EURASIP Journal on Wireless Communications and Networking 2004:2, 236–247 c 2004 Hindawi Publishing Corporation
A Hybrid Approach to Spatial Multiplexing in Multiuser MIMO Downlinks Quentin H. Spencer Department of Electrical and Computer Engineering, Brigham Young University, 459 Clyde Building, Provo, UT 84604, USA Email:
[email protected]
A. Lee Swindlehurst Department of Electrical and Computer Engineering, Brigham Young University, 459 Clyde Building, Provo, UT 84604, USA Email:
[email protected] Received 18 September 2003; Revised 14 May 2004 In the downlink of a multiuser multiple-input multiple-output (MIMO) communication system, simultaneous transmission to several users requires joint optimization of the transmitted signals. Allowing all users to have multiple antennas adds an additional degree of complexity to the problem. In this paper, we examine the case where a single base station transmits to multiple users using linear processing (beamforming) at each of the antenna arrays. We propose generalizations of several previous iterative algorithms for multiuser transmit beamforming that allow multiple antennas and multiple data streams for each user, and that take into account imperfect channel estimates at the transmitter. We then present a new hybrid algorithm that is based on coordinated transmit-receive beamforming, and combines the strengths of nonorthogonal iterative solutions with zero-forcing solutions. The problem of distributing power among the subchannels is solved by using standard bit-loading algorithms combined with the subchannel gains resulting from the zero-forcing solution. The result is a significant performance improvement over equal power distribution. At the same time, the number of iterations required to compute the final solution is reduced. Keywords and phrases: MIMO, multiuser, downlink, beamforming, SDMA, spatial multiplexing.
1.
INTRODUCTION
Multiple-input multiple-output (MIMO) wireless communications systems have attracted considerable attention recently because of their potential for improved reliability and capacity compared to single-input single-output (SISO) systems. The initial focus of research has been on point-to-point channels, but there is now increasing interest in multiuser MIMO channels, since many current applications involve multiuser networks. An excellent overview of recent results and open problems in the area of MIMO communications, including both the single-user and multiuser cases, can be found in [1]. The simplest multiuser MIMO case involves a base station (such as in a mobile telephone network or wireless LAN) equipped with an array of antennas that are used to communicate with a group of users, each of which has only one antenna [2, 3, 4, 5]. In this paper, we consider the more general and complex case where potentially all users have multiple antennas. We focus specifically on the downlink or broadcast channel where the base is simultaneously transmitting to a group of users. For simplicity, we consider a single base station.
Finding the theoretical limits of such a channel is still an area of active research [6, 7]. However, it is known that reaching these limits depends on a concept in information theory referred to as “writing on dirty paper” [8]. This concept refers to the fact that, given a channel consisting of additive noise and interference, if the interference is known exactly to the transmitter, the capacity is the same as in the channel with the interference absent. This has an obvious application to the multiuser communications problem since, if the transmitter has information about the channel transfer functions available in advance, it also knows the interuser interference it is causing when simultaneously transmitting to other users [3, 4, 5, 6, 7]. However, dirty paper coding requires a new class of codes that are interference-dependent. While of significant theoretical interest, this type of scheme is very complex and requires the use of codes that may be incompatible with existing communication systems and protocols. A suboptimal but simpler approach to this problem is to assume that each user’s data is encoded independently of the others, allowing the use of traditional modulation and coding schemes. In order to minimize hardware requirements, a common assumption in this situation is that no multiuser
Spatial Multiplexing in Multiuser MIMO Downlinks detection (MUD) is used at the receivers, and all interference due to signals intended for other users is treated as noise. This approach requires that the interference produced by the transmitter be balanced among all of the users in order to meet the quality of service (QoS) requirement of each. Like the dirty-paper approach, it also requires channel knowledge at the transmitter. The solutions to this problem that have been proposed to date can be put into two categories. The first consists of zero-forcing (ZF) solutions, where all interuser interference is driven to zero. This concept is introduced for receivers with only one antenna in [2, 3], and was first studied for multiantenna receivers in [9]. This problem has a straightforward closed-form solution when the transmitter’s array has proper dimensions [10, 11, 12, 13], while for other cases, it can be computed iteratively [14, 15]. Because these algorithms create a set of orthogonal subchannels, allocation of power to each subchannel can be done independently as a second step. This allows for flexibility in adapting the ZF solutions to solve multiple optimization problems, such as power-constrained throughput maximization or rate-constrained power minimization. The second category of algorithms are referred to here as “interference-balancing” (IB), since they attempt to trade off the interference against the additive noise in the system rather than forcing it to be identically zero. Since the resulting beamforming vectors are not necessarily orthogonal, adjusting the power transmitted to one user changes the amount of interference experienced by the other users. As a result, each optimization problem (power minimization, rate maximization, etc.) must be treated separately, and the solutions are in general iterative. For receivers with only one antenna, the optimal solution for minimizing power given a set of SINR constraints was first presented in [16, 17]. The same solution can also be computed using semidefinite matrix optimization methods [18]. More recently, solutions have been proposed for other optimization problems, such as maximization of the sum rate given a total power constraint, or the maximization of the SINR margin for all users given a minimum SINR requirement for each user and maximum total power [19]. For the case of single-antenna receivers, these algorithms are optimal among the class of linear processing methods, although nonlinear solutions such as dirty-paper coding can offer improved performance. Another related solution of interest is the “regularized channel inversion” approach of [4], a noniterative algorithm where the transmitter uses a structure similar to a minimum meansquared error (MMSE) receiver. Like the other IB solutions, it takes noise into account, and has been shown to outperform channel inversion (the ZF solution for single-antenna receivers). If we allow the receivers to have multiple antennas, the problem is much more complex, and finding a solution with globally minimal power or maximum rate is still an open problem. One approach is to assume that only one data stream is transmitted to each user, and jointly optimize the transmit and receive beamformers [20]. A jointly optimized beamforming approach has also been proposed for singleuser channels with multiple data streams, where the QoS re-
