Antenna Placement Optimization for Distributed Antenna Systems
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 7, JULY 2012
Antenna Placement Optimization for Distributed Antenna Systems Eunsung Park, Sang-Rim Lee, and Inkyu Lee, Senior Member, IEEE
Abstract—In this paper, we propose new algorithms to determine the antenna location for downlink distributed antenna systems (DASs) in single-cell and two-cell environments. We consider the composite fading channel which includes small and large scale fadings. First, for the single-cell DAS, we formulate the optimization problem of distributed antenna (DA) port locations by maximizing the lower bound of the expected signal to noise ratio (SNR). In comparison to the conventional algorithm based on the squared distance criterion which requires an iterative method, our problem generates a closed form solution. Next, for the two-cell DAS, we propose a gradient ascent algorithm which determines the optimum DA locations by maximizing the lower bound of the expected signal to leakage ratio (SLR). In our work, we consider selection transmission, maximal ratio transmission and zero-forcing beamforming (ZFBF) under sum power constraint and study equal gain transmission and scaled ZFBF under per-antenna power constraint. Simulation results show that our proposed algorithms based on both the SNR and the SLR criteria offer a capacity gain over the conventional centralized antenna systems. Index Terms—Distributed antenna systems, antenna placement.
I. I NTRODUCTION
I
N recent years, distributed antenna systems (DASs) have gained interests because of its ability to extend the cell coverage and increase the system capacity. The DAS was first introduced for indoor wireless communication systems to enhance the coverage [1]. Unlike conventional centralized antenna systems (CASs) where all antennas are co-located at the cell center, distributed antenna (DA) ports in the DAS are separated geographically. Thus, the DAS can reduce the access distance along with the transmit power and cochannel interference, which result in the improved cell-edge performance [2] [3]. Determining the locations of DA ports is an important issue in optimizing the system performance of DAS. Several papers have analyzed the performance of DAS with random or fixed antenna layout. In [4], the outage probability of DAS with the random antenna layout was derived. Also, the authors of [5] studied the capacity of DAS with the random antenna layout in a single-cell and proposed a sub-optimal power allocation scheme. In addition, the asymptotic ergodic capacity Manuscript received April 12, 2011; revised October 4, 2011 and February 17, 2012; accepted April 13, 2012. The associate editor coordinating the review of this paper and approving it for publication was S. Ghassemzadeh. The material in this paper was presented in part at the IEEE VTC, Budapest, Hungary, May 2011. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010- 0017909). The authors are with the School of Electrical Engineering, Korea University, Seoul, Korea (e-mail: {kupes2000, sangrim78, inkyu}@korea.ac.kr). Digital Object Identifier 10.1109/TWC.2012.051712.110670
was presented by applying random matrix theory for DAS with the random and circular antenna layout in [6] and [7], respectively. Furthermore, for DAS, performance analysis was conducted in [8] and transmission schemes based on sum rate analysis were proposed in [9]. Assuming that systems have two DA ports equipped with a single antenna, the optimal antenna layout was determined in [10] for space time block coded orthogonal frequency division multiplexing systems in the linear cell DAS. According to [10], the bit error rate can be minimized when two DA ports are placed symmetrically. For the uplink single-cell DAS, the squared distance criterion (SDC) was proposed in [11] in order to find antenna locations which maximize the lower bound of the cell averaged ergodic capacity. The algorithm based on the SDC is equivalent to a codebook design in vector quantization [12], and thus an iterative method is required. In [13], the SDC was applied to the downlink DAS with selection transmission (ST) where one DA is selected to transmit data to a mobile station (MS). However, this algorithm may not be suitable for DAS in multicell, since it does not take other-cell interference (OCI) into consideration. In [14], DA port location design was suggested for single-cell and two-cell DAS with ST. In this paper, we propose new algorithms to identify the antenna locations for the downlink DAS in single-cell and twocell environments. We study ST, maximal ratio transmission (MRT) [15] and zero-forcing beamforming (ZFBF) [16] under sum power constraint and examine equal gain transmission (EGT) [17] and scaled ZFBF [18] under per-antenna power constraint. It is assumed that DA ports have perfect channel state information between DA ports and MS in its own cell for beamforming schemes. In our study, we consider the composite fading channel which includes both small scale fadings and large scale fadings. First, for the single-cell DAS, we determine the location of DA ports which maximize the lower bound of the expected signal to noise ratio (SNR). This optimization problem is convex, and we can obtain a closed form solution for the DA locations. Next, for the two-cell DAS, we employ a criterion which maximizes the lower bound of the expected signal to leakage ratio (SLR). This SLR criterion is suitable for multi-cell environments, since it takes the leakage interference into consideration. In this case, the optimization problem becomes non-convex, and thus we develop an iterative algorithm which finds the locations of DA ports by applying a gradient ascent method. Simulation results demonstrate that DAS with the antenna locations obtained from the proposed algorithm based on both the SNR and the SLR criteria outperforms conventional CAS. Also, for DAS in the two-cell environment, the system
c 2012 IEEE 1536-1276/12$31.00
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
with the DA locations obtained from the proposed SLR based algorithm shows the higher cell averaged ergodic sum rate than that with DA ports obtained from the SNR criterion. The organization of this paper is as follows: Section II describes a system model and the cell averaged ergodic capacity for DAS in single-cell and two-cell environments. In Section III, we formulate the optimization problems and Section IV proposes new algorithms for antenna location designs. In Section V, we present the simulation results comparing with CAS. Finally, the paper is terminated with conclusions in Section VI. Throughout this paper, bold upper and lower case letters denote matrices and vectors, respectively, and the superscript (·)T , (·)H , (·)† and (·)∗ stand for transpose, Hermitian, pseudo inverse and conjugate, respectively. Also, E(·) represents the expectation and X(:, l) and X(m, :) indicate the l-th column and the m-th row of a matrix X, respectively. II. S YSTEM M ODEL In this section, we describe a system model for the downlink DAS. We assume that each cell has N DA ports and K MSs, and all DA ports and MSs are equipped with a single antenna. Also, it is assumed that all MSs are uniformly distributed. We consider a circular cell with the radius of R with circular antenna layout with or without a center antenna. √ Also, the distance between centers of two cells is set to 3R. Since DA ports are largely separated in DAS, the channel model encompasses not only the small scale fading (i.e. Rayleigh fadings) but also the large scale fading including shadowing and pathloss [2] [19]. (l ,l) (l ,l) (l ,l) (l ,l) with = gk,1 gk,2 · · · gk,N Let us denote gk (l ,l) (l ,l) (l ,l) (l ,l) (l ,l) (l ,l) gk,n = hk,n sk,n (dk,n )−α , where hk,n and sk,n equal the coefficients of the small scale fading and the shadowing fading from the n-th DA port in the l -th cell to the k-th (l ,l) MS in the l-th cell, respectively, dk,n stands for the distance between the n-th DA port in the l -th cell and the k-th MS in the l-th cell, and α is the path loss exponent. The (l ,l) coefficient of the small scale fading hk,n is an independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and unit variance and the shadowing (l ,l) (l ,l) fading sk,n has a log normal distribution, i.e., 10log10 sk,n becomes a Gaussian random variable with zero mean and standard deviation σs . Then, for a multi-user L-cell downlink DAS with a transmit beamforming scheme under sum power constraint, the received signal of the k-th MS in the l-th cell can be expressed (l) (l ,l) (l ) L P x + zk where P represents the as yk = K l =1 gk (l ,l)
total transmitted power in each cell, gk denotes the channel row vector of length N from the DA port in the l -th cell to the k-th MS in the l-th cell, x(l ) stands for the transmitted signal column vector of length N from the DA port in the l -th cell and zk indicates the additive complex Gaussian noise variable with zero mean and variance σz2 . Here, x(l ) is precoded as (l ) (l ) (l ) (l ) K x(l ) = k =1 fk ak where fk and ak are defined by column vector of length N with the k -th MS beamforming (l ) desired signal of the k -th unit norm (||fk || = 1) and the (l ) MS in the l -th cell with E |ak |2 = 1, respectively.
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The beamforming vector fk is determined by a transmission scheme. For the single-user DAS with MRT, the beam(l,l)H (l) g forming vector in the l-th cell is given as f1 = 1(l,l) . Also, g1
for the multi-user DAS with ZFBF, the k-th MS beamforming (l)† (l) (:,k) vector in the l-th cell is obtained as fk = G where ||G(l)† (:,k)|| T (l,l)T (l,l)T (l,l)T g2 · · · gK . G(l) = g1 For the single-user L-cell downlink DAS with the ST scheme where one DA port is selected which has the largest received signal strength (RSS) [4], the received signal (l) of the MS in the l-th cell can be expressed as y1 = √ L (l ,l) (l ) P l =1 g1,n(l ) a1 + z1 where n(l ) denotes the index of the DA port chosen in the l -th cell. Now, we will present the cell averaged ergodic capacity for the DAS under sum power constraint. For the single-cell (SC) DAS, the cell averaged ergodic capacity with ST and MRT, and the cell averaged ergodic sum rate with ZFBF can be written as ⎡ ⎛ ⎞⎤ (1,1) 2 P g 1,n(1) ⎟⎥ ⎢ ⎜ ST = Eh,s,p ⎣log2 ⎝1 + CSC ⎠⎦ , σz2 ⎡
M RT CSC
⎛
⎞⎤ (1,1) 2 P g1 ⎟⎥ ⎢ ⎜ = Eh,s,p ⎣log2 ⎝1 + ⎠⎦ , σz2 ⎡
ZF BF RSC
⎛ K ⎢ ⎜ = Eh,s,p ⎣ log2 ⎝1 + k=1
P K
⎞⎤ (1,1) (1) 2 gk fk ⎟⎥ ⎠⎦ (1) σz2
