Performance Analysis of Maximum Ratio Transmission Based Multi ...
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Transactions Papers Performance Analysis of Maximum Ratio Transmission Based Multi-Cellular MIMO Systems Yeliz Tokgoz, Member, IEEE, and Bhaskar D. Rao, Fellow, IEEE
Abstract— In this paper, we analyze the performance of the uplink of multi-cellular MIMO systems in flat Rayleigh fading. There is co-channel interference from users within the same cell as well as from other cell users. The channel model includes lognormal shadowing and path loss along with power control, resulting in a statistical model for user powers. Consistent with practical scenarios, the co-channel interference is categorized into two groups: intracell interference from users within the same cell as the desired user and intercell interference from outer cell users. We derive a compact, easily computable closed form outage probability expression in the form of finite sums. This expression allows for simpler and faster analysis of various MIMO configurations. It has been shown that using antennas on the receiver side results in better performance, since transmit diversity does not combat interference from same cell users. Index Terms— Antenna diversity, beamforming, multipleinput-multiple-output (MIMO) systems, outage probability, Rayleigh fading, co-channel interference, power control.
I. I NTRODUCTION
I
N WIRELESS communication systems, the use of space diversity is an effective technique for fading mitigation and interference suppression. By employing multiple antennas at the receiver and/or the transmitter, information is received through multiple independent channels. Through the use of intelligent transmission and receiver combining schemes, the overall system performance can be significantly improved. Although initial interest in smart antennas has mainly focused on receiver diversity, nowadays multiple-input-multiple-output (MIMO) systems with both transmit and receive diversity are drawing a lot of attention. A popular and simple scheme to maximize the signal-tonoise ratio (SNR) in MIMO systems is referred to as Maximum Ratio Transmission (MRT) [1]. In [2], the performance of MRT is analyzed in a Rayleigh fading environment, and the SNR characteristic function and the average symbol error rate (SER) expressions have been provided. In [3], the SNR Manuscript received September 9, 2003; revised August 27, 2004; accepted January 19, 2005. The associate editor coordinating the review of this letter and approving it for publication was H. Yanikomeroglu. This work was supportedby UC Discovery Grant com02-10119. Y. Tokgoz is with Qualcomm, Inc., San Diego, CA 92121 (e-mail: [email protected]). B. D. Rao is with the Center for Wireless Communications, University of California, San Diego, La Jolla, CA 92093-0407 (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2006.01005.
distribution and the moment generating function are derived for limited antenna systems, along with some SER and BER expressions. However, the impact of co-channel interference (CCI) is not considered in these papers. In the presence of CCI, the optimal strategy is to choose the transmission and combining weights to maximize the signal-to-interference plus noise ratio (SINR) [4], thereby achieving interference suppression. The SER expression of a K × 2 diversity system with CCI has been derived in [5] in integral form, whereas in [6] outage probability of systems with arbitrary number of antennas have been evaluated. This optimal technique does not provide significant performance improvement over MRT when the number of interferers is large, since the available diversity order is insufficient to cancel out all the interferers. In such systems, MRT is usually preferred because of its implementation simplicity and near optimal performance. In [3], the outage probability of MRT is derived for the case of equal power co-channel interferers. However, the assumption of equal power is limiting because in practical systems which consist of multiple cells, since all the users are not power controlled by the same base station (BS), their received powers levels would not be equal. In this paper, we analyze the uplink performance of a synchronous MIMO systems in a multi-cellular setting. There exists co-channel interference from users within the same cell (intracell interference), as well as from other cell users (intercell interference). In most studies, the intercell interference power is assumed to be proportional to the intracell interference [7]. This proportion is denoted by f and is assumed to be a constant value ∼0.55, determined through simulations. However in practice, the ratio of the two interference terms is not a constant, but a random variable itself due to the differences in the distribution statistics of intracell and intercell interference. Intracell interferers are power controlled by the same BS as the desired user (BS 0). Their transmit powers and weight vectors are adjusted to keep the average combined received power at BS 0 at the same desired level. In other words, the interference they cause to the desired user is controlled through the power control loop. However in the case of intercell interferers, the power control loop controls their average received power at another BS. The interference they
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cause at BS 0 (which is a function of the channel between the intercell users and the BS 0), is not controlled. Depending on the instantaneous fast fading, intercell users, especially the ones close to cell edges, can cause very significant interference. As discussed above, intracell and intercell interference terms have different characterizations due to the nature of the power controlling phenomenon in CDMA systems. Hence, for accurate characterization of interference in multi-cellular systems, it is very important to analyze and integrate intracell and intercell interference separately [8]–[11]. This approach is more crucial in MIMO systems, since the effects of transmit and receive antenna diversity on intracell and interference are different. In our earlier work [12], the performance of a multi-cellular system using MRT was analyzed. The analysis considers systems with both deterministic and statistical user power models in Rayleigh fading, and derives closed form outage probability expressions. However, the results are limited to systems where the number of antennas on either the transmitter or the receiver side does not exceed 2. Those results were generalized for systems with arbitrary number of antennas in [13], for the case of deterministic user power levels. In this paper, we continue the generalization by analyzing the performance of multi-cellular MIMO systems, where a power control scheme is implemented to compensate for path loss and shadowing, resulting in a statistical model for the power levels of the users. We assume that the same information bit is sent and received over an arbitrary number of antennas with certain transmission and combining weights. These weights are chosen to maximize the output SNR (MRT). We obtain a simple closed form outage probability expression in the form of finite sums valid for arbitrary number of antennas and is easily computable. It allows for simpler and faster analysis of various MIMO configurations. We use these expressions to analyze the performance improvements achieved by increasing the number of antennas at the receiver and the transmitter sides. The paper is organized as follows: The system model is introduced in Section II. In Section III, the outage probability is derived. Section IV contains the numerical results and discussion. II. S YSTEM M ODEL We consider the uplink of a multi-cellular communication system with n hexagonal cells of radius 1. The system operates in the presence of thermal noise and multi-access interference. The users are randomly distributed across the cell-site according to a uniform distribution with an average number of N users/cell. 120o sectorization is assumed so that the number of users per sector is Ns = N/3. Therefore, the effective multi-access interference is the sum of the interference from the users within the 120o sectors. We assume that there are t transmit antennas at the mobile and r receive antennas at the BS. The channel is spatially independent flat Rayleigh fading, which is a valid assumption when the antenna spacing is sufficiently large and the delay spread is small. It is also assumed to be slowly-varying, i.e., the channel remains unchanged over a frame. The users
experience path loss and lognormal shadowing. A power control (PC) scheme is employed to compensate for these effects and keep the average received power level from a user constant at unity at the servicing base-station. Base station selection by a user involves a search for the BS with the minimum attenuation caused by shadowing and path loss.
A. Received Signal The r × 1 received signal vector, r(t), consists of components from the desired user, the n Ns − 1 interfering users and thermal noise. r(t) =
nN s −1 t t P0 H0,0 w0,0 s0 (t)+ Pk Hk,0 wk,m s (t) k k k=1
+ n(t)
(1)
Since the fading coefficients are assumed to be the same over a frame, their time index have been dropped from the above expression. The received desired and interfering user powers are denoted by Pk , where k denotes the user index and index 0 corresponds to the desired user. Similarly, sk corresponds to the information bits with zero-mean and unit variance. The r × 1 noise vector n is complex white Gaussian with zerot , mean and covariance σ 2 I. For variables Hk,mk and wk,m k the first subscript k denotes the user index (k = 0 corresponds to the desired user), whereas the second subscript mk denotes the BS index that the k-th user is interacting with. Index t represents the mk = 0 denotes the desired user’s BS. wk,m k unit norm weight vector at the transmitter of the k-th mobile, determined according to the channel between the user and its t || = 1. The r × t channel gain controlling BS mk , and ||wk,m k matrix between user k and any BS mk is denoted by Hk,mk , and consists of independent complex Gaussian distributed elements with zero mean and unit variance, CN (0, 1). Hk,0 represents the channel matrix between user k and the desired user’s BS. The received signals at the multiple antennas are r to form combined with a spatial combining weight vector w0,0 r H r || = 1. the decision variable y(t) = (w0,0 ) r(t), where ||w0,0 A key consideration of this analysis is the more realistic and accurate modelling of the average received user power levels. These power levels, Pk ’s, are statistically modelled according to the PC mechanism, where the controlling BS tries to keep the received power level from a user constant at unity [7]. Pk = ν
dk,mk dk,0
β
10(k,0 −k,mk )/10
(2)
dk,mk is the distance between interferer k and its power controlling BS mk . β is the path loss exponent. The lognormal shadowing is modelled as 10k,mk /10 such that k,mk ∼ CN (0, σsh ). We have also included a voice activity term ν to take into account the fact that a voice user is not actively transmitting at all times during its service time. The random variable ν has a binomial distribution, and is equal to 1 with probability p. According to the defined PC mechanism, the active desired user’s received power level is unity, P0 = 1.
