennas as a closedform solution. Simulation results showed that the analysis based on approximation is valid and nearly optimal EE can be achieved in a closedform solution compared with an exhaustive search method. However, since the CLDAS is not general architecture, the results of this study have a limitation. Therefore, in future works, we will extend the research to a generalized system model, largescale random layoutDAS.
6 Methods/experimental
The purpose of this study is to optimize an antenna deployment of the largescale CLDAS in view of EE. The largescale CLDAS consists of one CU, M DAs, and K users with one receive antenna each. DAs are uniformly located on the circle with a radius of r and users are uniformly distributed within the cell coverage radius R. The channels between DAs and users are modeled based on Hadamard product of largescale fading and smallscale fading. For given system parameters including the number of DAs and users, the energy efficiency can be calculated by utilizing the solution of an approximated optimization problem.
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Abbreviations
 MIMO:

Multipleinput multipleoutput
 SE:

Spectral efficiency
 DAS:

Distributed antenna system
 DA:

Distributed antenna
 CU:

Central unit
 EE:

Energy efficiency
 CLDAS:

Circular layoutdistributed antenna system
 RF:

Radio frequency
 ZFBF:

Zeroforcing beamforming
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Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2020R1A6A1A12047945).
Funding
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2020R1A6A1A12047945).
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Han presented the basic idea, developed the theory, and performed the simulations. Sim verified the analytical methods and wrote the manuscript. All authors discussed the results and approved the final manuscript.
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Han, Y., Sim, D. The optimization of the number of deployed antennas in largescale CLDAS for energy efficiency. J Wireless Com Network 2023, 96 (2023). https://doi.org/10.1186/s13638023023086
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DOI: https://doi.org/10.1186/s13638023023086