In this chapter, techniques for combining mean and covariance information have been presented. Both sources of information can be either prior (e.g.
correlated channel with LOS) or posterior (given by a Bayesian approach).
We have provided a general precoding solution for PEP minimization when combining both sources of partial CSIT at the transmitter, and analyzed some cases of special interest. The simulation results have shown how mean and covariance information should be combined in order to exploit the avail-able sources of CSIT in point-to-point MIMO systems.
2.7 Conclusions 65
0 1 2 3 4 5 6 7 8 9 10
10−2 10−1 100
SNR [dB]
PEP
No Precoding Perfect CSIT Mean knowledge Covariance knowledge Mean & Covariance knowledge
Figure 2.2: PEP vs. SNR for different CSIT levels, ρ= 0.9 andγ = 20%
APPENDIX
2.A Derivation of Posterior Ditribution
Assuming the additive noise modelh=h+h, our goal is to find the posterior distribution ofh, given knowledge of the distributions of handhand perfect knowledge ofh. Let ph(h) andph(h) be the probability density functions of h andh, respectively. We are interested in finding the posterior distribution (or conditional distribution) ofhconditioned onh, denoted asph|h(h|h), which is completely specified by its mean and covariance since it is a Gaussian process.
We define the posterior mean as h =Eh|hh and the posterior covariance as Chh =Eh|h
h−Eh|hh h−Eh|hh H
.
The posterior mean is calculated as follows h=mh+ChhC−1
hh
"
h−mh
#
(2.28) where Chh = E[h−mh][h−mh]H, Chh =E[h−mh][h−mh]H and Chh = [h−mh][h−mh]H. The posterior covariance is given by
Chh =Chh−ChhC−1
hhChh . (2.29)
Based on the known distributions of h and h and due to independence between these random processes, we have that Chh = Chh = Chh and Chh =Chh+Chh. Substituting the obtained expressions for Chh, Chh and Chh into equations (2.28) and (2.29), and after standard matrix manipula-tions, we obtain the desired result for the posterior mean and covariance.
2.B Proof of Theorem 2.1
The optimization problem described in (2.16) can be expressed as
minΨ tr
mHCT−1
ηΨ +CT−1−1
CT−1mHH
−Nlog det(ηΨ +CT−1) s.t. tr(Ψ) = 1
2.C Proof of Corollary 2.1 67 The solution is obtained by means of the Karush-Kuhn-Tucker (KKT) con-ditions. Define the Lagrangian as
L
(Ψ, µ) = trwhere µ is the Lagrange multiplier associated with the equality constraint.
Differentiating
L
(Ψ, µ) w.r.t. Ψ, we getµΦΦ−ηNΦ−ηCT−1mHHmHCT−1 = 0 (2.31) where the change of variable Φ = ηΨ +CT−1 has been used for clarity. The solution for Ψ to the quadratic matrix equation described above is given by
Ψ =
where {·}+ takes the positive semidefinite (PSD) part.
2.C Proof of Corollary 2.1
Introducing the eigenvalue decompositions CT−1mHHmHCT−1 = UΣUH and IM have been used. Since the solution we seek assumes Ψ to be PSD, also Λ− 1ηVHCT−1V is assumed PSD. The optimum V that minimizes the first term can be chosen as V=U[20].
Let Σ = diag(σ1, , . . . , σM) and Λ = diag(λ1, . . . , λM). The Lagrangian in this case is given by
L
(λi, µ) =where µ is the Lagrange multiplier corresponding to the power constraint.
Differentiating
L
(λi, µ) w.r.t. λi, we get λi =N +
N2 + 4µσηi
2µ . (2.33)
Thus, the solution for Ψ is given by
Ψ =UΛUH − 1 ηCT−1.
Part II
MIMO Broadcast Channels
69
Chapter 3
Joint Linear Beamforming and Multiuser Scheduling
In this chapter, the problem of joint linear beamforming and multiuser schedul-ing is presented. A brief overview is provided, introducschedul-ing the most extended linear beamforming techniques and scheduling algorithms in the literature. In order to illustrate the challenges, advantages and disadvantages of such sys-tems, two low-complexity approaches are proposed. In the first approach, a scenario with perfect CSIT is considered, in which a simple multiuser schedul-ing strategy combined with a linear beamformschedul-ing technique based on orthog-onal beams are presented. This, as we show, leads to a dramatic complexity reduction in the multiuser scheduling stage. In the second approach, a low-complexity solution is proposed for a scenario with limited feedback. As we show, simple codebooks adapted to transmit spatial correlation can yield large performance gains. In addition, a bound on the multiuser interference ex-perienced by each user is derived, based on a geometric interpretation of the problem. This bound is of capital importance since, as we show in the next chapter, it can be used for the design of feedback measures for the purpose of user selection.
71
3.1 Introduction
MIMO systems can significantly increase the spectral efficiency by exploiting the spatial degrees of freedom created by multiple antennas. The capacity can be boosted by exploiting the spatial multiplexing capability of transmit antennas, transmitting to multiple users simultaneously by means of space division multiple access (SDMA), rather than maximizing the capacity of a single-user link, as shown in [10], [11]. It has recently been proven in [43]
that the sum capacity of the MIMO broadcast channel is achieved by dirty paper coding (DPC) [1]. However, the applicability of DPC is limited due to its computational complexity and the need for full CSIT.