TOKGOZ and RAO: PERFORMANCE ANALYSIS OF MAXIMUM RATIO TRANSMISSION BASED MULTI-CELLULAR MIMO SYSTEMS
B. Transmission and Receiver Weights We choose the transmit and receive weight vectors to maximize the overall output SNR of the system. This scheme is denoted as Maximum Ratio Transmission [1], where the transmit and receive weight vectors are the dominant right and left singular vectors of the channel matrix between the user and the corresponding BS. The main reason to choose MRT is the fact that the number of interferers is much larger than the order of diversity. The optimum scheme of choosing weights that maximize the output SINR would provide very little performance gain at the cost of increased complexity [14]. Another advantage is that, MRT does not require the mobiles to have full knowledge of the uplink channel to determine the transmit weights. Only the largest right singular vector is required, which can easily be sent through a feedback channel. The singular value decomposition of Hk,mk is given by min {r,t} i i Hk,mk = σk,mk uik,mk (vk,m )H , where the rank i=1 k of Hk,mk is the minimum of the number of transmit and k k , ukk,mk , vk,m are the singular value, receive antennas. σk,m k k left singular vector and the right singular vector respectively, min {r,t} 1 2 with |σk,m | ≥ |σk,m | ≥ . . . ≥ |σk,mk |. We note that the k k left and right singular vectors have the same distribution as normalized complex Gaussian random vectors [15], [16]. Users choose their transmit weight vectors according to the channel between them and the BS that they are power t 1 = vk,m ). The receive weight vector controlled by (wk,m k k is chosen to coherently combine the signals from the desired r = u10,0 ). Using these weights, the decision statistic user (w0,0 r )H · r(t)) can be expressed as (y(t) = (w0,0 y(t) = σ0,0 s0 (t) +
nN s −1
1 s (t) Pk (u10,0 )H Hk,0 vk,m k k
k=1 r + (w0,0 )H n(t).
(3) The output SINR is then given by SINR =
λ0 nN s −1
,
(4)
Ik,mk + σ 2
Iint corresponds to the sum of interference from users Ik,mk such that mk = 0, whereas Iext is the sum of Ik,mk such that mk = 0. III. O UTAGE P ROBABILITY Outage probability is one of the most common performance measures in wireless communication systems. It is defined as the probability that the system SINR falls below a certain quality of service threshold, . λ0 Pout () = Pr SINR = < . Iint + Iext + σ 2 Since Iint , Iext and λ0 are statistically independent (will be discussed in more detail in Section III-C), this probability can be rewritten in integral form as
fIint (z) fIext (y) Fλ (z + y + σ 2 ) dy dz, Pout () = (7) where fIint (z) and fIext (y) are the probability distribution functions of intracell and intercell interference respectively. Fλ (x) denotes the cumulative distribution function (cdf) of the largest eigenvalue of a Wishart matrix. In order to compute the outage probability, we need to identify these distributions, which is done next. A. Largest Eigenvalue of a Wishart matrix The joint distribution of the eigenvalues of a complex Wishart matrix, CWm (n, I), is given in [17]. The marginal distribution of the largest eigenvalue, λ1 , has been derived in [18] and [19]. The expressions provided are in terms of infinite sums of generalized Laguerre and Hermite polynomials with matrix arguments. Another expression for the same distribution is given in [20] in the form of determinants of matrices. However, none of these forms lead to analytically tractable computation of the outage probability integral (7). The expression that we will use for the distribution of λ1 is provided in [2] as a finite linear combination of elementary gamma densities, and allows us to compute the outage probability. Using that, the cdf of λ can be expressed as [21]
k=1
min{t,r} (t+r)n−2n2
1 2 |σi,0 |
denotes the largest eigenvalue of the comwhere λi = H plex Wishart matrix Hi,0 Hi,0 [15] of user i. The interference from a user k is expressed as 1 |2 . Ik,mk = Pk |(u10,0 )H Hk,0 vk,m k
(5)
The total interference consists of two components: intracell interference (Iint ) and intercell interference (Iext ). If an interfering user is controlled by the same BS as the desired user (mk = 0), it is called an intracell interferer as opposed to an intercell interferer controlled by other BS’s (mk = 0). The statistics of intracell and intercell interferers differ, therefore they need to be analyzed separately. Therefore we rewrite the SINR expression (4) as λ0 SINR = . Iint + Iext + σ 2
Fλ (a) =
n=1
m=|t−r|
gn,m γ(m + 1, na). m!
(8)
The exact values of the coefficients of the summands, gn,m , are computed and tabulated in [2] for most antenna configurations of interest. For m > 0, γ(m, x) is the incomplete gamma function defined as x m−1 xi −t m−1 −x . e t dt = (m − 1)! 1 − e γ(m, x) = i! 0 i=0 (9) The above finite sum representation of γ(m, x) is valid when m is an integer. The k-th order moment of λ is given as min{t,r} (t+r)n−2n2
E[λ ] = k
(6)
85
n=1
m=|t−r|
gn,m (m + k) ! . nk m !
(10)
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0.15
We first analyze the distribution of the intracell interference term caused by users power controlled by the desired user’s BS (mk = 0). Iint =
nN s −1
{Ik,mk : mk = 0}.
fIint(z)
B. Intracell Interference
0.1 0.05 0 0
(11)
5
10
The random variable ρk is defined as ρk = |(u10,0 )H u1k,0 |2 . Since the vectors u1i,0 have the same distribution as normalized complex Gaussian random vectors of size r×1, ρk is identified as a normalized correlation coefficient. When r = 1, ρk is equal to unity. For r ≥ 2, it is known to be a beta random variable with the following density function [22].
(r − 1) (1 − x)(r−2) if 0 ≤ x ≤ 1 fρk (x) = , (13) 0 else where E[ρk ] = 1/r and E[ρ2k ] = 2/[r (r+1)]. Hence, intracell interference is the sum of the products of largest eigenvalues of Wishart matrices and beta random variables, all of which are independent from one another. A closed form expression is not available for the distribution of this interference term. Therefore, we look for approximate distributions. Since each of the terms λk |(u10,0 )H u1k,0 |2 are positive, we make use of the central limit theorem for causal functions [23] and approximate the total intracell interference by a gamma distribution. fIint (z) ≈
z α−1 e−z/β , β α Γ(α)
fIint(z)
k=1
Using (2) with mk = 0, the expression for Ik,mk (5) simplifies to (12) Ik,0 = Pk λk |(u10,0 )H u1k,0 |2 = ν λk ρk .
2
(14)
The parameters α and β are defined as α = (E[z]) /Var(z) and β = Var(z)/E[z]. As can be seen from these expressions, we only need to know the first and second order moments of Iint to fully characterize the approximate distribution. Denoting the probability that a user is power controlled by the desired user’s BS as Pm0 = 1/n, we have
15 z
20
25
30
Simulation Gamma Fit
Nint=2
0.2 0.1 0 0
Fig. 1.
5
z
10
15
Intracell interference distribution - Nint = 2 and Nint = 6.
Having stated this, we still would like to note that in a CDMA system, the number of interferes per sector is usually high and the convergence of the central limit theorem is not of concern. C. Intercell Interference Using (2) and (5), the intercell interference term can be expressed as Iext =
nN s −1
{Ik,mk : mk = 0}
k=1
= z > 0, α, β > 0.