237 quirement for each data stream is known in advance [21, 22]. In this paper, we assume multiple antennas for each user and allow multiple data streams per user. The transmission of multiple data streams to a particular user is allowed if the user’s channel has sufficient rank, but the only QoS constraints we assume to be in force are the total rate (i.e., sum of the rates of all data streams) transmitted to that user and a specified bit error rate. We wish to achieve the QoS constraints for each user while minimizing the transmitted power. To solve this problem, we propose a new hybrid algorithm that combines ideas from the ZF and IB solutions. We first show how the IB algorithm of [16, 17, 18] can be generalized to coordinated beamforming such as in [20], and how it can make use of information about the accuracy of noisy channel estimates. We then describe a hybrid algorithm that begins by using a ZF approach to generate initial estimates of the receive beamformers and the subchannel gains. Given the subchannel gains, a bit-loading algorithm is then employed to determine the optimal power allocation. By using this information to initialize an IB algorithm, a solution is obtained that converges more quickly and achieves the QoS requirements with lower power than other approaches. We also show some simplifications that arise in the special case where only one data steam is transmitted to each user, and which allow a further reduction in computational cost. The paper will proceed as follows. Section 2 outlines the channel model and notation. Section 3 provides a brief summary of the generalized ZF (GZF) algorithm of [14, 15] and the IB power minimization algorithm of [16, 17, 18]. Generalizations to the IB algorithm are discussed in Section 4. The hybrid ZF/IB algorithm is then presented in Section 5, and Section 6 presents some simulation results that illustrate its performance. 2.
PROBLEM DEFINITION
Consider a narrowband multiuser downlink channel with K users and a single base station. The base has nT antennas, and the jth receiver has nR j antennas. The totalnumber of antennas at all receivers is defined to be nR = nR j . We will use the notation {nR1 , . . . , nRK } × nT to represent such a channel (as opposed to writing nR × nT as in a point-to-point MIMO channel). For example, a {2, 2} × 4 channel has a 4antenna base and two 2-antenna users. The channel matrix from the base to the jth user is denoted by H j , the associated transmit beamforming matrix by M j , and the transmitted data vector d j , which has dimension m j . The signal at the jth receiver is thus xj =
K
H j Mi di + n j .
(1)
i=1
The receiver’s estimate of the transmitted data is determined using the beamforming matrix W j : j = W∗ H j M j d j + d j
i= j
H j Mi di + n j .
(2)
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If H j has rank L j , we assume that m j ≤ L j . The total number of data streams or subchannels allocated by the transmitter is m = j m j . Any transmission method that attempts to cancel out all interuser interference will have the requirement that m ≤ nT , since a beamformer using nT antennas can null out at most nT − 1 interfering signals. For IB methods, it is theoretically possible to accommodate values of m > nT , but such an approach would likely require additional temporal coding. Since we do not consider this case, we take nT to be a practical upper bound on m for all the algorithms considered here. Clearly, if the number of users is larger than the number that can be supported using spatial multiplexing (SDMA), a likely scenario in cellular or wireless LAN systems, then other multiple access schemes must be used in conjunction with SDMA. An important question is how to allocate the users among the various available dimensions (space, time, frequency, etc.). For example, if two users are located close to each other or have highly correlated spatial channels, it would make sense to assign them separate time or frequency slots so that excess energy is not used in attempting to multiplex them spatially. Whereas finding solutions to this resource allocation problem is beyond the scope of this paper, we introduce it in order to compare the approaches of choosing m j = 1 versus using m j > 1 in channels that can support both. For example, suppose a base has nT antennas and there are nT users, each having two antennas. The single-channel approach could be used to communicate with all nT users simultaneously. On the other hand, the users could be split into two groups of nT /2 users and time-multiplexed (assuming nT /2 is an integer). The transmitter could use the multichannel approach to transmit two data streams to each user, but since only half the time is available, the bit rate constraints would need to be doubled. The optimal allocation of spatial and temporal channel resources in this situation can probably only be known by global search, and the cost function is expensive to compute. This question is studied to a limited extent in Section 6. In the rest of this paper, we assume that such allocation questions have already been answered and focus on spatially multiplexing a single group of users.
ZF transmit beamformers that result in interuser interference. For the moment, we assume that the channel is known perfectly, and consider noisy estimates in Section 4. The fundamental idea behind ZF for downlink transmission is that all interuser interference is forced to zero. The first type of ZF solutions that were proposed did this by forcing Hi M j = 0 for all i = j [10, 11, 12, 13]. This has a straightforward closed-form solution, but it imposes severe restrictions on the array sizes: in most cases, such a solution requires nT ≥ nR . In order to accommodate nR > nT for up to nT users, a generalized form of the ZF solution has been proposed [14, 15]. The larger sizes are accommodated by es j and applying the timating the receivers’ linear combiners W ZF solution to a set of “virtual channels” that represent the transfer function from the transmitter to the output of the receivers’ linear combiners. The virtual channels are defined as ∗j H j , so the ZF criterion becomes W ∗ W i Hi M j = 0 for i = j. We now briefly summarize the GZF solution. Let H j = ∗j H j and W
T T
= H ··· H H j 1 j −1 H j+1 · · · HK T
T
T
.
