Utilizing the Spatial Information Provided by Channel Norm Feedback in SDMA Systems Submitted to IEEE Transactions on Signal Processing c 2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
¨ DAVID HAMMARWALL, MATS BENGTSSON, AND BJORN OTTERSTEN
Stockholm October 2007
Signal Processing School of Electrical Engineering Royal Institute of Technology (KTH)
Utilizing the Spatial Information Provided by Channel Norm Feedback in SDMA Systems David Hammarwall*, Student Member, IEEE, Mats Bengtsson, Senior Member, IEEE, and Bj¨orn Ottersten, Fellow, IEEE
EDICS: MSP-MULT Abstract To achieve high performance, in terms of reliability and throughput, in future multiple-antenna communication systems, it is essential to fully exploit the spatial dimensions of the wireless propagation channel. In multiuser communication systems, the throughput can be significantly increased by simultaneously transmitting to several users in the same time-frequency slot by means of spatial-division multiaccess (SDMA). A major limiting factor for downlink SDMA transmission is the amount of channel-state information (CSI) that is available at the transmitter. In most cases, CSI can be measured/estimated only at the user terminals and must be fed back to the base station. This procedure typically constrains the amount of CSI that can be conveyed to the base station. Herein, we develop several low-complexity, as well as optimized, SDMA downlink resource-allocation schemes that are particularly suitable for systems utilizing statistical channel information and partial CSI feedback. A framework is proposed for combining statistical channel information with a class of instantaneous channel norms. It is shown that, in wide-area scenarios, the feedback of such a scalar norm provides sufficient information for the proposed resource-allocation algorithms to perform efficient SDMA beamforming (BF) and scheduling.1 Index Terms Resource management, communication systems, spatial-division multi-access, and scheduling. The authors are with the Signal Processing group of the School of Electrical Engineering, Royal Institute of Technology (KTH), SE-100 44, Stockholm, Sweden (fax: +46-8-790-7260). David Hammarwall is the corresponding author, e-mail:
[email protected], phone: +46-8-790-8434. Mats Bengtsson and Bj¨orn Ottersten are reached by (e-mail:
[email protected], phone +46-8-790-8463) and (e-mail:
[email protected], phone: +46-8-790-7239), respectively. 1
This work has been performed in the framework of the IST project IST-4-027756 WINNER II, which is partly funded by the
European Union. The authors would like to acknowledge the contributions of their colleagues. Parts of this work have previously appeared in [1], [2].
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I. I NTRODUCTION Downlink resource allocation in multiuser communication systems with multiple transmit antennas is a nontrivial task. The resource scheduler should maximize the system performance in terms of throughput and reliability without compromising performance fairness among the users. The system throughput can be significantly increased by utilizing the spatial dimensions provided by multiple transmit antennas [3], [4]—for example, using beamforming (BF) and spatial-division multi-access (SDMA) [5], [6], [7]. An essential part of downlink SDMA scheduling/user selection (US) is to determine a set of spatially compatible users—that is, finding users with almost orthogonal channels. For such sets of users, the base station (transmitter) can form non-interfering signal beams, without signal degradation, which results in high spectral efficiency. Finding such sets, and the associated beamformers, requires spatial channel information at the transmitter. Herein, three classes of channel information are of particular importance: channel-distribution information (CDI): information of the statistics of the wireless propagation channel (e.g., the mean and covariance matrix for complex Gaussian distributed channels) channel-state information (CSI): information of the current (instantaneous) state of the channel; that is, information of the realization of the stochastic channel. We differentiate between full CSI, and partial CSI: With full CSI, the realization of the channel is fully known; if, on the other hand, only some properties of the realization is known, we say that we have partial CSI. channel-gain information (CGI): information of the current (instantaneous) gain of the channel2 — that is, information of the current received signal power. Note that CGI is one example of partial CSI. A difficulty is that CSI can typically be estimated/measured only at the receivers, and must be fed back to the base station. Thus, in most cases, only partial CSI can be made available at the transmitter, because all users must share the limited capacity of the feedback link. The choice of feedback parameters from the users to the base station is important and defines the CSI available at the base station for downlink resource allocation. These parameters should minimize the feedback load, while providing sufficient information for the resource scheduler: •
The fading of the channel has to be tracked to exploit the multiuser diversity [8].
•
The scheduled users have to be spatially separable. Note that there is no need for spatial information of the users with unfavorable channel realizations, because they are not likely to be scheduled.
•
The instantaneous signal to interference-plus-noise ratio (SINR) of the scheduled users should be accurately estimated to allow rate adaptation [9], with reasonable outage probabilities.
2
In some contexts the terminology channel-quality information (CQI) is used in place of CGI.
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In [10], [11] it was shown that if the base station has CDI, then a scalar CGI feedback parameter (per user) achieves these criteria in wide-area, and metropolitan-area, scenarios. The Euclidean norm of the multi-antenna propagation channel was considered as CGI parameter, which was combined with CDI at the base station using a minimum mean squared-error (MMSE) estimation framework. The estimated instantaneous SINRs were used by the base station for scheduling and BF. It was further analytically shown that the combination of CGI and CDI substantially improves the spatial (directional) channel information, over using CDI alone—in particular for large realizations of the CGI feedback parameter; that is, CGI combined with CDI, provides accurate spatial information for the users that are likely candidates for scheduling. This is a favorable property that enables elaborate spatial processing, with limited feedback, in systems with channel dependent scheduling that exploits multiuser diversity. Herein, we extend the MMSE SINR estimation framework of [10], [11] to a larger class of CGI feedback parameters. Instead of limiting the feedback to the Euclidean norm of the full channel vector, as in [10], [11], we consider a weighted channel norm feedback. This can, for instance, be used to decrease the overhead by exciting only some of the antennas in the pilot signaling interval. The extended framework can also be used for significant computational complexity reductions when determining the MMSE estimates. Furthermore, herein we apply the MMSE SINR estimation framework in an SDMA setting, contrary to [10], [11] where a time-division multi-access (TDMA) system is considered. We propose several downlink SDMA resource-allocation algorithms, of varying computational complexity, that are designed to exploit channel information that arises when long-term CDI is combined with instantaneous partial CSI feedback. The throughput vs. fairness trade-off is balanced by considering a weighted sum-rate criterion function in the scheduling (and BF), which for instance can be used to implement proportional fair (PF) scheduling [12], [13]. A low complexity generalized zero-forcing (GZF) algorithm is proposed, which is more suitable for partial CSI than traditional zero forcing (ZF). Also, a BF algorithm based on the minimum-variance distortionless response (MVDR) criterion in the virtual-uplink domain is proposed that achieves near optimal performance at only a moderate complexity increase over GZF. In addition, an algorithm is proposed to optimize the beamformers, and the transmit power allocation, directly using a weighted sum-rate criterion. Even though the proposed optimization is computationally demanding, a re-parameterization of the problem significantly reduces the complexity. The proposed smart-antenna schemes do not solely rely on multiuser diversity for the BF, unlike the opportunistic (random BF) schemes proposed in [13], [14]. Instead the beamformers are optimized at the base station, given the information provided by the CDI and CGI feedback parameters, to suit any set of users of choice. October 26, 2007
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The combination of statistical channel information and Euclidean norm CGI has also been considered for scheduling in an SDMA setting in [15], where a maximum-likelihood (ML) estimator of the instantaneous channel vectors is proposed. However, it was observed that the ML framework did not provide sufficient accuracy for the actual data transmission, which was solved in [15] by requiring full CSI feedback of the scheduled users. In [16] the accuracy was addressed by introducing additional feedback of alignment parameters. Herein, we show that by using the MMSE framework, proposed in [10], [11], efficient SDMA BF and US can be implemented without any feedback in addition to the scalar CGI parameter. Fig. 1 shows an overview of the organization of the manuscript, in the context of the proposed system operation. The system model is described in Section II, and the extended MMSE estimation framework is considered in Section III. In Section IV we propose a greedy user selection algorithm and several low complexity BF algorithms. In Section V an optimized resource-allocation scheme is proposed. The performance of the MMSE SINR estimation framework with the proposed resource-allocation algorithms is evaluated in Section VI, where significant gains over other schemes, with similar CGI feedback, are demonstrated. Conclusions are made in Section VII. We adopt the notation of using boldface for column vectors (lower case), x, and matrices (upper case) X. For definitions, the notation is used. XT , XH and X∗ denote the transpose, Hermitian transpose, and
the conjugate of X, respectively. We denote the (i, j)th element of a matrix X by [X]i,j and diag {. . .} is a diagonal matrix with the indicated diagonal elements. The real and imaginary parts are denoted by and , respectively. The Kronecker product of two matrices (vectors) is denoted by ⊗ and vec(X) is
the vector obtained by stacking the columns of X. The identity matrix is denoted I and the matrix of all ones and the matrix of all zeros are denoted by 1 and 0, respectively. An optimal point or value is marked as {·} . No notational difference is made for a random variable and its realization. II. S YSTEM M ODEL The downlink of a single cell system is considered, where the base station is equipped with nT transmit antennas. The receivers each have a single receive antenna and the narrow band channel of user k ∈ {1, . . . , N } is modeled by the channel vector hk ∈ CnT .