Simulation Gamma Fit
Nint=6
nN s −1 k=1
β dk,mk (k,0 −k,mk )/10 2 ν 10 |bk | : mk = 0 dk,0 (17)
where the random variable bk is defined as bk = 1 . (u10,0 )H Hk,0 vk,m k 1 1 , the vector Hk,0 vk,m can easily Conditioned on vk,m k k be shown to be complex Gaussian distributed with zero mean and identity covariance matrix. Since this distribution is independent of the conditioning vector, the unconditional 1 ∼ CN (0, I). distribution is also the same. Hence, Hk,0 vk,m nNs − 1 k 1 p E[λk ] E[ρk ], Similarly, conditioned on u0,0 , the random variable bk = E[Iint ] = (nNs − 1) E[Ik,0 ] Pm0 = n 1 (15) (u10,0 )H Hk,0 vk,m can also be shown to be complex Gausk sian distributed with zero mean and unit variance. Since this 2 2 Var(Iint ) = (nNs − 1) E[Ik,0 ] Pm0 − (E[Ik,0 ] Pm0 ) conditional distribution is also independent of the conditioning 2 vector, u1 , we can conclude that b ∼ CN (0, 1). 2 2 k 0,0 p E[λk ] E[ρk ] p E[λk ] E[ρk ] = (nNs − 1) − The interference components I and Iext are functions int n n of independent variables with the exception of the common (16) term u10,0 in both expressions. Having shown that Iext is where the moments of λn and ρn have already been provided independent of u10,0 , we have also proven that intracell and in closed form. intercell interference are statistically independent. We do not know the distribution of the intercell interference This approximation is valid even for a small number of intracell interferers because the distribution of the product expression (17) in closed form. Therefore, we will again make λk ρk is in fact experimentally observed to be very similar to use of the central limit theorem for causal functions [23] and a gamma distribution, and the sum of i.i.d. gamma variables approximate it with a gamma distribution (14) with parameters is also a gamma random variable. In Figure 1, the intracell η and φ. This approximation requires the knowledge of the interference distribution is plotted through simulations for 2 first and second order moments of Iext . As can be seen from and 6 interferer systems. In both cases, the gamma distribution (17), the intercell interference is a function of the location provides a very close approximation to the actual distribution. of the users. Therefore, in order to calculate the moments of
TOKGOZ and RAO: PERFORMANCE ANALYSIS OF MAXIMUM RATIO TRANSMISSION BASED MULTI-CELLULAR MIMO SYSTEMS
TABLE I T HE VALUES OF IS0 ,q AND IS 0 ,q FOR β = 4
the interference from a single user, we first condition on the user’s location. We then integrate the conditional moments over the cell site to obtain the unconditional moments. A uniform distribution of users across the cell site is assumed, fX,Y (x, y) = 1/nA. The q-th moment of the interference from user k is given by E[ν q ] E[|bk |2q ] q : mk = 0] = E[Ik,m k nA q β dk,mk q(k,0 −k,mk )/10 10 : mk = 0 dx dy, × E dk,0 (18) where dk,mk and dk,0 , denoting the distance between user k and its power controlling BS mk and BS 0, are functions of the location variables x and y. In order to compute the above integral, we need to specify the BS selection criteria used in the system. 1) Base Station Selection: The users select the BS with the minimum attenuation due to path loss and shadowing. However, the use of this criteria makes the computation of the integral in (18) intractable. On the other hand, the minimum distance criterion does not provide a tight lower bound on interference although analytical computation of the moments is possible. Therefore, we consider an alternative method which is a close approximation to the minimum attenuation criteria and yet analytically tractable. We assume that the users choose the BS with the minimum attenuation among the Nc nearest BS’s. In order for this modified technique to yield results close to the minimum attenuation criterion, we further impose the following condition proposed in [9]. Even if the desired user’s BS is not among the Nc nearest cells, the user still considers it for selection. The resulting method is a very good approximation to the minimum attenuation criterion and yet analytically tractable. mk =
arg min i∈{0∪Nc nearest cells}
dβk,i 10−k,i /10
2) Moments of Intercell Interference: In order to compute (18), we divide the cell site into two regions S0 and S 0 , where S0 is the region where the Nc nearest cells include the desired user’s BS and S 0 is the complement of S0 . We can then rewrite the integral in (18) as the sum of two integrals over the regions S0 and S 0 , denoted by IS0 ,q and IS 0 ,q , respectively. The computation of IS0 ,q and IS 0 ,q have been discussed in detail and tabulated for certain values of σsh in [24]. These values 9, β = 4 and shadowing are presented in Table I for Nc = √ standard deviation of 8 dB and 8/ 2 dB. Using these, the first and second order moments of Iext can be written as p (IS0 ,1 + IS 0 ,1 ) E[Iext ] = (nNs − 1) nA Var(Iext ) = (nNs − 1) 2p p2 2 (IS0,2 + IS 0,2 ) − × (I + I ) S 0,1 S 0,1 nA (nA)2 D. Outage Probability We now utilize the distributions we have analyzed above to derive the outage probability of the system. We approximate
87
IS0 ,1 1.2026 1.2640
√ σs = 8/ 2 dB σs = 8 dB
IS 0 ,1 0.2198 0.4166
IS0 ,2 0.4845 0.5320
IS 0 ,2 0.0130 0.0666
the total intracell and intercell interference terms by the gamma distribution (14) and provide a closed form expression for the outage probability of the system. Using the distribution of the largest eigenvalue of a Wishart matrix (8), the approximate gamma distribution for the intracell and intercell interference with parameters (α, β) and (η, φ) respectively, the outage probability (7) can be written as min(t,r) (t+r)n−2n2
Pout () = × 0
∞
y
n=1
m=|t−r|
η−1 −y/φ
e φ Γ(η) η
gn,m m!
∞
0
z α−1 e−z/β β α Γ(α)
γ m + 1, n z + y + σ 2 dy dz
The values of the gamma distribution parameters (α, β) and (η, φ) can be calculated from the relation presented in Section III-B, using the first and second order moments of the interference terms. After lengthy manipulations which include making use of the finite sum representation of the incomplete gamma function (9) and the polynomial expansion, we evaluate the above integral. The resulting outage probability expression is given by min(t,r) (t+r)n−2n2
Pout () =
n=1
m=|t−r|
gm,n m!
× T m + 1, n, nσ 2 , [α, β, η, φ]
(19)
where the function T (.) is defined as T (n, c, d, [α, β, η, φ]) = ⎧ j n−1 i ⎨ i j (n − 1)! 1 − e−d ⎩ j k i=0 j=0 k=0
c Γ(α + k) Γ(η + i − j) β φ d × Γ(α) Γ(η) i! (1 + cβ)α+k (1 + cφ)η+i−j (20) k
i−j
j−k i−j+k
In (19), denotes the SINR threshold for outage and σ 2 is the thermal noise variance. (α,β) and (η,φ) are the gamma distribution parameters for the approximate intracell and intercell interference distributions respectively. This outage probability expression is in the form of finite sums and is easily computable, allowing for simpler and faster analysis of multi-cellular MIMO systems. E. SIMO/MISO Systems Let us now consider the MISO or the SIMO system. The matrices Hi,j in (1) become complex Gaussian vectors of size 1 × t for transmit diversity or r × 1 for receive diversity. To proceed with the analysis, we need to modify the distribution H Hi,j , since of the largest eigenvalue of the Wishart matrix Hi,j it has only one non-zero eigenvalue as opposed to two. For the
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case considered, it has a gamma distribution with parameter K (max(t, r) = K). The product of λn and ρn which appears in the intracell interference expression also has a gamma distribution with parameter t. Therefore, when calculating the moments of intracell interference, the product of the random variables λn and ρn has to be replaced by a gamma random variable with parameter t. Iext has the same statistics as Section III-C. By substituting the appropriate distributions into the outage probability expression (7) and carrying out the integral, we obtain − σ P
Pout () = 1 − e ×
0
j K−1 i
i j
k
i−j
−1
10
t=2 r=4− Analy. Sim. t=4 r=2− Analy. Sim. t=4 r=4− Analy. Sim. t=8 r=2− Analy. Sim.