(3)
∗ The ZF criterion W i Hi M j = 0 is satisfied if M j lies in the (0)
. Define V to be an orthonormal basis for null space of H j j
, and let the null space of H j
(0)
= U(1) U(0) Σ V(1) V(0) HjV j j j j j j
∗
(4)
(0)
, where U(1) and V (1) be the SVD of H j V j j correspond to j the nonzero singular values of Σ j . For user j, mutual information will be maximized (under the zero-interference constraint) when the transmitter uses V(1) j as its transmit vectors, the receiver uses U(1) as its linear combiner weights, and j the gains are allocated by water-filling on the singular val(0)
has been calculated, ues in Σ j [13, 23]. Note that after V j the interuser interference requirement has been satisfied, so other methods of transmitting to user j could be used such as space-time coding or non-ZF beamforming [21], treating (0)
3.
SUMMARY OF PREVIOUS RESULTS
3.1. Generalized zero-forcing The ZF approach to multiuser downlink processing [10, 11, 12, 13, 14, 15] is useful because it decomposes the channel into what is effectively a set of orthogonal SISO channels. Given the set of SISO channels and the gains associated with each one, power can be allocated to the channels to solve either the power control or the throughput maximization problems. ZF applied at the transmit side does not result in noise amplification as it does when used for receive beamforming; instead, a ZF transmit technique applied to a poorly conditioned channel can result in a high required transmit power to achieve the desired performance [4]. However, like the receive case, errors in the channel estimate will produce
as the transfer function of a single-user channel. HjV j
, optimizing the transmitter and receiver for Given H j j) user j is straightforward, but the other users’ receivers (W
must first be known. In turn, calculation needed to form H j of the transmit beamformers M j impacts the optimal values j . This suggests an iterative soof the receive beamformers W j and M j are alternately recomputed. To inilution, where W j is chosen as the first m j left singular tialize the iteration, W vectors of H j , and convergence is quantified by the properties of S=
W∗1 H1 .. . M1 · · · MK . ∗
WK HK
(5)
Spatial Multiplexing in Multiuser MIMO Downlinks
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Under the ZF criterion, S should converge to a diagonal matrix. In practice, however, the iterations are terminated when the off-diagonal elements are sufficiently small (as determined by the level of interuser interference that can be tolerated). In our simulations, we have set the threshold to 20 dB below the noise level to ensure that the effects are minimal, and in our experience, the algorithm usually converges in a few steps. Note that when the dimensions of the arrays satisfy the conditions for the closed-form ZF solution of [10, 11, 12, 13], the GZF solution described here achieves a perfectly diagonal S after the first iteration, and the two solutions can be shown to be equivalent.
case, it is reasonable to assume that the statistics of the esti j are known. In the literature, this problem has been mate H studied for some cases when nR j = 1. In [24, 25], it is assumed that either the mean or covariance of h j is known, but not both. In [18, 26], robust beamforming vectors are designed by using best and worst cases of the covariance matrices R j = E[hTj ∗ hTj ]. Here we take a slightly different approach, assuming that which we describe as we have an estimate of the channel H, the sum of the true channel and an error term: + N. H=H
(8)
3.2. Interference balancing The term IB is used here to refer to a general class of algorithms that take the effects of noise into account and allow for a manageable level of interuser interference. As an example, we describe the power minimization algorithm of [16], which was designed for the special case where all users have one antenna (nR j = 1). In such a case, the channel matrices are row vectors, which we will denote as hTj . When the channels are all known to the transmitter, the SINR at the jth receiver (γ j ) can be calculated by the transmitter as a function of the channels and the transmit vectors m j : γj = i= j
T 2 h m j j . T 2 h mi + σ 2 j
(6)
j
transmit vectors as m j = λ j u j , the product of a real scalar and a unit-norm vector. The algorithm of [16] attempts to minimize the total transmitted power ( λ j ), while satisfying (6) at the specified SINR or higher:
γ j λk uk R j uk ≥ γ j σ 2j .
n = vec(N),
(7)
∗
E nn
(9)
= Rc ⊗ Rr ,
(10)
where vec denotes the column stacking operation, ⊗ the Kronecker product, and Rc , Rr are covariance matrices corresponding, respectively, to the column and row statistics of N. Equation (10) indicates that these statistics are assumed to be separable:
Since the QoS requirement for each user can often be directly mapped to an equivalent SINR (γ j ) given the available signal and code designs, the approach is to find an optimal set of transmit vectors m j that satisfy all of the γ j requirements with minimum power. Define R j = hTj ∗ hTj and represent the
λ j u∗j R j u j −
We assume that the error term is zero mean with covariance
E [N]i, j [N]∗k,l = Rc
j,l
Rr
i,k .
(11)
Another way of interpreting this model is that it implies that N can be decomposed as N = Rr XRc∗ ,
(12)
where X contains independent, zero-mean, unit-variance, complex entries, and Rc , Rr are the square roots of Rc and Rr , respectively. The above model does not account for all possible cases, but it is appropriate for many situations encountered in practice, such as when H is estimated from training data. Let T be the matrix of training samples so that the received training signal Q is
j =k
Q = HT + Y, Related algorithms for solving this problem can be found in [17, 18]. In the next section, we focus on generalizations of the IB problem. 4.