A. Channel and Signaling Model The channel is modeled as a stochastic vector variable that is realized once in each block, in which it remains static (i.e., the channel is modeled as block fading). It is assumed that the receiver can perfectly estimate any required properties of hk (i.e., the feedback parameters), but at the transmitter, only the October 26, 2007
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distribution of hk is known, apart from the information that is fed back from the users. The symbol sampled, complex base band equivalent of the received narrowband signal, is modeled as rk (t) = hH k x(t) + nk (t),
(1)
where nk (t) is additive white Gaussian noise (AWGN) with power σk2 and x(t) ∈ CnT is the vector of transmitted signals. A narrowband channel assumption is thus made, but the proposed method can be extended to wideband systems using orthogonal frequency-division multiplexing (OFDM) where each narrowband subcarrier can be treated separately. A metropolitan or wide-area scenario is considered with the channel vector distributed as hk ∈ CN (hk , Rk ), where the covariance matrix, Rk , is assumed to be dominated by a single (or a few)
eigenmodes; that is, there is significant correlation, which is typical for cellular systems with elevated base stations [17]. Let S be the set of users that are scheduled for transmission in the current time slot. Hence, in an SDMA system, S typically contains more than one user. In systems relying only on CDI and CGI (channel norm) feedback, single stream BF is optimal with respect to ergodic capacity, for sufficiently large realizations of the CGI [19]; therefore, a BF system is considered where the signal, sk (t), intended for user k , is mapped onto the antenna array with the BF vector, wk , x(t) = wk sk (t),
(2)
k∈S
where the complex scalar sk (t) is normalized to unit average power. The beamformer is modeled as √ wk = pk uk , where uk is the normalized BF vector (uk = 1), and pk is the transmission power allocated to user k . For notational simplicity, the power allocation pk , beamformers uk , and wk , are collected in the vector p and the matrices U and W, respectively. The instantaneous SINR of user k for a given channel realization, ignoring inter-cell interference, is obtained by combining (1) and (2) as H 2 h wk SINRk = k . (3) hH wi 2 + σ 2 k k i∈S\{k}
B. Downlink Channel Knowledge at the Base Station To compute (optimize) the beamformers and to estimate the instantaneous SINR in (3), the base station must have information of the channels, hk . We assume that the base station is able to perfectly track the CDI (i.e., hk and Rk ). This assumption is motivated by the slow rate (compared to the small scale fading) at which the statistics of the channel change. The CDI can be made available at the base station with a slow rate feedback from user terminals, which results in little overhead. Alternatively, if the transmit and October 26, 2007
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New channel realization Pilot/Feedback
ρk
SINR Estimation
User Selection
(Section III)
(Section IV)
(optional) Post Optimization
Data
(Section V)
Low Complexity BF (Section IV) One-Shot GZF
or
GZF
or
One-Shot MVDR
or
MVDR
Performance/Complexity Fig. 1.
Illustration of the proposed system operation at the transmitter. The organization of the manuscript is also indicated.
receive chains are properly calibrated at the base station, the downlink channel statistics can be estimated directly at the base station from the reverse communication link, even if the downlink and uplink channels are separated in frequency [18]. However, to enable the base station to track how the instantaneous SINR changes with the smallscale fading, the user terminals continually feed back a CGI feedback parameter. Herein we let the CGI parameter, ρk , be the weighed Euclidean channel norm ρk Ak hk 2 ,
(4)
where Ak ∈ Cna ×nT , na ≤ nT is a weighting matrix that is chosen in the system design. This is an extension of the framework proposed in [11], where the special case, ρk = hk 2 , is considered. The weighting adds new degrees of freedom in the system design that, for example, can be used to limit downlink overhead by confining the pilot signaling to a subset of the antennas, see Section VI-C. To summarize, when the base station performs the user selection and beamforming, it is limited to the information provided by the CDI and the CGI feedback parameters. C. Overview of System Operation The proposed system design is illustrated in Fig. 1. By assumption, the channel remains constant within each block, which contains one or more frames. In the preamble of each channel realization block, the base station excites the channel with pilot signals that allow each user to estimate the desired properties of the channel. Then each user terminal feeds back the requested partial CSI, which herein is taken as a scalar CGI parameter, ρk (see Section III). The base station combines the feedback information with k (W), k = 1, . . . , N, which are functions of the CDI to form the instantaneous SINR estimates, SINR
beamformers, W. This concludes the preamble of the block. k (W), the The remainder of the downlink block contains data frames. Using the estimates, SINR
beamformers are computed (optimized) and an appropriate set of users, S , is selected. To this end, the US algorithm calls a low complexity BF function for each candidate user group. As an optional post October 26, 2007
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optimization step the beamformers of the selected users can be optimized using a more computationally demanding BF criterion. Then, data is transmitted to the scheduled users. In consecutive frames, new sets of users can be selected to achieve better fairness. Note that the beamformers, wk , k ∈ S , are not assumed known to the scheduled receivers. Thus the effective scalar channel, hH k wk , must be estimated at user k during the data transmission interval to perform coherent detection. This could for instance be done by initializing the data transmission to each user with a short, dedicated pilot signal. Next, we establish the MMSE SINR estimation and then proceed with US and beamforming. III. SINR E STIMATION
WITH
CGI F EEDBACK
The CGI feedback parameters, ρk , provide information about the instantaneous channels and are combined with the CDI (channel moments) known a priori to the transmitter. Using a MMSE framework the transmitter estimates the instantaneous moments of the channels as3 ˆ k E hk hH ρk , Q k
ˆ k E {hk | ρk } , h
(5)
where channel realizations of different users are assumed independent. Using the conditional correlation ˆ k , the MMSE estimate (for a given ρk ) of the signal/interference power, is given by matrix, Q 2 Hˆ E{|hH k wi | | ρk } = wi Qk wi . The instantaneous SINR can be estimated as
k (W) = SINR
ˆ k wk wkH Q . H ˆ k wi + σ 2 w Q
i∈S\{k}
i
(6)
k
ˆ k and Q ˆ k , are however in many cases non-trivial to compute. In [11] it was The conditional moments, h
shown how to efficiently compute these quantities for the special case, ρk = hk 2 , which is generalized ˆ k tends to rank one, as ρk = hk 2 tends in the following. An asymptotic analysis also showed that Q
to infinity; that is, the spatial (directional) channel information increases for large realization of ρk . This behavior is enhanced when Rk is dominated by a single eigenmode. A. Adapting for Estimation Errors k in (6) is an estimate of the actual instantaneous SINR, and thus It is important to note that SINR
includes estimation errors. Overestimating the SINR is not desirable because the channel will not support the allocated rate, resulting in an unacceptable packet/frame error rate. Overestimating the instantaneous SINR is referred to as an outage. The outage probability can be remedied by coding over several 3
Herein we apply the MMSE SINR estimation framework for the feedback parameter, ρk , but note that the framework is not
limited to this choice: The framework can be used with any type of partial CSI feedback, when the base station has CDI.
