schemes considered for future wireless standards, rely on simple beamform-ing techniques, focusbeamform-ing on the design of accurate and meanbeamform-ingful feedback measures. In this chapter, we highlight the importance of designing linear beamforming techniques robust to noisy CSIT. A linear beamforming tech-nique based on iterative optimization of unitary matrices is proposed, which achieves linear sum-rate growth with the number of transmit antennas and outperforms common linear beamforming techniques under imperfect CSIT conditions. As our results show, the performance of a system with joint lin-ear beamforming and multiuser scheduling in limited feedback scenarios can be improved by optimizing the linear beamformers, combined with simple feedback design and quantization techniques.
Finally, the general conclusions reached in this dissertation are presented, summarizing the main results obtained as well as future challenges of MIMO technologies.
1.2 Contributions
The contribution in Chapter 2 is the derivation of linear precoding schemes that minimize an upper bound on the pairwise error probability [18], [19].
Our work generalizes the work presented in [20], by averaging the PEP over a Gaussian distribution - prior or posterior - that can correspond to differ-ent scenarios, providing a solution to the general problem and a variety of particular cases.
In Chapter 3, the two main contributions are the low-complexity schemes proposed in [21] and [22], for perfect CSIT and limited feedback, respectively.
The former, coined as orthogonal linear beamforming (OLBF), consists of a joint solution for beamforming and scheduling which aims at reducing the complexity of exhaustive-search user selection algorithms, and builds upon the work presented in [2] for limited feedback scenarios. The second part of the chapter proposes a scheme which exploits the spatial correlation at the transmitter in a setting with limited feedback. The users are assigned a fixed channel quantization vector which is used as beamforming vector -and feed back information regarding the channel strength -and quantization error. We derive a useful upper bound on the multiuser interference, which is computed at the base station for the purpose of user selection.
The contributions in Chapter 4 are the result of a number of publications
in the quest for high-performance scalar feedback measures. In Chapter 4, a design framework is proposed [23] that generalizes the work in [24], [25], [26], [27], as well as other scalar measures used in well known approaches such as [2]
or [28]. After deriving an approximated cumulative distribution function for the proposed family of metrics and its associated sum rate approximation, our framework enables us to perform simple asymptotic analysis in different regimes, namely: large number of users, high SNR regime and low SNR regime. In addition, a clarifying comparison between TDMA and SDMA is given under different conditions, highlighting the importance of allowing a variable number of active beams at the transmitter. We identify the multiuser diversty vs. multiplexing gain tradeoff arising in scenarios where the total sum of bits for channel direction information and channel quality information - contained in the scalar feedback - is limited [29].
In Chapter 5, new codebook design approaches are proposed [30]. A Monte Carlo approach is used to generate optimized channel quantization codebooks in order to exploit the cell statistics, by minimizing the average sum rate distortion. In the first part, we stress the importance of adapting the codebook to non uniform user distributions. In the second part, predic-tive vector quantization is used to improve the performance by exploiting temporal correlations.
A novel method for iterative optimization of unitary beamformers is pro-posed in Chapter 6 [31], [32], based on successive optimization of Givens rotations. A convergence and complexity study is presented, evaluating the performance through simulations in several scenarios. As we show, the pro-posed technique achieves linear sum-rate increase with the number of trans-mit antennas and perfect channel knowledge at the transtrans-mitter side. More importantly, the proposed unitary beamforming approach proves to be very robust to channel estimation errors. When combined with simple vector quantization techniques for CSI feedback in MIMO broadcast channels, the proposed technique is shown to be well suited for limited feedback scenarios.
Other articles published in parallel during the course of this thesis, which have not been included in this dissertation, are the following. In [33], linear precoders that exploit the covariance information of the MIMO channel are presented, which are combined with spatio-temporal spreading. In the con-text of WCDMA systems, adaptive complexity equalizers have been proposed in [34].
Part I
Point-to-Point MIMO Channels
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Chapter 2
Linear Precoding
In this chapter techniques are proposed for combining information about the mean and the covariance of the channel for the purpose of MIMO trans-mission in point-to-point systems. Partial channel state information at the transmitter (CSIT) is typically used in MIMO systems for the design of spa-tial prefiltering and waterfilling. For the purpose of generating CSIT, the cases of mean or covariance information have generally been solved sepa-rately in the literature. A Bayesian approach is presented here incorporating both pieces of information. The proposed Bayesian approach encompasses the existing cases of mean or (transmit) covariance information as special instances. Various cases of mean and covariance information are discussed, including prior mean and covariance (Ricean channel distribution) and pos-terior mean and covariance (based on a noisy channel estimate and prior covariance information). For a given Gaussian channel distribution (prior or posterior), an optimized linear precoding solution is derived, which mini-mizes an upper bound on the pairwise error probability in a space-time coded system. In addition, several particular cases of practical interest are studied, namely: zero mean information, unit rank mean and singular covariance in-formation. Simulation results illustrate the performance benefits that can be reached by effectively exploiting the available mean and covariance informa-tion in point-to-point MIMO systems.
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2.1 Introduction
In practical wireless systems, training sequences or pilot symbols are in-corporated in the transmitted signal to allow for channel estimation at the receiver. The density of training data needs to increase as the mobility and the channel variation increases. Nevertheless, even with training data available, the channel estimate can only be of limited quality, and the chan-nel estimation errors reduce the chanchan-nel capacity. Furthermore, the fact of substituting information symbols by training symbols obviously limits the capacity. Channel knowledge at the transmitter helps improving the system throughput in wireless systems. A means to obtain CSIT consists of feeding back to the transmitter channel estimates obtained at the receiver. Since the bandwidth available is limited, all statistical information about the channel should be taken into account. For instance, a priori information on the chan-nel distribution can be used to yield improved chanchan-nel estimates, leading to a posterior channel distribution, as we show in this chapter.
The presence of severe correlations has important detrimental effects on the capacity and performance of MIMO systems. In fact, most space-time code designs assume independent Rayleigh fading for each stream, which in practice is not true as shown in [35]. The problem has been addressed by transmitting on the eigen-modes of the transmit antenna correlation ma-trix [36], which yields better performance and capacity gains. In [37], a prefiltering approach is proposed assuming partial CSIT, where knowledge of the transmit antenna correlations is successfully exploited to improve the pairwise error probability (PEP) of a space-time (ST) coded system.
In order to exploit partial CSIT in point-to-point MIMO systems, most of the current precoding schemes exploit either information about the mean [38]
or the covariance [37]. A combination of the two can improve exploitation of channel knowledge by weighting them according to certain criteria. In [20], the combination of mean and covariance information at the transmitter side of a point-to-point MIMO system is considered, for the purpose of PEP min-imization. In that work, the transmitter concatenates orthogonal space-time block coding (O-STBC) and linear precoding. The available CSIT is used to design an optimized linear precoder, which adapts the transmitted ST codewords to the channel statistics. However, no closed form solution is pro-vided to the general problem, and the optimal precoder is solved numerically.
In our work, we consider a similar scenario, in which the linear precoder is optimized in order to minimize an upper bound on the PEP. Following the
2.2 MIMO Channel Model 55