GENERALIZATIONS OF INTERFERENCE BALANCING
To begin, we generalize to the case where H j is a matrix (i.e., multiple antennas at each receiver) and a single beamformer w j is used at the receiver (i.e., one data stream per user). We also assume that the estimate of H j is not perfect. If H j is obtained by feedback from the receiver, the primary causes of channel estimation error will be the finite length of the training signals, and time variation in the channel. In either
(13)
where Y represents an additive noise term. Then the least = QT∗ (TT∗ )−1 , and the error squares estimate of H is H ∗ in the estimate is YT (TT∗ )−1 . If the elements of Y are uncorrelated, then Rr = ρInR for some ρ that depends on noise variance and the training data length, and Rc = TT∗ (since T is a design parameter, it can be chosen such that TT∗ = ξInT ). One situation that is more likely to produce colored estimates of H is when significant time variation in the channel is present. If the statistics of the time variations are separable in time and space (not an unreasonable assumption), then (12) will hold in this case as well. Let d j be the symbol transmitted to user j. If user j uses a predetermined unit-norm beamforming vector w j , then the
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estimate of the transmitted symbol at the receiver will be
dj = w∗j
K
H j mi di + n j
i=1
j + N j m j d j + w∗j H j + N j mi di + w∗j n j . = w∗j H i= j signal (dj(sig) ) noise (dj(noise) )
requirements for each subchannel have been specified; that is, let γ j,k represent the SINR required for the kth data stream and let w j,k be the corresponding column of W j for user j. If we define
∗j w j,k w∗j,k H j + nT w∗j,k Rr j w j,k Rc j , R j,k = H
(17)
the SINR is thus
(14) The transmitter views the “channel” as being the transfer function from the output of the transmitting array to the output of the receive beamformer w j , and the covariance of this channel is
∗j w j w∗j H j + E N∗j w j w∗j N j R j = E H∗j w j w∗j H j = H
∗j w j w∗j H j + nT w∗j Rr j w j Rc j . =H
(15) Given R j , and assuming the noise vector n has covariance σn2 I, the SINR for user j becomes E d2j(sig)
λ j u∗j R j u j γj = 2 . = 2 σn + i= j λi u∗i R j ui E dj(noise)
(16)
The result is that the problem is now identical to that posed in (7), except that the definition of R j now includes information regarding channel estimation error and user j’s beamformer. Solving this system of equations would require of course that the receiver weight vectors w j be known. Since the transmitter knows what the statistics of the received signal at each user will be, it can predict w j for some receiver structures (e.g., MMSE, maximal ratio combining (MRC), etc.). If the transmitter first guesses an initial set of w j vectors, it is then possible to calculate all R j and solve (7) for the m j vectors. The transmitter’s estimate of each w j can then be updated according to the known receiver structure. Repeated alternating calculation of w j and m j will reduce the required transmitted power until it converges to a minimum [20, 21, 27]. There are situations where (7) does not have a solution for the given values of γ j (see [16, 17, 18]). Care must be taken in choosing the initial set of w j vectors because it is possible to choose an initial vector that does not lead to a solution, even when one exists. Because the problem can be cast as a semidefinite optimization with convex constraints [18], any valid initial values for w j will lead to the same solution, but a poorly chosen initialization point can increase the number of iterations required for convergence. In our simulation results, we test two possibilities: using the left singular j , or computing the GZF solution and using the vectors of H w j vectors that result. The coordinated beamforming scheme described above can be generalized one step further to accommodate the transmission of multiple data streams per user, assuming the channel dimensions and rank allow it. Assume that the SINR
γ j,k =
σn2 +
λ j,k u∗j,k R j,k u j,k . ∗ j =l, k=m λl,m ul,m R j,k ul,m
(18)
Given a set of SINR requirements γ j,k , the same iterative scheme of alternatively updating the transmitter and receiver vectors can be applied. The challenge in this case is determining the optimal SINR requirements. A simple solution would be to divide the rate requirement for each user into m j channels with equal SINR, but this could easily result in a situation where subchannels with low gain have unusually high amounts of power forced into them (especially in highly correlated channels). In iterative IB algorithms, this results in more interference for other channels to deal with, and therefore solutions that require higher total power, as will be seen in the simulation results. If the gains of each of the spatial subchannels are known, a better power distribution among the subchannels could be calculated using water-filling. However, the gains of the subchannels cannot be known until the iterative power minimization is computed, and the solution will change if the values of γi, j are changed. Rather than attempting an additional iteration loop, we propose using the ZF solution as a means to estimate γi, j . This hybrid algorithm is discussed in the next section. 5.
HYBRID ZERO-FORCING/INTERFERENCEBALANCING ALGORITHM
We propose as an alternative approach a hybrid between the GZF and IB power minimization algorithms. The basic idea of the algorithm is the following: the GZF solution is found as the first step, the resulting subchannel gains are used to compute a power distribution among the subchannels for j matrices that result are used as the inieach user, and the W tial values for the IB iteration. The GZF solution does not provide the optimal power distribution, but in practice, it provides a reasonable estimate, and using it as an initialization significantly reduces the number of iterations in the IB step. Since the GZF solution is itself iterative, a simplified ap j for the same proach would be to use the SVD of each H purpose: use the left singular vectors to initialize W j and use water-filling on the singular values to estimate the power distribution. The problem with using either the SVD or GZF initializations to determine power allocation is that the subsequent IB step will change the gains for each subchannel, making any previous power distribution suboptimal. However,
Spatial Multiplexing in Multiuser MIMO Downlinks it is important to note that the classic water-filling solution assumes an infinite granularity in the signal constellations available for transmission. In practical applications, the system designer is likely constrained to a predefined discrete set of available constellations and code designs, and must resort to “bit loading” [28, 29, 30, 31]. For the hybrid algorithm, the consequence of this fact is that there are a finite number of bit allocations, which can be mapped directly to SINR requirements for each channel. This makes the approach of estimating the power distribution without an outer iteration loop feasible. We briefly explain the bit-loading algorithm used here. Bit loading is a well-studied technique that is most well known for its application to multicarrier modulation schemes, but that has recently been applied to the singleuser MIMO channel [32, 33]. Solutions have been proposed for both power minimization and rate maximization. We focus on power minimization here, which means determining the optimal bit distribution over the available subchannels, given the subchannel gains and a total required transmission rate and bit error rate. The original algorithm for this problem is described in [28, 29] and requires O(L log L) computations for L available channels. Faster algorithms have since been proposed [30], but for relatively small arrays, the cost will be similar for all algorithms. The algorithm operates as follows: let Pn,l be the power required to transmit n bits through the lth subchannel given the SINR of the subchannel, and define P0,l = 0. Initialize all channels to zero bits, and repeat the following steps until the desired total rate is achieved. (1) Compute ∆Pl = PN+1,l − PN,l , where N is the number of bits currently assigned to subchannel l. (2) Let k = arg minl=1,...,L ∆Pl , increment N for channel l, and recompute ∆Pk . Note that this method could also work for a code with noninteger rate R, as long as there exist codes of rate 2R, 3R, and so forth, up to the maximum possible rate. The remaining problem that has not been addressed is choosing m j . Like the problem of power allocation, the optimal choice of m j will depend on the interaction between all the users’ channels and their SINR requirements, and is difficult to find without an exhaustive search. This has been demonstrated by the experimental results in [34]. One way of estimating m j has recently been proposed [35]. In the simulation results presented later, we have used examples where the number of transmit antennas is large enough to assign m j = nR j . The following is a summary of the complete proposed hybrid algorithm: Hybrid ZF/IB algorithm (1) Given R j and m j for each user, compute the generalized iterative ZF solution and use bit loading to determine the subchannel power distribution subject to meeting the rate constraints. Let γ j,1 , . . . , γ j,m j represent the resulting subchannel SINRs for user j and let w j,l be the lth column of W j .