where h, s and p account for the small scale fading, the shadowing fading and the MS position, repectively. Also, for the two-cell (T C) DAS, the cell averaged ergodic sum rates with ST, MRT and ZFBF can be represented as ⎡ ⎛ ⎞⎤ (l,l) 2 2 P g 1,n(l) ⎢ ⎜ ⎟⎥ log2 ⎝1 + RTSTC =Eh,s,p ⎣ ¯ 2 ⎠⎦ , ( l,l) l=1 σz2 + P g1,n(¯l) ⎡ ⎛ ⎞⎤ (l,l) 2 2 P g 1 ⎢ ⎜ ⎟⎥ log2 ⎝1 + RTMCRT =Eh,s,p ⎣ ¯ ¯ 2 ⎠⎦ , (l,l) (l) l=1 σz2 + P g1 f1 ⎡ ⎛ ⎞⎤ 2 P (l,l) (l) 2 K f g k k K ⎢ ⎜ ⎟⎥ RTZFC BF =Eh,s,p ⎣ log2⎝1+ ¯ ¯ 2⎠⎦(2) ( l,l) ( l) K P l=1 k=1 σz2+ k =1K gk fk where ¯l = 1 if l = 2 and ¯l = 2 if l = 1. Next, we describe a beamforming scheme for DAS under per-antenna power constraint. In this case, we assume that each antenna has separate power constraint P . For multiple-input single-output systems with per-antenna power constraint, EGT is known as the optimal beamforming scheme in terms of the capacity [17], and thus we employ EGT for the single-user DAS. Let us denote (l,l) (l,l) (l,l) (l,l) (l,l) (l,l) (l,l) g1 = g1,1 ej∠g1,1 g1,2 ej∠g1,2 · · · g1,N ej∠g1,N (l,l) (l,l) where g1,n and ∠g1,n indicate the magnitude and the (l,l)
phase, respectively, for the channel coefficient g1,n . Then,
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the EGT beamforming vector can be given as f1 = √ j∠g(l,l) j∠g(l,l) (l,l) H j∠g1,N 1,1 1,2 P e e ··· e . For the muti-user case, the optimal beamforming matrix in terms of the cell averaged ergodic sum rate can be obtained by an iterative method [22]. However, it is complicated to derive a lower bound of the expected SNR and SLR since we cannot compute a closed-form solution. Thus, we consider scaled ZFBF [18] which supports K MSs for a simple derivation. (l) Defining F(l) as the beamforming √ matrix in the l-th cell, F P (l) (l)† can be given as F = max G . (l)† (n,:) | 1≤n≤N |G III. P ROBLEM F ORMULATION In this section, we formulate a problem in order to optimize the antenna locations for the downlink DAS in single-cell and two-cell. In general, it is quite complicated to derive a solution which maximizes the cell averaged ergodic capacity and sumrate (1) and (2) in the previous section. Instead, to simplify the analysis, we attempt to maximize the lower bound of the expected SNR and SLR, for the single-cell case and the twocell case, respectively. A. Single-Cell DAS ST M RT , CSC and In (1), applying the Jensen’s inequality, CSC ZF BF RSC are bounded by ⎛ ⎡ ⎤⎞ (1,1) 2 P g 1,n(1) ⎥⎟ ⎜ ⎢ ST CSC ≤ log2 ⎝1 + Eh,s,p ⎣ ⎦⎠ , σz2 ⎛
M RT CSC
⎡
⎤⎞ (1,1) 2 ⎜ ⎢ P g1 ⎥⎟ ≤ log2 ⎝1 + Eh,s,p ⎣ ⎦⎠ , σz2 ⎛
ZF BF SC
R
≤
K k=1
⎡ P
⎜ ⎢K log2 ⎝1 + Eh,s,p ⎣
⎤⎞ (1,1) (1) 2 gk fk ⎥⎟ ⎦⎠ .(3) σz2
Thus, we will maximize the expected SNR in order to come up with the DA locations for the single-cell DAS. We assume that a cell is divided into N regions with the same physical area. Also we consider circular antenna layout with or without a center antenna. Then, the i-th DA location (i = 1, · · · , N ) can be expressed as DAi = Ri exp (jθi ) for Ra,i ≤ Ri ≤ Rb,i , θa,i ≤ θi ≤ θb,i where Ri and θi are the magnitude and the phase of the i-th DA, Ra,i and Rb,i denote the minimum and the maximum value of the magnitude, and θa,i and θb,i represent the minimum and the maximum value of the phase in the i-th DA region, respectively. For example, when N = 7, for the layout with a center antenna, the parameters for the√region of the center antenna equal Ra,i = 0, Rb,i = R/ 7, θa,i = 0 and θb,i√= 2π, while those for other antennas are set to Ra,i = R/ 7, Rb,i = R and θb,i − θa,i = π/3. On the other hand, the regions of antennas for the layout without a center antenna have Ra,i = 0, Rb,i = R and θb,i − θa,i = 2π/7. Now, we introduce the optimization problem which determines the DA locations for DAS with ST, MRT and ZFBF. Then, we will show thatthe lower bound of the expected SNR 2 (1,1) for all cases. is related to Ep d1,i
1) Single-user DAS with ST and MRT: For MRT, the expected SNR is obtained as ⎡ ⎡ ⎤ ⎤ (1,1) (1) 2 (1,1) 2 ⎢ P g1 f1 ⎥ ⎢ P g1 ⎥ Eh,s,p [SNR]=Eh,s,p⎣ ⎦=Eh,s,p⎣ ⎦ 2 σz σz2 ⎡
2 ⎤ ⎡ ⎤ N s(1,1) h(1,1) N 1,n 1,n 1 P ⎢ ⎥ P β ⎣ α ⎦≥ 2 Ep α ⎦ (4) = 2 Eh,s,p⎣ (1,1) (1,1) σz σz d n=1 n=1 d 1,n
1,n
2 2 2 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) where β= min s1,1 h1,1 , s1,2 h1,2 ,· · ·, s1,N h1,N . Assuming that MS is located in the i-th DA region, and we focus on the distance MS and the i-th DA port for α2 1 between is convex for α ≥ 0, by applying a derivation. Since X the Jensen’s inequality, we compute the lower bound of Eh,s,p [SNR] as Pβ σz2
Eh,s,p [SNR] ≥
1 . 2 α/2 (1,1) Ep d1,i
(5)
Next, for ST, without loss of generality, we assume that the i-th DA port is chosen for transmission. Then, we can consider (1) that the i-th element of f1 is 1 and the other elements are all set to 0. Thus, regarding ST as a special case of MRT, we can formulate the lower bound of Eh,s,p [SNR] for ST as 2 1 P (1,1) (1,1) Eh,s,p [SNR] ≥ 2 Eh,s s1,i h1,i . 2 α/2 σz (1,1) Ep d1,i In (6), we can see that Eh,s
(6) 2 (1,1) (1,1) s1,i h1,i is a constant value
since it does not depend on the MS and DA locations. 2) Multi-user DAS with ZFBF: In (3), in order to enhance ZF BF , we should maximize all the expected SNRs of MSs. RSC However, we can determine the DA locations by maximizing the expected SNR of only one MS since the optimizing problem is equivalent for all MSs. Thus, without loss of generality, we consider only MS 1 for the formulation. Then, the expected SNR of MS 1 is written as P (1,1) (1) 2 f1 Eh,s,p [SNR1 ] = Eh,s,p g Kσz2 1 ⎡ 2 ⎤ N P (1,1) (1) ⎦ ⎣ (7) = E g f h,s,p 1,n n,1 Kσz2 n=1
(1)
(1)
where fn,1is the n-th element of f1 . (1,1) (1,1)∗ = 0 for i = j, Eh,s,p [SNR1 ] can be Since E g1,i g1,j represented as N P (1,1) (1) 2 Eh,s,p [SNR1 ] = Eh,s,p g1,n fn,1 Kσz2 n=1 ⎡ ⎤ ⎡ ⎤ (1,1) N N s1,n (1,1) (1) 2 P β 1 P α h1,n fn,1 ⎦≥ α⎦ Eh,s,p⎣ Ep⎣ = 2 (1,1) (1,1) Kσz2n=1 Kσ z d d n=1 1,n
1,n
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
2 2 (1,1) (1,1) (1) (1,1) (1,1) (1) where β = min s1,1 h1,1 f1,1 , s1,2 h1,2 f2,1 , · · · , 2 (1,1) (1,1) (1) s1,N h1,N fN,1 . Similar to the MRT case, assuming that MS 1 is located in the i-th DA region and applying the Jensen’s inequality, the lower bound of Eh,s,p [SNR1 ] can be written as 1 P β . (8) Eh,s,p [SNR1 ] ≥ Kσz2 (1,1) 2 α/2 Ep d1,i From (5), (6) and (8), the lower bounds of the expected SNR 2 −α/2 (1,1) . for all ST, MRT and ZFBF are related to Ep d1,i Thus, the i-th DA can be optimized by minimizing location 2 (1,1) (1,1) for DAS. Here, the distance d1,i the value of Ep d1,i (1,1) can be obtained as d1,i = Ri2 + r12 − 2Ri r1 cos (φ1 − θi ) where r1 and φ1 indicate the magnitude and of the the phase 2 (1,1) i MS position. Finally, defining ΓSC as Ep d1,i , we can formulate the antenna location problem for single-cell as Rˆi , θˆi for i = 1, · · · , N (9) = arg min ΓiSC {Ri ,θi }
Ra,i ≤ Ri ≤ Rb,i , θa,i ≤ θi ≤ θb,i .
subject to
Problem (9) involves minimization of the expectation of the squared distance between a DA port and an MS. Intuitively, if the location of an MS is fixed, all DA ports should be closely located to an MS in order to maximize performance. However, we assume that an MS is uniformly distributed, not fixed in a cell, and thus our antenna location problem (9) is formulated in an average sense. B. Two-Cell DAS Similar to the single-cell case, in order to optimize the cell averaged ergodic sum rate for the two-cell DAS, we will maximize the expected signal to interference plus noise ratio (SINR). However, this criterion is still complicated due to the coupled nature of the corresponding metric. Thus, we adopt an alternative approach based on the expected SLR which considers the leakage interference and offers simple analysis. 1) Single-user DAS with ST and MRT: For the single-user two-cell DAS with ST and MRT, the cell averaged ergodic sum rate can be approximated at high SNR as 2 2 (l) ST , M RT ≈E =E log2 1+SINR log2 SINR(l) RT C l=1
(l,l) 2
since SLNR(l) ≈ SLR(l) where SLR(l) is given by (l,l) (l,¯ l) (l,l) (l,¯ l) (l) |g1,n(l) |2 /|g1,n(l) |2 for ST and g1 2 /|g1 f1 |2 for MRT. Finally, since log2 X is concave for X > 0, the upper bound of RTSTC, M RT can be expressed as RTSTC, M RT ≤ 2 (l) . Thus, we attempt to enhance the cell l=1 log2 E SLR averaged ergodic sum rate by maximizing the expected SLR. In MRT, the expected SLR at cell 1 is given as ⎡ " " ⎤ " (1,1) "2 " ⎥ "g 1 ⎢ Eh,s,p1 ,p2 SLR(1) = Eh,s,p1 ,p2 ⎣ ⎦ (1,2) (1) 2 g1 f1 ⎡ ⎤ N (1,1) 2 g n=1 1,n ⎢ ⎥ = Eh,s,p1 ,p2 ⎣ 2 ⎦ N (1,2) (1) n=1 g1,n fn,1 ⎡ ⎤ N (1,1) 2 g n=1 1,n ⎢ ⎥ ≥ Eh,s,p1 ,p2 ⎣ ⎦ (1,2) 2 N (1) 2 N n=1 g1,n n=1 fn,1 ⎡ ⎤ N (1,1) 2 1 ⎢ ⎥ = Eh,s,p1 g1,n Eh,s,p2⎣ 2 ⎦(10) (1,2) N n=1 n=1 g1,n where Ep1 and Ep2 denote the expectation with respect to the MS positions in cell 1 and 2, respectively, and the inequality follows from the Cauchy-Schwarz inequality. Let us define β and γ as 2 2 2 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) β min s1,1 h1,1 , s1,2 h1,2 , · · ·, s1,N h1,N 2 2 2 (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) γ max s1,1 h1,1 , s1,2 h1,2 , · · ·, s1,N h1,N . is Eh,s,p1 ,p2 SLR(1) N β 1 α Ep N (1,1) 2 n=1 γ Ep1
lower
Then,
d1,n
n=1
bounded by Assuming (1,2) α .
1 1/ d1,n
that the MS is located in the i-th and applying the DA region Jensen’s inequality, Eh,s,p1 ,p2 SLR(1) can be written as E h,s,p1 ,p2 SLR(1) ≥
β γ
Ep1
1 (1,1)
d1,i
⎡ 2 α/2
Ep2 ⎣
⎤ N n=1
1 α ⎦ . (1,2) 1/ d1,n
l=1
where the superscript l indicates the cell index, and SINR(l) (l,l) (¯ l,l) is equal to P |g1,n(l) |2 /(σz2 + P |g1,n(¯l) |2 ) for ST and (¯ l,l) (¯ l)
/(σz2 +!P |g1 f1 |2 ) for MRT. Using the fact that (l) = 2l=1 SLNR(l) where SLNR(l) is denoted by l=1 SINR (l,l) (l,¯ l) (l,l) P |g1,n(l) |2 /(σz2 + P |g1,n(l) |2 ) for ST and P g1 2 /(σz2 + (l,¯ l) (l) 2 (l) is equal to P |g1 f1 |2 ) for MRT, l=1 log2 SINR 2 (l) . l=1 log2 SLNR ST , M RT Assuming the interference limited regime, R is TC 2 (l) ST , M RT approximated as RT C ≈ E l=1 log2 SLR P g1 ! 2
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To simplify the derivation, we assume that all DA ports in cell 2 are equi-distant from the MS 1. Then, we in cell α (1,2) 1 1 . to N Ep2 d1,i can approximate Ep2 N (1,2) α n=1 1/ d1,n Finally, it follows that Eh,s,p1 ,p2 SLR(1) for MRT is bounded by α (1,2) d E p2 1,i β Eh,s,p1 ,p2 SLR(1) ≥ . (11) 2 α/2 γ N (1,1) Ep1 d1,i
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For ST, similar to the single-cell case, we can formulate the lower bound of Eh,s,p1 ,p2 SLR(1) as ⎡
⎤
2 α (1,1) (1,1) (1,2) d s E h p 1,i 1,i ⎢ 1,i ⎥ 2 Eh,s,p1 ,p2 SLR(1) ≥Eh,s⎣ . 2⎦ 2α/2 (1,2) (1,2) s1,i h1,i E d(1,1) p1 1,i Here, we can see that Eh,s
(1,1) (1,1) 2 s1,i h1,i (1,2) (1,2) 2 s1,i h1,i
(12)
≈
Eh,s,p
is a constant value.