−2
10
j k
i=0 j=0 k=0 α+η+k−j β θ P0 Γ(α + k) β)α+k (P0 + θ)η+i−j Γ(α)
2(j−k) i
σ i! (P0 +
2
0
10
Outage Probability
88
Γ(η + i − j) Γ(η) (21)
IV. N UMERICAL R ESULTS AND D ISCUSSION The outage capacity of a system is defined as the number of users that can be supported such that the outage probability does not exceed a certain value. In this example, we analyze the uplink outage capacity improvements in a MIMO CDMA system with increasing number of antennas. The user power levels are statistically modelled according to a power control scheme that keeps track of lognormal shadowing and path loss as discussed in Section II-A and (2). We also compare the simulation results to the analytical expressions derived in Section III-D. The system consists of 59 hexagonal cells of radius 1. √ The area of a single cell is then 3 3/2. The users are randomly distributed across the cell-site according to a uniform distribution with an average number of N users/cell. 120o sectorization is assumed so that the number of users per sector is Ns = N/3. The required SINR level after de-spreading is = 0.0464, and the thermal noise power is 101/10 . We assume that the path loss coefficient β is 4 and the voice activity factor ν is equal to 1 with probability p = 0.38. The standard deviation of the lognormal shadowing term is set to 8 dB. For BS selection, Nc = 9 nearest cells are considered. In Figure 2, the outage probability is plotted versus user density. It can be seen that the analytical and simulation results are in very close agrement. When we compare the different curves, we observe that the 2 × 4 (t × r) system performs better than the 4 × 2 and even the 8 × 2 configurations. In other words, using multiple antennas at the receiver side results in better performance. This is a very interesting observation since the system with 8 transmit 2 receive antennas has twice the diversity of the 2 × 4 system, but still performs worse than that. The reason for this result is the fact that transmit diversity does not combat interference from intracell users. The intracell users optimize their transmission weights according to the channel between them and the desired user’s BS, and therefore they do benefit from the transmit diversity that the system has to offer. On the other hand, receiver diversity enhances only the desired user’s signal power, but not the intracell users’ powers. Therefore only receiver diversity does help against the intracell interference in the system.
−3
10
0
10
20
30 40 50 60 70 Number of Users/Sector
80
90
100
Fig. 2. Outage probability vs. user density - = 0.0464, σsh = 8 dB, σ 2 = 101/10 .
This reasoning can be supported analytically by observing that the desired user’s signal power is λ0 , whereas an active intracell user’s interference is λk ρk (12). The difference in these two powers is the term ρk . We note that the distribution and the moments of ρk , as given by (13), are only functions of the number of receive antennas, r. They do not depend on the number of transmit antennas used in the system. V. C ONCLUSION In this paper, we analyzed the performance of multi-cellular MIMO systems with co-channel interference from both same cell and other cell users. We derived easy to compute closed form outage probability expressions in the form of finite sums for MIMO systems with an arbitrary number of antennas (20). We have shown that using antennas on the receiver side results in better performance since unlike transmit diversity, the receive diversity does combat interference from same cell users. R EFERENCES [1] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47, no. 10, pp. 1458-1461, Oct. 1999. [2] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of transmitreceive diversity in Rayleigh fading,” IEEE Trans. Commun., vol. 51, no. 4, pp. 694-703, Apr. 2003. [3] M. Kang and M.-S. Alouini, “A comparative study in the performance of MIMO MRC systems with and without co-channel interference,” in Proc. IEEE International Conference on Communications, vol. 3, pp. 2154-2158, 2003. [4] K. K. Wong, K. B. Letaief, and R. D. Murch, “Investigating the performance of smart antenna systems at the mobile and base stations in the down and uplinks, in Proc. IEEE Vehicular Technology Conference, vol. 2, pp. 880-884, 1998. [5] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of K-transmit dual-receive diversity with cochannel interferers over a Rayleigh fading channel,” Wireless Pers. Commun., vol. 25, no. 2, pp. 87-100, Feb. 2003. [6] M. Kang and M.-S. Alouini, “Performance analysis of MIMO systems with cochannel interference over Rayleigh fading channels,” in Proc. IEEE International Conference on Communications, vol. 1, pp. 391395, 2002. [7] K. S. Gilhoussen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver, Jr., and C. E. Wheatley III, “On the capacity of a cellular CDMA system,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 303-312, May 1991.
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[8] A. J. Viterbi and A. M. Viterbi, “Other-cell interference in cellular power-controlled CDMA,” IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1501-1504, Feb./Mar./Apr. 1994. [9] G. E. Corazza, G. D. Maio, and F. Vatalaro, “CDMA cellular systems performance with fading, shadowing, and imperfect power control,” IEEE Trans. Commun., vol. 47, no. 2, pp. 450-459, May 1998. [10] B. Hashem and E. S. Sousa, “On the capacity of cellular DS/CDMA systems under slow Rician/Rayleigh-fading channels,” IEEE Trans. Veh. Technol., vol. 49, no. 5, pp. 1752-1759, Sept. 2000. [11] J. Zhang and V. Aalo, “Performance analysis of a multicell DS-CDMA system with base station diversity,” IEE Proc.-Commun., vol. 148, no. 2, pp. 112-118, Apr. 2001. [12] Y. Tokgoz and B. D. Rao, “Outage capacity of maximal ratio transmission-based multi-cellular MIMO systems,” in Proc. IEEE Wireless Communications and Networking Conference, 2004. [13] Y. Tokgoz and B. D. Rao, “Outage probability of multi-cellular MIMO systems In Rayleigh fading,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, 2004. [14] Y. Akyildiz and B. D. Rao, “Statistical performance analysis of optimum combining with co-channel interferers and flat Rayleigh fading,” in Proc. IEEE Global Telecommunications Conference, vol. 6, pp. 3663-3667, 2001. [15] R. J. Muirhead, Aspects of Multivariate Statistical Theory. Wiley, 1982. [16] N. R. Goodman, “Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction),” Annals of Mathematical Statistics, vol. 34, pp. 152-177, 1963. [17] A. T. James, “Distribution of matrix variates and latent roots derived from normal samples,” Annals of Mathematical Statistics,” vol. 35, pp. 476-501, 1964. [18] T. Hayakawa, “On the distribution of the latent roots of a complex Wishart matrix (non-central case),” Annals of the Institute of Statistical Mathematics, vol 24, pp. 1-17, 1972. [19] F. Hirakawa, “Some distributions of the latent roots of a complex Wishart matrix variate,” Annals of the Institute of Statistical Mathematics, vol. 27, no. 2, pp. 357-363, 1975. [20] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems,” IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 418-426, Apr. 2003.
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[21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Inc., 1980. [22] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 1, Second Edition. Wiley. [23] A. Papoulis, The Fourier Integral and its Applications. New York: McGraw-Hill, 1962. [24] Y. Tokgoz and B. D. Rao, “Outage capacity in CDMA systems using receive antenna diversity and fast power control,” in Proc. IEEE International Conference on Communications, vol. 5, pp. 3493-3497, 2003. Yeliz Tokgoz (BS’98-MS’00-PhD’04) received the B.S. degree in electrical and electronics engineering from Bilkent University, Turkey in 1998 and the M.S. degree from Ohio State University in 2000. She then received the Ph.D. degree in Electrical and Computer Engineering from the University of California, San Diego in 2004. She is currently working as a system engineer at Qualcomm, Inc. Her research interests include multiple antenna systems, spread spectrum communication systems, communication theory and statistical signal processing. Bhaskar D. Rao (S’80-M’83-SM’91-F’00) received the B.Tech. degree in electronics and electrical communication engineering from the Indian Institute of Technology, Kharagpur, in 1979 and the M.S. and Ph.D. degrees from the University of Southern California, Los Angeles, in 1981 and 1983, respectively. Since 1983, he has been with the University of California at San Diego, La Jolla, where he is currently a Professor with the Electrical and Computer Engineering Department. His interests are in the areas of digital signal processing, estimation theory, and optimization theory, with applications to digital communications, speech signal processing, and human-computer interactions. Dr. Rao is a member of the Statistical Signal and Array Processing Technical Committee of the IEEE Signal Processing Society. He is also a member of the Signal Processing Theory and Methods Technical Committee.
