Constrained power allocation schemes for coordinated base station transmission using block diagonalization
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In this study, we propose several power allocation schemes in a coordinated base station downlink transmission with per antenna and per base station power constraints. Block Diagonalization is employed to remove interference among users. For each set of power constraints, two schemes based on the waterfilling distribution are proposed and compared to the optimal solution, which can only be obtained numerically by using convex optimization. We show that the proposed schemes achieve a performance, in terms of weighted sum rate, very close to the optimal, without the heavy computational complexity required by the latter. The sum rates are compared first in a simplified two-user two-cell case where we also compare our approach to the previous solutions available in the literature. Then, we examine the performance in a multi-cell scenario where we also evaluate the degradation of the performance caused by imperfect channel state information.
Keywordscoordinated base stations multiple-antennas network MIMO block diagonalization
Space-division multiplexing (SDM) based on multiple input-multiple output (MIMO) techniques emerged as a means of achieving high-capacity communications . However, the introduction of MIMO processing in cellular networks does not offer the expected benefits, the main reason being the interference that characterizes these environments. SDM requires high signal-to-noise-plus-interference ratios (SINR) to leverage its capacity-achieving potential. Unfortunately, the interference in cellular systems lowers the operating point toward low SINR, thus making MIMO processing not so advantageous. Recently some study has been devoted to manage interference in cellular systems with reuse one, where all cells are allowed to use the same frequencies, also known as universal frequency reuse . In , the other-cell interference (OCI) is considered when designing the transmission for a multi-user MIMO downlink. In , the authors analyze several approaches for overcoming interference in SDM MIMO cellular networks. If the interference is known by the transmitters, then cooperative encoding among base stations using dirty paper coding (DPC) can suppress OCI . In , several strategies are proposed to perform coordinated base station transmission (CBST). Interference is eliminated by jointly and coherently coordinating the transmission from the base stations in the network, assuming that base stations know all downlink signals. Besides DPC, they propose a zero-forcing (ZF) scheme that, although suboptimal, does not involve the complexity of DPC. The capacity of MIMO benefits from CBST not only because of the rise of the operating SINR point, but also from the better rank condition of the joint channel matrix resulting from non-collocated base stations .
Similar to multi-user MIMO, block diagonalization (BD) [8, 9] may be applied for CBST as a good compromise between complexity and performance. In , BD is applied in a multicell scenario in combination with the OCI reduction scheme of . Alternatively, in , a singular value decomposition (SVD) approach is proposed that simplifies the channel estimation requirements at the expense of a performance degradation.
In this article, we focus on BD-based CBST with different power constraints at the transmission side, with the aim of maximizing the weighted sum rate (WSR) of the users in a cellular network. A first reasonable assumption for power constraints is to consider that each base station (BS) has restricted its total transmission power; this was used, for example, in [6, 10, 12]. Alternatively, per antenna constraints may be more realistic, since each transmission antenna is usually driven by its own high-power amplifier .
In this article, we consider both per base station and per antenna restrictions. For each of them, we will formulate the optimization problem and derive two power allocation schemes that resemble the well-known waterfilling (WF) distribution. While WF is known to achieve capacity in single-user frequency-selective transmission , modified versions of WF also give the capacity-achieving power allocation in multi-user communications [15, 16]. In , a scaled WF (SWF) scheme is heuristically proposed for the case of per base station power constraints to avoid a lengthy numerical optimization. However, its performance is not discussed nor compared to optimal approaches. In , a BD scheme denoted as JT-decomp is proposed where the powers are assigned to the users' transmissions with the only aim of insuring that per base station power constraints are fulfilled. No optimization is performed on the transmit powers to maximize the achievable rates. Consequently, the obtained rates are lower. Also, some partial results of the study shown here, again only for the case of per base station power constraints, have been presented in . We will show that the schemes that we are proposing, although suboptimal, perform very close to the optimum power allocation--obtained by numerical convex optimization--with a reduced complexity.