(l) log2 SINRk
l=1 k=1
= ≈
Eh,s,p Eh,s,p
2 K
l=1 k=1 2 K
(l) log2 SLNRk (l) log2 SLRk
l=1 k=1
(l)
need to maximize the expected SLRk for k = 1, · · · , K and l = 1, 2 in order to enhance the cell averaged ergodic sum rate. However, similar to the single-cell ZFBF case, we can determine the DA locations in the l-th cell by maximizing the expected SLR of only one MS. Thus, without loss of generality, we consider only MS1 for the optimizing problem. (1,1) (1,1)∗ = 0 for i = j, the expected At cell 1, since E g1,i g1,j (1)
⎢ ⎢ 1 (1,1) (1) 2 ⎢ Eh,s,p1 g1,n fn,1 Eh,s,p2⎢ K N ≥ N ⎢ (1,2) 2 (1) n=1 ⎣ g f
is computed as
⎡ 2 ⎤ N (1,1) (1) g f ⎢ n=1 1,n n,1 ⎥ (1) E h,s,p1 ,p2 SLR1 = Eh,s,p1 ,p2 ⎣ ⎦ (1,2) (2) 2 K g f 1 k =1 k ⎡ ⎤ ⎡ ⎤ 2 N 1 ⎢ ⎥ (1,1) (1) = Eh,s,p1 ⎣ g1,n fn,1 ⎦ Eh,s,p2 ⎣ 2 ⎦ (1,2) (1) K n=1 fk k =1 g1 ⎡ ⎤ N (1,1) (1) 2 1 ⎢ ⎥ = Eh,s,p1 g1,n fn,1 Eh,s,p2 ⎣ 2 ⎦ . (1,2) (1) K n=1 fk k =1 g1 (1) Applying the Cauchy-Schwarz inequality, Eh,s,p1,p2 SLR1
⎥ ⎥ ⎥ ⎥ 2⎥ ⎦ n,k
N
1,n
n=1
⎤ N 2 1 (1,1) (1) ⎢ ⎥ = Eh,s,p1 g1,n fn,1 Eh,s,p2⎣ 2 ⎦ . (13) (1,2) n=1 K N n=1 g1,n Defining β and γ as β 2 2 2 (1,1) (1,1) (1) (1,1) (1,1) (1) (1,1) (1,1) (1) min s1,1 h1,1 f1,1 , s1,2 h1,2 f2,1 , · · ·, s1,N h1,N fN,1 2 2 2 (1,2) (1,2) (1,2) (1,2) (1,2) (1,1) γ max s1,1 h1,1 , s1,2 h1,2 , · · ·, s1,N h1,N , (1) the lower bound of Eh,s,p1 ,p2 SLR1 can be expressed as (1) E h,s,p1 ,p2 SLR1 ⎡ ⎤ ⎡ N β 1 α ⎦ Ep2 ⎣ ≥ Ep ⎣ (1,1) N γ K n=1 1 d 1,n
(l) (l,l) (l) (¯ l,l) (¯ l) 2 P P K where SINRk = K |gk fk |2 /(σz2 + K k =1 |gk fk | ), ¯ (l) (l,l) (l) (l,l) (l) 2 K P P SLNRk = K |gk fk |2 /(σz2 + K k =1 |gk fk | ) and ¯ (l) (l,l) (l) (l,l) (l) K SLRk = |gk fk |2 / k =1 |gk fk |2 . ZF BF By applying the Jensen’s inequality, RT C is bounded (l) 2 K . Thus, we by RTZFC BF ≤ l=1 k=1 log2 Eh,s,p SLRk
SLR1
⎤
⎡
(l)
RTZFC BF
⎡
k =1n=1
2) Multi-user DAS with ZFBF: Let us define SLRk as the ratio of the signal power of the k-th MS in the l-th cell to the leakage power from all MSs in the l-th cell to the k-th MS (l) (l) in the ¯ l-th cell. SINRk and SLNRk are similarly defined. Similar to the single-user case, assuming high SNR and the interference limited regime, RTZFC BF is given as 2 K
is represented as (1) E h,s,p1 ,p2 SLR1
⎤ 1 α ⎦ . (1,2) n=1 1/ d1,n
Again we assume that MS 1 at cell 1 is located in the ith DA region. Then, by applying theJensen’s inequality and α (1,2) 1 approximating Ep2 N to N1 Ep2 d1,i (1,2) α n=1 1/ d1,n (1) for simple analysis, the lower bound of Eh,s,p1 ,p2 SLR1 can be written as α (1,2) Ep2 d1,i β (1) Eh,s,p1 ,p2 SLR1 ≥ . (14) 2 α/2 γ KN (1,1) Ep1 d1,i From (11), (12) and (14), we can see that (1) Eh,s,p1 ,p2 SLR1 for all ST, MRT, and ZFBF are related to Υ
(1,2) α Ep2 d1,i α/2 . (1,1) 2 Ep1 d1,i
Thus, the DA location optimization
of ST, MRT and ZFBF for the i-th DA at cell 1 is equivalent to maximizing Υ. Note that the cost function Υ cannot be computed for α > 2 in general. Thus, we change the cost function to ln Υ in order to make it easy to deal with an arbitrary path loss exponent α. Again applying the Jensen’s inequality, the lower bound of ln Υ can be given as α 2 α/2 (1,2) (1,1) − ln Ep1 d1,i ln Υ = ln Ep2 d1,i 2 α/2 2 α/2 (1,2) (1,1) d1,i ≥ Ep2 ln − ln Ep1 d1,i =
α 2
2 2 (1,2) (1,1) − ln Ep1 d1,i . Ep2 ln d1,i
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS (1,2)
(1,1)
Defining ΓiT C as Ep2 [ln(d1,i )2 ] − ln Ep1 [(d1,i )2 ], the problem for i-th DA is now independent of α and can be reformulated as = arg max ΓiT C for i = 1, · · · , N (15) Rˆi , θˆi {Ri ,θi }
subject to
Ra,i ≤ Ri ≤ Rb,i , θa,i ≤ θi ≤ θb,i . (1,2)
Here, the distance d1,i
is calculated as
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bounded by
⎡ 2 ⎤ N (1,1) (1) g f n,1 ⎥ n=1 1,n ⎢ (1) E h,s,p1 ,p2 SLR1 = Eh,s,p1 ,p2 ⎣ ⎦ (1,2) (2) 2 K g f 1 k k =1 ⎡ ⎤ N 1 1 (1,1) (1) 2 ⎢ ⎥ ≥ Eh,s,p1 g1,n fn,1 Eh,s,p2⎣ 2⎦ (.17) N P Kn=1 N (1,2) n=1 g1,n
Note that (16) and (17) have the same form with (7) and (13), √ (1,2) d1,i = Ri2+r22+3R2−2Ri r2 cos(φ2−θi)+2 3R(r2cosφ2−Ri cosθi ) respectively. Thus, we confirm that the problem which determines the DA locations for DAS under per-antenna power constraint is where r2 and φ2 indicate the magnitude and the phase of the equivalent to the case with sum power constraint. MS position in cell 2. Problem (15) identifies the location of the i-th DA port at IV. A NTENNA L OCATION OPTIMIZATION cell 1 in an average sense. Note that the i-th DA port at cell 1 causes interference to MSs at cell 2. Thus, in contrast to the In this section, we introduce new algorithms to solve (9) single-cell case, an interference term is considered to enhance and (15). First, we propose a non-iterative algorithm which performance for the two-cell case. maximizes the lower bound of the expected SNR and present a layout selection method in single-cell. Then, in two-cell, an iterative algorithm which maximizes the lower bound of the expected SLR is illustrated. C. Per-Antenna Power Constraint For DAS, per-antenna power constraint may be more realistic than sum power constraint in practice. In this section, we formulate a problem for the DA locations under per-antenna power constraint. First, we present the single-user DAS with EGT. For the single-cell case, Eh,s,p [SNR] is bounded by Eh,s,p [SNR]
=
≥
⎡# $ ⎤ N 2 P (1,1) ⎦ Eh,s,p ⎣ g1,n σz2 n=1 N (1,1) 2 P E g1,n h,s,p σz2 n=1
where P denotes a power constraint for each antenna. Also, for the two-cell DAS, by applying the Cauchy-Schwarz inequality, the bound of Eh,s,p SLR(1) can be represented as
E h,s,p1 ,p2 SLR(1)
⎡
⎤ (1,1) 2 g1,n ⎥ ⎦ (1,2) (1) 2 g1 f1 ⎡ ⎤
⎢P = Eh,s,p1 ,p2 ⎣
N n=1
N (1,1) 2 1 ⎢ ≥ Eh,s,p1 g1,n Eh,s,p2 ⎣ N N n=1
1 ⎥ ⎦. (1,2) 2 n=1 g1,n
Thus, for DAS with EGT, we can obtain the same form as DAS with MRT in (4) and (10). For the muti-user DAS with scaled ZFBF in single-cell, Eh,s,p [SNR1 ] can be written as ⎡ 2 ⎤ N 1 (1,1) (1) g1,n fn,1 ⎦ . Eh,s,p [SNR1 ] = 2 Eh,s,p ⎣ σz
(16)
n=1
(1) Also, for the two-cell case, Eh,s,p1 ,p2 SLR1 can be lower-
A. Single-Cell DAS In [11] and [13], the SDC algorithm for antenna location designs was studied in the uplink and downlink single-cell DAS. The SDC algorithm tries to optimize the cell averaged ergodic capacity by minimizing the expectation of the squared distance from a randomly distributed MS to the nearest antenna port. This criterion is similar to a codebook design in vector quantization [12], and needs an iterative method. In order to reduce the computational complexity, we propose a new algorithm which does not require an iterative procedure. In (9), since the MS is uniformly distributed in the i-th DA region, the cost function can be computed by % Rb,i% θb,i 2 2 1 (1,1) d1,i r1 2 dφ1 dr1 ΓiSC = 2 θ R − R − θa,i Ra,i θa,i a,i b,i b,i 2 R 2 2Ri 3 2 3 Rb,i −Ra,i (θb,i −θa,i )− · = A i Rb,i −Ra,i 2 3 4 4 Rb,i −Ra,i (θb,i −θa,i ) (18) (sin(θb,i−θi )−sin(θa,i−θi ))+ 4 2 2 . − Ra,i where A = 2/ (θb,i − θa,i ) Rb,i Then, the first and second partial derivatives of ΓiSC are expressed as 2 2 3 i 2 3 ∇Ri ΓSC =A Ri Rb,i (θb,i − θa,i )− Rb,i · − Ra,i − Ra,i 3 (sin (θb,i − θi ) − sin (θa,i − θi )) 3 2 3 ∇θi ΓiSC = ARi Rb,i −Ra,i (cos (θb,i −θi )−cos (θa,i −θi )) 3 2 i 2 ∂ ΓSC 2 (θb,i − θa,i ) =A Rb,i − Ra,i 2 ∂Ri 3 ∂ 2 ΓiSC 2 3 (sin (θb,i −θi )+sin (θi −θa,i )) . = ARi Rb,i −Ra,i ∂θi2 3
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 7, JULY 2012 ∂ 2 Γi
&
cos θb,i − cos θa,i sin θa,i − sin θb,i
' .
B. Two-Cell DAS Since the conventional SDC algorithm does not take the OCI into account, the DA locations from the SDC algorithm may not be suitable for the two-cell DAS. In this subsection, we propose an iterative algorithm which maximizes the lower bound of the expected SLR. Since the MSs in cell 2 are 2 (1,2) uniformly distributed, we can calculate Ep2 ln d1,i similar to the single-cell case as 2 1 % R % 2π 2 (1,2) (1,2) Ep2 ln d1,i r2 ln d1,i dφ2 dr2 . (21) = 2 πR 0 0 √ ( 2π 2 −b2 Using 0 ln (1 + a cos φ + b sin φ) dφ = 2π ln 1+ 1−a 2 for 1 − a2 − b2 ≥ 0 [20], (21) is computed as 2 γ2 (1,2) ln R2 − 1 + R for γ < R 2 Ep2 ln d1,i = 2 for γ ≥ R ln γ √ Ri2 + 3R2 − 2 3RRi cos θi . Then, the cost where γ = function ΓiT C in (15) can be represented as γ2 i ln R2 − 1 + R for γ < R i 2 − ln ΓSC (22) ΓT C = 2 i for γ ≥ R. ln γ − ln ΓSC Here, γ is √ greater than or √ equal to R if 30◦ ≤ θi ≤ 330◦ and 0 ≤ Ri ≤ 3R cos θi − R 3 cos2 θi − 2. Otherwise, we have γ < R. Due to non-convexity of ΓiT C , we identify the local optimal DA locations by applying a gradient ascent algorithm which iteratively updates the DA locations in the direction of the gradient of the cost function. Then, the gradients of the cost function with respect to Ri and θi are derived as √ 2 i i 2 Ri− 3R cos θi −∇Ri ΓSC /ΓSC for γ < R i R √ ∇Ri ΓT C = 2 i i γ 2 Ri− 3R cos θi −∇Ri ΓSC /ΓSC for γ ≥ R ∇θi ΓiT C
=
√ 2√3 RRi sin θi − ∇θi ΓiSC /ΓiSC i i i 2 3 RR γ 2 sin θi − ∇θi ΓSC /ΓSC
for γ < R for γ ≥ R.
With the derived gradient expressions, our algorithm which solves (15) is summarized as follows:
−0.5 −1
0.5 0 −0.5 −1
0.5 0 −0.5 −1 −0.5 0 0.5 1 Position in x axis
Fig. 1.
1 0.5 0 −0.5 −1
−0.5 0 0.5 1 Position in x axis
−0.5 0 0.5 1 Position in x axis
N=7
N=6 1
N=5 Position in y axis
Position in y axis
Position in y axis
0
−0.5 0 0.5 1 Position in x axis
(20)
Finally, comparing circular antenna layouts with and without a center antenna, we choose the one which has a larger value N −α/2 of Ω = i=1 ΓiSC .
0.5
1
1
N=8 Position in y axis
θˆi = arctan
1
Position in y axis
and
N=4
N=3
Position in y axis
∂ 2 Γi
We simply prove that ∂RSC and ∂θSC are always non2 2 i i negative, and thus the cost function is convex with respect to Ri and θi . Therefore, the minimum value of (18) is obtained at either the critical points of ∇Ri ΓiSC = 0 and ∇θi ΓiSC = 0, or the boundary points. As a result, for the single-cell DAS, a closed form solution for the i-th DA location can be obtained by 3 3 2 Rb,i − Ra,i sin θb,i − θˆi + sin θˆi − θa,i Rˆi = (19) 2 − R2 3 Rb,i a,i (θb,i − θa,i )
0.5 0 −0.5 −1
1 0.5 0 −0.5 −1
−0.5 0 0.5 1 Position in x axis
−0.5 0 0.5 1 Position in x axis
Locations of DA ports for DAS in single-cell
Initialization: 1) Initialize Ri and θi for the i-th DA in cell 1 as Ra,i ≤ Ri ≤ Rb,i and θa,i ≤ θi ≤ θb,i Main Loop: 2) Update Ri ← Ri +δRi·∇Ri ΓiT C and θi ← θi +δθi·∇θi ΓiT C 3) Calculate ΓiT C with the updated values 4) Go back to 2) until convergence In [21], several line search methods were presented for the selection of the step size δRi and δθi . In our algorithm, we employ Armijo’s rule [21] which provides provable convergence. Using the Armijo’s rule, we can obtain a non-decreasing ΓiT C value with respect to the number of iterations. After the DA locations are computed for cell 1, the DA locations at cell 2 can be simply determined by using the rotational symmetric property between cell 1 and 2. V. S IMULATION R ESULTS In this section, simulation results are presented to demonstrate the efficacy of our proposed algorithms. In the simulation, the cell radius, the standard deviation for shadowing and the path loss exponent are set to R = 1, σsh = 4 dB and α = 3.75, respectively. Also, we define SNR as P/σz2 . First, we present some simulation results under sum power constraint. For a fair comparison between DAS and CAS, the total transmitted power in each cell for DAS and CAS is set to P . In Figure 1, we first plot the locations of DA ports obtained from the proposed SNR based algorithm for DAS in single-cell with various numbers of DA ports. In the single-cell DAS for N = 3, 4, · · · , 8, the locations of DA ports are calculated from (19) and (20). In the figure, the dots and the asterisks indicate the locations of DA ports from the proposed algorithm for the layout with a center antenna and without a center antenna, respectively. In Table I, the values of Ω are listed for both layouts. The table shows that for N = 3, 4, and 5, the circular layout without a center antenna is preferred, while when N = 6, 7, and 8, a center DA port is helpful in terms of the expected SNR. The results of our proposed algorithm are almost the same as the conventional SDC case in [13]. Note that the SDC
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
TABLE I Ω FOR DIFFERENT CIRCULAR ANTENNA LAYOUTS Ω w/ a center antenna 39.8 104.7 185.0 503.4 861.4 1299.5
Cell Averaged Ergodic Sum Rate for DAS and CAS at SNR=20dB 40
Ω w/o a center antenna 63.7 160.3 307.8 498.3 718.3 958.2
3 MSs Cell Averaged Ergodic Sum Rate [bps/Hz]
N 3 4 5 6 7 8
Cell Averaged Ergodic Capacity for DAS and CAS at SNR=20dB 20 18 16
35 30 25 20 15 2 MSs 10 DAS w/ ZFBF CAS w/ ZFBF
5
14 0 12
8
4
5 6 Number of DA
DAS w/ MRT DAS w/ ST CAS w/ MRT CAS w/ ST
7
8
3
4
5 6 Number of DA
7
8
Position in y axis
algorithm requires an iterative method, whereas our problem has closed form solutions. In Figure 2, we present the cell averaged ergodic capacity curves as a function of the number of DA ports for the singleuser DAS and CAS with SNR=20 dB in single-cell. For both DAS and CAS, we employ ST and MRT. For the CAS with ST, a MS chooses one antenna which has the largest RSS and only the selected antenna transmits the signal to the MS. When N = 8, compared to CAS, DAS with ST and MRT exhibit the capacity gains of 26% and 17%, respectively, since the DAS can reduce the access distance by separating the DA ports geographically. Especially, we observe that DAS with ST has a capacity gain of 14% over CAS with MRT. It should be emphasized that MRT requires full channel state information at the transmitter, and thus is much more complex than ST in terms of the feedback mechanism. In addition, we illustrate the cell averaged ergodic sum rate curve for the multi-user DAS and CAS with ZFBF in singlecell in Figure 3. When N = 8, DAS with ZFBF provide a performance gain of 18% compared to CAS with ZFBF for K = 2 and 3. Again, the performance gap between the DAS and the CAS increases with the number of DA. In Figure 4, we show the locations of DA ports for DAS in two-cell with various numbers of DA obtained from our proposed solution in Section IV. In the figure, the dots and the asterisks denote the locations from the proposed algorithms based on the SNR and the SLR criteria, respectively. Since our proposed SLR based algorithm takes the leakage interference into consideration, the DA ports for the two-cell DAS are shifted against each other in order to reduce the interference. In Figure 5, we exhibit the cell averaged ergodic capacity curves as a function of the number of DA for the single-user
Position in y axis
Fig. 2. The cell averaged ergodic capacity for the single-user DAS and CAS with ST and MRT in single-cell at SNR= 20dB
Fig. 4.