83
Transactions Papers Performance Analysis of Maximum Ratio Transmission Based Multi-Cellular MIMO Systems Yeliz Tokgoz, Member, IEEE, and Bhaskar D. Rao, Fellow, IEEE
Abstract— In this paper, we analyze the performance of the uplink of multi-cellular MIMO systems in flat Rayleigh fading. There is co-channel interference from users within the same cell as well as from other cell users. The channel model includes lognormal shadowing and path loss along with power control, resulting in a statistical model for user powers. Consistent with practical scenarios, the co-channel interference is categorized into two groups: intracell interference from users within the same cell as the desired user and intercell interference from outer cell users. We derive a compact, easily computable closed form outage probability expression in the form of finite sums. This expression allows for simpler and faster analysis of various MIMO configurations. It has been shown that using antennas on the receiver side results in better performance, since transmit diversity does not combat interference from same cell users. Index Terms— Antenna diversity, beamforming, multipleinput-multiple-output (MIMO) systems, outage probability, Rayleigh fading, co-channel interference, power control.
I. I NTRODUCTION
I
N WIRELESS communication systems, the use of space diversity is an effective technique for fading mitigation and interference suppression. By employing multiple antennas at the receiver and/or the transmitter, information is received through multiple independent channels. Through the use of intelligent transmission and receiver combining schemes, the overall system performance can be significantly improved. Although initial interest in smart antennas has mainly focused on receiver diversity, nowadays multiple-input-multiple-output (MIMO) systems with both transmit and receive diversity are drawing a lot of attention. A popular and simple scheme to maximize the signal-tonoise ratio (SNR) in MIMO systems is referred to as Maximum Ratio Transmission (MRT) [1]. In [2], the performance of MRT is analyzed in a Rayleigh fading environment, and the SNR characteristic function and the average symbol error rate (SER) expressions have been provided. In [3], the SNR Manuscript received September 9, 2003; revised August 27, 2004; accepted January 19, 2005. The associate editor coordinating the review of this letter and approving it for publication was H. Yanikomeroglu. This work was supportedby UC Discovery Grant com02-10119. Y. Tokgoz is with Qualcomm, Inc., San Diego, CA 92121 (e-mail: [email protected]). B. D. Rao is with the Center for Wireless Communications, University of California, San Diego, La Jolla, CA 92093-0407 (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2006.01005.
distribution and the moment generating function are derived for limited antenna systems, along with some SER and BER expressions. However, the impact of co-channel interference (CCI) is not considered in these papers. In the presence of CCI, the optimal strategy is to choose the transmission and combining weights to maximize the signal-to-interference plus noise ratio (SINR) [4], thereby achieving interference suppression. The SER expression of a K × 2 diversity system with CCI has been derived in [5] in integral form, whereas in [6] outage probability of systems with arbitrary number of antennas have been evaluated. This optimal technique does not provide significant performance improvement over MRT when the number of interferers is large, since the available diversity order is insufficient to cancel out all the interferers. In such systems, MRT is usually preferred because of its implementation simplicity and near optimal performance. In [3], the outage probability of MRT is derived for the case of equal power co-channel interferers. However, the assumption of equal power is limiting because in practical systems which consist of multiple cells, since all the users are not power controlled by the same base station (BS), their received powers levels would not be equal. In this paper, we analyze the uplink performance of a synchronous MIMO systems in a multi-cellular setting. There exists co-channel interference from users within the same cell (intracell interference), as well as from other cell users (intercell interference). In most studies, the intercell interference power is assumed to be proportional to the intracell interference [7]. This proportion is denoted by f and is assumed to be a constant value ∼0.55, determined through simulations. However in practice, the ratio of the two interference terms is not a constant, but a random variable itself due to the differences in the distribution statistics of intracell and intercell interference. Intracell interferers are power controlled by the same BS as the desired user (BS 0). Their transmit powers and weight vectors are adjusted to keep the average combined received power at BS 0 at the same desired level. In other words, the interference they cause to the desired user is controlled through the power control loop. However in the case of intercell interferers, the power control loop controls their average received power at another BS. The interference they
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cause at BS 0 (which is a function of the channel between the intercell users and the BS 0), is not controlled. Depending on the instantaneous fast fading, intercell users, especially the ones close to cell edges, can cause very significant interference. As discussed above, intracell and intercell interference terms have different characterizations due to the nature of the power controlling phenomenon in CDMA systems. Hence, for accurate characterization of interference in multi-cellular systems, it is very important to analyze and integrate intracell and intercell interference separately [8]–[11]. This approach is more crucial in MIMO systems, since the effects of transmit and receive antenna diversity on intracell and interference are different. In our earlier work [12], the performance of a multi-cellular system using MRT was analyzed. The analysis considers systems with both deterministic and statistical user power models in Rayleigh fading, and derives closed form outage probability expressions. However, the results are limited to systems where the number of antennas on either the transmitter or the receiver side does not exceed 2. Those results were generalized for systems with arbitrary number of antennas in [13], for the case of deterministic user power levels. In this paper, we continue the generalization by analyzing the performance of multi-cellular MIMO systems, where a power control scheme is implemented to compensate for path loss and shadowing, resulting in a statistical model for the power levels of the users. We assume that the same information bit is sent and received over an arbitrary number of antennas with certain transmission and combining weights. These weights are chosen to maximize the output SNR (MRT). We obtain a simple closed form outage probability expression in the form of finite sums valid for arbitrary number of antennas and is easily computable. It allows for simpler and faster analysis of various MIMO configurations. We use these expressions to analyze the performance improvements achieved by increasing the number of antennas at the receiver and the transmitter sides. The paper is organized as follows: The system model is introduced in Section II. In Section III, the outage probability is derived. Section IV contains the numerical results and discussion. II. S YSTEM M ODEL We consider the uplink of a multi-cellular communication system with n hexagonal cells of radius 1. The system operates in the presence of thermal noise and multi-access interference. The users are randomly distributed across the cell-site according to a uniform distribution with an average number of N users/cell. 120o sectorization is assumed so that the number of users per sector is Ns = N/3. Therefore, the effective multi-access interference is the sum of the interference from the users within the 120o sectors. We assume that there are t transmit antennas at the mobile and r receive antennas at the BS. The channel is spatially independent flat Rayleigh fading, which is a valid assumption when the antenna spacing is sufficiently large and the delay spread is small. It is also assumed to be slowly-varying, i.e., the channel remains unchanged over a frame. The users
experience path loss and lognormal shadowing. A power control (PC) scheme is employed to compensate for these effects and keep the average received power level from a user constant at unity at the servicing base-station. Base station selection by a user involves a search for the BS with the minimum attenuation caused by shadowing and path loss.
A. Received Signal The r × 1 received signal vector, r(t), consists of components from the desired user, the n Ns − 1 interfering users and thermal noise. r(t) =
nN s −1 t t P0 H0,0 w0,0 s0 (t)+ Pk Hk,0 wk,m s (t) k k k=1
+ n(t)
(1)
Since the fading coefficients are assumed to be the same over a frame, their time index have been dropped from the above expression. The received desired and interfering user powers are denoted by Pk , where k denotes the user index and index 0 corresponds to the desired user. Similarly, sk corresponds to the information bits with zero-mean and unit variance. The r × 1 noise vector n is complex white Gaussian with zerot , mean and covariance σ 2 I. For variables Hk,mk and wk,m k the first subscript k denotes the user index (k = 0 corresponds to the desired user), whereas the second subscript mk denotes the BS index that the k-th user is interacting with. Index t represents the mk = 0 denotes the desired user’s BS. wk,m k unit norm weight vector at the transmitter of the k-th mobile, determined according to the channel between the user and its t || = 1. The r × t channel gain controlling BS mk , and ||wk,m k matrix between user k and any BS mk is denoted by Hk,mk , and consists of independent complex Gaussian distributed elements with zero mean and unit variance, CN (0, 1). Hk,0 represents the channel matrix between user k and the desired user’s BS. The received signals at the multiple antennas are r to form combined with a spatial combining weight vector w0,0 r H r || = 1. the decision variable y(t) = (w0,0 ) r(t), where ||w0,0 A key consideration of this analysis is the more realistic and accurate modelling of the average received user power levels. These power levels, Pk ’s, are statistically modelled according to the PC mechanism, where the controlling BS tries to keep the received power level from a user constant at unity [7]. Pk = ν
dk,mk dk,0
β
10(k,0 −k,mk )/10
(2)
dk,mk is the distance between interferer k and its power controlling BS mk . β is the path loss exponent. The lognormal shadowing is modelled as 10k,mk /10 such that k,mk ∼ CN (0, σsh ). We have also included a voice activity term ν to take into account the fact that a voice user is not actively transmitting at all times during its service time. The random variable ν has a binomial distribution, and is equal to 1 with probability p. According to the defined PC mechanism, the active desired user’s received power level is unity, P0 = 1.
