Abstract
A multiuser multiple-input multiple-output (MIMO) downlink communication system is analyzed in a Rayleigh fading environment. The approximate closed-form expressions for the probability density function (PDF) of the signal-to-leakage ratio (SLR), its average, and the outage probability have been derived in terms of the transmit beamformer weight vector. With the help of some conservative derivations, it has been shown that the transmit beamformer which maximizes the average SLR also minimizes the outage probability of the SLR. Computer simulations are carried out to compare the theoretical and simulation results for the channels whose spatial correlations are modeled with different methods.
, 
, 
, 
, 
and 
denote the Hermitian transpose, mathematical expectation, 
identity matrix, Euclidean norm, trace operator, and the space of 
matrices with complex entries, respectively.
sensors communicating with 
multi-antenna users. (If there are multiple BSs and they have also the channel information of users assigned to other BSs, the SLR-based method needs to be modified in such a way that each BS takes into account the power leaked by it to the users of other BSs. The necessary modifications, in our case, can be done with some straightforward steps.) The block diagram is shown in Figure 
and 
are, respectively, the signal stream and the transmit beamformer weight vector for 
th user. It is assumed that 
and 
for 
(We consider equal power allocations to all users. Note that power control can be included in the design of beamformers by using a two-step approach, that is, by optimizing the beamformers first and then the powers or vice-versa [
. Let 
denote the number of receive antennas at 
th user. The signal vector received by 
th user is
is a constant that includes the effect of distance-dependent path loss factor and the distance-independent mean-channel power gain, 
is the spatially correlated MIMO channel matrix, and 
denotes the additive noise. It is assumed that each user is surrounded by a large number of scatterers whereas the BS, which is generally located at larger heights from the ground level, does not observe rich scattering. In this scenario, the MIMO channel as seen from the user/BS is spatially uncorrelated/correlated. Thus, the 
th MIMO channel can be given by replacing the receive correlation matrix with an identity matrix in the famous Kronecker-model [
, where the entries of 
are assumed to be zero-mean circularly symmetric complex Gaussian (ZMCSCG) random variables with unit variance such that 
, and 
represents the spatial correlation matrix at the BS corresponding to the 
th user channel. It is important to emphasize here that the derivations for the SLR mean and SLR ouatge probability can be easily extended to double-sided correlated MIMO channels (including the user side correlation), and thus, our main results are also valid for such MIMO channels. Note that 
are symmetric positive semidefinite matrices and are a function of the antenna spacing, average direction of arrival of the scattered signal from 
th user, and the corresponding angular spread [
. Furthermore, without loss of generality, the elements of 
in (2) are considered to be ZMCSCG with the variance 
, that is, 
, where 
denotes 
identity matrix. Inserting (1) into (2) and applying the statistical expectation over signal and noise realizations, the SLNR for 
th user can be expressed as [
is the power of the desired signal for user 
whereas 
is the power of interference that is caused by user 
on the signal received by some other user 
. The leakage for user 
is thus the total power leaked from this user to all other users which is 
. The objective of beamformer is to make 
as large as possible when compared to the leakage power 
. (The performance of the beamformer can be boosted by taking into account the noise term 
which acts as a diagonal loading factor [
th user uses a matched filter to recover its signal. The detected signal of this user can be given by 
where 
is the matched filter response. Then, using (1) and (2), 
can be written as
th user is
for all 
, and assuming that the leakage power (The derivation of outage probability expression and its minimization become too involved if the noise power is not negligible. However, noting that the cellular systems such as UMTS with beamforming techniques can support a significant number of cochannel users per cell [
are statistically independent, and each row has an 
-variate complex Gaussian distribution with the mean vector 
and the covariance matrix 
. According to [
are complex Wishart distributed with the scaling matrix 
and the degrees of freedom parameter 
. For conciseness and simplified mathematical presentation, in the rest of this paper, we assume that 
. Here, we also stress that our results can be easily extended to the general case where 
are different. Mathematically, we can thus write 
, where 
represents the complex Wishart matrix of size 
. Let us use the notations 
and 
. According to the results of [
, we get 
. We note that for any 
, 
, because 