A promising low complexity alternative for the downlink of MIMO sys-tems is linear beamforming. As opposed to DPC, linear beamforming does not perform any interference pre-subtraction, and thus multiuser interference is treated as noise. Finding the optimal beamforming vectors is a non-convex optimization problem, and the optimal solution for a downlink channel with K users can only be given by exhaustive search. Evidently, the complex-ity of this problem becomes prohibitively high for large K. Hence, we fo-cus instead on suboptimal linear beamforming strategies which, combined with efficient multiuser scheduling algorithms to exploit the multiuser diver-sity [17], achieve high sum rates. Downlink linear beamforming has been shown to achieve a large portion of DPC capacity, exhibiting the best trade-off between complexity and performance [13], [14], [16]. When combined with multiuser scheduling, zero-forcing beamforming [44] has been shown to achieve the same asymptotic sum rate as that of DPC [15].
In point-to-point MIMO systems, the capacity increases linearly with the minimum of the number of transmit/receive antennas, irrespective of the availability of channel state information (CSIT) [4], [5]. In multiuser MIMO systems, the capacity gain is highly dependent on the available CSIT. If a base station with M transmit antennas, communicating with K single-antenna receivers, has perfect CSIT, a multiplexing gain of min(M, K) can be achieved. The approximation of close to perfect CSI at the receiver (CSIR) is often reasonable; however, this assumption is unrealistic at the transmit-ter. Recently, it was shown that if the base station has imperfect channel knowledge, the full multiplexing gain is reduced at high SNR [45], whereas if there is complete lack of CSI knowledge, the multiplexing gain collapses to one [46]. Hence, as the broadcast channel’s capacity is sensitive to the accuracy of CSIT, it is of particular interest to identify what kind of partial
3.1 Introduction 73 CSIT can be conveyed to the base station in order to achieve data rates rea-sonably close to the optimum. In this disertation, we give special interest to this issue, defining and optimizing the feedback that the user terminals need to compute and send to the base station. This feedback must be designed with the purpose of not only yielding good channel estimates for linear beam-forming design, but also enabling the base station to perform efficient user selection.
Several limited feedback approaches, imposing a bandwidth constraint on the feedback channel, have been studied in point-to-point MIMO sys-tems [47], [48], [49], [50]. In this context, each user feeds back finite precision (quantized) CSIT on its channel direction by quantizing its normalized chan-nel vector to the closest vector contained in a predetermined codebook. An extension of the limited feedback model for multiple antenna broadcast chan-nels for the case of K ≤M is made in [51], [52]. In [51] it was shown that the feedback load per user must increase approximately linearly with M and the transmit power (in dB), in order to achieve the full multiplexing gain. These schemes rely only on channel direction information and no information on the channel magnitude is provided.
Recently, MIMO broadcast channels with limited feedback and more users than transmit antennas (i.e. K ≥M) have attracted particular interest and several joint beamforming and scheduling schemes aiming to maximize the sum rate have been proposed. A popular, very low-rate feedback technique, coined as random beamforming (RBF), is proposed in [2], where M random orthonormal beamforming vectors are generated and the best user on each beam is scheduled. By exploiting multiuser diversity, this scheme is shown to yield the optimal capacity growth of Mlog logK for K → ∞. However, the sum rate of this scheme degrades quickly with decreasing number of users.
A different type of limited feedback approaches considers that each user reports channel direction information (CDI) related to a codebook back to the transmitter, as well as some form of scalar channel quality informa-tion (CQI). Several beamforming methods have been investigated, including unitary beamforming [53], transmit matched-filtering [54], and zero-forcing beamforming [54], [55], [26], [56]. Note that in these contributions, the CQI feedback is considered unquantized for analytical simplicity. Considering the more realistic constraint of finite CSI feedback rate, [29] studies the problem of optimal bit rate allocation between CQI and CDI, and the resulting mul-tiuser diversity - multiplexing gain tradeoff in limited feedback MIMO sys-tems. On the other hand, [57] proposes a threshold-based feedback scheme
for SDMA systems under a sum feedback rate constraint.
The benefit of using as CQI a measure related to the signal-to-interference-plus-noise ratio (SINR) was shown in [28]. One challenge of designing feed-back metrics is that the SINR measurement depends, among others, on the channel as well as on the number of simultaneously scheduled users. Since user cooperation is not allowed, the number of simultaneous users and the available power for each of them is generally unknown at the mobile. In addition, the feedback metrics must be computed at the user end prior to computation of the beamforming vectors at the transmitter side. Thus, es-timation of the multiuser interference at the receiver side is a difficult and critical task. A principal drawback of previous works is that the proposed metrics assume a fixed number of scheduled SDMA users, based on non-achievable SINR upper bounds. Instead, it may be interesting to consider schemes allowing adaptive transition between SDMA and time division mul-tiple access (TDMA) modes, as well as more practical received SINR esti-mates. This problem is addressed in Chapter 4, in a scenario with separate CDI and CQI feedback. In Chapter 5, a different limited feedback scenario is considered, in which the users quantize directly their vector channels by using optimized channel quantization codebooks, thus embedding channel direction and quality information in a single codebook.
The remainder of this chapter is organized as follows. Initially, an intro-duction to linear beamforming schemes with perfect CSIT is provided. Com-mon linear beamforming techniques used in the multiuser MIMO literature are formally presented, namely zero-forcing beamforming, minimum mean squared error beamforming and transmit matched filtering. In the following section, the problem of multiuser selection comes into play. Standard user selection algorithms are reviewed, which, together with the linear beamform-ing techniques previously mentioned, will be often referred to throughout this dissertation. In the last two sections of this chapter, reduced-complexity approaches combining linear beamforming and multiuser scheduling are pro-posed, first in a scenario with perfect CSIT and next in a more realistic scenario with limited feedback.