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transmission blocks that experience independent channel realizations; for example, over several well separated subcarriers in an OFDM system. To further reduce the outage probability to an acceptable level, the SINR may be pessimisticly estimated to allow an estimation error margin. This could be a fixed back off, as in [10], [21], where a margin of 1.5-2 dB resulted in an outage probability below 5%. Having a non-adaptive back off margin does however have the disadvantage that the outage probability may vary greatly for different realizations of the feedback parameters, ρk , and the margin is therefore set unnecessarily high for most of the realizations. For example, with ρk = hk 2 , it was shown in the asymptotic analysis in [11] that the relative estimation error will tend to zero as hk 2 tends to infinity. This motivates an adaptive back off margin, based on the actual accuracy of the estimate. The mean ˆ k wi , is obtained as squared error (MSE) of the signal/interference gain estimate, wiH Q 2 2 ˆ k wi | ρk = E wH hk 4 |ρk − (wH Q ˆ k wi )2 . ρk E hH wi 2 − wiH Q MSE hH w i i i k k
(7)
The MSE is thus readily obtained from the fourth order moment, E{|wiH hk |4 |ρk }, which can be efficiently computed, see Section III-B. The MSE can be used to include an adaptive back off margin in the SINR estimate as ˆ k wk − α RMSE |wH hk |2 ρk max 0, wkH Q k SINRMSE =
, (8) k 2 ˆ k wi + α RMSE |wH hk |2 ρk wiH Q + σ i k i∈S\{k}
where RMSE (·) is the root mean squared error and α is a tuning parameter that is chosen to achieve a target frame error rate. There is essentially a one-to-one mapping between α and the outage probability; hence α balances the tradeoff between outage and delay. k and the form of (8) Computing SINRMSE is more computationally demanding than computing SINR k
is typically unsuitable for BF optimization. Therefore, the BF computations in BFAPPEND are performed k . The adaptive back off should however be included in the rate adaptation and the with respect to SINR
US, see Section IV. Next, we show how to efficiently compute the required conditional moments. B. Computing the Conditional Moments Consider the conditional moments of hk for a given ρk , as defined in (4). For brevity the index k is omitted, without loss of generality. To obtain the desired first and second conditional moments, ˆ h(ρ) E {h | ρ} ,
ˆ Q(ρ) E hhH ρ ,
(9)
it is useful to separate the active and passive parts of the channel by a change of coordinates. Define the ⎡
⎤ VH A ⎦, At ⎣ UH p
coordinate change matrix,
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⎡ ⎤ ha ht ⎣ ⎦ At h, hp
(10)
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where V is an arbitrarily rotated unitary matrix and the columns of Up form an orthonormal basis for the null space of A; that is, Up , spans the dimensions that do not affect the feedback parameter (the passive part). The coordinate change is linear; thus the complex Gaussian distribution is preserved. Recollect that the channel, h, is distributed as CN (h, R), and consequently the transformed channel, ht , is distributed as ht ∈ CN (ht , Rt ), where ⎡ ⎤ ha ht = ⎣ ⎦ = At h, hp
⎡
Ra
Rt = ⎣ Rpa
RH pa Rp
⎤ ⎦ = At RAH t.
(11)
Next, consider the distribution of the active part, ha ∈ CN (ha , Ra ). The feedback parameter, ρ, depends only on ha and can be expressed as ρ = Ah2 = Vha 2 = ha 2 .
The conditional PDF of ha for a given ρ, f (ha |ρ), is thus fully characterized by the results of [11], and we can efficiently compute the conditional moments of ha for a given ρ; for example, ˆ a (ρ) E ha hH ha 2 = ρ . ˆ a (ρ) E ha ha 2 = ρ and Q v a These computations are significantly simplified for diagonal Ra . Fortunately, the unitary matrix V can be chosen as to impose the diagonal structure on Ra . The feedback parameter, ρ, satisfies the conditions in the following theorem that characterizes the first and second order conditional moments of ht . Theorem 1: If η is a stochastic variable satisfying E {g(ht )|ha , η} = E {g(ht )|ha } ,
(12)
ˆ t (η) E {ht |η} and for any function, g(·), then the conditional first and second order moments, h ˆ t (η) E ht hH |η , are given by Q t ⎡ ⎤ I ˆ a (η) − ha ) ˆ t (η) = ht + ⎣ ⎦ (h (13) h −1 Rpa Ra
and
⎡
Ra (η)
ˆ t (η)h ˆ H (η) + ⎣ ˆ t (η) = h Q t Rpa (η)
RH pa (η) Rp (η)
⎤ ⎦,
(14)
where ˆ a (η)h ˆ H (η), Rpa (η) = Rpa R−1 Ra (η) ˆ a (η) − h Ra (η) Q a a −1 H Rp (η) = Rp + Rpa R−1 a Ra (η) − Ra Ra Rpa .
Proof: The proof is given in Appendix I-A. October 26, 2007
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ˆ t (ρ) and Q ˆ t (ρ), and the sought Using Theorem 1 we can compute the transformed conditional moments, h
conditional moments are obtained by transforming back to the original coordinates as ˆ ˆ h(ρ) = A−1 t ht (ρ),
−H ˆ ˆ Q(ρ) = A−1 t Qt (ρ)At .
Note that At is invertible (A can be assumed full rank). Also the fourth order moment can be computed from the conditional moments of ha . The following corollary characterizes this relation. Corollary 2: For η satisfying (12), the conditional fourth order moment is given by,
4 2 4 ∗ 2 ∗ E wH h |η = 2(σ 2 + hp + m2 )2 + m4 − hp − 2m22 + 2 2hp (hp + 2σ 2 )m1 + m3 + (hp )2 m2T , H H −1 where σ 2 = wpH Rp − Rpa R−1 a Rpa wp , hp = wp (hp − Rpa Ra ha ) and the remaining parameters are H a wa + R−1 computed using w a Rpa wp as
ˆ a (η), aH h m1 = w ∗ a, aH E ha hTa |η w m2T = w
and
ˆ a (η)w aH Q a, m2 = w aH ) aH E hTa ⊗ ha hH aw m3 = w a η vec(w
∗ T H H H a ) E ha ha ⊗ ha ha η vec(w aH ) aw aw m4 = vec(w
Proof: The proof is given in Appendix I-B. ¯ p , m1 , m2T , and m3 , are all zero for the considered feedback Note that for zero-mean channels, h
parameter, ρ. C. Approximation for Reduced Complexity Even though the conditional moments of ha ∈ Cna can be efficiently computed using the algorithms developed in [11], the computational complexity grows rapidly with the dimension of ha . This is particularly true for Ricean fading channels, where the complexity of computing, for instance, the 4 aH ha ρ , grows as O n5a . Thus, there is much to gain by reducing the fourth order moment, E w ¯ a , Ra ) by the dominating dimensionality of the problem. This can be done by approximating ha ∈ CN (h
eigenmodes of Ra , similar to the approximation made in [21]. For simplicity we let V = [VL VS ] in (10) be chosen such that Ra is diagonal, and partition the eigenvalues as ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ h hL ΛL 0 ⎦ , and ha = ⎣ L ⎦ , ha = ⎣ ⎦ , Ra = ⎣ 0 ΛS hS hS where ΛL are the large eigenvalues, and ΛS the small. The first elements of ha , H hL = VL Ah AL h,
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thus have large variations, whereas hS represents more static eigenmodes. We can estimate the norm of hL as
2 AL h = hL 2 ≈ ρ − hS − Tr {ΛS } ρL ,
where the approximation is accurate for large values of ρ—that is, for users that are likely candidates for scheduling. The approximation is obtained by noting 2 2 (a) ρ − hS − Tr {ΛS } = hL 2 + hS 2 − hS − Tr {ΛS } (b) 2 ≈ hL 2 + E hS 2 − hS − Tr {ΛS } = hL 2 ,
where (a) follows from ρ = ha 2 = hL 2 + hS 2 and the approximation in (b) is obtained by noting that the variations of the “static” elements, hS 2 (corresponding to ΛS ), are small relative to hL 2 for large values of ρ, see [11]. Since ρL can be accurately estimated from ρ, and vice versa, they almost carry the same information about the channel realization. To reduce the computational burden, the transmitter can therefore reduce the dimensionality by using ρL and AL in place of ρ and A. If the size of ΛL is not set too small, the loss in accuracy introduced by the approximation is negligible. IV. G REEDY U SER S ELECTION
WITH
L OW C OMPLEXITY B EAMFORMING
Allocating system resources to the users is a balance between total system throughput and fairness among the users. Also, the multiuser diversity should be exploited to increase the throughput while ensuring that the delays are kept at a reasonable level. A scheduling (and BF) criterion that is able to take all these factors into account is the maximization of the weighted sum-rate4 : RΣ MSE (W, S) =
MSE k (W) , βk r SINR
(15)
k∈S
where r(SINR) is the rate function (assumed non-decreasing) that maps an SINR to transmission rate. The weights, βk , determine the priority of users and may be chosen according to, for instance, the PF scheduling criterion [12], [13]. However, maximizing the weighted sum rate is, in general, a non-convex optimization problem and is therefore non-trivial. Especially finding the optimal user group, S , is computationally complex; therefore, 4
The notation RΣ MSE is used to indicate that the estimation error (back off) margin should be included in the rate evaluation;
for the proposed framework, SINRMSE is used to evaluate the rate, for other schemes the SINR back off is not necessarily k based on the MSE
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TABLE I O UTLINE OF THE RESOURCE - ALLOCATION ALGORITHM WITH GREEDY US 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15:
C ← set of all users Wnew ← [ ], Snew ← ∅, RΣnew ← 0, PSnew ← ∅ repeat W ← Wnew , S ← Snew , RΣ ← RΣnew , PS ← PSnew for all k ∈ C do Sc ← S ∪ {k} (Wc , PSc ) ← BFAPPEND (k, S, PS ) RΣ [k] ← RΣ MSE (Wc , Sc ) if RΣ [k] > RΣnew then RΣnew ← RΣ [k], Snew ← Sc , Wnew ← Wc , PSnew ← PSc end if end for C ← i i ∈ C, RΣ [i] > RΣ \ Snew until S = Snew return S, W
we next propose a greedy suboptimal resource allocation algorithm that relies on low complexity BF algorithms. A. Resource Allocation with Greedy US Finding the optimal user group to schedule, typically involves evaluating the performance for all conceivable user groupings, which becomes computationally prohibitive even for a modest number of users. Instead we propose a suboptimal greedy US scheme, similar to the approach in [20], but we extended it to allow for an arbitrary BF criterion. The proposed greedy US can thus be combined with any of the BF techniques discussed in the sequel. For now, the arbitrary beamformer function is denoted as WS = BF(S), and determines the beamformers for the set of scheduled users, S , using some criterion. The complexity of the algorithms can be significantly reduced if the beamformers can be readily updated for a user set that is appended by a single user k . To emphasize this we define the BF function, (WS∪{k} , PS∪{k} ) = BFAPPEND (k, S, PS ),
that computes the beamformers for the user set S ∪ {k} by utilizing parameters that have already been computed for the set S ; these parameters are collected in the set, PS .
