1360 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 7, JULY 2013 A Moment Matching Technique to Estimate the Achievable Ergodic Rate for Limited Feedback Distributed Antenna Systems in the Presence of Out-of-Cell Interference Kasun T. Hemachandra, Student Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE Abstract—The performance of a random vector quantization based transmit precoding distributed antenna system is inves- tigated in terms of the achievable ergodic rate. An approxi- mate expression is derived for the ergodic rate of the system considering the effects of path loss, Rayleigh fading and out- of-cell interference. A moment matching technique is used to approximate the desired signal power and the total interference power distributions of the system using a Gamma distribution. The proposed approximation estimates the performance of the system accurately and saves computational time in simulations. Index Terms—Distributed antenna systems, ergodic rate, lim- ited feedback, random vector quantization. I. I NTRODUCTION D ISTRIBUTED antenna systems (DASs) where users are served using a set of geographically distributed antenna units (DAUs) connected to a central unit (CU), were first introduced as a solution to remove coverage dead-spots in indoor locations [1]. Later DASs emerged as a viable solution to provide reduced outages and higher throughputs in wire- less networks [2]–[5]. A thorough analysis of the achievable ergodic rate of multiuser multiple antenna DASs with out-of- cell interference was presented in [6]. In the majority of the works examining the performance of DASs in multi-cell networks, it has been assumed that perfect channel state information (CSI) of all in-cell channels is available at the CU. However, this may impose heavy feedback requirements on the DASs. Reference [7] gave an overview of the feedback requirements and proposed a limited feedback precoding scheme for DASs. Random vector quantization (RVQ) limited feedback precoding, proposed in [8], has been identified as a less complex, suboptimal technique to reduce feedback in co-located multiple antenna systems. Although, RVQ limited feedback precoding is not optimal for DASs, the performance of DASs with RVQ based precoding can be used as a benchmark to compare other limited feedback precoding techniques. There are a limited number of studies that investi- gate RVQ based precoding in DASs [9], [10], considering only the cases where there are no sources of interference other than additive Gaussian noise. Since DASs have crucial application in heterogeneous networks, it is important to investigate their performance in the presence of out-of-cell interference. In this letter, we analyze the performance of DASs with RVQ limited feedback precoding in the presence of out-of- cell interference, in terms of achievable ergodic rate. Since an exact analysis is not mathematically tractable, we propose an approximate solution for the ergodic rate. We justify our Manuscript received February 6, 2013. The associate editor coordinating the review of this letter and approving it for publication was Y. Li. The authors are with iCORE Wireless Communications Laboratory, Uni- versity of Alberta, Edmonton, AB, Canada, T6G2V4 (e-mail: {hemachan, beaulieu}@icoremail.ece.ualberta.ca). Digital Object Identifier 10.1109/LCOMM.2013.052413.130298 approximations using extensive Monte Carlo simulations. The proposed approximations will be useful for practicing engi- neers to obtain realistic estimates for system design parameters such as codebook size, number of DAUs, and number of antennas at each DAU. The remainder of this letter is organized as follows. In Section II, we present the DAS model used for our analysis. Section III presents the new performance approximations for DAS. Numerical and simulation results are given in Section IV, while Section V concludes this letter. The following notations will be used throughout this paper. The probability density function (PDF) and the cumulative distribution function (CDF) of a random variable (RV) X are denoted as f X (x) and F X (x), respectively. The symbol E[·] denotes expectation while the probability of an event A is denoted Pr(A). A complex Gaussian distribution with mean μ and variance σ 2 is denoted as CN (μ, σ 2 ). A Gamma distribution with parameters k and θ is denoted by Γ(k,θ). Eu- ler’s Gamma function is denoted as Γ(·), while the Whittaker function [11, 9.220.4] is denoted by W λ,ν (·). II. SYSTEM MODEL Consider a multi-cellular DAS with N DAUs and a central base station. Universal frequency reuse is assumed among the L neighboring cells. All the base stations and DAUs are equipped with N t transmit antennas while the user equipments are single antenna devices. It is assumed that in each cell, there is only one active user. Without loss of generality, we consider a typical cell (cell 0 in Fig. 1) as our reference cell for the analysis, while the transmissions of neighboring cells are treated as co-channel interference. The DAS uses blanket transmission [3] to serve the users and the macroscopic channel vector for the user in cell 0 (user 0) can be given as h =   L (0) 0 h (0) 0  L (0) 1 h (0) 1 ··· ,  L (0) N h (0) N  (1) where h (j) i denotes the 1×N t channel vector with independent and identically distributed (i.i.d.) CN (0, 1) components from the i th DAU in the j th cell and L (j) i denotes the propagation path loss from the i th DAU in the j th cell. For simplicity, in theoretical computations and simulations, we use a free-space path loss model to compute L (j) i as L (j) i = P (j) i c 2 (4πf c d (j) i ) 2 (2) where P (j) i is the power allocated to the i th DAU in the j th cell, c =3 × 10 8 m/s, f c is the carrier frequency and d (j) i is the distance from the i th DAU in the j th cell. Per cell power constraints are assumed such that ∑ N i=0 P (j) i = P c , where P c is the power constraint at each cell. The received signal at user 1089-7798/13$31.00 c  2013 IEEE