241 ∗j w j,k w∗j,k H j +nT (w∗j,k Rr j w j,k )Rc j . Find (2) Define R j,k = H
the unit vectors u j,k and power coefficients λ j,k such that λ j,k is minimized and λ j,k u∗j,k R j,k u j,k −
j =l, k=m
γ j,k λl,m u∗l,m R j,k ul,m ≥ γ j,k σ 2j,k , (19)
using, for example, an available algorithm from [18]. (3) Repeat until convergence (optional). (i) Recalculate the predicted receiver weights according to the algorithm used at the receiver. For MMSE, this is j w j,l = H
−1
∗j + σ 2j I λ j,k u j,k u∗j,k H
j u j,l , H
(20)
j,k
assuming that the noise at the receiver is spatially white with variance σ 2j . If the receiver has an MRC structure, the weights are j u j,l . w j,l = H
(21)
In either case, the weight vectors must be normalized so that w∗j,l w j,l = 1. (ii) Recalculate u j,k and λ j,k for all j and k (repeat step (2)). (iii) Let Pn be the total transmitted power for iteration n. If
10 log10
Pn−1 Pn
< dB,
(22)
for some , stop. The problem in step (3) is convex, so it is guaranteed to converge. The reason that step (3) can be considered optional is that after step (2) is completed, a feasible solution exists that satisfies all constraints. In the simulation results that follow, the reduction in power due to step (3) is often small, so some performance could be sacrificed to reduce the computational cost. An important special case of the hybrid algorithm is when m j = 1 for all users. This can be a natural consequence of channel dimensions or rank, but it may also be introduced by design in order to simplify the system. Furthermore, while it has not been studied specifically for the downlink case, studies on uplink channels [36] have shown that coordinated beamforming approaches the maximum available throughput of the channel as the number of users becomes very large. We discuss this case separately here because there are some natural simplifications that can substantially reduce the cost of the hybrid algorithm. The first simplification is in the ZF step used for initialization. When all users are limited to one data stream, the block ZF step of (3) and (4) reduces to a simple pseudoin-
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verse: MS = HS (the ZF solution in [3] is computed this way). In the case where there are no channel estimation er j m j d j + n j , simplifies rors, the received signal for user j, H to u j β j d j + n j , where u j is the first column of U j and β j is the first singular value of B j . The optimal receive beamformer for user j is u j , which is equivalent to MRC (note that this does not hold when m j > 1). Thus is it possible to j m j and avoid computing an SVD to find w j . define w j = H However, in order to repeat this process iteratively without causing the rows of HS to diverge, it is necessary to normalize w j at each step. With multiple data streams, the generalized ZF solution also provides an improved power distribution among the subchannels for each user, which is no longer needed in this case. In this simplified version, we use the first j as H j to initialize the algorithm. The fast initializarow of H tion, which replaces step (1) in the hybrid algorithm is summarized below. Fast initialization for hybrid ZF/IB (1) Initialization. For j=1, . . . , K, set w j = [1 0 · · · 0]T . (2) Repeat until convergence. (i) HS =
1 w1∗ H .. . .
K wK H ∗
(23)
†
(ii) MS = HS . j m j / m∗j H ∗j H jmj. (iii) For j = 1, . . . , K, set w j = H Step (2) does not necessarily converge to a diagonal solution for HS MS , but it generally converges to a diagonally dominant solution in just a few iterations. For situations where computational cost is critical, one could use a “fixedcost” approach, where step (2) is repeated a fixed number of times and step (4) is omitted. This results in a feasible solution that comes at a known cost. 6.
SIMULATION RESULTS
In this section, we present simulation results in three different areas. We first present performance data comparing the hybrid ZF/IB algorithm with other ways of initializing IB algorithms. We then test the generalized power minimization algorithm in the presence of noisy channel estimates. Finally, we include performance results on allocating multiple data streams per user when the option is available. For simulation results that involve bit loading, we assume that the required rate for each user is an integer number of bits/use. The available set of signals are the QAM constellations from 1–8 bits/symbol (including BPSK and QPSK as special cases). The power requirement is based on the upper bound on the symbol error rate from [37, equation 5-2-80]:
PM ≤ 2 erfc
3k γb , 2(M − 1)
(24)
where k is the number of bits, M = 2k , and γb is the average SNR per bit. The required bit error rate is fixed at 10−5 , assuming no additional coding. Except where otherwise noted, the channels used in the simulations were randomly generated from an uncorrelated complex Gaussian distribution. 6.1.
