The hostility of the wireless fading environment and channel variation makes the design of high rate communication systems very challenging.  To this end, multiple-antenna systems have shown to be effective in fading environments by providing significant performance improvements and achievable data rates in comparison to single antenna systems.  Wireless communication systems employing multiple antennas both at the transmitter and the receiver demonstrate tremendous potential to meet the spectral efficiency requirements for next generation wireless applications.  This has spurred research in the efficient design and deployment of various multiple-input-multiple-output (MIMO) configurations for practical systems.  The collection of papers in IEEE Transactions on Information Theory, vol.  49, Issue 10, October 2003, represents a sample of the wide array of research in MIMO systems. 
Moreover, multiple transmit and receive antennas have become an integral part of the standards of many wireless systems such as cellular systems and wireless LANs.  In particular, the recent development of UMTS Terrestrial Radio Access Network (UTRAN) and Evolved-UTRA has raised the need for multiple antenna systems to reach higher user data rates and better quality of service, thereby resulting in an improved overall throughput and better coverage.  A number of proposals have discussed and concluded the need for multiple antenna systems to achieve the target spectral efficiency, throughput, and reliability of EUTRA.  These proposals have considered different modes of operation applicable to different scenarios.  The basic assumptions that vary among such proposals include: (i) using a single stream versus multiple streams; (ii) scheduling one user at a time versus multiple users; (iii) having multiple streams per user versus a single stream per user; and (iv) coding across multiple streams versus using independent streams.  A basic common factor, however, among the various downlink physical layer MIMO proposals is a feedback strategy to control the transmission rate and possibly a variation in transmission strategy. 
While the proposals for the use of multiple antenna systems in downlink EUTRA such as per antenna rate control (PARC), per stream rate control (PSRC), per group rate control (PGRC), per user and stream rate control (PUSRC), per user unitary rate control (PU2RC), single codeword/multiple codeword transmission (SCW/MCW), spatial domain multiplex/spatial domain multiple access (SDM/SDMA), and current transmit diversity scheme in release 6 such as selection transmit diversity (STD), space-time transmit diversity (STTD), and transmit adaptive antennas (TxAA) differ in terms of the system description, they all share the following features: (i) possible multiplexing of streams to multiple streams; (ii) possible use of linear precoding of streams before sending to antennas; (iii) possible layering of the streams between the antennas; and (iv) rate control per stream or multiple jointly coded streams. 
The performance gain achieved by multiple antenna system increases when the knowledge of the channel state information (CSI) at each end, either the receiver or transmitter, is increased.  Although perfect CSI is desirable, practical systems are usually built only on estimating the CSI at the receiver, and possibly feeding back the CSI to the transmitter through a feedback link with a very limited capacity.  Using CSI at the transmitter, the transmission strategy is adapted over space (multiple antennas) and over time (over multiple blocks). 
The performance of multiple antenna systems with or without knowledge of the channel state information has been extensively analyzed over the last decade.  Partial feedback models have been considered due to the limitations of the feedback channel from the receiver to the transmitter.  Different partial feedback models include: channel mean feedback (see, e.g., A. Narula et al., “ Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE Journal on Selected Areas of Communications, vol.  16, no.  8, pp.  1423-1436, October 1998); channel covariance feedback (E. Visotsky et al., “ Space-time precoding with imperfect feedback,” in Proceedings ISIT 2000, Sorrento, Italy, June 2000); feedback of k-out-of-min(M,N) eigenvectors and eigenvalues of an M×N multiple antenna channel (J. Roh et al., “ Multiple antenna channels with partial channel state information at the transmitter,” Wireless Communications, IEEE Transactions on, vol.  3, pp.  677-688, 2004); partial feedback based on statistical model and robust design (A. Abdel-Samad et al., “ Robust transmit eigen-beamforming based on imperfect channel state information,” in Smart Antennas, 2004.  ITG Workshop on, 2004); and quantized feedback. 
Beamforming introduces an alternative use of quantized feedback bits in which the design is almost independent of the average received SNR and constant average transmit power is assumed at the transmitter.  In beamforming, independent streams are transmitted along different eigenmodes of the channel resulting in high transmission rates without the need to perform space-time coding. 
Beamforming has received considerable attention for the case of multiple transmit antennas and a single receive antenna. ( See, e.g., Narula et al.  cited above.)  Rate regions for the optimality of dominent eigen-beamformers (rank-one beamformers) in the sense of maximizing mutual information has also been studied (see, e.g., S. A. Jafar, et al., “ Throughput maximization with multiple codes and partial outages,” in Proceedings of Globecom Conference, San Antonio, Tex.,  USA, 2001), as have systematic constructions for finite-size beamformer codebooks for multiple transmit single receive antenna systems resulting in near optimal performance.  Similarly, a design criterion for a dominant eigen-beamformer, for use with both single and multiple receive antenna systems, has been proposed. ( See D. Love et al., “ Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Transactions on Information Theory, vol.  49, pp.  2735-2747, 2003.)  As discovered by the inventors of the present invention, however, such unit rank beamformers can result in significant performance degradation with MIMO systems for certain transmission rates, thus requiring higher rank transmission schemes. 
The design of higher rank beamformers for MIMO systems has also been studied in the past.  Roh et al.  addressed the problem of MIMO beamforming in the presence of perfect knowledge about a subset of the channel eigenvectors.  Knopp et al.  explored the design of joint power control and beamforming when the eigenvectors and the eigenvalues are completely known at the transmitter. ( See R. Knopp et al., “ Power control and beamforming for systems with multiple transmit and receive antennas,” IEEE Transactions on Wireless Communications, vol.  1, pp.  638-648, October 2002.)  Jafar et al.  derived the conditions for the optimality of MIMO beamforming (in the sense of achieving capacity) when the channel covariance is fed back to the transmitter. ( See, S. A. Jafar et al. “ Throughput maximization with multiple codes and partial outages,” in Proceedings of Globecom Conference, San Antonio, Tex.,  USA, 2001.)  The design of MIMO systems using multiple simultaneous streams for transmission when finite rate feedback is available in the system has also been studied (see Love et al.,  cited above.)  The design criterion therein sought to quantize the set of active eigenvectors such that the loss in SNR compared to perfect channel feedback is minimized. 
In addition to the aforementioned considerations, it is desirable to achieve the highest possible spectral efficiencies in MIMO systems with reasonable receiver and transmitter complexity.  Though theoretically space-time codes are capable of delivering very high spectral efficiencies, e.g. 100 s of megabits per second, their implementation becomes increasingly prohibitive as the bandwidth of the system increases.