In brief, the innovative contributions of this article are the following. We develop closed-form and implementable solutions for the power allocation in a BD-based CBST system with realistic power constraints at the transmission side. These solutions are not empirical, but they are obtained, starting from the optimal allocation, using only few approximations that allow us to understand why they perform close to the optimum. In the case of per base station power constraints, one of our proposals gives the same result as SWF , while the others are new. We show also that our approaches reduce dramatically the complexity with respect to the optimal search. Moreover, we consider also the effect of errors in the channel estimation and of a time-varying channel, in which the use of outdated channel state information due to the feed-back delay reduces the achievable rates.
The remainder of this article is structured as follows. In the next section, the system model is presented; in the "Constrained optimization and optimal power allocation" section, the optimization problem is described; while in the "Waterfilling distributions for suboptimal power allocation schemes" section the proposed power allocation schemes with per base station and per antenna constraints are developed. The "Numerical results" section discusses some performance results and the "Complexity" section explores the complexity of the proposed solutions. The article concludes with some concluding remarks.
Notations: In this article, the following notations will be used. Boldface symbols will be used for matrices and vectors, while italic letters will be used for scalars. Superscripts T and H denote the transpose and the Hermitian transpose of a matrix, respectively; superscript * refers to an optimal solution; [·]+ denotes the maximum between zero and the argument; and ||·||| F denotes de Frobenius norm of a matrix.
The system model assumes a coordinated transmission downlink scenario, where M base stations serve N users. Each base station has t transmit antennas, and each user has r receive antennas. Although our analysis is general, the performance will be illustrated for BS-user pairs; therefore the case M = N will be considered in the "Results" section.
Assuming narrowband transmission (if the channel is frequency selective, it can be decomposed into a number of parallel non-interfering subchannels, each experiencing approximately frequency-flat fading), the channel may be modeled by a Nr × Mt channel matrix H where each matrix coefficient represents the gain from each transmit antenna in the BS to each receive antenna at the user side.
where b = [b11,..., b1r,..., b Nr ] T , b ij represents the j th symbol for user i transmitted with power P ij , the precoding matrix is defined as W = [w11,..., w1r,..., w Nr ] and are the precoding vectors.
Constrained optimization and optimal power allocation
Under the BD-CBST strategy, it can be observed from (9) that the overall system is then turned into a set of parallel non-interfering channels. Therefore, the achievable rate of user i is .
where the values , can be seen as indicating the priorities of the users: the closer α i is to 1, the higher the priority given to user i. In the particular case of α i = 1/N, for all i, the solution of the above problem maximizes the sum rate.
for each BS k = 1,..., M.
Maximizing a weighted sum of the rates R i under any of the two proposed constraints is a convex problem, since the logarithmic function is concave in the power assignments: the additional operation preserves concavity, and the constraints (11) are linear. Therefore, the optimal solution may eventually be derived by numerical convex optimization techniques [18, 19]. However, closed-form solutions, even if suboptimal, are highly preferable, to reduce the computational time and resources required by the CBST for the power allocation. Thus, we approach a closed-form solution of the problem by applying the Lagrange duality theory.
In order to solve the problem (10) with constraints (11) or (12), we can introduce a Lagrangian Λ(P, μ ) where P is the vector collecting all the powers P ij , i = 1,..., N, j = 1,..., r, and μ contains the (non-negative) Lagrange multipliers.
and μ= [μ1,..., μ M ].
where the vector of Lagrange multipliers μ*, which defines , should be chosen so that each set of power constraints is satisfied. It can be observed that in both cases, the solution resembles the well-known WF distribution. However, here the waterlevel is given by , that is, the waterlevel is different for each symbol j to be transmitted to each user i.
We have obtained an expression for the power allocation that is still highly complex. However, this procedure gives us an insight on how to build alternative simplified schemes based on the same idea of the well-known WF. Although suboptimal, they may perform close to the optimal solution, with the advantage of a much lower optimization burden.