The proposed greedy resource-
allocation algorithm is given in Table I. In summary, each iteration of the algorithm is composed of three stages. 1) Try adding each user in the set of candidates, C , to the set of scheduled users, S , and evaluate the resulting weighted sum rate (Line 6 – 8 in Table I). 2) If the most favorable candidate increases the total weighted sum rate, add it to the set of scheduled users, S (Line 9 – 11 in Table I). 3) Remove incompatible users from the set of candidates, C (Line 13 in Table I). These steps are repeated until no more users are selected (i.e., until S = Snew ).
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The computational complexity of the greedy US algorithm depends on the choice of the BF function, because the calls to BFAPPEND dominate the computational burden. However, the number of calls to BFAPPEND bounded, from above, by N |S|, where N is the total number of users. In the following,
several low complexity algorithms are proposed that can be used in the BF allocation. B. Low Complexity Beamforming In this section we propose low complexity BF algorithms that are able to utilize the spatial information provided by the MMSE SINR estimation framework. Most of the currently available low-complexity BF ˆk, algorithms (e.g., traditional ZFBF) are designed to operate on estimates of the channel vector itself, h
that are typically based on quantized channel feedback, which is unavailable in the considered system. ˆ k wi , However, from an estimation error point of view, it is preferable to use the MMSE estimator, wiH Q ˆ k , as to estimate the received signal/interference power, in favor of using a channel vector estimate, h ˆ H wi |2 . This becomes particularly apparent when wi is chosen orthogonal to h ˆ k (as in ZFBF) because |h k
any such direction is rated as equally good and optimistically estimated to yield zero interference by the ˆ H wi |2 . The MMSE estimator, wH Q ˆ k wi , does however track the expected interference in any criterion |h i k ˆ k , in which it is likely to cause direction, which makes it possible to avoid directions, in addition to h
significant interference. Next, the ZFBF criterion is modified to take better advantage of the information ˆ k . We also propose a BF algorithm, based on the MVDR criterion, that achieves better performance in Q
than GZF at the cost of increased computational complexity. 1) GZF Beamforming: Traditional ZFBF is a computationally efficient SDMA BF solution that, in case of full CSI, yields near optimal performance, especially when combined with user selection [22]. However, traditional zero forcing is based on estimates of the channel vectors, hk . The design goal of the GZF algorithm, which is proposed next, is to maintain the low computational complexity of traditional ZFBF, but adapt it to better suite the CSI provided by the MMSE SINR estimation framework. Using the MMSE estimation framework, the co-channel interference at user k is estimated at the base ˆ k wj . Since Q ˆ k is of full rank, in general, the design criterion of zero intra-cell station as Iˆk = j=k wjH Q interference, used in ZFBF, is too strict and cannot be applied. Instead an interference sensitive subspace is identified for each user, where the interference is known to significantly degrade the performance. This ˆ k as, subspace is identified from the spectral (eigenvalue) decomposition of Q ˆ k = vk V(S) V(I) Λk vk V(S) V(I) H , Q k k k k
where the eigenvalues are ordered decreasingly along the diagonal of Λk . Hence, vk is the dominating ˆ k , which is the BF direction that is most favorable for user k . Thus, vk takes the role of eigenmode of Q October 26, 2007
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(I)
(I)
the channel vector hk in classical zero-forcing. Similarly, Vk ∈ CnT ×nk spans the subspace associated (I)
with the nk smallest eigenvalues, that is, the space in which the user is insensitive to interference. The (S)
intermediate nk
(I)
(S)
= nT −nk −1 eigenmodes are represented by Vk
(S)
∈ CnT ×nk , which spans the space
in which the user remains sensitive to interference. The design criterion of GZF is to allow interference only in the interference insensitive subspace of the co-channel users, or equivalently, to require (A) (A) (S) ∀k ∈ S : wk ∈ VS\{k} , where VS˜ w wH [vj Vj ] = 0, ∀j ∈ S˜ ,
(16)
is the aggregate allowed subspace associated with the users in S˜. A question that arises is how to choose (I)
(I)
the dimension of the interference insensitive subspace, nk ∈ {0, 1, . . . , nT − 1}. The dimension, nk , associated with user k balances the trade-off: (I)
1) The larger nk , the less constrained the other scheduled users become; that is, the performance of the co-channel users increases and potentially more users can be scheduled simultaneously (i.e., the dimension of the aggregate allowed subspace increases for the other users). (I)
2) The larger nk , the larger interference at user k ; that is, the performance of user k , when scheduled, decreases. (I)
Herein, we let nk
= n(I) be a fixed system design parameter, even though more elaborate adaptive
methods that take user and/or channel realization specific factors into account can be considered. With this choice, GZF can allocate at most 1 + (nT − 1)/(nT − n(I) ) simultaneous users. In particular, with (I)
(I)
nk = 0 only a single users can be scheduled, whereas with nk = nT − 1, GZF becomes equivalent to
traditional ZF, and up to nT users can be scheduled simultaneously. (A)
Let ΠS˜
(A)
be the projection matrix that projects onto VS˜ . As mentioned above, we treat vk as a
channel (direction) estimate and consider signal power only along vk as useful. The beamformer that maximize the signal power along vk , while satisfying (16), is obtained (up to a scaling) as (A)
k = ΠS\{k} vk . w
(17)
k according to the transmit power allocation. The beamformer, wk , is finally obtained by rescaling w
Herein, the GZF algorithm distributes the power equally among the scheduled users, but other choices [23] can also be implemented. (A)
The computational complexity of GZF arises from the computation of ΠS\{k} for each scheduled user (A)
k ∈ S . In general, ΠS˜
is computed as (A)
(R)
ΠS˜ = I − VS˜
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TABLE II O UTLINE OF THE GZF ALGORITHM (a) Sequential implementation of GZF 1: 2: 3: 4: 5: 6:
(b) Implementation of GZFAPPEND (A)
(A)
1: input k, S, PS = {ΠS } ∪ {ΠS\{kc } |kc ∈ S}
(A)
Sc ← ∅, P ← {ΠSc = I} for all k ∈ S do (W, P) ← GZFAPPEND (k, Sc , P) Sc ← Sc ∪ {k} end for return wk , ∀k ∈ S
2: 3: 4: 5: 6: 7: 8: 9:
(A)
(A)
(A)
(A)
set Πk ← ΠS and Πkc ← ΠS\{kc } , ∀kc ∈ S Sc ← S ∪ {k} for all kc ∈ Sc do ← Π(A) [vk V(S) ] V kc k using Gram-Schmidt V(o) ← Orthonormalize V (A) (A) (o) (o) H Πkc ← Πkc − V (V ) end for (A) (A) (A) (A) ΠSc ← Πk , Πk ← ΠS
10: ukc ←
(A)
Π vkc kc , (A) Πk vkc c
(A)
(A)
ΠSc \{kc } ← Πkc , ∀kc ∈ Sc (A)
(A)
11: return ukc , ∀kc ∈ Sc and PSc = {ΠSc } ∪ {ΠSc \{kc } |kc ∈ Sc } (R)
where the column orthonormal matrix, VS˜ , is obtained by applying the Gram-Schmidt orthonormaliza(S)
(S)
(R)
tion procedure to the columns of [vS(1) . . . vS( ˜ VS(1) ˜ |S˜|) VS( ˜ ˜ |S˜|) ]; that is, the columns of VS˜ span the aggregate restricted subspace. However, the GZF algorithm is particularly well suited for a sequential implementation, where a single user is added to the set of scheduled users in each iteration: The projection (A)
matrices, ΠS\{k} , can be updated efficiently in each iteration. This procedure is outlined in Table II(a). In each iteration the function, (W, PS∪{k} ) = GZFAPPEND (k, S, PS ),
is called that updates the GZF beamformers to the user set, S ∪ {k}, by utilizing projection matrices previously computed for S . These projection matrices are collected in the parameter set, PS . In each (A)
iteration GZFAPPEND (k, S, PS ) computes the projection matrix, ΠS∪{k} that projects onto the aggregated allowed signaling space that all subsequent candidates are constrained to. This is efficient because this subspace, or rather its spanning basis vectors, are computed only once for all subsequent users. The GZFAPPEND algorithm can be summarized as follows: (A)
1) Set the matrices Πkc , ∀kc ∈ Sc S ∪{k} to the projection matrices in PS (Line 2 of Table II(b)). (A)
(S)
2) Update each projection matrix Πkc such that the range is orthogonal to [vk Vk ] (Lines 4 to 8 of (A)
(A)
(A)
Table II(b)). The updated matrices satisfies Πkc = ΠSc \{kc } , ∀kc ∈ S and Πk
(A)
= ΠS c .
