A Multiuser MIMO Transmit Beamformer Based on the Statistics of the Signal-to-Leakage Ratio
© B.K. Chalise and L. Vandendorpe. 2009
Received: 23 February 2009
Accepted: 3 June 2009
Published: 5 July 2009
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.
The capacity of a wireless cellular system is limited by the mutual interference among simultaneous users. Using multiple antenna systems, and in particular, the adaptive beamforming, this problem can be minimized, and the system capacity can be improved. In recent years, the optimum downlink beamforming problem (including power control) has been extensively studied in [1–3] where the signal-to-interference-plus-noise ratio (SINR) is used as a quality of service (QoS) criterion. After it has been found that the multiple-input multiple-output (MIMO) techniques significantly enhance the performance of wireless communication systems [4, 5], the joint optimization of the transmit and receive beamformers  has also been investigated for MIMO systems. Motivated by the fact that the optimum transmit beamformers [1–3] and the joint optimum transmit-receive beamformers  can be obtained only iteratively due to the coupled nature of the corresponding optimization problems, recently, the concept of leakage and subsequently the signal-to-leakage-plus-noise ratio (SLNR) as a figure of merit have been introduced in [7, 8]. (Note that SLNR as a performance criterion has been considered in [9–11] for multiple-input-single-output (MISO) systems.) Although the latter approach only gives suboptimum solutions, it leads to a decoupled optimization problem and admits closed-form solutions for downlink beamforming in multiuser MIMO systems.
While investigating multiuser systems from a system level perspective, in many cases, the outage probability has also been widely used as a QoS parameter. The closed-form expressions of the outage probability with equal gain and optimum combining have been derived in [12, 13], respectively, in a flat-fading Rayleigh environment with cochannel interference. The latter work has been extended in  to a Rician-Rayleigh environment where the desired signal and interferers are subject to Rician and Rayleigh fading, respectively. However, in all of the above-mentioned papers, investigations have been limited to the derivations of the outage probability expressions for specific types of receivers. The outage probability of the signal-to-interference ratio is used to formulate the optimum power control problem for interference limited wireless systems in [15, 16] where the total transmit power is minimized subject to outage probability constraints. However, both of the these works [15, 16] are limited to systems with single antenna at transmitters and receivers.
In this paper, we consider the downlink of a multiuser MIMO wireless communication system in a Rayleigh fading environment. The base station (BS) communicates with several cochannel users in the same time and frequency slots. In our method, we use the average signal-to-leakage ratio (SLR) and the outage probability of SLR as performance metrics which are based on the concept of leakage power [7, 8]. In particular, the novelty of our work lies on the facts that we first derive an approximation of the statistical distribution of SLR  for each cochannel user of the MIMO system in terms of transmit beamforming weight vector. Second, the approximate closed-form expression for the outage probability of SLR is derived. Then, we obtain the solution for the transmit beamformer that minimizes the aforementioned outage probability. According to our best source of knowledge, this approach has not been previously considered for the multiuser MIMO downlink beamforming. With some conservative derivations, we also demonstrate that the beamformer which minimizes the outage probability is same as the one which maximizes the average SLR. Note that similar conclusion has been made in  where the downlink beamforming for multiuser MISO systems is analyzed using the SINR and its outage probability as the performance criteria. In contrast to , we consider that the BS has only the knowledge of the second-order statistics such as the covariance matrix of the downlink user-channels. The motivation behind this assumption is that the knowledge of instantaneous channel information can be available at the BS only through the feedback from users. The drawbacks of the feedback approach are the reduction of the system capacity because of the frequent channel usage required for the transmission of the feedback information from users to the BS, and inherent time delays, errors, and extra costs associated with such a feedback. Furthermore, if the channel varies rapidly, it is not reasonable to acquire the instantaneous feedback at the transmitter, because the optimal transmitter designed on the basis of previously acquired information becomes outdated quickly (see  and the references therein). Thus, we consider that no full-rate feedback information is available at the BS.
The remainder of this paper is organized as follows. The system model is presented in Section 2. The probability density function (PDF) of SLR, its mean, and the outage probability of SLR are derived in terms of the beamformer weight vector in Section 3. In Section 4, the transmit beamformer which maximizes the average SLR and minimizes the outage probability is obtained. In Section 5, analytical and numerical results are compared. Finally, conclusions are drawn in Section 6.
Upper (lower) bold face letters will be used for matrices (vectors); , , , , and denote the Hermitian transpose, mathematical expectation, identity matrix, Euclidean norm, trace operator, and the space of matrices with complex entries, respectively.
2. System Model
It is considered that the transmitter (also the BS) does not know user's receiver, and thus, the SINR (5) is not available at the transmitter. In this case, the transmitter optimizes its beamforming vector to maximize the SLNR (3) thereby assisting the user's receiver in its task of improving the SINR (5). The latter fact can be verified numerically. Note that the beamformer based on maximization of (3) can also be designed for the cases where only the knowledge of second-order statistics of downlink channels is available at the BS. In such cases, the advantages are twofold; the BS and receivers can work in a distributed manner (since the criterion is SLNR), and the BS needs only a limited feedback information from the receivers. To facilitate the aforementioned scheme, we first analyze the statistics of SLNR (3) in the following section.
3. Average SLR and the Outage Probability
Using the notations 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  (this number can be further increased if more scrambling codes can be allocated for each cell ), the assumption that the multiuser leakage power dominates the thermal noise power at each user is not a stringent one.) is large compared to the noise power, we get the SLR from (3) as
We first note that the rows of are statistically independent, and each row has an -variate complex Gaussian distribution with the mean vector and the covariance matrix . According to , in this case, 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  and since , 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:
where , and is the Gamma function. Comparing the PDF of (7) to the standard form of Chi-square PDF , can be alternatively expressed as
where 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
where 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.
4. Maximize the Average SLR and Minimize the Outage Probability
In this section, our objective is to find the optimum 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.
4.1. Maximize the Average SLR
Thus, the optimum weight vector 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).
4.2. Minimize the SLR Outage
Mathematically, this problem has the following unconstrained minimization form: . 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
which is also in the familiar Rayleigh quotient form. (Since we replace the exact cost function by its upper bound, the minimization problem becomes independent of .) 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).
5. Numerical Results and Discussions
In this section, we first verify the correctness of the analytically derived PDF (17) of SLR by comparing the analytical results with the Monte-Carlo simulation results. Next, we investigate the tightness of the bounds in (29) and (36). The outage probability of SLR for the 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.
5.1. Exponential Correlation Model
In this example, the amplitudes of the spatial correlations among the elements of the BS antenna array are considered to be exponentially related. With this assumption, the correlation matrices are defined as
5.2. Spatial Correlation Model-Gaussian Angle of Arrival (AoA)
In this example, the spatial correlation among elements of the BS antenna array is modeled according to the distribution of the AoA of the incoming plane waves at the BS from the 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 
where 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 6 where is varied) and for all .
A fine agreement between the theoretical and simulation results for the PDF of SLR and its outage probability confirms the correctness of the proposed analysis for a multiuser MIMO downlink beamforming in a Rayleigh fading environment. The results also show that the spatial correlation between the antenna elements significantly helps to increase the performance of the SLR-based transmit beamformer in terms of the SLR outage probability. It has been found via some approximations that the transmit beamformer which maximizes the average SLR also minimizes the outage probability of the SLR.
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