is a positive semidefinite matrix. Since 
is a Chi-square distribution, the random variable 
has the following PDF:
, 
and 
is the Gamma function. Comparing the PDF of (7) to the standard form of Chi-square PDF [
can be alternatively expressed as
is the Chi-square distribution with degrees of freedom 
. Using (8), 
can be written as
is a weighted sum of statistically independent Chi-square random variables, where the weights 
since 
are positive semidefinite. The exact and closed-form solution for the PDF of 
is not known. However, according to [
can be found by approximating 
as a random variable with the Chi-square distribution having degrees of freedom 
and the scaling factor 
as
and 
can be determined by equating the first- and second-order moments of the left-and right-hand sides of relation (10). (This approximation is very accurate and widely adopted in statistics and engineering. The accuracy of the approximation will be confirmed later through numerical simulation results.) Evaluation of the first-order moment (mean) of the both sides of (10) gives
and 
can be expressed as
given in (7), the PDF of 
is well known to be [
. For the sake of better exposition, let 
, where 
is the ratio of two statistically independent random variables. The PDF of 
can be thus written as
from (17), applying [
is the Beta function. Noting that 
and 
, (19) can be further simplified as
th user is defined as
is the system specific threshold value. Note that (21) represents the probability of the transmit beamformer failing to perform its beamforming task properly. Hence, the concept of the SLR outage is analogous to the probability of receiver failing to work properly but is only applicable from a transmitter's point of view. Since the PDF of SLR is already known, the outage probability of (21) can be expressed as
and 
is the Gauss hypergeometric function (see [
(see [
= 
, and after some simple manipulations, (23) can also be expressed in the following alternative form:
, 
(7) becomes exponentially distributed whereas 
(9) becomes a weighted sum of independent exponentially distributed random variables. In this case, the outage probability expression of [
in (23) due to the approximation (10). Also, note that the proposed outage probability analysis can be applied to frequency-selective fading channels where we can consider that the orthogonal frequency division multiplexing (OFDM) is used as a modulation technique. In this context, the MIMO channel for each subcarrier can be considered to be a flat-fading channel. Considering that all users can access a given subcarrier and that the lengths of channel impulse responses for all receive-transmit antenna combinations of all users are shorter than the cyclic prefix [
th user and 
th subcarrier can be expressed as
is the MIMO channel in frequency domain for the 
th user and 
th subcarrier, and 
is the corresponding gain. Let 
be the 
th row and 
th column entry of 
, and be given by
is the total number of subcarriers, 
is the number of independently fading channel-taps, and 
is the impulse response for 
th tap of the channel between 
th receive and 
th transmit antenna. If 
are ZMCSCG, it is very easy to note that 
is a ZMCSCG. Furthermore, if the average sum of the tap-powers for the channel between the 
th receive and 
th transmit antennas is same, that is, if 
for all 
, after some straightforward steps, we can easily verify that the distribution of 
remains complex Wishart with the same scaling matrix 
and the degrees of freedom parameter 
. This shows that the statistics of the signal and leakage powers for a given subcarrier and user remain unchanged.
which maximizes the average SLR and minimizes the outage probability of the SLR observed by 
th user. Note that due to the fact that we use the average SLR and SLR outage as the criteria, the beamformer design is a decoupled problem and can be carried out separately for each user.
which is a difficult optimization problem as 
and 
are complicated functions of 
, although 
is a quadratic function of 
. In order to make this optimization problem tractable, we make certain assumptions which will be clear in the sequel. We can write (20) as
for all 
, where 
. Then, with the help of a well-known power-mean inequality, we can write
are all equal. Applying the above inequality to the expression of 
in (13), we can get an upper bound for 
and more specifically we can write 
. With this observation, the average SLR (27) can be lowerbounded as
and 
are separately nonquadratic functions of 
, their products 
is quadratic in 
. The latter fact can be observed from (13), and thus the product 
can be expressed as
in terms of 
, (29) can be expressed as
(the useful power directed to the 
th user) while keeping the denominator 
(the leakage power) constant. This gives the well-known solution
is the eigenvector associated with the largest eigenvalue (generalized eigenvalue problem) of the characteristic equation given by (32). Later, our numerical results confirm the tightness of the lower bound (31) of average SLR for the weight obtained from (32).