0
−1 −1
0 1 2 Position in x axis N=5
1
0
−1 −1
0 1 2 Position in x axis N=7
1
0
−1 −1
N=4 Position in y axis
2
N=3 1
Position in y axis
4
Position in y axis
6
0
3
Fig. 3. The cell averaged ergodic sum rate for the multi-user DAS and CAS with ZFBF in single-cell at SNR= 20dB
10
Position in y axis
Cell Averaged Ergodic Capacity [bps/Hz]
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0 1 2 Position in x axis
1
0
−1 −1
0 1 2 Position in x axis N=6
1
0
−1 −1
0 1 2 Position in x axis N=8
1
0
−1 −1
0 1 2 Position in x axis
The locations of DA ports for DAS in two-cell with N = 7
DAS and CAS in two-cell at SNR=20 dB. The same transmission schemes as in the single-cell case for the single-user DAS and CAS are employed. Since the optimized locations of DA ports from the proposed SLR based algorithm minimize the OCI, DAS with ST and MRT show the capacity gains of 36% and 23% over CAS with ST and MRT, respectively. Note that DAS with ST shows the performance almost identical to DAS with MRT. This is due to a fact that for DAS with ST, the OCI can be lower than DAS with MRT in multicell environments. Considering the higher feedback overhead required in the MRT scheme, DAS with ST may be preferable in the multi-cell case. In Figure 6, we exhibit the cell averaged ergodic sum rate for the multi-user DAS and CAS with ZFBF in two-cell at SNR=20 dB. Similar to the single-user case, DAS with ZFBF outperforms the CAS with ZFBF. When N = 8, DAS shows a performance gain of 17% and 19% over CAS for K = 2 and 3, respectively. When N = 3, the cell averaged ergodic sum rate curves
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 7, JULY 2012
Cell Averaged Ergodic Capacity for DAS and CAS at SNR=20dB
Cell Averaged Ergodic Sum Rate for DAS with N=3
20
20 DAS w/ MRT DAS w/ ST CAS w/ MRT CAS w/ ST
16
18 Cell Averaged Ergodic Sum Rate [bps/Hz]
Cell Averaged Ergodic Capacity [bps/Hz]
18
14 12 10 8 6 4 2 0
3 MSs 16 14 12 10 2 MSs 8 6 4 DAS based on SLR criterion DAS based on SNR criterion
2 3
4
5 6 Number of DA
7
0
8
Fig. 5. The cell averaged ergodic capacity for the single-user DAS and CAS with ST and MRT in two-cell at SNR= 20dB
0
15
20 SNR [dB]
25
30
35
40
Cell Averaged Ergodic Capacity for DAS and CAS at SNR=20dB
Cell Averaged Ergodic Sum Rate for DAS and CAS at SNR=20dB 20 3 MSs
18 Cell Averaged Ergodic Capacity [bps/Hz]
Cell Averaged Ergodic Sum Rate [bps/Hz]
10
Fig. 7. The cell averaged ergodic sum rate for the multi-user DAS with N = 3 based on the SNR and the SLR criteria in two-cell
30
25
20
15
10 2 MSs 5
0
5
DAS w/ ZFBF CAS w/ ZFBF
16 14 12 10 8 6 4
0 3
4
5 6 Number of DA
7
DAS w/ EGT CAS w/ EGT
2
8
3
4
5 6 Number of DA
7
8
Fig. 6. The cell averaged ergodic sum rate for the multi-user DAS and CAS with ZFBF in two-cell at SNR= 20dB
Fig. 8. The cell averaged ergodic capacity for the single-user DAS and CAS with per-antenna power constraint in single-cell at SNR= 20dB
are exhibited in Figure 7 for DAS with ZFBF in two-cell. At SNR=40 dB, DAS based on the proposed SLR criterion provide about 8% and 11% performance gains compared to DAS based on the SNR criterion for K = 2 and 3, respectively. Thus, we confirm that the proposed SLR based algorithm is suitable for the two-cell DAS compared to the algorithm based on the SNR criterion. Next, we present the cell averaged ergodic capacity and sum rate curves with per-antenna power constraint. For a fair comparison between DAS and CAS, each antenna has separate power constraint P for DAS and CAS. In Figure 8, we plot the cell averaged ergodic capacity curves as a function of the number of DA ports for the single-user DAS and CAS with EGT in single-cell at SNR=20 dB. When N = 8, DAS with EGT exhibits a capacity gain of 7% compared to CAS with EGT. For the multi-user DAS and CAS with scaled ZFBF in two-cell at SNR=20 dB, the cell averaged ergodic sum rate curves are illustrated in Figure 9. When N = 8, DAS has a performance gain of 22% and 21% over CAS for K = 2 and 3, respectively. Similar to the case with sum power constraint, DAS outperforms CAS under per-antenna power constraint.
In this paper, we have proposed new algorithms which identify antenna locations for DAS in single-cell and two-cell. While the conventional SDC algorithm is only appropriate for the single-cell DAS with ST, our algorithm is applicable for the single-cell and two-cell DAS with ST, MRT and ZFBF under sum power constraint and EGT and scaled ZFBF under per-antenna power constraint. For the single-cell DAS, we obtain the locations of DA ports which maximizes the lower bound of the expected SNR. In comparison to the conventional SDC algorithm which needs an iterative method, the proposed SNR based algorithm provides a closed form solution. For the two-cell DAS, we have proposed an iterative algorithm which finds the locations of DA ports by applying the gradient ascent algorithm. Since the proposed SLR based algorithm takes the leakage interference into consideration, our method is suitable for DAS in multi-cell. The simulation results confirm that DAS with antenna locations obtained from the proposed algorithm offers a capacity gain over CAS in single-cell and two-cell environments. Also, for the multi-user DAS in twocell, the simulation results show that DAS with DA locations obtained from the proposed SLR based algorithm outperforms the system with DA locations obtained from the SNR criterion.
VI. C ONCLUSION
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
Cell Averaged Ergodic Sum Rate for DAS and CAS at SNR=20dB
[17] M. Vu, “MISO capacity with per-antenna power constraint,” IEEE Trans. Commun., vol. 59, pp. 1268–1274, May 2011. [18] J. Zhang, R. Chen, J. G. Andrews, A. Ghosh, and R. W. Heath, “Networked MIMO with clustered linear precoding,” IEEE Trans. Wireless Commun., vol. 8, pp. 1910–1921, Apr. 2009. [19] W. Roh and A. Paulraj, “Outage performance of the distributed antanna systems in a composite fading channel,” in Proc. 2002 IEEE VTC – Fall, Sep. 2002, pp. 1520–1524.
Cell Averaged Ergodic Sum Rate [bps/Hz]
30
25 3 MSs 20
[20] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edition. Academic Press, 2007. [21] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd edition. John Wiley & Sons, 2006. [22] W. Yu and T. Lan, ”Transmitter optimization for the multi-antenna downlink with per-antenna power constraints”, IEEE Trans. Signal Process., vol. 55, pp. 2646-2660, Jun. 2007.
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10 2 MSs 5
0
DAS w/ scaled ZFBF CAS w/ scaled ZFBF 3
4
5 6 Number of DA
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Fig. 9. The cell averaged ergodic sum rate for the multi-user DAS and CAS with per-antenna power constraint in two-cell at SNR= 20dB
R EFERENCES [1] A. A. M. Saleh, A. J. Rustako, and R. Roman, “Distributed antennas for indoor radio communications,” IEEE Trans. Commun., vol. 35, pp. 1245–1251, Dec. 1987. [2] W. Roh and A. Paulraj, “MIMO channel capacity for the distributed antenna systems,” in Proc. 2002 IEEE VTC – Fall, pp. 706–709. [3] D. Wang, X. You, J. Wang, Y. Wang, and X. Hou, “Spectral efficiency of distributed MIMO cellualr systems in a composite fading channel,” in Proc. 2008 IEEE ICC, pp. 1259–1264. [4] J. Zhang and J. G. Andrews, “Disributed antenna systems with randomness,” IEEE Trans. Wireless Commun., vol. 7, pp. 3636–3646, Sep. 2008. [5] H. Zhuang, L. Dai, and Y. Yao, “Spectral efficiency of distributed antenna system with random antenna layout,” Electron. Lett., vol. 39, pp. 495–496, Mar. 2003. [6] W. Feng, X. Zhang, S. Zhou, J. Wang, and M. Xia, “Downlink power allocation for distributed antenna systems with random antenna layout,” in Proc. 2009 IEEE VTC – Fall, pp. 1–5. [7] J. Gan, Y. Li, S. Zhou, and J. Wang, “On sum rate of multi-user distributed antenna system with circular antenna layout,” in Proc. 2007 IEEE VTC – Fall, pp. 596–600. [8] S.-R. Lee, S.-H. Moon, J.-S. Kim, and I. Lee, “Capacity analysis of distributed antenna systems in a composite fading channel,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 1076–1086, Mar. 2012. [9] H. Kim, S.-R. Lee, K.-J. Lee, and I. Lee, “Transmission schemes based on sum rate analysis in distributed antenna systems,” IEEE Trans. Wireless Commun., vol. 11, no.3, pp. 1201–1209, Mar. 2012. [10] J. Gan, S. Zhou, J. Wang, and K. Park, “On the sum-rate capacity of multi-user distributed antenna systems,” IEICE Trans. Commun., vol. E89-B, pp. 2612–2616, Sep. 2006. [11] X. Wang, P. Zhu, and M. Chen, “Antenna location design for generalized distributed antenna systems,” IEEE Commun. Lett., vol. 13, pp. 315–317, May 2009. [12] J. Makhoul, S. Roucos, and H. Gish, “Vector quantization in speech coding,” Proc. IEEE, vol. 73, pp. 1551–1587, Nov. 1985. [13] Y. Qian, M. Chen, X. Wang, and P. Zhu, “Antenna location design for distributed antenna systems with selective transmission,” in Proc. 2009 International Conference on WCSP, pp. 3893–3897. [14] E. Park and I. Lee, “Antenna placement for downlink distributed antenna systems with selection transmission,” in Proc. 2011 IEEE VTC – Spring. [15] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47, pp. 1458–1461, Oct. 1999. [16] C. B. Peel, B. M. Hochwald, and A. L. Swindelhurst, “A vector-perturbation technique for near-capacity multiantenna multiuser communication—part I: channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 1, pp. 195–202, Jan. 2005.