Hybrid ZF/IB performance
We first compare the hybrid ZF/IB algorithm with three other algorithms. The first is the ZF algorithm alone (without any subsequent IB). To maintain a fair comparison, we use bit loading to determine the power distribution among the subchannels instead of the water-filling approach used in [13]. The second is IB with the SVD initialization described in Section 5, and is referred to in the plots as “SVD/IB.” The third algorithm uses IB with equal power distribution among the subchannels and the SVD initialization. We also include results for the hybrid solution and SVD initialization after only one iteration of the IB step, which yields a feasible solution with reduced computation. We consider two scenarios. The first involves a {3, 3} × 6 channel, where the channel coefficients are complex Gaussian, uncorrelated at the transmitter, but correlated at the receivers. The receiver interelement correlation matrix was a Toeplitz matrix with [1 0.5 0.25] as its first row. The two users are assumed to have equal gains. The second scenario is a {1, 2, 3} × 6 channel, where the channel coefficients are also complex Gaussian, but this time they are all uncorrelated, and each user is subject to a channel attenuation randomly varying between 0 and 10 dB. In both cases, the rate requirements for the users are integers chosen randomly in the range of 2–8 bits/use per user. These same channel dimensions, attenuations, and transmission rates are used in Figures 1, 2, 3, and 4. In these simulations, we do not include channel estimation error, so Rc and Rr are both zero. Figure 1 shows the total power relative to noise required to transmit at the requested QoS for both scenarios. The hybrid algorithm achieves a consistent performance improvement over all other solutions. The difference is particularly dramatic in the two-user case where the receivers are partially correlated. This is to be expected, since this would lead to a more uneven distribution of singular values of the channel matrices. The hybrid solution is the best in both cases, but it has a larger performance improvement over the SVD/IB and ZF solutions in the three-user case. This is likely due to the variable channel attenuations, since the results in [13] show that the ZF solution works best when the users have similar power levels. Figure 2 illustrates the number of iterations required for convergence in the scenarios considered in Figure 1 for the hybrid algorithm, SVD initialization, and equal power distribution. In this plot and other subsequent plots illustrating convergence speed, convergence was defined as occurring when two iterations produce a reduction in transmitted power of less than 0.1 dB. Convergence requires only a few steps in all cases. The advantage of the hybrid scheme in this case is that there are fewer cases that require a large number of iterations.
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Figure 3: Performance comparison of MMSE and MRC receivers for correlated {3, 3} × 6 and uncorrelated {1, 2, 3} × 6 channels.
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Figure 1: A comparison of the power minimization capability of the algorithms for correlated {3, 3}× 6 and uncorrelated {1, 2, 3}× 6 channels.
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Figure 2: A comparison of the number of iterations required for convergence for correlated {3, 3} × 6 and uncorrelated {1, 2, 3} × 6 channels.
Figure 4: Comparison of the required iterations for convergence for MMSE and MRC receivers for correlated {3, 3}× 6 and uncorrelated {1, 2, 3} × 6 channels.
6.1.1. A comparison of MMSE and MRC receivers The hybrid ZF/IB algorithm can be configured for any receiver structure that is a function of information available to the transmitter. Figure 3 compares the performance of the algorithm with MMSE and MRC receivers, using the same two-user and three-user setup as in the previous example. The MMSE receiver performs better in both cases, with an improvement in the correlated channel case of only about half that of the uncorrelated case. The increased cost in power for MRC becomes significant only in a small percentage of cases with difficult channels that lead to higher power requirements. For some applications, this increase in required power may be acceptable in order to reduce computation at the receiver.
Figure 4 shows the number of iterations required for the second stage of the algorithm to converge for both MMSE and MRC receivers. In the uncorrelated three-user scenario, the MMSE structure reaches convergence after the first step in almost all cases. In both cases, the MMSE structure always converges after a few iterations, while MRC requires 10 or more iterations in a small percentage of cases. The relative computational cost per iteration of MMSE and MRC at the transmitter is a function of array size, but these results indicate that while MRC will result in a reduced computational cost at the receiver, the cost at the transmitter can potentially be higher due to an increase in the number of iterations required for convergence.
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Figure 5: Performance comparison of single-channel algorithms.
6.1.2. Single-channel performance In order to examine the performance of the single-beam algorithm, we use a {2, 2, 2, 2} × 4 channel and compare four scenarios. First, the conventional single-beam algorithm was run until convergence for the SVD initialization and for the fast single-beam initialization. We also tested the SVD initialization after a single iteration, and examined the “fixed cost” approach in which the iterations in step (4) are omitted and the number of iterations in step (2) is fixed. If step (2) is allowed to converge, it does so typically in around 5 iterations, and in virtually all cases, less than 10 iterations. While these iterations require much less computation than those of step (4), it may still be desirable to keep computational cost fixed by repeating step (2) a predetermined number of times, rather than allowing it to repeat until convergence. Perfect convergence is not necessary since the purpose of step (2) is only to find an intelligent guess at an initial set of w j vectors. Here we fixed the first step at 5 iterations. In Figure 5, we compare the fixed-cost approach for 1, 2, and 3 iterations with the SVD initialization and the final convergence point of both initializations. Since the hybrid and SVD cases both have the same SINR requirements for the users, and the IB step is a convex optimization problem, they will converge to the same point given enough iterations, but here they do not because the step size for determining convergence was fixed at 0.1 dB. The hybrid algorithm converges to a point that is on average more than 0.5 dB better than the SVD initialization. Furthermore, all of the fixed-cost algorithms are substantially better than the SVD approach after 1 iteration, and it appears that 3 iterations are sufficient to reach nearoptimal performance almost all the time. The required number of iterations for convergence for the two initialization approaches are compared in Figure 6. The cost difference is quite dramatic: the hybrid algorithm requires more than 2 iterations only a tiny fraction of the
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Figure 6: Computational cost of single-channel algorithms.
time, while the SVD approach requires more than 10 iterations 50% of the time. 6.2.