Waterfilling distributions for suboptimal power allocation schemes
where KMWF must be found to fulfill the constraints (11) or (12). This corresponds again to a WF distribution with variable waterlevel. However here, and unlike the optimal solution in (20), in the variable waterlevel, we have decoupled the term containing the Lagrange multiplier KMWF from and α i . That is, the problem reduces to finding the only unknown value KMWF in (26), while the variability in the waterlevel is confined to the known parameters and α i . This can be solved with the same type of algorithms that solve standard WF .
where again KWF must be found to fulfill the constraints (11) or (12). This corresponds to a WF distribution with the waterlevel modified only by the user priorities. In particular, for equal priorities, α i = 1/N, which corresponds to a standard WF.
In this section we compare the performance of the proposed modified waterfilling (MWF) of (26), waterfilling (WF) of (27), and the optimum solution found by numerical convex optimization (CVX) . For the sake of comparison, we also include the rates achieved when using the scaled WF (SWF) proposed in  for per base station constraints, the results of , also for per base station constraints, and a uniform power distribution (UP). In the case of UP, the power allocated to each user stream is the same and corresponds to the maximum value that fulfills either constraints (11) or (12).
In the following subsection, we analyze the achievable rates for each scheme in a simple scenario to understand how close they perform without the influence of the fading model. Then, in the subsequent subsections, we analyze a more realistic scenario with the effects of imperfect channel estimation and of the feed-back delay in a time-varying channel, which outdates the current channel with respect to the one used for precoding and power assignment.
It is also interesting to analyze with more detail the behavior of UP in Figures 2 and 3 for t = r = 2 and t = 2, r = 1. CBST is transmitting as many data streams per user as the number of receive antennas r (10), each multiplied by the elements of the diagonal matrix after the compound effect of transmit, channel, and receive processing. This means that one stream for r = 1 and two streams for r = 2 are transmitted using, therefore, 1 or 2 values of per user i. For each user, in these channel conditions, one of these values is generally considerably higher than the other, and so sharing the transmission power between two streams (r = 2) in the case of UP results in a waste of power that renders a lower rate than just using the entire available power in one stream (r = 1). One illustrative example from a particular channel realization: for r = 1 we have ,, while for r = 2 we obtained , ,,. This is a well-known effect leading to the dominant eigenmode transmission concept described in .
Effect of an erroneous or outdated channel estimation
In the results of previous subsection, we assumed that the channel was perfectly estimated at each receiver and instantaneously fed back to the base stations so that BD insured that perfect cancelation of the interference was achieved. However, the channel is usually estimated at the receivers using the information conveyed by pilot symbols, and this estimation will normally be corrupted by additive white Gaussian noise (AWGN). Moreover, sending the estimated channel state information (CSI) to the base stations will require some time, and therefore a delayed version of the estimated CSI will be available there. If BD is performed with erroneous or outdated CSI, then the diagonalization will not be perfect and some interference will remain. The power will be subsequently allocated using the wrong estimates. With the results shown in the subsequent figures, we discuss theses two effects.
which is coincident with the normalized MSE of  where we can see [, Figure 4] that values of MSE lower than 10-1 can be achieved for operational numbers of antennas and signal-to-noise ratio (SNR) values.
In this definition of SNRs, only the effect of path loss is included, not of the shadowing or fading, according to . Also, it should be noted that the receivers placed closer to the BS will experience a higher value of SNR. Figures 6, 7, and 8 show the CDF of the rates achieved in this scenario when t = r = 1, t = r = 2, and t = r = 4, respectively. The SNR at the cell boundary is 18 dB. The parameter D indicates the delay between the actual CSI and the CSI being used for the BD. That is, the CSI is outdated by a delay D with respect to ideal CSI. In the figures, we show the performance for different values of D conveniently normalized with respect to the channel coherence time (T c = 1/f D ). We can observe that the delay must be very small compared to the coherence time of the channel to cancel effectively the interference. When D = 0.001T c , the degradation of the rates is already substantial. The degradation is more accentuated when transmitter and receiver have a higher number of antennas. Actually, with D = 0.005T c , the advantages of increasing the number of antennas are lost, and the performance is basically the same for all the number of antennas considered in these figures.