Eunsung Park (S’09) received the B.S. and M.S. degrees in electrical engineering from Korea University, Seoul, Korea in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree at Korea University, Seoul, Korea. During the spring in 2009, he visited University of Southern California, Los Angeles, CA, USA to conduct collaborative research under the Brain Korea 21 (BK21) Program. His research interests are communication theory and signal processing techniques for multi-user MIMO wireless networks and distributed antenna systems. Sang-Rim Lee (S’06) Sang-Rim Lee received the B.S. and M.S. degrees in electrical engineering from Korea University, Seoul, Korea, in 2005 and 2007. From 2007 to 2010, he was a research engineer in the Samsung Electronics, where he conducted research on WiMAX system. Currently, he is working toward the Ph.D. degree at Korea University, Seoul, Korea. He was awarded the Silver and Bronze Prizes respectively in the 2011 Samsung Humantech Paper Contest in February 2012. His research topics include communication theory and signal processing techniques for multi-user MIMO wireless networks and distributed antenna systems. He is currently interested in convex optimization, random matrix theory and stochastic geometry. Inkyu Lee (S’92-M’95-SM’01) received the B.S. degree (Hon.) in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1990, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1992 and 1995, respectively. From 1995 to 2001, he was a Member of Technical Staff at Bell Laboratories, Lucent Technologies, where he conducted research on high-speed wireless system designs. He later worked for Agere Systems (formerly Microelectronics Group of Lucent Technologies), Murray Hill, NJ, as a Distinguished Member of Technical Staff from 2001 to 2002. In September 2002, he joined the faculty of Korea University, Seoul, Korea, where he is currently a Professor in the School of Electrical Engineering. During 2009, he visited University of Southern California, LA, USA, as a visiting Professor. He has published around 80 journal papers in IEEE, and has 30 U.S. patents granted or pending. His research interests include digital communications and signal processing techniques applied for next generation wireless systems. Dr. Lee currently serves as an Associate Editor for IEEE T RANSACTIONS ON C OMMUNICATIONS and the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS. Also, he has been a Chief Guest Editor for the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS (Special Issue on 4G Wireless Systems). He received the IT Young Engineer Award as the IEEE/IEEK joint award in 2006, and received the Best Paper Award at APCC in 2006 and IEEE VTC in 2009. Also he was a recipient of the Hae-Dong Best Research Award of the Korea Information and Communications Society (KICS) in 2011.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 7, JULY 2012
Antenna Placement Optimization for Distributed Antenna Systems Eunsung Park, Sang-Rim Lee, and Inkyu Lee, Senior Member, IEEE
Abstract—In this paper, we propose new algorithms to determine the antenna location for downlink distributed antenna systems (DASs) in single-cell and two-cell environments. We consider the composite fading channel which includes small and large scale fadings. First, for the single-cell DAS, we formulate the optimization problem of distributed antenna (DA) port locations by maximizing the lower bound of the expected signal to noise ratio (SNR). In comparison to the conventional algorithm based on the squared distance criterion which requires an iterative method, our problem generates a closed form solution. Next, for the two-cell DAS, we propose a gradient ascent algorithm which determines the optimum DA locations by maximizing the lower bound of the expected signal to leakage ratio (SLR). In our work, we consider selection transmission, maximal ratio transmission and zero-forcing beamforming (ZFBF) under sum power constraint and study equal gain transmission and scaled ZFBF under per-antenna power constraint. Simulation results show that our proposed algorithms based on both the SNR and the SLR criteria offer a capacity gain over the conventional centralized antenna systems. Index Terms—Distributed antenna systems, antenna placement.
I. I NTRODUCTION
I
N recent years, distributed antenna systems (DASs) have gained interests because of its ability to extend the cell coverage and increase the system capacity. The DAS was first introduced for indoor wireless communication systems to enhance the coverage [1]. Unlike conventional centralized antenna systems (CASs) where all antennas are co-located at the cell center, distributed antenna (DA) ports in the DAS are separated geographically. Thus, the DAS can reduce the access distance along with the transmit power and cochannel interference, which result in the improved cell-edge performance [2] [3]. Determining the locations of DA ports is an important issue in optimizing the system performance of DAS. Several papers have analyzed the performance of DAS with random or fixed antenna layout. In [4], the outage probability of DAS with the random antenna layout was derived. Also, the authors of [5] studied the capacity of DAS with the random antenna layout in a single-cell and proposed a sub-optimal power allocation scheme. In addition, the asymptotic ergodic capacity Manuscript received April 12, 2011; revised October 4, 2011 and February 17, 2012; accepted April 13, 2012. The associate editor coordinating the review of this paper and approving it for publication was S. Ghassemzadeh. The material in this paper was presented in part at the IEEE VTC, Budapest, Hungary, May 2011. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010- 0017909). The authors are with the School of Electrical Engineering, Korea University, Seoul, Korea (e-mail: {kupes2000, sangrim78, inkyu}@korea.ac.kr). Digital Object Identifier 10.1109/TWC.2012.051712.110670
was presented by applying random matrix theory for DAS with the random and circular antenna layout in [6] and [7], respectively. Furthermore, for DAS, performance analysis was conducted in [8] and transmission schemes based on sum rate analysis were proposed in [9]. Assuming that systems have two DA ports equipped with a single antenna, the optimal antenna layout was determined in [10] for space time block coded orthogonal frequency division multiplexing systems in the linear cell DAS. According to [10], the bit error rate can be minimized when two DA ports are placed symmetrically. For the uplink single-cell DAS, the squared distance criterion (SDC) was proposed in [11] in order to find antenna locations which maximize the lower bound of the cell averaged ergodic capacity. The algorithm based on the SDC is equivalent to a codebook design in vector quantization [12], and thus an iterative method is required. In [13], the SDC was applied to the downlink DAS with selection transmission (ST) where one DA is selected to transmit data to a mobile station (MS). However, this algorithm may not be suitable for DAS in multicell, since it does not take other-cell interference (OCI) into consideration. In [14], DA port location design was suggested for single-cell and two-cell DAS with ST. In this paper, we propose new algorithms to identify the antenna locations for the downlink DAS in single-cell and twocell environments. We study ST, maximal ratio transmission (MRT) [15] and zero-forcing beamforming (ZFBF) [16] under sum power constraint and examine equal gain transmission (EGT) [17] and scaled ZFBF [18] under per-antenna power constraint. It is assumed that DA ports have perfect channel state information between DA ports and MS in its own cell for beamforming schemes. In our study, we consider the composite fading channel which includes both small scale fadings and large scale fadings. First, for the single-cell DAS, we determine the location of DA ports which maximize the lower bound of the expected signal to noise ratio (SNR). This optimization problem is convex, and we can obtain a closed form solution for the DA locations. Next, for the two-cell DAS, we employ a criterion which maximizes the lower bound of the expected signal to leakage ratio (SLR). This SLR criterion is suitable for multi-cell environments, since it takes the leakage interference into consideration. In this case, the optimization problem becomes non-convex, and thus we develop an iterative algorithm which finds the locations of DA ports by applying a gradient ascent method. Simulation results demonstrate that DAS with the antenna locations obtained from the proposed algorithm based on both the SNR and the SLR criteria outperforms conventional CAS. Also, for DAS in the two-cell environment, the system
c 2012 IEEE 1536-1276/12$31.00
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
with the DA locations obtained from the proposed SLR based algorithm shows the higher cell averaged ergodic sum rate than that with DA ports obtained from the SNR criterion. The organization of this paper is as follows: Section II describes a system model and the cell averaged ergodic capacity for DAS in single-cell and two-cell environments. In Section III, we formulate the optimization problems and Section IV proposes new algorithms for antenna location designs. In Section V, we present the simulation results comparing with CAS. Finally, the paper is terminated with conclusions in Section VI. Throughout this paper, bold upper and lower case letters denote matrices and vectors, respectively, and the superscript (·)T , (·)H , (·)† and (·)∗ stand for transpose, Hermitian, pseudo inverse and conjugate, respectively. Also, E(·) represents the expectation and X(:, l) and X(m, :) indicate the l-th column and the m-th row of a matrix X, respectively. II. S YSTEM M ODEL In this section, we describe a system model for the downlink DAS. We assume that each cell has N DA ports and K MSs, and all DA ports and MSs are equipped with a single antenna. Also, it is assumed that all MSs are uniformly distributed. We consider a circular cell with the radius of R with circular antenna layout with or without a center antenna. √ Also, the distance between centers of two cells is set to 3R. Since DA ports are largely separated in DAS, the channel model encompasses not only the small scale fading (i.e. Rayleigh fadings) but also the large scale fading including shadowing and pathloss [2] [19]. (l ,l) (l ,l) (l ,l) (l ,l) with = gk,1 gk,2 · · · gk,N Let us denote gk (l ,l) (l ,l) (l ,l) (l ,l) (l ,l) (l ,l) gk,n = hk,n sk,n (dk,n )−α , where hk,n and sk,n equal the coefficients of the small scale fading and the shadowing fading from the n-th DA port in the l -th cell to the k-th (l ,l) MS in the l-th cell, respectively, dk,n stands for the distance between the n-th DA port in the l -th cell and the k-th MS in the l-th cell, and α is the path loss exponent. The (l ,l) coefficient of the small scale fading hk,n is an independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and unit variance and the shadowing (l ,l) (l ,l) fading sk,n has a log normal distribution, i.e., 10log10 sk,n becomes a Gaussian random variable with zero mean and standard deviation σs . Then, for a multi-user L-cell downlink DAS with a transmit beamforming scheme under sum power constraint, the received signal of the k-th MS in the l-th cell can be expressed (l) (l ,l) (l ) L P x + zk where P represents the as yk = K l =1 gk (l ,l)
total transmitted power in each cell, gk denotes the channel row vector of length N from the DA port in the l -th cell to the k-th MS in the l-th cell, x(l ) stands for the transmitted signal column vector of length N from the DA port in the l -th cell and zk indicates the additive complex Gaussian noise variable with zero mean and variance σz2 . Here, x(l ) is precoded as (l ) (l ) (l ) (l ) K x(l ) = k =1 fk ak where fk and ak are defined by column vector of length N with the k -th MS beamforming (l ) desired signal of the k -th unit norm (||fk || = 1) and the (l ) MS in the l -th cell with E |ak |2 = 1, respectively.