TOKGOZ and RAO: PERFORMANCE ANALYSIS OF MAXIMUM RATIO TRANSMISSION BASED MULTI-CELLULAR MIMO SYSTEMS
B. Transmission and Receiver Weights We choose the transmit and receive weight vectors to maximize the overall output SNR of the system. This scheme is denoted as Maximum Ratio Transmission [1], where the transmit and receive weight vectors are the dominant right and left singular vectors of the channel matrix between the user and the corresponding BS. The main reason to choose MRT is the fact that the number of interferers is much larger than the order of diversity. The optimum scheme of choosing weights that maximize the output SINR would provide very little performance gain at the cost of increased complexity [14]. Another advantage is that, MRT does not require the mobiles to have full knowledge of the uplink channel to determine the transmit weights. Only the largest right singular vector is required, which can easily be sent through a feedback channel. The singular value decomposition of Hk,mk is given by min {r,t} i i Hk,mk = σk,mk uik,mk (vk,m )H , where the rank i=1 k of Hk,mk is the minimum of the number of transmit and k k , ukk,mk , vk,m are the singular value, receive antennas. σk,m k k left singular vector and the right singular vector respectively, min {r,t} 1 2 with |σk,m | ≥ |σk,m | ≥ . . . ≥ |σk,mk |. We note that the k k left and right singular vectors have the same distribution as normalized complex Gaussian random vectors [15], [16]. Users choose their transmit weight vectors according to the channel between them and the BS that they are power t 1 = vk,m ). The receive weight vector controlled by (wk,m k k is chosen to coherently combine the signals from the desired r = u10,0 ). Using these weights, the decision statistic user (w0,0 r )H · r(t)) can be expressed as (y(t) = (w0,0 y(t) = σ0,0 s0 (t) +
nN s −1
1 s (t) Pk (u10,0 )H Hk,0 vk,m k k
k=1 r + (w0,0 )H n(t).
(3) The output SINR is then given by SINR =
λ0 nN s −1
,
(4)
Ik,mk + σ 2
Iint corresponds to the sum of interference from users Ik,mk such that mk = 0, whereas Iext is the sum of Ik,mk such that mk = 0. III. O UTAGE P ROBABILITY Outage probability is one of the most common performance measures in wireless communication systems. It is defined as the probability that the system SINR falls below a certain quality of service threshold, . λ0 Pout () = Pr SINR = < . Iint + Iext + σ 2 Since Iint , Iext and λ0 are statistically independent (will be discussed in more detail in Section III-C), this probability can be rewritten in integral form as
fIint (z) fIext (y) Fλ (z + y + σ 2 ) dy dz, Pout () = (7) where fIint (z) and fIext (y) are the probability distribution functions of intracell and intercell interference respectively. Fλ (x) denotes the cumulative distribution function (cdf) of the largest eigenvalue of a Wishart matrix. In order to compute the outage probability, we need to identify these distributions, which is done next. A. Largest Eigenvalue of a Wishart matrix The joint distribution of the eigenvalues of a complex Wishart matrix, CWm (n, I), is given in [17]. The marginal distribution of the largest eigenvalue, λ1 , has been derived in [18] and [19]. The expressions provided are in terms of infinite sums of generalized Laguerre and Hermite polynomials with matrix arguments. Another expression for the same distribution is given in [20] in the form of determinants of matrices. However, none of these forms lead to analytically tractable computation of the outage probability integral (7). The expression that we will use for the distribution of λ1 is provided in [2] as a finite linear combination of elementary gamma densities, and allows us to compute the outage probability. Using that, the cdf of λ can be expressed as [21]
k=1
min{t,r} (t+r)n−2n2
1 2 |σi,0 |
denotes the largest eigenvalue of the comwhere λi = H plex Wishart matrix Hi,0 Hi,0 [15] of user i. The interference from a user k is expressed as 1 |2 . Ik,mk = Pk |(u10,0 )H Hk,0 vk,m k
(5)
The total interference consists of two components: intracell interference (Iint ) and intercell interference (Iext ). If an interfering user is controlled by the same BS as the desired user (mk = 0), it is called an intracell interferer as opposed to an intercell interferer controlled by other BS’s (mk = 0). The statistics of intracell and intercell interferers differ, therefore they need to be analyzed separately. Therefore we rewrite the SINR expression (4) as λ0 SINR = . Iint + Iext + σ 2
Fλ (a) =
n=1
m=|t−r|
gn,m γ(m + 1, na). m!
(8)
The exact values of the coefficients of the summands, gn,m , are computed and tabulated in [2] for most antenna configurations of interest. For m > 0, γ(m, x) is the incomplete gamma function defined as x m−1 xi −t m−1 −x . e t dt = (m − 1)! 1 − e γ(m, x) = i! 0 i=0 (9) The above finite sum representation of γ(m, x) is valid when m is an integer. The k-th order moment of λ is given as min{t,r} (t+r)n−2n2
E[λ ] = k
(6)
85
n=1
m=|t−r|
gn,m (m + k) ! . nk m !
(10)
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0.15
We first analyze the distribution of the intracell interference term caused by users power controlled by the desired user’s BS (mk = 0). Iint =
nN s −1
{Ik,mk : mk = 0}.
fIint(z)
B. Intracell Interference
0.1 0.05 0 0
(11)
5
10
The random variable ρk is defined as ρk = |(u10,0 )H u1k,0 |2 . Since the vectors u1i,0 have the same distribution as normalized complex Gaussian random vectors of size r×1, ρk is identified as a normalized correlation coefficient. When r = 1, ρk is equal to unity. For r ≥ 2, it is known to be a beta random variable with the following density function [22].
(r − 1) (1 − x)(r−2) if 0 ≤ x ≤ 1 fρk (x) = , (13) 0 else where E[ρk ] = 1/r and E[ρ2k ] = 2/[r (r+1)]. Hence, intracell interference is the sum of the products of largest eigenvalues of Wishart matrices and beta random variables, all of which are independent from one another. A closed form expression is not available for the distribution of this interference term. Therefore, we look for approximate distributions. Since each of the terms λk |(u10,0 )H u1k,0 |2 are positive, we make use of the central limit theorem for causal functions [23] and approximate the total intracell interference by a gamma distribution. fIint (z) ≈
z α−1 e−z/β , β α Γ(α)
fIint(z)
k=1
Using (2) with mk = 0, the expression for Ik,mk (5) simplifies to (12) Ik,0 = Pk λk |(u10,0 )H u1k,0 |2 = ν λk ρk .
2
(14)
The parameters α and β are defined as α = (E[z]) /Var(z) and β = Var(z)/E[z]. As can be seen from these expressions, we only need to know the first and second order moments of Iint to fully characterize the approximate distribution. Denoting the probability that a user is power controlled by the desired user’s BS as Pm0 = 1/n, we have
15 z
20
25
30
Simulation Gamma Fit
Nint=2
0.2 0.1 0 0
Fig. 1.
5
z
10
15
Intracell interference distribution - Nint = 2 and Nint = 6.
Having stated this, we still would like to note that in a CDMA system, the number of interferes per sector is usually high and the convergence of the central limit theorem is not of concern. C. Intercell Interference Using (2) and (5), the intercell interference term can be expressed as Iext =
nN s −1
{Ik,mk : mk = 0}
k=1
= z > 0, α, β > 0.