3) Compute the GZF beamformers of the users (Line 10 of Table II(b)). Since the beamformers are readily updated when a single user is appended to S (GZFAPPEND has a complexity of the order O(n(S) nT 2 |S|) ), the GZF algorithm is well suited for the proposed greedy US scheme, a combination that share many similarities with the greedy US with traditional ZF proposed in [20]. October 26, 2007
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2) Virtual-Uplink MVDR BF: It is well known that beamformer optimization in the downlink is more intricate than the corresponding uplink problem. This can be overcome by transforming the downlink problem into an equivalent uplink formulation. This virtual-uplink – downlink duality [5], [6] is used to obtain the virtual-uplink MVDR BF algorithm proposed in the following, as well as to efficiently optimize the weighted sum rate as proposed in Section V. Define the virtual-uplink SINR as, SINRVUL k
ˆ qk uH k Qk uk , ˆ j + I)uk Q uH ( q j j∈S\{k} k
(18)
where qj are the virtual-uplink powers, which are collected in the vector q. The virtual-uplink powers are limited by the weighted sum-power constraint, k∈S σk2 qk ≤ Pmax . By virtual-uplink duality, for any given q and U, there exists a downlink power allocation, p, that satisfies k∈S pi = k∈S σk2 qk , for which k (U, p) = SINRVUL SINR k (uk , q), ∀k ∈ S
(19)
and vice versa [6], [24], [25]; that is, the beamformers, U, achieves the same SINRs in both the virtualuplink and the downlink. Hence, the downlink beamformers can be computed in the virtual-uplink domain. The main advantage of transforming the problem to the virtual-uplink domain, is that the beamformer of each user can be optimized separately. Since SINRVUL depends only on uk and not on uj , j = k , k the optimal uk (for a given power allocation) is obtained as uk = arg max SINRVUL k (uk ), uk
(20)
which is a generalized eigenvalue problem and can be solved using standard techniques. The virtualuplink SINRs do however remain coupled by the power allocation. Finding an optimal power allocation, therefore, typically requires more involved optimization procedures, see Section V. Next, we proceed with a fixed power allocation, where the weighted virtual-uplink powers, σk2 qk , are allocated equally as qk =
Pmax . σk2 |S|
When the beamformers have been computed, SINRVUL k (uk , q) is known, and the downlink power allocation is computed from (19), which can be rearranged into a linear system of equations in p [7]. To achieve a low complexity beamforming algorithm it is desirable to reduce the computational complexity of (20), which is achieved by the virtual-uplink MVDR algorithm proposed next. In many cases it is unknown whether signal power along non-dominating eigenmodes will add constructively or destructively with the dominating eigenmode [10]. Therefore, if signal power only along the dominating
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ˆ k is considered useful, the SINR expression of user k can be modified as eigenmode of Q ˆ k ) uH vk 2 qk λmax (Q k MVDR SINRk = H , ˆ j + I)uk uk ( j∈S\{k} qj Q
where λmax (A) denotes the largest eigenvalue of A. The optimal SINR maximizing beamformer can be obtained by the well known MVDR criterion ˆ j + I)˜ ˜H min u qj Q uk k( ˜k u
˜H u k vk = 1.
subject to
j∈S\{k}
The optimal (with respect to the MVDR criterion) normalized beamformer is given by [26]
where Bk
B−1 vk uk = k−1 , B vk k
ˆ + I. A beneficial property of the MVDR BF design is that we avoid the
j∈S\{k} qj Qj
strict classification of a interference insensitive subspace and ad-hoc decisions of subspace dimensions; in addition, unlike traditional ZF and GZF, MVDR beamforming does not have a strict upper bound on the number of scheduled users. The computational complexity of the MVDR beamformer is considerably less than solving the generalized eigenvalue problem in (20). Ignoring signal power along non-dominating eigenmodes is thus also motivated from a computational complexity point of view. There is no straightforward way to append a user to the set of scheduled users, S , by utilizing previously computed quantities. Therefore, the complexity of using virtual-uplink MVDR BF in the US algorithm as proposed in Table I is higher than using GZF. C. One-Shot BF Much of the complexity of both GZF and virtual-uplink MVDR BF arise because the computations must be reevaluated for each scheduled user. More specifically, in GZF a separate projection matrix, (A)
ΠS\{k} , is required for each user k , and virtual-uplink MVDR requires a separate matrix inversion of Bk for each user. Next, we propose modifications to both of these algorithms allowing the beamformers
to be computed in a single shot. This is achieved by considering signal power along non-dominating eigenmodes as self interference. (S)
By modifying the GZF interference constraint (16) to also require wkH Vk
= 0 (i.e., by also requiring
the beamformer to be orthogonal to its own interference sensitive subspace), it is possible to compute all beamformers from a single matrix (pseudo) inversion. The generalized zero-forcing constraint of beamformer k is thus given by (S)
∀k ∈ S : wkH vj = 0 ∀j ∈ S \ {k} and wkH Vj October 26, 2007
= 0, ∀j ∈ S.
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˜ = [w S(1) . . . w S(|S|) ], are obtained from the pseudo With this modified constraint, the beamformers, W (S)
(S)
inverse of VS(tot) = [vS(1) . . . vS(|S|) VS(1) . . . VS(|S|) ] as
−1 ˜ Wdummy ] = (V(tot) )H† = V(tot) (V(tot) )H V(tot) [W , S S S S
which are scaled according to the power allocation. Hence, by also restricting self interference, all the beamformers can be obtained from a single matrix inversion. Note that also one-shot GZF is suitable for sequential implementation, which is beneficial in the joint BF and US algorithm given in Table I. For any matrices A, B and C, related as C = [A B]P, where P is a column permutation matrix, the inverse, (CH C)−1 , can be expressed as,
⎡
⎡
⎤−1
−1 AH A A H B ⎦ =PT ⎣ CH C H H B A B B
P PT ⎣
H 11 C C 21 21 C 22 C
⎧ ⎪ −1 = BH [I − A(AH A)−1 AH ]B ⎪ C ⎪ ⎨ 22 ⎦ P, 22 BH A(AH A)−1 21 = −C C ⎪ ⎪ ⎪ ⎩ C 11 = (AH A)−1 + C −1 H C 21 22 C21 , ⎤
mn are given by the matrix block inversion lemma. Since V(tot) = [V(tot) vk V(S) ]P, the where C S k S∪{k} (tot) H (tot) −1 (tot) H (tot) −1 inverse, (VS ) VS , is efficiently updated to (VS∪{k} ) VS∪{k} using the above procedure.
Similarly to one-shot GZF, also the MVDR BF can be obtained from a single matrix inversion if signal power along non-dominating eigenmodes is considered as self interference. The modified MVDR virtual-uplink SINR of user k is given by ˆ k ) uH vk 2 qk λmax (Q k , ˆ k )vk vH )uk uH (BS − qk λmax (Q k
where
BS
k
It is easy to show that the beamformer, uk
= arg max
H 2 u vk
k uH k BS uk
ˆ j + I. qj Q
(22)
j∈S
B−1 vk = S−1 , B vk S
also maximizes (22). Hence all beamformers, wk , k ∈ S , are obtained from a single matrix inversion of BS . Note however that one-shot MVDR BF does not gain from a sequential implementation, because
the inversion of BS∪{k} is not aided by knowing B−1 S . V. O PTIMIZED W EIGHTED S UM -R ATE BF The BF techniques proposed in the preceding sections are based on heuristic arguments and optimality cannot be claimed. Next, we propose a technique to optimize the BF vectors and power allocation directly using the weighted sum-rate criterion: max W
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k (W) , βk r SINR
(23)
k∈S
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where it should be emphasized that the estimation error margin is not included, contrary to (15): Similar to the GZF and MVDR BF algorithms, it is not analytically tractable to include the error margins in the BF design. Due to the non-convex nature of this criterion, it is not suitable to include all users in the optimization and implicitly determine the set of scheduled users from the power allocation. In particular local extrema will cause serious difficulties for such a scheme. A typical scenario is that allocating zero power to a user often results in higher weighted sum rate, than a very small power allocation to the user. Only when the power is sufficiently increased does the weighted sum rate increase. Hence, there is often a barrier restricting a search algorithm to go from and to the zero power allocation state of a user. Therefore, the weighted sum-rate maximization is applied as a post processing (optimization) to the suboptimal US scheme in Table I, which is matched with one of the suboptimal BF schemes proposed in the previous sections, as is illustrated by Fig. 1. Hence, only the power allocation and the beamformers of the initially scheduled users are optimized. A. Re-parameterization of the Weighted Sum-Rate Optimization Optimizing the beamformers and power allocation directly using the highly non-convex weighted sumrate criterion (23) will in most cases be computationally infeasible. To decrease the number of unstructured optimization parameters, we express the SINRs in terms of a common gain factor, Γ, and the relative gains, γk (collected in the vector γ ), as k = Γγk , ∀k SINR
with
1T γ = 1.