2469 (l)
The beamforming vector fk is determined by a transmission scheme. For the single-user DAS with MRT, the beam(l,l)H (l) g forming vector in the l-th cell is given as f1 = 1(l,l) . Also, g1
for the multi-user DAS with ZFBF, the k-th MS beamforming (l)† (l) (:,k) vector in the l-th cell is obtained as fk = G where ||G(l)† (:,k)|| T (l,l)T (l,l)T (l,l)T g2 · · · gK . G(l) = g1 For the single-user L-cell downlink DAS with the ST scheme where one DA port is selected which has the largest received signal strength (RSS) [4], the received signal (l) of the MS in the l-th cell can be expressed as y1 = √ L (l ,l) (l ) P l =1 g1,n(l ) a1 + z1 where n(l ) denotes the index of the DA port chosen in the l -th cell. Now, we will present the cell averaged ergodic capacity for the DAS under sum power constraint. For the single-cell (SC) DAS, the cell averaged ergodic capacity with ST and MRT, and the cell averaged ergodic sum rate with ZFBF can be written as ⎡ ⎛ ⎞⎤ (1,1) 2 P g 1,n(1) ⎟⎥ ⎢ ⎜ ST = Eh,s,p ⎣log2 ⎝1 + CSC ⎠⎦ , σz2 ⎡
M RT CSC
⎛
⎞⎤ (1,1) 2 P g1 ⎟⎥ ⎢ ⎜ = Eh,s,p ⎣log2 ⎝1 + ⎠⎦ , σz2 ⎡
ZF BF RSC
⎛ K ⎢ ⎜ = Eh,s,p ⎣ log2 ⎝1 + k=1
P K
⎞⎤ (1,1) (1) 2 gk fk ⎟⎥ ⎠⎦ (1) σz2
where h, s and p account for the small scale fading, the shadowing fading and the MS position, repectively. Also, for the two-cell (T C) DAS, the cell averaged ergodic sum rates with ST, MRT and ZFBF can be represented as ⎡ ⎛ ⎞⎤ (l,l) 2 2 P g 1,n(l) ⎢ ⎜ ⎟⎥ log2 ⎝1 + RTSTC =Eh,s,p ⎣ ¯ 2 ⎠⎦ , ( l,l) l=1 σz2 + P g1,n(¯l) ⎡ ⎛ ⎞⎤ (l,l) 2 2 P g 1 ⎢ ⎜ ⎟⎥ log2 ⎝1 + RTMCRT =Eh,s,p ⎣ ¯ ¯ 2 ⎠⎦ , (l,l) (l) l=1 σz2 + P g1 f1 ⎡ ⎛ ⎞⎤ 2 P (l,l) (l) 2 K f g k k K ⎢ ⎜ ⎟⎥ RTZFC BF =Eh,s,p ⎣ log2⎝1+ ¯ ¯ 2⎠⎦(2) ( l,l) ( l) K P l=1 k=1 σz2+ k =1K gk fk where ¯l = 1 if l = 2 and ¯l = 2 if l = 1. Next, we describe a beamforming scheme for DAS under per-antenna power constraint. In this case, we assume that each antenna has separate power constraint P . For multiple-input single-output systems with per-antenna power constraint, EGT is known as the optimal beamforming scheme in terms of the capacity [17], and thus we employ EGT for the single-user DAS. Let us denote (l,l) (l,l) (l,l) (l,l) (l,l) (l,l) (l,l) g1 = g1,1 ej∠g1,1 g1,2 ej∠g1,2 · · · g1,N ej∠g1,N (l,l) (l,l) where g1,n and ∠g1,n indicate the magnitude and the (l,l)
phase, respectively, for the channel coefficient g1,n . Then,
2470
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 7, JULY 2012 (l)
the EGT beamforming vector can be given as f1 = √ j∠g(l,l) j∠g(l,l) (l,l) H j∠g1,N 1,1 1,2 P e e ··· e . For the muti-user case, the optimal beamforming matrix in terms of the cell averaged ergodic sum rate can be obtained by an iterative method [22]. However, it is complicated to derive a lower bound of the expected SNR and SLR since we cannot compute a closed-form solution. Thus, we consider scaled ZFBF [18] which supports K MSs for a simple derivation. (l) Defining F(l) as the beamforming √ matrix in the l-th cell, F P (l) (l)† can be given as F = max G . (l)† (n,:) | 1≤n≤N |G III. P ROBLEM F ORMULATION In this section, we formulate a problem in order to optimize the antenna locations for the downlink DAS in single-cell and two-cell. In general, it is quite complicated to derive a solution which maximizes the cell averaged ergodic capacity and sumrate (1) and (2) in the previous section. Instead, to simplify the analysis, we attempt to maximize the lower bound of the expected SNR and SLR, for the single-cell case and the twocell case, respectively. A. Single-Cell DAS ST M RT , CSC and In (1), applying the Jensen’s inequality, CSC ZF BF RSC are bounded by ⎛ ⎡ ⎤⎞ (1,1) 2 P g 1,n(1) ⎥⎟ ⎜ ⎢ ST CSC ≤ log2 ⎝1 + Eh,s,p ⎣ ⎦⎠ , σz2 ⎛
M RT CSC
⎡
⎤⎞ (1,1) 2 ⎜ ⎢ P g1 ⎥⎟ ≤ log2 ⎝1 + Eh,s,p ⎣ ⎦⎠ , σz2 ⎛
ZF BF SC
R
≤
K k=1
⎡ P
⎜ ⎢K log2 ⎝1 + Eh,s,p ⎣
⎤⎞ (1,1) (1) 2 gk fk ⎥⎟ ⎦⎠ .(3) σz2
Thus, we will maximize the expected SNR in order to come up with the DA locations for the single-cell DAS. We assume that a cell is divided into N regions with the same physical area. Also we consider circular antenna layout with or without a center antenna. Then, the i-th DA location (i = 1, · · · , N ) can be expressed as DAi = Ri exp (jθi ) for Ra,i ≤ Ri ≤ Rb,i , θa,i ≤ θi ≤ θb,i where Ri and θi are the magnitude and the phase of the i-th DA, Ra,i and Rb,i denote the minimum and the maximum value of the magnitude, and θa,i and θb,i represent the minimum and the maximum value of the phase in the i-th DA region, respectively. For example, when N = 7, for the layout with a center antenna, the parameters for the√region of the center antenna equal Ra,i = 0, Rb,i = R/ 7, θa,i = 0 and θb,i√= 2π, while those for other antennas are set to Ra,i = R/ 7, Rb,i = R and θb,i − θa,i = π/3. On the other hand, the regions of antennas for the layout without a center antenna have Ra,i = 0, Rb,i = R and θb,i − θa,i = 2π/7. Now, we introduce the optimization problem which determines the DA locations for DAS with ST, MRT and ZFBF. Then, we will show thatthe lower bound of the expected SNR 2 (1,1) for all cases. is related to Ep d1,i
1) Single-user DAS with ST and MRT: For MRT, the expected SNR is obtained as ⎡ ⎡ ⎤ ⎤ (1,1) (1) 2 (1,1) 2 ⎢ P g1 f1 ⎥ ⎢ P g1 ⎥ Eh,s,p [SNR]=Eh,s,p⎣ ⎦=Eh,s,p⎣ ⎦ 2 σz σz2 ⎡
2 ⎤ ⎡ ⎤ N s(1,1) h(1,1) N 1,n 1,n 1 P ⎢ ⎥ P β ⎣ α ⎦≥ 2 Ep α ⎦ (4) = 2 Eh,s,p⎣ (1,1) (1,1) σz σz d n=1 n=1 d 1,n
1,n
2 2 2 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) where β= min s1,1 h1,1 , s1,2 h1,2 ,· · ·, s1,N h1,N . Assuming that MS is located in the i-th DA region, and we focus on the distance MS and the i-th DA port for α2 1 between is convex for α ≥ 0, by applying a derivation. Since X the Jensen’s inequality, we compute the lower bound of Eh,s,p [SNR] as Pβ σz2
Eh,s,p [SNR] ≥
1 . 2 α/2 (1,1) Ep d1,i
(5)
Next, for ST, without loss of generality, we assume that the i-th DA port is chosen for transmission. Then, we can consider (1) that the i-th element of f1 is 1 and the other elements are all set to 0. Thus, regarding ST as a special case of MRT, we can formulate the lower bound of Eh,s,p [SNR] for ST as 2 1 P (1,1) (1,1) Eh,s,p [SNR] ≥ 2 Eh,s s1,i h1,i . 2 α/2 σz (1,1) Ep d1,i In (6), we can see that Eh,s
(6) 2 (1,1) (1,1) s1,i h1,i is a constant value
since it does not depend on the MS and DA locations. 2) Multi-user DAS with ZFBF: In (3), in order to enhance ZF BF , we should maximize all the expected SNRs of MSs. RSC However, we can determine the DA locations by maximizing the expected SNR of only one MS since the optimizing problem is equivalent for all MSs. Thus, without loss of generality, we consider only MS 1 for the formulation. Then, the expected SNR of MS 1 is written as P (1,1) (1) 2 f1 Eh,s,p [SNR1 ] = Eh,s,p g Kσz2 1 ⎡ 2 ⎤ N P (1,1) (1) ⎦ ⎣ (7) = E g f h,s,p 1,n n,1 Kσz2 n=1
(1)
(1)
where fn,1is the n-th element of f1 . (1,1) (1,1)∗ = 0 for i = j, Eh,s,p [SNR1 ] can be Since E g1,i g1,j represented as N P (1,1) (1) 2 Eh,s,p [SNR1 ] = Eh,s,p g1,n fn,1 Kσz2 n=1 ⎡ ⎤ ⎡ ⎤ (1,1) N N s1,n (1,1) (1) 2 P β 1 P α h1,n fn,1 ⎦≥ α⎦ Eh,s,p⎣ Ep⎣ = 2 (1,1) (1,1) Kσz2n=1 Kσ z d d n=1 1,n
1,n
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
2 2 (1,1) (1,1) (1) (1,1) (1,1) (1) where β = min s1,1 h1,1 f1,1 , s1,2 h1,2 f2,1 , · · · , 2 (1,1) (1,1) (1) s1,N h1,N fN,1 . Similar to the MRT case, assuming that MS 1 is located in the i-th DA region and applying the Jensen’s inequality, the lower bound of Eh,s,p [SNR1 ] can be written as 1 P β . (8) Eh,s,p [SNR1 ] ≥ Kσz2 (1,1) 2 α/2 Ep d1,i From (5), (6) and (8), the lower bounds of the expected SNR 2 −α/2 (1,1) . for all ST, MRT and ZFBF are related to Ep d1,i Thus, the i-th DA can be optimized by minimizing location 2 (1,1) (1,1) for DAS. Here, the distance d1,i the value of Ep d1,i (1,1) can be obtained as d1,i = Ri2 + r12 − 2Ri r1 cos (φ1 − θi ) where r1 and φ1 indicate the magnitude and of the the phase 2 (1,1) i MS position. Finally, defining ΓSC as Ep d1,i , we can formulate the antenna location problem for single-cell as Rˆi , θˆi for i = 1, · · · , N (9) = arg min ΓiSC {Ri ,θi }
Ra,i ≤ Ri ≤ Rb,i , θa,i ≤ θi ≤ θb,i .
subject to
Problem (9) involves minimization of the expectation of the squared distance between a DA port and an MS. Intuitively, if the location of an MS is fixed, all DA ports should be closely located to an MS in order to maximize performance. However, we assume that an MS is uniformly distributed, not fixed in a cell, and thus our antenna location problem (9) is formulated in an average sense. B. Two-Cell DAS Similar to the single-cell case, in order to optimize the cell averaged ergodic sum rate for the two-cell DAS, we will maximize the expected signal to interference plus noise ratio (SINR). However, this criterion is still complicated due to the coupled nature of the corresponding metric. Thus, we adopt an alternative approach based on the expected SLR which considers the leakage interference and offers simple analysis. 1) Single-user DAS with ST and MRT: For the single-user two-cell DAS with ST and MRT, the cell averaged ergodic sum rate can be approximated at high SNR as 2 2 (l) ST , M RT ≈E =E log2 1+SINR log2 SINR(l) RT C l=1
(l,l) 2
since SLNR(l) ≈ SLR(l) where SLR(l) is given by (l,l) (l,¯ l) (l,l) (l,¯ l) (l) |g1,n(l) |2 /|g1,n(l) |2 for ST and g1 2 /|g1 f1 |2 for MRT. Finally, since log2 X is concave for X > 0, the upper bound of RTSTC, M RT can be expressed as RTSTC, M RT ≤ 2 (l) . Thus, we attempt to enhance the cell l=1 log2 E SLR averaged ergodic sum rate by maximizing the expected SLR. In MRT, the expected SLR at cell 1 is given as ⎡ " " ⎤ " (1,1) "2 " ⎥ "g 1 ⎢ Eh,s,p1 ,p2 SLR(1) = Eh,s,p1 ,p2 ⎣ ⎦ (1,2) (1) 2 g1 f1 ⎡ ⎤ N (1,1) 2 g n=1 1,n ⎢ ⎥ = Eh,s,p1 ,p2 ⎣ 2 ⎦ N (1,2) (1) n=1 g1,n fn,1 ⎡ ⎤ N (1,1) 2 g n=1 1,n ⎢ ⎥ ≥ Eh,s,p1 ,p2 ⎣ ⎦ (1,2) 2 N (1) 2 N n=1 g1,n n=1 fn,1 ⎡ ⎤ N (1,1) 2 1 ⎢ ⎥ = Eh,s,p1 g1,n Eh,s,p2⎣ 2 ⎦(10) (1,2) N n=1 n=1 g1,n where Ep1 and Ep2 denote the expectation with respect to the MS positions in cell 1 and 2, respectively, and the inequality follows from the Cauchy-Schwarz inequality. Let us define β and γ as 2 2 2 (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) β min s1,1 h1,1 , s1,2 h1,2 , · · ·, s1,N h1,N 2 2 2 (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) γ max s1,1 h1,1 , s1,2 h1,2 , · · ·, s1,N h1,N . is Eh,s,p1 ,p2 SLR(1) N β 1 α Ep N (1,1) 2 n=1 γ Ep1
lower
Then,
d1,n
n=1
bounded by Assuming (1,2) α .
1 1/ d1,n
that the MS is located in the i-th and applying the DA region Jensen’s inequality, Eh,s,p1 ,p2 SLR(1) can be written as E h,s,p1 ,p2 SLR(1) ≥
β γ
Ep1
1 (1,1)
d1,i
⎡ 2 α/2
Ep2 ⎣
⎤ N n=1
1 α ⎦ . (1,2) 1/ d1,n
l=1
where the superscript l indicates the cell index, and SINR(l) (l,l) (¯ l,l) is equal to P |g1,n(l) |2 /(σz2 + P |g1,n(¯l) |2 ) for ST and (¯ l,l) (¯ l)
/(σz2 +!P |g1 f1 |2 ) for MRT. Using the fact that (l) = 2l=1 SLNR(l) where SLNR(l) is denoted by l=1 SINR (l,l) (l,¯ l) (l,l) P |g1,n(l) |2 /(σz2 + P |g1,n(l) |2 ) for ST and P g1 2 /(σz2 + (l,¯ l) (l) 2 (l) is equal to P |g1 f1 |2 ) for MRT, l=1 log2 SINR 2 (l) . l=1 log2 SLNR ST , M RT Assuming the interference limited regime, R is TC 2 (l) ST , M RT approximated as RT C ≈ E l=1 log2 SLR P g1 ! 2
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To simplify the derivation, we assume that all DA ports in cell 2 are equi-distant from the MS 1. Then, we in cell α (1,2) 1 1 . to N Ep2 d1,i can approximate Ep2 N (1,2) α n=1 1/ d1,n Finally, it follows that Eh,s,p1 ,p2 SLR(1) for MRT is bounded by α (1,2) d E p2 1,i β Eh,s,p1 ,p2 SLR(1) ≥ . (11) 2 α/2 γ N (1,1) Ep1 d1,i
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For ST, similar to the single-cell case, we can formulate the lower bound of Eh,s,p1 ,p2 SLR(1) as ⎡
⎤
2 α (1,1) (1,1) (1,2) d s E h p 1,i 1,i ⎢ 1,i ⎥ 2 Eh,s,p1 ,p2 SLR(1) ≥Eh,s⎣ . 2⎦ 2α/2 (1,2) (1,2) s1,i h1,i E d(1,1) p1 1,i Here, we can see that Eh,s
(1,1) (1,1) 2 s1,i h1,i (1,2) (1,2) 2 s1,i h1,i
(12)
≈
Eh,s,p
is a constant value.