Simulation Gamma Fit
Nint=6
nN s −1 k=1
β dk,mk (k,0 −k,mk )/10 2 ν 10 |bk | : mk = 0 dk,0 (17)
where the random variable bk is defined as bk = 1 . (u10,0 )H Hk,0 vk,m k 1 1 , the vector Hk,0 vk,m can easily Conditioned on vk,m k k be shown to be complex Gaussian distributed with zero mean and identity covariance matrix. Since this distribution is independent of the conditioning vector, the unconditional 1 ∼ CN (0, I). distribution is also the same. Hence, Hk,0 vk,m nNs − 1 k 1 p E[λk ] E[ρk ], Similarly, conditioned on u0,0 , the random variable bk = E[Iint ] = (nNs − 1) E[Ik,0 ] Pm0 = n 1 (15) (u10,0 )H Hk,0 vk,m can also be shown to be complex Gausk sian distributed with zero mean and unit variance. Since this 2 2 Var(Iint ) = (nNs − 1) E[Ik,0 ] Pm0 − (E[Ik,0 ] Pm0 ) conditional distribution is also independent of the conditioning 2 vector, u1 , we can conclude that b ∼ CN (0, 1). 2 2 k 0,0 p E[λk ] E[ρk ] p E[λk ] E[ρk ] = (nNs − 1) − The interference components I and Iext are functions int n n of independent variables with the exception of the common (16) term u10,0 in both expressions. Having shown that Iext is where the moments of λn and ρn have already been provided independent of u10,0 , we have also proven that intracell and in closed form. intercell interference are statistically independent. We do not know the distribution of the intercell interference This approximation is valid even for a small number of intracell interferers because the distribution of the product expression (17) in closed form. Therefore, we will again make λk ρk is in fact experimentally observed to be very similar to use of the central limit theorem for causal functions [23] and a gamma distribution, and the sum of i.i.d. gamma variables approximate it with a gamma distribution (14) with parameters is also a gamma random variable. In Figure 1, the intracell η and φ. This approximation requires the knowledge of the interference distribution is plotted through simulations for 2 first and second order moments of Iext . As can be seen from and 6 interferer systems. In both cases, the gamma distribution (17), the intercell interference is a function of the location provides a very close approximation to the actual distribution. of the users. Therefore, in order to calculate the moments of
TOKGOZ and RAO: PERFORMANCE ANALYSIS OF MAXIMUM RATIO TRANSMISSION BASED MULTI-CELLULAR MIMO SYSTEMS
TABLE I T HE VALUES OF IS0 ,q AND IS 0 ,q FOR β = 4
the interference from a single user, we first condition on the user’s location. We then integrate the conditional moments over the cell site to obtain the unconditional moments. A uniform distribution of users across the cell site is assumed, fX,Y (x, y) = 1/nA. The q-th moment of the interference from user k is given by E[ν q ] E[|bk |2q ] q : mk = 0] = E[Ik,m k nA q β dk,mk q(k,0 −k,mk )/10 10 : mk = 0 dx dy, × E dk,0 (18) where dk,mk and dk,0 , denoting the distance between user k and its power controlling BS mk and BS 0, are functions of the location variables x and y. In order to compute the above integral, we need to specify the BS selection criteria used in the system. 1) Base Station Selection: The users select the BS with the minimum attenuation due to path loss and shadowing. However, the use of this criteria makes the computation of the integral in (18) intractable. On the other hand, the minimum distance criterion does not provide a tight lower bound on interference although analytical computation of the moments is possible. Therefore, we consider an alternative method which is a close approximation to the minimum attenuation criteria and yet analytically tractable. We assume that the users choose the BS with the minimum attenuation among the Nc nearest BS’s. In order for this modified technique to yield results close to the minimum attenuation criterion, we further impose the following condition proposed in [9]. Even if the desired user’s BS is not among the Nc nearest cells, the user still considers it for selection. The resulting method is a very good approximation to the minimum attenuation criterion and yet analytically tractable. mk =
arg min i∈{0∪Nc nearest cells}
dβk,i 10−k,i /10
2) Moments of Intercell Interference: In order to compute (18), we divide the cell site into two regions S0 and S 0 , where S0 is the region where the Nc nearest cells include the desired user’s BS and S 0 is the complement of S0 . We can then rewrite the integral in (18) as the sum of two integrals over the regions S0 and S 0 , denoted by IS0 ,q and IS 0 ,q , respectively. The computation of IS0 ,q and IS 0 ,q have been discussed in detail and tabulated for certain values of σsh in [24]. These values 9, β = 4 and shadowing are presented in Table I for Nc = √ standard deviation of 8 dB and 8/ 2 dB. Using these, the first and second order moments of Iext can be written as p (IS0 ,1 + IS 0 ,1 ) E[Iext ] = (nNs − 1) nA Var(Iext ) = (nNs − 1) 2p p2 2 (IS0,2 + IS 0,2 ) − × (I + I ) S 0,1 S 0,1 nA (nA)2 D. Outage Probability We now utilize the distributions we have analyzed above to derive the outage probability of the system. We approximate
87
IS0 ,1 1.2026 1.2640
√ σs = 8/ 2 dB σs = 8 dB
IS 0 ,1 0.2198 0.4166
IS0 ,2 0.4845 0.5320
IS 0 ,2 0.0130 0.0666
the total intracell and intercell interference terms by the gamma distribution (14) and provide a closed form expression for the outage probability of the system. Using the distribution of the largest eigenvalue of a Wishart matrix (8), the approximate gamma distribution for the intracell and intercell interference with parameters (α, β) and (η, φ) respectively, the outage probability (7) can be written as min(t,r) (t+r)n−2n2
Pout () = × 0
∞
y
n=1
m=|t−r|
η−1 −y/φ
e φ Γ(η) η
gn,m m!
∞
0
z α−1 e−z/β β α Γ(α)
γ m + 1, n z + y + σ 2 dy dz
The values of the gamma distribution parameters (α, β) and (η, φ) can be calculated from the relation presented in Section III-B, using the first and second order moments of the interference terms. After lengthy manipulations which include making use of the finite sum representation of the incomplete gamma function (9) and the polynomial expansion, we evaluate the above integral. The resulting outage probability expression is given by min(t,r) (t+r)n−2n2
Pout () =
n=1
m=|t−r|
gm,n m!
× T m + 1, n, nσ 2 , [α, β, η, φ]
(19)
where the function T (.) is defined as T (n, c, d, [α, β, η, φ]) = ⎧ j n−1 i ⎨ i j (n − 1)! 1 − e−d ⎩ j k i=0 j=0 k=0
c Γ(α + k) Γ(η + i − j) β φ d × Γ(α) Γ(η) i! (1 + cβ)α+k (1 + cφ)η+i−j (20) k
i−j
j−k i−j+k
In (19), denotes the SINR threshold for outage and σ 2 is the thermal noise variance. (α,β) and (η,φ) are the gamma distribution parameters for the approximate intracell and intercell interference distributions respectively. This outage probability expression is in the form of finite sums and is easily computable, allowing for simpler and faster analysis of multi-cellular MIMO systems. E. SIMO/MISO Systems Let us now consider the MISO or the SIMO system. The matrices Hi,j in (1) become complex Gaussian vectors of size 1 × t for transmit diversity or r × 1 for receive diversity. To proceed with the analysis, we need to modify the distribution H Hi,j , since of the largest eigenvalue of the Wishart matrix Hi,j it has only one non-zero eigenvalue as opposed to two. For the
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 1, JANUARY 2006
case considered, it has a gamma distribution with parameter K (max(t, r) = K). The product of λn and ρn which appears in the intracell interference expression also has a gamma distribution with parameter t. Therefore, when calculating the moments of intracell interference, the product of the random variables λn and ρn has to be replaced by a gamma random variable with parameter t. Iext has the same statistics as Section III-C. By substituting the appropriate distributions into the outage probability expression (7) and carrying out the integral, we obtain − σ P
Pout () = 1 − e ×
0
j K−1 i
i j
k
i−j
−1
10
t=2 r=4− Analy. Sim. t=4 r=2− Analy. Sim. t=4 r=4− Analy. Sim. t=8 r=2− Analy. Sim.