(24)
The sum-power constraint translates into a constraint on Γ, which is confined to the set Gγ , defined as Gγ {Γ | ∃ W ∈ W, s.t. SINRk (W) = Γγk ∀k} = 0, max Gγ ,
where W is the set of beamformers and power allocations satisfying the normalization and sum power √ constraints—that is, beamformers, wk = pk uk , satisfying pk ≥ 0 and 1T p ≤ Pmax . By the assumption of non-decreasing rate functions it follows that the maximization in (23) can be expressed as RΣ = where
Γ (γ)
= max Gγ . For a given γ ,
max
γ, 1T γ=1
Γ (γ)
βk r Γ (γ)γk ,
k
is given by the max-min problem,
Γ (γ) max min W∈W
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(25)
k
SINRk (W) , γk
(26)
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which is thoroughly analyzed in [6], where an efficient algorithm for the optimization is derived. It was further shown that at the optimum, all the SINRs are balanced; that is, SINRk /γk = Γ (γ) ∀k . It follows that the max-min problem is equivalent to max Gγ . The maximization in (25) does however remain non-convex, but the number of optimization parameters is substantially reduced. The non-convex optimization is implemented as a gradient search starting at a heuristically chosen γ 0 . Next, it is shown that this search can be efficiently implemented because also the gradient can be computed from the optimization parameters of (26), with negligible additional complexity. B. Gradients Define the optimal weighted sum rate for given relative gains, γ , as RΣ (γ)
k
βk r Γ (γ)γk .
Hence, the gradient of RΣ (γ) is given by ∇ RΣ (γ) = Γ (γ)r (γ) + γ T r (γ) ∇Γ (γ),
(27)
where [r (γ)]k = βk r Γ (γ)γk . The gradient of the weighted sum-rate function is thus obtained from
the gradient of Γ (γ). Interestingly, this gradient can be expressed explicitly in terms of the optimization variables of (26). The solver of (26) utilizes virtual-uplink duality and first solves for uk , ∀k , and the virtual-uplink powers q, from which the optimal p is obtained [6], [7]. The next theorem states that the gradient, ∇Γ (γ), can be computed from the optimal q and W. Theorem 3: The gradient of Γ (γ), can be expressed in terms of virtual-uplink powers, q , and the optimal beamformers, W (i.e., p and uk ). The gradient is defined if the optimal point, (W , q ), is unique (up to phase rotations) and is given by ∂Γ (γ) ∂γk
=
ˆ k u q p uH Q k , − 2 kTk k γk (q Ψp + Pmax )
where
⎧ ⎨ uH Q ˆ i uj j [Ψ]ij = ⎩ 0
j = i, j = i,
is the cross-talk matrix. Proof: See Appendix II. VI. P ERFORMANCE E VALUATION Consider a single cell, where the base station is equipped with a uniform circular array (UCA) of transmit antennas. The diameter of the array is set to 1 wavelength. The performance of the MMSE SINR estimation framework, when combined with various BF/US schemes discussed herein, is evaluated in terms of system (cell) throughput. The SINR to throughput mapping is modeled as log2 (1+SINRMSE ). In all considered schemes, the US is made according to the greedy approach given in Table I. The channel is modeled as block fading, and in each block a single set of users is scheduled using the PF criterion October 26, 2007
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[12], [13], where the short term average throughput of each user is evaluated using an exponentially weighted low pass filter, ∝ exp(−t/a), where a = 100 blocks. The covariance matrix of the channel fading, Rk , was computed using the assumption of Gaussian distributed scattering with angular spread σθ (standard deviation of the distribution) that is centered at the direction of the user. When not stated otherwise, a Rayleigh fading channel with σθ = 10 degrees is considered, which is typical for wide area scenarios. Note that this corresponds to an angular spread sector of 2σθ = 20 degrees. For Ricean fading channels, the mean component, hk , is aligned with the array response in the direction of the user. A circular cell is considered in which the user positions are realized according to a uniform distribution over the cell, but not closer to the base station than 20% of the cell radius. A realization of the user positions is denoted a scenario. The average cell throughput in each scenario is numerically evaluated over 500 channel realizations (scheduling decisions). Note that the average cell throughput is a random variable, changing with the user positions. The SNR of a simulation refers to the average single antenna SNR as perceived by a single user at the cell edge, SNR Pmax E {[h]i } σ 2 . For users within the cell the SNR is scaled proportional to r−2 , where r is the distance to the mobile station. When not explicitly stated otherwise, the users feed back only the instantaneous channel norm, hk 2 , to the transmitter, which in addition is assumed to have perfect information of the channel statistics. For (S)
GZF BF, the interference sensitive subspace was set to the two dominating eigenmodes (i.e., nk
= 1).
In simulations with adaptive estimation error margin, the back off, α, is aggressively set to α = 2, which results in an outage probability below 5% in all considered simulation settings. Note that in a system that allows coding over several independent channel realizations, such as an OFDM system, the outage could be overcome by a significantly less aggressive margin. The presented figures do not consider a package sent in outage as lost: This information can still be utilized using, for example, a type-II hybrid automatic-repeat request (ARQ) (see e.g., [27], [28], and references therein). A. Comparison of GZF and MVDR BF Here we focus on the ability of the GZF and MVDR BF algorithms to maximize the estimated SINR as given in (6), and therefore do not include any adaptive estimation error margin in the rate adaptation. Instead it is assumed that the outage probability is addressed by other means, as discussed above. The cumulative distribution function of the average cell throughput for GZF and MVDR BF is illustrated in Fig. 2. For comparison, also the performance of traditional ZF with full CSI with greedy US as proposed in Table I is given. It is observed that both schemes achieve high throughput, and achieves a significant
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GZF
GZF, Optimized
22
MVDR
MVDR, Optimized 1
0.8
0.8
ZF, Full CSI
0.4
0.4
0.2 0
s
0.6
5U ser
0.6
5U ser
s
1
0.2 3
4
5
6
7
Cell throughput [bits/s/Hz]
8
9
0
6
8
0.8
5U ser
0.6 0.4
0.2 0
16
18
5U ser
0.8
s
1
0.4
14
(b) SNR = 10 dB, 3 Antennas
1 0.6
12
s
(a) SNR = 0 dB, 3 Antennas
10
Cell throughput [bits/s/Hz]
0.2 6
8
10
12
14
Cell throughput [bits/s/Hz]
(c) SNR = 0 dB, 8 Antennas
16
18
0
10
15
20
25
30
Cell throughput [bits/s/Hz]
35
(d) SNR = 10 dB, 8 Antennas
Fig. 2. Numerical evaluation of the GZF and MVDR BF strategies when combined with the proposed greedy US algorithm. The CDF of the average cell throughput, as realized over the random scenarios, is shown for 5 and 40 users in the cell (increasing performance).
fraction of the full CSI case; the performance gap increases with the number of cell users, N , and antennas: For nT = 3 and N = 5, MVDR BF achieves more than 90 percent of the performance of the full CSI case, but for N = 40 and nT = 8, MVDR achieves approximately two thirds of the performance of full CSI. In all cases, MVDR BF outperforms GZF, at the cost of increased computational complexity; with few users, and few antennas, the performance difference is small, but with many cell users and many antennas, the performance difference is more notable. The performance when post optimization of the power allocation and beamformers are applied, as proposed in Section V, is also shown. The optimization is performed as a post processing step to both the MVDR and GZF US. Interestingly, post optimization of MVDR BF, results in only a negligible performance increase. This suggests that the fixed power allocation is indeed close to optimal for PF US. Contrary to MVDR, GZF is aided by the post processing optimization, especially if there are many cell users. This is explained by the optimization which improves the crude BF direction produced by the GZF algorithm, whereas the MVDR BF criteria results in beamformers closely aligned with the optimized beamformers. It should also be noted that more simultaneous users can be selected using MVDR BF than with GZF. A performance loss for the one-shot implementations of GZF and MVDR, considered in Section IV-C, is expected compared with the regular versions. This performance loss is illustrated in Fig. 3. It is observed
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GZF
GZF one−shot
MVDR
MVDR one−shot
0.8 0.6 0.4
0.8 0.6 0.4
0.2
4 Ant ennas
1
4 Ant ennas
1
23
0.2
0
8
10
12
14
16
18
20
Cell throughput [bits/s/Hz]
22
24
26
0
8
10
(a) No Estimation Error Margin Fig. 3.