(l) log2 SINRk
l=1 k=1
= ≈
Eh,s,p Eh,s,p
2 K
l=1 k=1 2 K
(l) log2 SLNRk (l) log2 SLRk
l=1 k=1
(l)
need to maximize the expected SLRk for k = 1, · · · , K and l = 1, 2 in order to enhance the cell averaged ergodic sum rate. However, similar to the single-cell ZFBF case, we can determine the DA locations in the l-th cell by maximizing the expected SLR of only one MS. Thus, without loss of generality, we consider only MS1 for the optimizing problem. (1,1) (1,1)∗ = 0 for i = j, the expected At cell 1, since E g1,i g1,j (1)
⎢ ⎢ 1 (1,1) (1) 2 ⎢ Eh,s,p1 g1,n fn,1 Eh,s,p2⎢ K N ≥ N ⎢ (1,2) 2 (1) n=1 ⎣ g f
is computed as
⎡ 2 ⎤ N (1,1) (1) g f ⎢ n=1 1,n n,1 ⎥ (1) E h,s,p1 ,p2 SLR1 = Eh,s,p1 ,p2 ⎣ ⎦ (1,2) (2) 2 K g f 1 k =1 k ⎡ ⎤ ⎡ ⎤ 2 N 1 ⎢ ⎥ (1,1) (1) = Eh,s,p1 ⎣ g1,n fn,1 ⎦ Eh,s,p2 ⎣ 2 ⎦ (1,2) (1) K n=1 fk k =1 g1 ⎡ ⎤ N (1,1) (1) 2 1 ⎢ ⎥ = Eh,s,p1 g1,n fn,1 Eh,s,p2 ⎣ 2 ⎦ . (1,2) (1) K n=1 fk k =1 g1 (1) Applying the Cauchy-Schwarz inequality, Eh,s,p1,p2 SLR1
⎥ ⎥ ⎥ ⎥ 2⎥ ⎦ n,k
N
1,n
n=1
⎤ N 2 1 (1,1) (1) ⎢ ⎥ = Eh,s,p1 g1,n fn,1 Eh,s,p2⎣ 2 ⎦ . (13) (1,2) n=1 K N n=1 g1,n Defining β and γ as β 2 2 2 (1,1) (1,1) (1) (1,1) (1,1) (1) (1,1) (1,1) (1) min s1,1 h1,1 f1,1 , s1,2 h1,2 f2,1 , · · ·, s1,N h1,N fN,1 2 2 2 (1,2) (1,2) (1,2) (1,2) (1,2) (1,1) γ max s1,1 h1,1 , s1,2 h1,2 , · · ·, s1,N h1,N , (1) the lower bound of Eh,s,p1 ,p2 SLR1 can be expressed as (1) E h,s,p1 ,p2 SLR1 ⎡ ⎤ ⎡ N β 1 α ⎦ Ep2 ⎣ ≥ Ep ⎣ (1,1) N γ K n=1 1 d 1,n
(l) (l,l) (l) (¯ l,l) (¯ l) 2 P P K where SINRk = K |gk fk |2 /(σz2 + K k =1 |gk fk | ), ¯ (l) (l,l) (l) (l,l) (l) 2 K P P SLNRk = K |gk fk |2 /(σz2 + K k =1 |gk fk | ) and ¯ (l) (l,l) (l) (l,l) (l) K SLRk = |gk fk |2 / k =1 |gk fk |2 . ZF BF By applying the Jensen’s inequality, RT C is bounded (l) 2 K . Thus, we by RTZFC BF ≤ l=1 k=1 log2 Eh,s,p SLRk
SLR1
⎤
⎡
(l)
RTZFC BF
⎡
k =1n=1
2) Multi-user DAS with ZFBF: Let us define SLRk as the ratio of the signal power of the k-th MS in the l-th cell to the leakage power from all MSs in the l-th cell to the k-th MS (l) (l) in the ¯ l-th cell. SINRk and SLNRk are similarly defined. Similar to the single-user case, assuming high SNR and the interference limited regime, RTZFC BF is given as 2 K
is represented as (1) E h,s,p1 ,p2 SLR1
⎤ 1 α ⎦ . (1,2) n=1 1/ d1,n
Again we assume that MS 1 at cell 1 is located in the ith DA region. Then, by applying theJensen’s inequality and α (1,2) 1 approximating Ep2 N to N1 Ep2 d1,i (1,2) α n=1 1/ d1,n (1) for simple analysis, the lower bound of Eh,s,p1 ,p2 SLR1 can be written as α (1,2) Ep2 d1,i β (1) Eh,s,p1 ,p2 SLR1 ≥ . (14) 2 α/2 γ KN (1,1) Ep1 d1,i From (11), (12) and (14), we can see that (1) Eh,s,p1 ,p2 SLR1 for all ST, MRT, and ZFBF are related to Υ
(1,2) α Ep2 d1,i α/2 . (1,1) 2 Ep1 d1,i
Thus, the DA location optimization
of ST, MRT and ZFBF for the i-th DA at cell 1 is equivalent to maximizing Υ. Note that the cost function Υ cannot be computed for α > 2 in general. Thus, we change the cost function to ln Υ in order to make it easy to deal with an arbitrary path loss exponent α. Again applying the Jensen’s inequality, the lower bound of ln Υ can be given as α 2 α/2 (1,2) (1,1) − ln Ep1 d1,i ln Υ = ln Ep2 d1,i 2 α/2 2 α/2 (1,2) (1,1) d1,i ≥ Ep2 ln − ln Ep1 d1,i =
α 2
2 2 (1,2) (1,1) − ln Ep1 d1,i . Ep2 ln d1,i
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS (1,2)
(1,1)
Defining ΓiT C as Ep2 [ln(d1,i )2 ] − ln Ep1 [(d1,i )2 ], the problem for i-th DA is now independent of α and can be reformulated as = arg max ΓiT C for i = 1, · · · , N (15) Rˆi , θˆi {Ri ,θi }
subject to
Ra,i ≤ Ri ≤ Rb,i , θa,i ≤ θi ≤ θb,i . (1,2)
Here, the distance d1,i
is calculated as
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bounded by
⎡ 2 ⎤ N (1,1) (1) g f n,1 ⎥ n=1 1,n ⎢ (1) E h,s,p1 ,p2 SLR1 = Eh,s,p1 ,p2 ⎣ ⎦ (1,2) (2) 2 K g f 1 k k =1 ⎡ ⎤ N 1 1 (1,1) (1) 2 ⎢ ⎥ ≥ Eh,s,p1 g1,n fn,1 Eh,s,p2⎣ 2⎦ (.17) N P Kn=1 N (1,2) n=1 g1,n
Note that (16) and (17) have the same form with (7) and (13), √ (1,2) d1,i = Ri2+r22+3R2−2Ri r2 cos(φ2−θi)+2 3R(r2cosφ2−Ri cosθi ) respectively. Thus, we confirm that the problem which determines the DA locations for DAS under per-antenna power constraint is where r2 and φ2 indicate the magnitude and the phase of the equivalent to the case with sum power constraint. MS position in cell 2. Problem (15) identifies the location of the i-th DA port at IV. A NTENNA L OCATION OPTIMIZATION cell 1 in an average sense. Note that the i-th DA port at cell 1 causes interference to MSs at cell 2. Thus, in contrast to the In this section, we introduce new algorithms to solve (9) single-cell case, an interference term is considered to enhance and (15). First, we propose a non-iterative algorithm which performance for the two-cell case. maximizes the lower bound of the expected SNR and present a layout selection method in single-cell. Then, in two-cell, an iterative algorithm which maximizes the lower bound of the expected SLR is illustrated. C. Per-Antenna Power Constraint For DAS, per-antenna power constraint may be more realistic than sum power constraint in practice. In this section, we formulate a problem for the DA locations under per-antenna power constraint. First, we present the single-user DAS with EGT. For the single-cell case, Eh,s,p [SNR] is bounded by Eh,s,p [SNR]
=
≥
⎡# $ ⎤ N 2 P (1,1) ⎦ Eh,s,p ⎣ g1,n σz2 n=1 N (1,1) 2 P E g1,n h,s,p σz2 n=1
where P denotes a power constraint for each antenna. Also, for the two-cell DAS, by applying the Cauchy-Schwarz inequality, the bound of Eh,s,p SLR(1) can be represented as
E h,s,p1 ,p2 SLR(1)
⎡
⎤ (1,1) 2 g1,n ⎥ ⎦ (1,2) (1) 2 g1 f1 ⎡ ⎤
⎢P = Eh,s,p1 ,p2 ⎣
N n=1
N (1,1) 2 1 ⎢ ≥ Eh,s,p1 g1,n Eh,s,p2 ⎣ N N n=1
1 ⎥ ⎦. (1,2) 2 n=1 g1,n
Thus, for DAS with EGT, we can obtain the same form as DAS with MRT in (4) and (10). For the muti-user DAS with scaled ZFBF in single-cell, Eh,s,p [SNR1 ] can be written as ⎡ 2 ⎤ N 1 (1,1) (1) g1,n fn,1 ⎦ . Eh,s,p [SNR1 ] = 2 Eh,s,p ⎣ σz
(16)
n=1
(1) Also, for the two-cell case, Eh,s,p1 ,p2 SLR1 can be lower-
A. Single-Cell DAS In [11] and [13], the SDC algorithm for antenna location designs was studied in the uplink and downlink single-cell DAS. The SDC algorithm tries to optimize the cell averaged ergodic capacity by minimizing the expectation of the squared distance from a randomly distributed MS to the nearest antenna port. This criterion is similar to a codebook design in vector quantization [12], and needs an iterative method. In order to reduce the computational complexity, we propose a new algorithm which does not require an iterative procedure. In (9), since the MS is uniformly distributed in the i-th DA region, the cost function can be computed by % Rb,i% θb,i 2 2 1 (1,1) d1,i r1 2 dφ1 dr1 ΓiSC = 2 θ R − R − θa,i Ra,i θa,i a,i b,i b,i 2 R 2 2Ri 3 2 3 Rb,i −Ra,i (θb,i −θa,i )− · = A i Rb,i −Ra,i 2 3 4 4 Rb,i −Ra,i (θb,i −θa,i ) (18) (sin(θb,i−θi )−sin(θa,i−θi ))+ 4 2 2 . − Ra,i where A = 2/ (θb,i − θa,i ) Rb,i Then, the first and second partial derivatives of ΓiSC are expressed as 2 2 3 i 2 3 ∇Ri ΓSC =A Ri Rb,i (θb,i − θa,i )− Rb,i · − Ra,i − Ra,i 3 (sin (θb,i − θi ) − sin (θa,i − θi )) 3 2 3 ∇θi ΓiSC = ARi Rb,i −Ra,i (cos (θb,i −θi )−cos (θa,i −θi )) 3 2 i 2 ∂ ΓSC 2 (θb,i − θa,i ) =A Rb,i − Ra,i 2 ∂Ri 3 ∂ 2 ΓiSC 2 3 (sin (θb,i −θi )+sin (θi −θa,i )) . = ARi Rb,i −Ra,i ∂θi2 3
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 7, JULY 2012 ∂ 2 Γi
&
cos θb,i − cos θa,i sin θa,i − sin θb,i
' .
B. Two-Cell DAS Since the conventional SDC algorithm does not take the OCI into account, the DA locations from the SDC algorithm may not be suitable for the two-cell DAS. In this subsection, we propose an iterative algorithm which maximizes the lower bound of the expected SLR. Since the MSs in cell 2 are 2 (1,2) uniformly distributed, we can calculate Ep2 ln d1,i similar to the single-cell case as 2 1 % R % 2π 2 (1,2) (1,2) Ep2 ln d1,i r2 ln d1,i dφ2 dr2 . (21) = 2 πR 0 0 √ ( 2π 2 −b2 Using 0 ln (1 + a cos φ + b sin φ) dφ = 2π ln 1+ 1−a 2 for 1 − a2 − b2 ≥ 0 [20], (21) is computed as 2 γ2 (1,2) ln R2 − 1 + R for γ < R 2 Ep2 ln d1,i = 2 for γ ≥ R ln γ √ Ri2 + 3R2 − 2 3RRi cos θi . Then, the cost where γ = function ΓiT C in (15) can be represented as γ2 i ln R2 − 1 + R for γ < R i 2 − ln ΓSC (22) ΓT C = 2 i for γ ≥ R. ln γ − ln ΓSC Here, γ is √ greater than or √ equal to R if 30◦ ≤ θi ≤ 330◦ and 0 ≤ Ri ≤ 3R cos θi − R 3 cos2 θi − 2. Otherwise, we have γ < R. Due to non-convexity of ΓiT C , we identify the local optimal DA locations by applying a gradient ascent algorithm which iteratively updates the DA locations in the direction of the gradient of the cost function. Then, the gradients of the cost function with respect to Ri and θi are derived as √ 2 i i 2 Ri− 3R cos θi −∇Ri ΓSC /ΓSC for γ < R i R √ ∇Ri ΓT C = 2 i i γ 2 Ri− 3R cos θi −∇Ri ΓSC /ΓSC for γ ≥ R ∇θi ΓiT C
=
√ 2√3 RRi sin θi − ∇θi ΓiSC /ΓiSC i i i 2 3 RR γ 2 sin θi − ∇θi ΓSC /ΓSC
for γ < R for γ ≥ R.
With the derived gradient expressions, our algorithm which solves (15) is summarized as follows:
−0.5 −1
0.5 0 −0.5 −1
0.5 0 −0.5 −1 −0.5 0 0.5 1 Position in x axis
Fig. 1.
1 0.5 0 −0.5 −1
−0.5 0 0.5 1 Position in x axis
−0.5 0 0.5 1 Position in x axis
N=7
N=6 1
N=5 Position in y axis
Position in y axis
Position in y axis
0
−0.5 0 0.5 1 Position in x axis
(20)
Finally, comparing circular antenna layouts with and without a center antenna, we choose the one which has a larger value N −α/2 of Ω = i=1 ΓiSC .