A value D = 0.001T c is in line with the feed-back delay used in  to evaluate the performance of closed-loop MIMO systems (D = 0.1 ms with T c = 167 ms, so D = 0.0006T c ), while smaller values of delay seem infeasible in practice. Therefore, these results confirm that the use of outdated CSI can seriously degrade the performance of BD, and therefore efficient feed-back mechanisms must be designed, which are beyond the scope of this article. We can note that UP and WF suffer approximately the same degradation (and also MWF not shown), while CVX is more prone to the effects of the outdated CSI, which makes sense, since it strongly relays on the channel information to optimize the power allocation.
The optimum power distribution can be obtained through a convex optimization procedure, while WF approaches allow a much reduced complexity at the expense of some performance degradation. In this section, we examine the difference in terms of complexity between both approaches for the power optimization procedure.
Since the power is distributed over Nr user transmissions, the complexity does not depend on the number of transmit antennas or base stations (as long as Mt ≥ Nr as required for BD). Therefore, the complexity of WF, MWF, and CVX does not increase with the number of antennas per BS, which is a preferable characteristic, since often t > r in practice.
A thorough comparison of complexity of the methods is not easy, since the optimization procedures are adaptive with a number of operations that can vary according to the channel realization. In general, the convex optimization by using interior-point methods implements a Newton search with a number of iterations, which is slightly dependent on the problem size, and in most of the cases can be considered limited to few tens, while inside each Newton iteration, the complexity is dominated by the determination of the so-called Newton step which has a complexity order of about (Nr)3/3 . For the WF, again, we can have a number of iterations variable with the channel conditions and the required accuracy; however, a theoretical number of operations for each iteration is on the order of Nr log(Nr). In the specific case of modified WF, the search procedure cannot be optimized as in the WF, because of the variable waterlevel, and the complexity saving with respect to the convex optimization is lower.
We have to keep in mind that the specific code implementation of the convex optimization and WF will have an impact on the measured times, and so what we give here is just an idea of their relative execution times. Having said that, we can observe that both WF and MWF are always more than three orders of magnitude faster than CVX. More specifically, WF is between 6,000 and 12,000 times faster, while MWF is between 1,000 and 2,000 times faster. Therefore, even if they are suboptimum, WF and MWF make a good choice in terms of the balance between complexity and achievable rates.
We have proposed two power optimization schemes (WF and MWF) for the CBST downlink based on BD with different transmit power constraints. Both are derived with a technique similar to the WF distribution: the first (WF) has the lowest complexity and reduces to the standard WF if the user priorities are the same. In the case of per base station constraints, it achieves the same performance as SWF of . The second (MWF) shows a better performance, more noticeable for the case of per antenna constraints, with a small increase in complexity. They both perform close to the optimal solution. However, the optimum can be derived only by resorting to the numerical solution of the convex optimization problem, with a heavy computational complexity, much higher than the proposed schemes of WF and MWF. Also, the degradation in terms of mean rates caused by imperfect channel estimation is small for reasonable values of the MSE of the channel estimation. However, our simulation results confirm the need of a fast feed-back of the estimated CSI to the base stations to avoid a severe degradation of the rates.
We have observed that the rates achieved with the more realistic per antenna constraints are lower compared to a per base station one. In general, the proposed schemes allow us to obtain the capacity improvements of MIMO, canceling the high amount of interference which characterizes cellular environments.
In , it is shown that, in the context of the Broadcast Channel, the performance of BD gets very close to DPC with a proper selection of the user scheduling. In further study, we will cope with the joint optimization of the power allocation, the precoding scheme, and the user scheduling.
a We assume, without loss of generality, that all the base stations have the same maximum available power.
b Again, we assume, without loss of generality, that all the antennas in all the base stations have the same maximum available power.
This study has been partly funded by projects TEC2008-06327-C03-02, CCG10-UC3M/TIC-4620, and CSD2008-00010.
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