12
14
16
18
20
Cell throughput [bits/s/Hz]
22
24
26
(b) Adaptive Estimation Error Margin
Evaluation of the one-shot implementations of GZF and MVDR. The SNR is 10 dB and there is 20 users in the cell.
The CDF of the average cell throughput is shown for 4 and 8 antennas (increasing performance).
Interference bounded ZF
Opportunistic BF 1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
1.5
2
2.5
3
3.5
4
4.5
Cell throughput [bits/s/Hz]
5
5.5
6
0
4
5
(a) SNR = 0 dB, 3 Antennas 1
1 0.8
0.6
0.6
0.4
0.4
0.2
0.2 2
4
6
8
10
Cell throughput [bits/s/Hz]
12
(c) SNR = 0 dB, 8 Antennas Fig. 4.
6
7
8
9
10
Cell throughput [bits/s/Hz] (b) SNR = 10 dB, 3 Antennas
0.8
0
MVDR, MMSE framework
0
5
10
15
Cell throughput [bits/s/Hz]
20
(d) SNR = 10 dB, 8 Antennas
System comparison of multi-beam opportunistic BF, interference bounded ZF and the proposed virtual-uplink MVDR
BF, combined with the MMSE SINR estimation framework. The performance is given for 5 cell users (dash-dotted) and 40 cell users (solid).
that the performance difference of one-shot and regular MVDR is small in all considered settings. For GZF, the performance loss of the one-shot implementation is more notable when no estimation margin is included, as in Fig. 3(a), whereas it remains small when an adaptive estimation error is included, as in Fig. 3(b). B. System Evaluation In Fig. 4 the performance of an SDMA system utilizing the MMSE SINR estimation framework is compared to other systems with similar feedback strategies (feedback loads): multi-beam opportunistic BF [14], and interference bounded ZF [16], [29]. October 26, 2007
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Multi-beam opportunistic BF: In each realization a set of random orthogonal beamformers is generated. Each user feeds back the index, and instantaneous SINR of the best beam, which is estimated from common pilot signaling. The number of beams was chosen for each setting (i.e., for each SNR level, number of transmit antennas, and number of cell users), to yield the highest performance. The user which maximizes the PF criteria for each beam is scheduled. Interference bounded ZF (IBZF): In the IBZF scheme each user, k , feeds back an alignment 2 parameter, ξk = vkH hk / hk 2 , in addition to the channel gain parameter, ρk = hk 2 . In line with [29], the transmitter computes the beamformers using the ZF criteria, with the channel estimates, ˆ k = √ξ hk vk . Using the alignment, a lower bound on the instantaneous SINR of each user is h computed [29] that is used in the rate adaptation. The greedy US of Table I was used, which is similar to the greedy-US algorithm considered in [29]. The US algorithms differ in that incompatible users are eliminated and a PF criteria is used herein contrary to the sum-rate criteria in [29]. MVDR BF with MMSE SINR estimation framework: The performance of the MMSE SINR estimation framework is illustrated using US with MVDR BF. Since the other compared schemes, in theory, have zero outage probability, an adaptive estimation error margin is appended. No post optimization of the beamformers is performed. It is observed in Fig. 4 that opportunistic beamforming is the overall weakest performer. Only for the scenario with low SNR, few transmit antennas, and a handful of cell users, is IBZF able to marginally outperform the proposed MMSE SINR estimation framework with MVDR BF, see Fig. 4(a). The weak performance of the proposed MMSE estimation framework in this scenario, is explained by the lack of multiuser diversity with few users: The scheduler is forced to allocate recourses also to users with relatively small ρk for which the spatial information is limited, which causes large estimation errors. However, note that IBZF requires the additional feedback of ξk , and that the proposed MVDR beamforming, in most cases, significantly outperforms IBZF. The performance gain of MVDR BF is particularly significant in settings with high SNR and many transmit antennas. The performance of the MMSE SINR estimation framework does however depend on how correlated the channel is. The correlation decreases when the angular spread increases. The performance for different angular spreads (Ricean K-factor is set to zero) is illustrated in Fig. 5(a). It is observed that the MMSE SINR estimation framework maintains its performance advantage over the compared schemes for angular spreads as large as σθ = 20 degrees. In Fig. 5(b) the average cell throughput is illustrated for different 2 Ricean K-factors, K hk Tr {Rk }. With Ricean fading the performance advantage for the proposed framework is maintained, and for large K-factors the throughput of the MVDR algorithm approaches that October 26, 2007
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ZF, Full CSI
MVDR
20
10
0 0
5
10
15
Angular spread [degrees]
20
25
(a) Average cell throughput for different angular spreads.
Cell throughput [bits/s/Hz]
Cell throughput [bits/s/Hz]
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GZF
IBZF
Opp. BF
20
10
0 0
2
4
6
8
10
Ricean K-factor
12
14
16
(b) Average cell throughput for different Ricean K factors.
Fig. 5. Evaluation of the mean (over scenarios) average cell throughput of the proposed MVDR scheme for parameters affecting the channel distribution (angular spread, σθ , and Ricean K-factor). An adaptive estimation error margin is used and the system parameters are set as, SNR = 10dB, 6 transmit antennas, and 20 cell users. 1 0.8 0.6 0.4
1 Antenna 2 Antennas 4 Antennas 8 Antennas
0.2 0
12
14
16
18
Cell throughput [bits/s/Hz]
20
Fig. 6. Performance of the MMSE SINR estimation framework with MVDR beamforming, when only a subset of the antennas are active in the pilot signaling. The figure shows the CDF of the average cell throughput for a MVDR system with SNR = 10dB, 8 Tx Antennas, 20 Users, and adaptive estimation error margin.
of ZF with full CSI. C. Pilot Signaling on Selected Antennas To accurately estimate hk ∈ CnT or hk 2 at the receivers, all nT complex dimensions of the propagation channel must be excited, which requires a pilot signaling interval of at least nT symbols due to orthogonality constraints. In multi cell systems the overhead is even more severe because pilot sequences of neighboring cells are often required to be orthogonal. For large nT this, otherwise significant, overhead can be remedied by exciting only na nT antennas and feed back the weighted norm CGI parameter, ρk = i∈A |[hk ]i |2 , where A is the set of active antennas. The transmitter can then combine ρk with the CDI and utilize the full antenna array in the data frames.
This scheme is evaluated in Fig. 6, where a system with an 8 element UCA is considered, but only 1, 2 or 4 elements are active in the plot signaling. The active pilot antennas are distributed evenly over the UCA. As observed in the figure, the performance loss of using only one pilot antenna is significant, whereas 2 pilot antennas result in only a small performance loss over 4 and 8 pilot antennas, where the performance difference is negligible. October 26, 2007
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VII. C ONCLUSIONS An MMSE SINR estimation framework has been established that is suitable for resource allocation in multi antenna, wide-area systems. A greedy user selection scheme and several beamforming techniques, suitable for the MMSE framework were proposed and evaluated. It was demonstrated that, in wide area systems, it is sufficient to feed back a single scalar gain parameter from each users to achieve good performance for SDMA systems, when combined with CDI. The proposed framework was further shown to significantly outperform other techniques based on similar feedback. Similar performance was also achieved when common pilot signaling is restricted to only a small subset of the transmit antennas. A PPENDIX I D ERIVATIONS
OF
C ONDITIONAL M OMENTS
A. Proof of Theorem 1 Proof: Let Mx (y) denote a conditional moment (of arbitrary order) of x for a given y. The moment, Mht (η), can be expressed in terms of Mht (ha ) as Mht (η) = Eha {Mht (ha )|η} .