0.5
1
1
N=8 Position in y axis
θˆi = arctan
1
Position in y axis
and
N=4
N=3
Position in y axis
∂ 2 Γi
We simply prove that ∂RSC and ∂θSC are always non2 2 i i negative, and thus the cost function is convex with respect to Ri and θi . Therefore, the minimum value of (18) is obtained at either the critical points of ∇Ri ΓiSC = 0 and ∇θi ΓiSC = 0, or the boundary points. As a result, for the single-cell DAS, a closed form solution for the i-th DA location can be obtained by 3 3 2 Rb,i − Ra,i sin θb,i − θˆi + sin θˆi − θa,i Rˆi = (19) 2 − R2 3 Rb,i a,i (θb,i − θa,i )
0.5 0 −0.5 −1
1 0.5 0 −0.5 −1
−0.5 0 0.5 1 Position in x axis
−0.5 0 0.5 1 Position in x axis
Locations of DA ports for DAS in single-cell
Initialization: 1) Initialize Ri and θi for the i-th DA in cell 1 as Ra,i ≤ Ri ≤ Rb,i and θa,i ≤ θi ≤ θb,i Main Loop: 2) Update Ri ← Ri +δRi·∇Ri ΓiT C and θi ← θi +δθi·∇θi ΓiT C 3) Calculate ΓiT C with the updated values 4) Go back to 2) until convergence In [21], several line search methods were presented for the selection of the step size δRi and δθi . In our algorithm, we employ Armijo’s rule [21] which provides provable convergence. Using the Armijo’s rule, we can obtain a non-decreasing ΓiT C value with respect to the number of iterations. After the DA locations are computed for cell 1, the DA locations at cell 2 can be simply determined by using the rotational symmetric property between cell 1 and 2. V. S IMULATION R ESULTS In this section, simulation results are presented to demonstrate the efficacy of our proposed algorithms. In the simulation, the cell radius, the standard deviation for shadowing and the path loss exponent are set to R = 1, σsh = 4 dB and α = 3.75, respectively. Also, we define SNR as P/σz2 . First, we present some simulation results under sum power constraint. For a fair comparison between DAS and CAS, the total transmitted power in each cell for DAS and CAS is set to P . In Figure 1, we first plot the locations of DA ports obtained from the proposed SNR based algorithm for DAS in single-cell with various numbers of DA ports. In the single-cell DAS for N = 3, 4, · · · , 8, the locations of DA ports are calculated from (19) and (20). In the figure, the dots and the asterisks indicate the locations of DA ports from the proposed algorithm for the layout with a center antenna and without a center antenna, respectively. In Table I, the values of Ω are listed for both layouts. The table shows that for N = 3, 4, and 5, the circular layout without a center antenna is preferred, while when N = 6, 7, and 8, a center DA port is helpful in terms of the expected SNR. The results of our proposed algorithm are almost the same as the conventional SDC case in [13]. Note that the SDC
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
TABLE I Ω FOR DIFFERENT CIRCULAR ANTENNA LAYOUTS Ω w/ a center antenna 39.8 104.7 185.0 503.4 861.4 1299.5
Cell Averaged Ergodic Sum Rate for DAS and CAS at SNR=20dB 40
Ω w/o a center antenna 63.7 160.3 307.8 498.3 718.3 958.2
3 MSs Cell Averaged Ergodic Sum Rate [bps/Hz]
N 3 4 5 6 7 8
Cell Averaged Ergodic Capacity for DAS and CAS at SNR=20dB 20 18 16
35 30 25 20 15 2 MSs 10 DAS w/ ZFBF CAS w/ ZFBF
5
14 0 12
8
4
5 6 Number of DA
DAS w/ MRT DAS w/ ST CAS w/ MRT CAS w/ ST
7
8
3
4
5 6 Number of DA
7
8
Position in y axis
algorithm requires an iterative method, whereas our problem has closed form solutions. In Figure 2, we present the cell averaged ergodic capacity curves as a function of the number of DA ports for the singleuser DAS and CAS with SNR=20 dB in single-cell. For both DAS and CAS, we employ ST and MRT. For the CAS with ST, a MS chooses one antenna which has the largest RSS and only the selected antenna transmits the signal to the MS. When N = 8, compared to CAS, DAS with ST and MRT exhibit the capacity gains of 26% and 17%, respectively, since the DAS can reduce the access distance by separating the DA ports geographically. Especially, we observe that DAS with ST has a capacity gain of 14% over CAS with MRT. It should be emphasized that MRT requires full channel state information at the transmitter, and thus is much more complex than ST in terms of the feedback mechanism. In addition, we illustrate the cell averaged ergodic sum rate curve for the multi-user DAS and CAS with ZFBF in singlecell in Figure 3. When N = 8, DAS with ZFBF provide a performance gain of 18% compared to CAS with ZFBF for K = 2 and 3. Again, the performance gap between the DAS and the CAS increases with the number of DA. In Figure 4, we show the locations of DA ports for DAS in two-cell with various numbers of DA obtained from our proposed solution in Section IV. In the figure, the dots and the asterisks denote the locations from the proposed algorithms based on the SNR and the SLR criteria, respectively. Since our proposed SLR based algorithm takes the leakage interference into consideration, the DA ports for the two-cell DAS are shifted against each other in order to reduce the interference. In Figure 5, we exhibit the cell averaged ergodic capacity curves as a function of the number of DA for the single-user
Position in y axis
Fig. 2. The cell averaged ergodic capacity for the single-user DAS and CAS with ST and MRT in single-cell at SNR= 20dB
Fig. 4.
0
−1 −1
0 1 2 Position in x axis N=5
1
0
−1 −1
0 1 2 Position in x axis N=7
1
0
−1 −1
N=4 Position in y axis
2
N=3 1
Position in y axis
4
Position in y axis
6
0
3
Fig. 3. The cell averaged ergodic sum rate for the multi-user DAS and CAS with ZFBF in single-cell at SNR= 20dB
10
Position in y axis
Cell Averaged Ergodic Capacity [bps/Hz]
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0 1 2 Position in x axis
1
0
−1 −1
0 1 2 Position in x axis N=6
1
0
−1 −1
0 1 2 Position in x axis N=8
1
0
−1 −1
0 1 2 Position in x axis
The locations of DA ports for DAS in two-cell with N = 7
DAS and CAS in two-cell at SNR=20 dB. The same transmission schemes as in the single-cell case for the single-user DAS and CAS are employed. Since the optimized locations of DA ports from the proposed SLR based algorithm minimize the OCI, DAS with ST and MRT show the capacity gains of 36% and 23% over CAS with ST and MRT, respectively. Note that DAS with ST shows the performance almost identical to DAS with MRT. This is due to a fact that for DAS with ST, the OCI can be lower than DAS with MRT in multicell environments. Considering the higher feedback overhead required in the MRT scheme, DAS with ST may be preferable in the multi-cell case. In Figure 6, we exhibit the cell averaged ergodic sum rate for the multi-user DAS and CAS with ZFBF in two-cell at SNR=20 dB. Similar to the single-user case, DAS with ZFBF outperforms the CAS with ZFBF. When N = 8, DAS shows a performance gain of 17% and 19% over CAS for K = 2 and 3, respectively. When N = 3, the cell averaged ergodic sum rate curves
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Cell Averaged Ergodic Capacity for DAS and CAS at SNR=20dB
Cell Averaged Ergodic Sum Rate for DAS with N=3
20
20 DAS w/ MRT DAS w/ ST CAS w/ MRT CAS w/ ST
16
18 Cell Averaged Ergodic Sum Rate [bps/Hz]
Cell Averaged Ergodic Capacity [bps/Hz]
18
14 12 10 8 6 4 2 0
3 MSs 16 14 12 10 2 MSs 8 6 4 DAS based on SLR criterion DAS based on SNR criterion
2 3
4
5 6 Number of DA
7
0
8
Fig. 5. The cell averaged ergodic capacity for the single-user DAS and CAS with ST and MRT in two-cell at SNR= 20dB
0
15
20 SNR [dB]
25
30
35
40
Cell Averaged Ergodic Capacity for DAS and CAS at SNR=20dB
Cell Averaged Ergodic Sum Rate for DAS and CAS at SNR=20dB 20 3 MSs
18 Cell Averaged Ergodic Capacity [bps/Hz]
Cell Averaged Ergodic Sum Rate [bps/Hz]
10
Fig. 7. The cell averaged ergodic sum rate for the multi-user DAS with N = 3 based on the SNR and the SLR criteria in two-cell
30
25
20
15
10 2 MSs 5
0
5
DAS w/ ZFBF CAS w/ ZFBF
16 14 12 10 8 6 4
0 3
4
5 6 Number of DA
7
DAS w/ EGT CAS w/ EGT
2
8
3
4
5 6 Number of DA
7
8
Fig. 6. The cell averaged ergodic sum rate for the multi-user DAS and CAS with ZFBF in two-cell at SNR= 20dB
Fig. 8. The cell averaged ergodic capacity for the single-user DAS and CAS with per-antenna power constraint in single-cell at SNR= 20dB
are exhibited in Figure 7 for DAS with ZFBF in two-cell. At SNR=40 dB, DAS based on the proposed SLR criterion provide about 8% and 11% performance gains compared to DAS based on the SNR criterion for K = 2 and 3, respectively. Thus, we confirm that the proposed SLR based algorithm is suitable for the two-cell DAS compared to the algorithm based on the SNR criterion. Next, we present the cell averaged ergodic capacity and sum rate curves with per-antenna power constraint. For a fair comparison between DAS and CAS, each antenna has separate power constraint P for DAS and CAS. In Figure 8, we plot the cell averaged ergodic capacity curves as a function of the number of DA ports for the single-user DAS and CAS with EGT in single-cell at SNR=20 dB. When N = 8, DAS with EGT exhibits a capacity gain of 7% compared to CAS with EGT. For the multi-user DAS and CAS with scaled ZFBF in two-cell at SNR=20 dB, the cell averaged ergodic sum rate curves are illustrated in Figure 9. When N = 8, DAS has a performance gain of 22% and 21% over CAS for K = 2 and 3, respectively. Similar to the case with sum power constraint, DAS outperforms CAS under per-antenna power constraint.
In this paper, we have proposed new algorithms which identify antenna locations for DAS in single-cell and two-cell. While the conventional SDC algorithm is only appropriate for the single-cell DAS with ST, our algorithm is applicable for the single-cell and two-cell DAS with ST, MRT and ZFBF under sum power constraint and EGT and scaled ZFBF under per-antenna power constraint. For the single-cell DAS, we obtain the locations of DA ports which maximizes the lower bound of the expected SNR. In comparison to the conventional SDC algorithm which needs an iterative method, the proposed SNR based algorithm provides a closed form solution. For the two-cell DAS, we have proposed an iterative algorithm which finds the locations of DA ports by applying the gradient ascent algorithm. Since the proposed SLR based algorithm takes the leakage interference into consideration, our method is suitable for DAS in multi-cell. The simulation results confirm that DAS with antenna locations obtained from the proposed algorithm offers a capacity gain over CAS in single-cell and two-cell environments. Also, for the multi-user DAS in twocell, the simulation results show that DAS with DA locations obtained from the proposed SLR based algorithm outperforms the system with DA locations obtained from the SNR criterion.
VI. C ONCLUSION
PARK et al.: ANTENNA PLACEMENT OPTIMIZATION FOR DISTRIBUTED ANTENNA SYSTEMS
Cell Averaged Ergodic Sum Rate for DAS and CAS at SNR=20dB
[17] M. Vu, “MISO capacity with per-antenna power constraint,” IEEE Trans. Commun., vol. 59, pp. 1268–1274, May 2011. [18] J. Zhang, R. Chen, J. G. Andrews, A. Ghosh, and R. W. Heath, “Networked MIMO with clustered linear precoding,” IEEE Trans. Wireless Commun., vol. 8, pp. 1910–1921, Apr. 2009. [19] W. Roh and A. Paulraj, “Outage performance of the distributed antanna systems in a composite fading channel,” in Proc. 2002 IEEE VTC – Fall, Sep. 2002, pp. 1520–1524.
Cell Averaged Ergodic Sum Rate [bps/Hz]
30
25 3 MSs 20
[20] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edition. Academic Press, 2007. [21] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd edition. John Wiley & Sons, 2006. [22] W. Yu and T. Lan, ”Transmitter optimization for the multi-antenna downlink with per-antenna power constraints”, IEEE Trans. Signal Process., vol. 55, pp. 2646-2660, Jun. 2007.
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DAS w/ scaled ZFBF CAS w/ scaled ZFBF 3
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Fig. 9. The cell averaged ergodic sum rate for the multi-user DAS and CAS with per-antenna power constraint in two-cell at SNR= 20dB
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Eunsung Park (S’09) received the B.S. and M.S. degrees in electrical engineering from Korea University, Seoul, Korea in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree at Korea University, Seoul, Korea. During the spring in 2009, he visited University of Southern California, Los Angeles, CA, USA to conduct collaborative research under the Brain Korea 21 (BK21) Program. His research interests are communication theory and signal processing techniques for multi-user MIMO wireless networks and distributed antenna systems. Sang-Rim Lee (S’06) Sang-Rim Lee received the B.S. and M.S. degrees in electrical engineering from Korea University, Seoul, Korea, in 2005 and 2007. From 2007 to 2010, he was a research engineer in the Samsung Electronics, where he conducted research on WiMAX system. Currently, he is working toward the Ph.D. degree at Korea University, Seoul, Korea. He was awarded the Silver and Bronze Prizes respectively in the 2011 Samsung Humantech Paper Contest in February 2012. His research topics include communication theory and signal processing techniques for multi-user MIMO wireless networks and distributed antenna systems. He is currently interested in convex optimization, random matrix theory and stochastic geometry. Inkyu Lee (S’92-M’95-SM’01) received the B.S. degree (Hon.) in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1990, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1992 and 1995, respectively. From 1995 to 2001, he was a Member of Technical Staff at Bell Laboratories, Lucent Technologies, where he conducted research on high-speed wireless system designs. He later worked for Agere Systems (formerly Microelectronics Group of Lucent Technologies), Murray Hill, NJ, as a Distinguished Member of Technical Staff from 2001 to 2002. In September 2002, he joined the faculty of Korea University, Seoul, Korea, where he is currently a Professor in the School of Electrical Engineering. During 2009, he visited University of Southern California, LA, USA, as a visiting Professor. He has published around 80 journal papers in IEEE, and has 30 U.S. patents granted or pending. His research interests include digital communications and signal processing techniques applied for next generation wireless systems. Dr. Lee currently serves as an Associate Editor for IEEE T RANSACTIONS ON C OMMUNICATIONS and the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS. Also, he has been a Chief Guest Editor for the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS (Special Issue on 4G Wireless Systems). He received the IT Young Engineer Award as the IEEE/IEEK joint award in 2006, and received the Best Paper Award at APCC in 2006 and IEEE VTC in 2009. Also he was a recipient of the Hae-Dong Best Research Award of the Korea Information and Communications Society (KICS) in 2011.