Imperfect CSI performance
The IB algorithm for coordinated Tx-Rx processing was tested on a set of randomly generated {4, 4} × 4 channels. In each case, a true channel matrix and a corrupted channel matrix were generated according to the additive noise model in (8). We chose the noise term N to be uncorrelated, with Rc ⊗ Rr = 0.1I. The corrupted channel matrix, along with the matrices Rc and Rr , were made available to the transmitter for designing the transmit-side beamformers. The receive beamformers were calculated given the transmit beamformers and the true channel, and the SINR of the link to each user was computed. The SINR gain in Figure 7 is the mean difference between the actual SINR and the original SINR design requirement (10 dB in all cases). We show the SINR gain with and without taking into account Rc and Rr in the calculations. We also plot a curve showing the additional power required at the transmitter. The robust technique allows the SINR requirement to be met in virtually all cases, although it comes at a higher cost in power. 6.3.
Multichannel/single-channel comparison
In this section, we compare the multichannel and singlechannel approaches using a {2, 2, 2, 2} × 4 channel. The users can all be accommodated simultaneously with m j = 1, or they can be handled as two {2, 2} × 4 channels with m j = 2. For the m j = 2 case, three distinct groupings are possible, and the performance of all three was evaluated. Figures 8 and 9 show results derived by choosing the best of the three, and a second curve representing the performance of one of the groupings chosen at random. The plotted performance numbers are derived by generating 1000 random channels and computing the median power required to transmit at the desired rates to all of the users. All channels are assumed to be simple Rayleigh fading channels. Figure 8 compares the
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Figure 9: A comparison of channel allocation schemes as a function of correlation between receiver antennas.
a covariance matrix R defined by E{[R]i,k [R]∗j,k } = α|i− j | , where α is the correlation parameter on the horizontal axis of Figure 9. In completely uncorrelated channels, the singlebeam approach is about 1.5 dB better, while it is over 2 dB worse in highly correlated channels. For large α, it is expected that the gain of the secondary subchannels would be reduced. However, this example shows that the greater flexibility of using the secondary subchannels results in better performance as the channel becomes more correlated.
40 35 30 25 20 15 10
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Figure 8: A comparison of channel allocation schemes as a function of required transmission rate.
channel allocation strategies as a function of the transmission rate, which is assumed to be equal for all users. For this figure, all channels are uncorrelated. The performance gap between the optimal and randomly chosen groupings for the multichannel approach is less than 0.5 dB at low transmission rates, and almost 1 dB at 5 bits/user. The single-channel approach outperforms the multichannel approach only at the lowest rate of 2 bits/user. Thus, while the single-channel approach is attractive because of its reduced computational cost, the use of multiple subchannels per user comes at a significantly lower cost in transmitted power as the transmission rate increases. Figure 9 compares the channel allocation approaches as a function of channel correlation when all users have a transmission rate of 2 bits/user. In this example, the H j matrices are independent of one another, the individual columns of H j are independent, but the rows are assumed to have
7.
CONCLUSION
We have presented a new hybrid algorithm for designing transmit vectors in multiuser MIMO downlink channels. By combining the properties of generalized iterative ZF with bit loading, a better initialization point for iterative IB algorithms is obtained that improves both the performance and the number of iterations required for convergence. We have also generalized the iterative solution to include imperfect channel information with known covariance at the transmitter. The simulation results have revealed that the hybrid approach requires less transmitted power than all other approaches considered here, and is particularly better than distributing the power equally among the subchannels. In terms of computational cost, the hybrid algorithm does not converge as quickly as IB with an equal power distribution, but more quickly than other schemes that attempt to find a better initial power distribution. The use of MRC rather than MMSE receivers increases the required power slightly, and may increase computational cost at the transmitter, but comes with the benefit of reduced cost of the receiver. ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under Information Technology Research Grant CCR-0313056.
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Spatial Multiplexing in Multiuser MIMO Downlinks [35] K.-K. Wong, “Adaptive space-division-multiplexing and bitand-power allocation in multiuser MIMO flat fading broadcast channels,” in IEEE 58th Vehicular Technology Conference (VTC ’03), vol. 1, pp. 447–451, Orlando, Fla, USA, October 2003. [36] W. Rhee and J. M. Cioffi, “On the asymptotic optimality of beam-forming in multi-antenna Gaussian multiple access channels,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’01), vol. 2, pp. 891–895, San Antonio, Tex, USA, November 2001. [37] J. G. Proakis, Digital Communications, McGraw Hill, New York, NY, USA, 3rd edition, 1989. Quentin H. Spencer received the B.S., M.S., and Ph.D. degrees in electrical engineering from Brigham Young University (BYU), Provo, Utah, USA, in 1994, 1996, and 2004, respectively. From 1995 to 1996, he was a Research Assistant with the Department of Electrical Engineering, BYU, where he worked on propagation modeling for indoor wireless communication channels. From 1996 to 1997, he was an engineer with L-3 Communications, Salt Lake City, Utah, working on broadband military communications systems. From 1998 to 2003, he was a Research Assistant at BYU, where he worked on signal processing algorithms for MIMO wireless communication systems. During that time, he also worked concurrently as a control systems engineer at UniDyn Corporation, Orem, Utah, from 1999 to 2001, and as a signal processing engineer at Inari Corporation, Draper, Utah, in 2001. During 2002, he was a Guest Researcher at the Technical University of Ilmenau, Germany, and Royal Institute of Technology, Stockholm, Sweden. Since 2003, he has been at Distribution Control Systems, Inc., Hazelwood, Missouri, USA, where he leads research efforts in powerline communication systems. His research interests are in array signal processing and other applications of signal processing to communication systems. He is a Member of the IEEE. A. Lee Swindlehurst received the B.S. and M.S. degrees in electrical engineering from Brigham Young University (BYU) in 1985 and 1986, respectively, and the Ph.D. degree in electrical engineering from Stanford University in 1991. From 1986 till 1990, he was employed at ESL, Inc., Sunnyvale, Calif, where he worked on the design of algorithms and architectures for radar and sonar signal processing systems. He joined the faculty of the Department of Electrical and Computer Engineering, BYU in 1990, where he is a Full Professor and is currently serving as Department Chair. During 1996-1997, he was a Visiting Scholar at both Uppsala University and the Royal Institute of Technology, Sweden. His research interests lie primarily in the application of sensor array signal processing to radar and wireless communications problems. Dr. Swindlehurst is a Fellow of the IEEE, is currently serving as a Member of the Editorial Board for the EURASIP Journal on Wireless Communications and Networking and is a former Associate Editor for the IEEE Transactions on Signal Processing. He is a recipient of the 2000 IEEE W.R.G. Baker Prize Paper Award, and is the coauthor of a paper that received the IEEE Signal Processing Society Young Author Best Paper Award in 2001.