−2
10
j k
i=0 j=0 k=0 α+η+k−j β θ P0 Γ(α + k) β)α+k (P0 + θ)η+i−j Γ(α)
2(j−k) i
σ i! (P0 +
2
0
10
Outage Probability
88
Γ(η + i − j) Γ(η) (21)
IV. N UMERICAL R ESULTS AND D ISCUSSION The outage capacity of a system is defined as the number of users that can be supported such that the outage probability does not exceed a certain value. In this example, we analyze the uplink outage capacity improvements in a MIMO CDMA system with increasing number of antennas. The user power levels are statistically modelled according to a power control scheme that keeps track of lognormal shadowing and path loss as discussed in Section II-A and (2). We also compare the simulation results to the analytical expressions derived in Section III-D. The system consists of 59 hexagonal cells of radius 1. √ The area of a single cell is then 3 3/2. The users are randomly distributed across the cell-site according to a uniform distribution with an average number of N users/cell. 120o sectorization is assumed so that the number of users per sector is Ns = N/3. The required SINR level after de-spreading is = 0.0464, and the thermal noise power is 101/10 . We assume that the path loss coefficient β is 4 and the voice activity factor ν is equal to 1 with probability p = 0.38. The standard deviation of the lognormal shadowing term is set to 8 dB. For BS selection, Nc = 9 nearest cells are considered. In Figure 2, the outage probability is plotted versus user density. It can be seen that the analytical and simulation results are in very close agrement. When we compare the different curves, we observe that the 2 × 4 (t × r) system performs better than the 4 × 2 and even the 8 × 2 configurations. In other words, using multiple antennas at the receiver side results in better performance. This is a very interesting observation since the system with 8 transmit 2 receive antennas has twice the diversity of the 2 × 4 system, but still performs worse than that. The reason for this result is the fact that transmit diversity does not combat interference from intracell users. The intracell users optimize their transmission weights according to the channel between them and the desired user’s BS, and therefore they do benefit from the transmit diversity that the system has to offer. On the other hand, receiver diversity enhances only the desired user’s signal power, but not the intracell users’ powers. Therefore only receiver diversity does help against the intracell interference in the system.
−3
10
0
10
20
30 40 50 60 70 Number of Users/Sector
80
90
100
Fig. 2. Outage probability vs. user density - = 0.0464, σsh = 8 dB, σ 2 = 101/10 .
This reasoning can be supported analytically by observing that the desired user’s signal power is λ0 , whereas an active intracell user’s interference is λk ρk (12). The difference in these two powers is the term ρk . We note that the distribution and the moments of ρk , as given by (13), are only functions of the number of receive antennas, r. They do not depend on the number of transmit antennas used in the system. V. C ONCLUSION In this paper, we analyzed the performance of multi-cellular MIMO systems with co-channel interference from both same cell and other cell users. We derived easy to compute closed form outage probability expressions in the form of finite sums for MIMO systems with an arbitrary number of antennas (20). We have shown that using antennas on the receiver side results in better performance since unlike transmit diversity, the receive diversity does combat interference from same cell users. R EFERENCES [1] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Trans. Commun., vol. 47, no. 10, pp. 1458-1461, Oct. 1999. [2] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of transmitreceive diversity in Rayleigh fading,” IEEE Trans. Commun., vol. 51, no. 4, pp. 694-703, Apr. 2003. [3] M. Kang and M.-S. Alouini, “A comparative study in the performance of MIMO MRC systems with and without co-channel interference,” in Proc. IEEE International Conference on Communications, vol. 3, pp. 2154-2158, 2003. [4] K. K. Wong, K. B. Letaief, and R. D. Murch, “Investigating the performance of smart antenna systems at the mobile and base stations in the down and uplinks, in Proc. IEEE Vehicular Technology Conference, vol. 2, pp. 880-884, 1998. [5] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of K-transmit dual-receive diversity with cochannel interferers over a Rayleigh fading channel,” Wireless Pers. Commun., vol. 25, no. 2, pp. 87-100, Feb. 2003. [6] M. Kang and M.-S. Alouini, “Performance analysis of MIMO systems with cochannel interference over Rayleigh fading channels,” in Proc. IEEE International Conference on Communications, vol. 1, pp. 391395, 2002. [7] K. S. Gilhoussen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver, Jr., and C. E. Wheatley III, “On the capacity of a cellular CDMA system,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 303-312, May 1991.
TOKGOZ and RAO: PERFORMANCE ANALYSIS OF MAXIMUM RATIO TRANSMISSION BASED MULTI-CELLULAR MIMO SYSTEMS
[8] A. J. Viterbi and A. M. Viterbi, “Other-cell interference in cellular power-controlled CDMA,” IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1501-1504, Feb./Mar./Apr. 1994. [9] G. E. Corazza, G. D. Maio, and F. Vatalaro, “CDMA cellular systems performance with fading, shadowing, and imperfect power control,” IEEE Trans. Commun., vol. 47, no. 2, pp. 450-459, May 1998. [10] B. Hashem and E. S. Sousa, “On the capacity of cellular DS/CDMA systems under slow Rician/Rayleigh-fading channels,” IEEE Trans. Veh. Technol., vol. 49, no. 5, pp. 1752-1759, Sept. 2000. [11] J. Zhang and V. Aalo, “Performance analysis of a multicell DS-CDMA system with base station diversity,” IEE Proc.-Commun., vol. 148, no. 2, pp. 112-118, Apr. 2001. [12] Y. Tokgoz and B. D. Rao, “Outage capacity of maximal ratio transmission-based multi-cellular MIMO systems,” in Proc. IEEE Wireless Communications and Networking Conference, 2004. [13] Y. Tokgoz and B. D. Rao, “Outage probability of multi-cellular MIMO systems In Rayleigh fading,” in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, 2004. [14] Y. Akyildiz and B. D. Rao, “Statistical performance analysis of optimum combining with co-channel interferers and flat Rayleigh fading,” in Proc. IEEE Global Telecommunications Conference, vol. 6, pp. 3663-3667, 2001. [15] R. J. Muirhead, Aspects of Multivariate Statistical Theory. Wiley, 1982. [16] N. R. Goodman, “Statistical analysis based on a certain multivariate complex gaussian distribution (an introduction),” Annals of Mathematical Statistics, vol. 34, pp. 152-177, 1963. [17] A. T. James, “Distribution of matrix variates and latent roots derived from normal samples,” Annals of Mathematical Statistics,” vol. 35, pp. 476-501, 1964. [18] T. Hayakawa, “On the distribution of the latent roots of a complex Wishart matrix (non-central case),” Annals of the Institute of Statistical Mathematics, vol 24, pp. 1-17, 1972. [19] F. Hirakawa, “Some distributions of the latent roots of a complex Wishart matrix variate,” Annals of the Institute of Statistical Mathematics, vol. 27, no. 2, pp. 357-363, 1975. [20] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems,” IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 418-426, Apr. 2003.
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[21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Inc., 1980. [22] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 1, Second Edition. Wiley. [23] A. Papoulis, The Fourier Integral and its Applications. New York: McGraw-Hill, 1962. [24] Y. Tokgoz and B. D. Rao, “Outage capacity in CDMA systems using receive antenna diversity and fast power control,” in Proc. IEEE International Conference on Communications, vol. 5, pp. 3493-3497, 2003. Yeliz Tokgoz (BS’98-MS’00-PhD’04) received the B.S. degree in electrical and electronics engineering from Bilkent University, Turkey in 1998 and the M.S. degree from Ohio State University in 2000. She then received the Ph.D. degree in Electrical and Computer Engineering from the University of California, San Diego in 2004. She is currently working as a system engineer at Qualcomm, Inc. Her research interests include multiple antenna systems, spread spectrum communication systems, communication theory and statistical signal processing. Bhaskar D. Rao (S’80-M’83-SM’91-F’00) received the B.Tech. degree in electronics and electrical communication engineering from the Indian Institute of Technology, Kharagpur, in 1979 and the M.S. and Ph.D. degrees from the University of Southern California, Los Angeles, in 1981 and 1983, respectively. Since 1983, he has been with the University of California at San Diego, La Jolla, where he is currently a Professor with the Electrical and Computer Engineering Department. His interests are in the areas of digital signal processing, estimation theory, and optimization theory, with applications to digital communications, speech signal processing, and human-computer interactions. Dr. Rao is a member of the Statistical Signal and Array Processing Technical Committee of the IEEE Signal Processing Society. He is also a member of the Signal Processing Theory and Methods Technical Committee.