(28)
This follows from the expansion, Mht (η) Eht {Mht (ht )|η} = Eha Ehp {Mht (ht )|ha , η} |η , and
by noting that the conditioning on η in the inner expectation can be dropped because ht , by assumption, depends only on η through ha , which is fully determined in this expectation. Hence, we start by evaluating Mht (ha ) and next proceed with the outer expectation. For a given ha the stochastic behavior of ht is fully described by the distribution of hp|a which is interpreted as hp for a given ha . Also hp|a is complex Gaussian [30] and distributed as, ⎧ ⎨ R R − R R−1 RH , p pa a p|a pa hp|a ∈CN (hp|a , Rp|a ), ⎩ h h + R R−1 (h − h ¯ a ), p pa a p|a
(29)
a
where Rp|a is the Schur complement of Ra in (11). Note that hp|a = hp|a (ha ), whereas the conditional covariance, Rp|a , is unaffected by the actual realization of ha . It follows that the first order conditional moment or ht for a given ha is given by
⎡
⎤
⎡
⎤
h I ˆ t (ha ) E {ht |ha } = ⎣ a ⎦ = ht + ⎣ ⎦ (ha − ha ), h −1 hp|a Rpa Ra ˆ t (η) = E {ht |η} is obtained as in (13) by applying the outer expectation described in (28). from which h
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ˆ t (η), we start with the expansion To obtain the second order conditional moment, Q
H H H ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ E ht (ha )ht (ha )|η = ht (η)ht (η) + E ht (ha ) − ht (η) ht (ha ) − ht (η) η ⎡
⎡
⎤
⎤H
I I ˆ H (η) + ⎣ ˆ t (η)h ⎦ Ra (η) ⎣ ⎦ , =h t −1 −1 Rpa Ra Rpa Ra
(30)
where Ra (η) is defined in the lemma. From (29) it follows that the conditional moment, E ht hH t |ha ,
is given by
E ht hH t |ha
⎤ ⎡ 0 0 ˆ H (ha ) + ⎣ ˆ t (ha )h ⎦ =h t 0 Rp|a
ˆ t (η) in (14). which, if combined with (28) and (30), yields the expression for Q
B. Proof of Corollary 2 Proof: Define the stochastic variable h wH h = waH ha + wpH hp . If follows from (29), that for a given ha , the scalar, h, is complex Gaussian distributed as ⎧ ⎨ σ 2 = wH Rp − Rpa R−1 RH wp , p pa a ¯ σ 2 ), h ∈ CN (h, ⎩ h = wH h + wH h . a
Note that
σ2
a
p
p|a
is a constant, independent of ha . The mean component, h, does however depend on ha , but by
H a wa + R−1 aH ha , and w writing it as, h = hp + ha , where hp = wpH (hp − Rpa R−1 a ha ), ha = w a Rpa wp ,
the dependence can be isolated to ha . The fourth order moment, for a given ha , is obtained as [31], 4 4 2 2 2 E |h|4 ha = 2σ 4 + ¯ h + 4σ 2 ¯ h = 2σ 4 + hp + 4σ 2 hp + |ha |4 + 4(hp + σ 2 ) |ha |2 ∗ 2 ∗ ∗ + 2 2hp (hp + 2σ 2 )ha + 2hp (|ha |2 ha ) + (hp )2 h2a , where the identity |a + b|4 = |a|4 + |b|4 + 4 |a|2 |b|2 + 2 2 |a|2 ab∗ + 2 |b|2 ba∗ + a2 b∗2 , ∀a, b ∈ C, 4 4 was used to expand h = ha + hp . In this expression, only ha depends on ha . By averaging over ha the sought fourth order moment is obtained as, 4 4 2 2 E wH h |ρ = Eha E |h|4 ha |ρ = 2σ 4 + hp + 4σ 2 hp + m4 + 4(hp + σ 2 )m2
∗ 2 ∗ + 2 2hp (hp + 2σ 2 )m1 + m3 + (hp )2 m2T , (31) where m4 E |ha |4 |η , m1 E {ha |η}, m2 E |ha |2 η , m2T E h2a η and m3 2 E |ha | ha η . The theorem follows by completing the squares in (31) and by expanding the conditional aH ha . moments of ha = w October 26, 2007
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A PPENDIX II P ROOF
OF
T HEOREM 3
The derivation of the gradient of Γ (γ) is based on optimality conditions of the defining max-min problem in (26). The optimal downlink and virtual-uplink powers, p and q , solves the eigenvalue problems [6], [7] Υpext =
1 pext , Γ
pext = [p , 1]T ,
and
Λqext =
1 qext , Γ
qext = [q , 1]T ,
(32)
respectively. The optimal powers are obtained from the first elements of the extended power vectors, pext and qext , by normalizing them such that the last element is 1. The matrices, Υ and Λ, are the extended downlink and virtual-uplink coupling matrices, respectively, ⎡ ⎤ ⎡ DΨ Dσ DΨT ⎦ Υ ⎣ Λ ⎣ T 1 1 1 T T Pmax 1DΨ Pmax 1 Dσ Pmax σ DΨ
⎤
D1 1
⎦
T Pmax σ D1
ˆ where D diag{. . . , γk /uH k Qk uk , . . .}, Ψ is the cross-talk matrix (defined in Theorem 3), and the
notation of [7] is adopted. Observe that the left eigenvector of Λ is given by ⎤ ⎡ Ψ 0 1 ⎦. ext , where p ext = p ext Tpext , T ⎣ ΛT p Γ 0 Pmax This follows directly from (32) because ΛT can be factorized as ΛT = TΥT−1 . Next, consider the derivative of Γ (γ) with respect to γn . Define λ(γn ) as the eigenvalue of optimality condition (32), λ(γn ) 1/Γ Γ (γ)−1 . We can express Γ (γ) in terms of the derivative of Λ, ⎤ ⎤ ⎡ ⎡
I I ∂ ∂ Λ Λ= ⎣ T ⎦ DΨT D1 = ⎣ T ⎦ D ΨT + D(Ψ )T D 1 , σ σ ∂γn ∂λn !" # Pmax
as,
∂ ∂γn Γ (γ)
n) 2 2 = − λλ2(γ (γn ) = −Γ λ (γn ) = −Γ
Pmax
Text Λ qext p ext qText p
(33)
An
, where the last identity follows from [32, Theorem
ext is a left eigenvector of Λ. This holds whenever λ(γn ) is a simple eigenvalue (i.e., 6.3.12] because p
the solution is unique). By expanding Λ as in (33), the above expression yields pT D−1 An qext ∂ (Ψp + σ)T An qext Γ (γ) = −Γ2 = −Γ , ∂γn qT Ψp + Pmax qT Ψp + Pmax
(34)
where the last identity is obtained by applying the relation ( Γ1 I − DΨ)p = Dσ , which follows by rearranging (24) [7]. The k th element of the vector, An qext , is obtained straight forwardly by explicitly
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writing out the derivatives as, [An qext ]k = [D (ΨT q + 1) + DT ]k q
=qk /SINRVUL = Γγk k
# &' " $ % % ! & H H ˆ ˆ j u Hˆ u q + I u q uk Q Q uk Qk uk j j j k j=k j=k k k = δkn − 2γk +2γk HQ HQ ˆ k uk ˆ k uk ˆ k uk uH u u Q k k k qk ˆ ˆ 2γk uH j=k qj Qj − Γγk Qk uk k qk qk δkn + = δkn , (35) = H ˆ Γγk Γγk uk Qk uk k
where δkn is the Kronecker delta function, δkn = 1 for k = n and zero otherwise. The last identity follows by noting that the optimal beamformers, uk , which solve (20) satisfy the generalized eigenvalue problem, ˆ k uk = Γγk ( ˆ qk Q j=k qj Qj + I)uk , which by rearranging the terms and multiplying with uk yields H qk ˆ ˆ 0 = uH k ( j=k qj Qj − Γγk Qk )uk , where the normalization, uk = 1, ensures uk Iuk = 0. Theorem 3 follows by combining (34) and (35). R EFERENCES [1] D. Hammarwall, M. Bengtsson, and B. Ottersten, “Beamforming and user selection in SDMA systems utilizing channel statistics and instantaneous SNR feedback,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Apr. 2007. [2] D. Hammarwall and B. Ottersten, “Spatial transmit processing using long-term channel statistcs and pilot signaling on selected antennas,” in Proc. Asilomar Conf. on Signals, Systems and Computers, Oct. 2006. [3] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, 1998. [4] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecommun., vol. 10, no. 6, pp. 585–595, 1999. [5] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmit beamforming and power control for cellular wireless systems,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1437–1450, Oct. 1998. [6] M. Schubert and H. Boche, “Solution of the multi-user downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. Technol., vol. 53, no. 1, Jan. 2004. [7] D. Hammarwall, M. Bengtsson, and B. Ottersten, “On downlink beamforming with indefinite shaping constraints,” IEEE Trans. Signal Processing, vol. 54, no. 9, pp. 3566–3580, Sept. 2006. [8] R. Knopp and P. A. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proc. IEEE Int. Conf. on Commun., vol. 1, June 1995, pp. 331–335. [9] A. J. Goldsmith and S. G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. 46, no. 5, May 1998. [10] D. Samuelsson, M. Bengtsson, and B. Ottersten, “Improved multiuser diversity using smart antennas with limited feedback,” in Proc. European Signal Processing Conf., Sept. 2005. [11] D. Hammarwall, M. Bengtsson, and B. Ottersten, “Acquiring partial CSI for spatially selective transmission by instantaneous channel norm feedback,” IEEE Trans. Signal Processing, June 2007, accepted for publication. October 26, 2007
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