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EURASIP Journal on Advances in Signal Processing
Special Issue on Applications of Time-Frequency Signal Processing in Wireless Communications and Bioengineering Call for Papers Time-frequency signal processing is a well-established area with applications ranging from bioengineering and wireless communications to earthquake engineering and machine monitoring. Signals in these applications are typically nonstationary and as such require joint time-frequency analysis. The objective of this special issue is to bring together theoretical results and application of time-frequency methodologies from investigators in the wireless communications and bioengineering disciplines. While novel theoretical results and applications of timefrequency signal processing in wireless communications and biomedical systems will be preferred, applications in other areas will also be considered. Likewise, this issue will emphasize methodologies related to Priestley’s evolutionary spectrum and the fractional Fourier transform, but other methodologies will also be considered. The intended focus of this issue will be on presenting time-frequency signal processing applications to wireless communications and biomedical systems using evolutionary spectral techniques and fractional Fourier transform. Topics of interest include, but are not limited to: • Biomedical systems: EEG, ECG waveforms and heart
sound, vibroartrographic signals emitted by human knee joints, EEG signals, and various other biomedical waveforms analyzed by time-frequency techniques • Wireless communications: time-frequency receivers, channel characterization, channel diversity, time-varying modulation schemes, and suppressing nonstationary interference as chirp jammers
Manuscript Due
January 1, 2010
First Round of Reviews
April 1, 2010
Publication Date
July 1, 2010
Lead Guest Editor Luis F. Chaparro, Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA, USA;
[email protected] Guest Editors Aydin Akan, Department of Electrical and Electronics Engineering, Istanbul University, Istanbul, Turkey;
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Special Issue on Dynamic Spectrum Access: From the Concept to the Implementation Call for Papers We are today witnessing an explosive growth in the deployment of wireless communication services. At the same time, wireless system designers are facing the continuously increasing demand for capacity and mobility required by the new user applications. The scarcity of the radio spectrum, densely allocated by the regulators, is a major bottleneck in the development of new wireless communications systems. However actual spectrum occupancy measurements show that the frequency band scarcity is not a result of the heavy usage of the spectrum, but is rather due to the inefficient static frequency allocation pursued by the regulators. Dynamic spectrum access has been proposed as a new technology to resolve this paradox. Sparse assigned frequency bands are opened to secondary users, provided that interference generated on the primary licensee is negligible. Even if the concept constitutes a real paradigm shift, it is still unclear how the dynamic spectrum access can operate efficiently and how it can be implemented cost-effectively. Scope. Original contributions are solicited in all aspects of dynamic spectrum access related to the integration of the technology in a real communication system. The special issue should give clear advice on how to make dynamic spectrum access work in practice. Topics of interest include, but are not limited to: • Spectrum sensing and access: ◦ Spectrum sensing mechanisms and protocol sup-
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◦ Interference modeling and avoidance ◦ Adaptive coding and modulation for interference ◦ Beamforming and MIMO for interference avoid-
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André Bourdoux, Interuniversity Micro-Electronics Center (IMEC), Leuven, Belgium;
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Special Issue on Advances in Quality and Performance Assessment for Future Wireless Communication Services Call for Papers Wireless communication services are evolving rapidly in tandem with developments and vast growth of heterogeneous wireless access and network infrastructures and their potential. Many new, next-generation, and advanced future services are being conceived. New ideas and innovation in performance and QoS, and their assessment, are vital to the success of these developments. These should be open and transparent, with not only network-provider-driven but also service-provider-driven and especially user-driven, options on management and control to facilitate always best connected and served (ABC&S), in whatever way this is perceived by the different stake holders. To wireless communication services suppliers and users, alike the complexity and integrability of the immense, diverse, heterogeneous wireless networks’ infrastructure should add real benefits and always appear as an attractive user-friendly wireless services enabler, as a wireless services performance enhancer and as a stimulant to wireless services innovation. Effecting the integration of services over a converged IP platform supported by this diverse and heterogeneous wireless infrastructure presents immense QoS and traffic engineering challenges. Within this context, a special issue is planned to address questions, advances, and innovations in quality and performance assessment in heterogeneous wireless service delivery. Topics of interest include, but are not limited to: • Performance evaluation and traffic modelling • Performance assessments and techniques at system/ • • •
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flow level, packet level, and link level Multimedia and heterogeneous service integrationperformance issues, tradeoffs, user-perceived QoS, and quality of experience Network planning; capacity; scaling; and dimensioning Performance assessment, management, control, and solutions: user-driven; service-provider-driven; network-provider-driven; subscriber-centric and consumer-centric business model dependency issues Wireless services in support of performance assessment, management, and control of multimedia service delivery
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