. We note that 
is a complicated function of 
and 
which in turn depend on 
. Therefore, the standard way of finding the first-order derivative of the outage probability with respect to 
and equating the corresponding result to zero does not enable us to solve the problem in closed-form. Here, our approach is to first intituitively find the limiting values of 
and 
for which the outage in (24) approaches to zero. The second step is to find 
in order to achieve those limiting values of 
and 
. After simple manipulation, the outage probability (24) can also be written as
converges for arbitrary 
, 
and 
if 
(see [
]). This is the case in (33) since 
for any 
. It is also not difficult to see from the series form of 
(see [
which can be achieved if 
and 
. As 
, the term 
approaches to zero whereas when 
and 
, the term 
tends to be zero. Hence, it can be concluded that if 
and 
can be minimized with respect to 
, the outage expression (33) can also be minimized. Here, we want to emphasize that the analytical proof for the optimality of the above mentioned approach is still an open issue. Now, the outage probability minimization problem can be turned to the problem of minimizing 
and 
simultaneously with respect to 
, that is, 
, which is a multicriterion optimization problem [
, this multicriterion minimization problem can by scalarized by forming the weighted objective function [
and 
, respectively. Here, we can interpret 
as the relative importance of the second objective function with respect to the first one. Note that (34) is a difficult optimization problem. The following inequality can be easily shown:
are equal. Using the above upper bound and resubstituting for 
and 
, the minimization problem (34) takes the following form:
.) With the help of Lagrangian multiplier method, we can show that the optimum weight vector that minimizes (37) is given by (32) which is just the solution of the transmit beamformer that maximizes the average SLR. Hence, it is clear from (32) that the minimum outage probability and maximum average SLR transmit beamformer require only the knowledge of correlation matrices and average channel power gains. We will later demonstrate, with the numerical results, that the upper bounds in (35), (28), and (36) are relatively tight for the beamformer weight derived from (32).
th user (for conciseness, the results are shown for 
) obtained via theory (23) and Monte-Carlo simulations are also shown for different parameters and correlation models. However, these results are not intended to illustrate the outage performance of a particular system. This would require additional assumptions regarding power control, modulation, and channel coding. Finally, we also demonstrate that the maximum average SLR or minimum outage probability transmit beamformer also helps to significantly improve the user SINR when the user employs linear operation such as matched filtering. We consider MIMO channels in which the transmit correlations are modeled with two different methods; exponential correlation and Gaussian angle of arrival (AoA) models. Throughout all examples, we take 
, 
, 
, 
, 
, and 
. Note that this is purely by way of example, and other values could just have easily been considered. The outage probability of SLR is presented using Monte-Carlo simulation runs during which the channels (
change independently and randomly. For each channel realization, the SLR for 
th user is computed and compared with the threshold value 
for determining the outage probability.
represent the 
th row and 
th column of 
, 
are the amplitudes of correlation coefficients and 
is the AoA of the plane wave from the 
th point source.
for (a) the optimized 
from (32), (b) the non-optimized 
(
), and (c) 
which is the eigenvector corresponding to the maximum eigenvalue of 
. Note that the last method simply tries to maximize the signal power toward the user of interest without even trying to suppress the leakage power toward the other users. Although this approach is highly suboptimal, it is very simple to implement, and its performance can be encouraging especially in UMTS cellular networks [
and 
are compared in Figure 
and in Figures 
, 
, 
, 
and 
.
is obtained from (32), and the exponential correlation model is used).
for user 
(exponential correlation model).
for the user 
(exponential correlation model).
th user. The AoA is assumed to be Gaussian distributed with a standard deviation 
of angular spreading. For this case, we consider a uniform linear array with the half-wavelength spacing. The correlation is thus given by [
is the central angle of the incoming rays to the BS from the 
th user. We assume that the first user is located at 
relative to the BS array broadside, and the other two users are located at 
where we take 
(except in Figure 
is varied) and 
for all 
.
are compared in Figure 
for this figure. It can be seen from Figure 
which in fact confirms that the beamformer (32) maximizes the average SLR with a very fine accuracy. The exact functions in (28) and (35), their corresponding upper bounds, the sum function (36) (with 
), and its upper bound are displayed in Figure 
where the beamformer is derived from (32). It can be observed from this figure that the bound in (28) is very tight for all values of 
whereas that in (35) is tight for the medium and larger values of 
. In fact, the gap between the overall exact function (36) and its upper bound is sufficiently small for all values of 
. Figure 
obtained via theory and simulations for different values of 
and 
. The average SINR (5) and the average SLNR (3) of 
th user versus the receiver noise power 
are displayed in Figure 
of (32), (b) the non-optimized 
(
), and (c) 
which is the eigenvector corresponding to the maximum eigenvalue of 
. In this figure, the SINR and SLNR are averaged over 
independent channel realizations, and it is considered that the receiver has perfect knowledge of instantaneous channels. It can be seen from Figure 
. 
for the user 
(
is obtained from (32), and Gaussian AoA model is used).
for the user 
(
is obtained from (32), and Gaussian AoA model is used, 
, 
, 
, and 
).
for user 
(
is obtained from (32) and Gaussian AoA model is used).
(Gaussian AoA model with 
).