Feedbackinacommunicationssystemcan enablethetransmittertoexploitchannelcondi - tionsandavoidinterference.Inthecaseofa multiple-inputmultiple-outputchannel,feedback canbeusedtospecifyaprecodingmatrixatthe transmitter,whichactivatesthestrongestchan - nelmodes.Insituationswherethefeedbackis severelylimited,importantissuesarehowto quantizetheinformationneededatthetransmitterandhowmuchimprovementinassociated performancecanbeobtainedasafunctionof theamountoffeedbackavailable.Wegivean overviewofsomerecentworkinthisarea.Meth - odsarepresentedforconstructingasetofpossibleprecodingmatrices, fromwhichaparticular choicecanberelayedtothetransmitter.Perfor - manceresultsshowthatevenafewbitsoffeedbackcanprovideperformanceclosetothatwith fullchannelknowledgeatthetransmitter.

Abstract—Multiple-input multiple-output (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space–time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to illconditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter—which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a low-delay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleigh-fading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection two-norm and Fubini–Study distances. Simulation results showthat the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback. Index Terms—Diversity methods, Grassmannian subspace packing, multiple-input multiple-output (MIMO) systems, quantized precoding, Rayleigh channels, spatial multiplexing, vertical Bell Labs layered space– time (V-BLAST) architecture. I.

Abstract — This paper considers the quantization problem on the Grassmann manifold with dimension n and p. The unique contribution is the derivation of a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results are only valid for either p =1or a fixed p with asymptotically large n. Based on the volume formula, the Gilbert-Varshamov and Hamming bounds for sphere packings are obtained. Assuming a uniformly distributed source and a distortion metric based on the squared chordal distance, tight lower and upper bounds are established for the distortion rate tradeoff. Simulation results match the derived results. As an application of the derived quantization bounds, the information rate of a Multiple-Input Multiple-Output (MIMO) system with finite-rate channel-state feedback is accurately quantified for arbitrary finite number of antennas, while previous results are only valid for either Multiple-Input Single-Output (MISO) systems or those with asymptotically large number of transmit antennas but fixed number of receive antennas. I.

Abstract—This paper investigates the problem of transmit beamforming in multiple-antenna spatial multiplexing (SM) systems employing a finite-rate feedback channel. Assuming a fixed number of spatial channels and equal power allocation, we propose a new criterion for designing the codebook of beamforming matrices that is based on minimizing an approximation to the capacity loss resulting from the limited rate in the feedback channel. Using the criterion, we develop an iterative design algorithm that converges to a locally optimum codebook. Under the independent identically distributed channel and high signal-to-noise ratio (SNR) assumption, the effect on channel capacity of the finite-bit representation of the beamforming matrix is analyzed. Central to this analysis is the complex multivariate beta distribution and tractable approximations to the Voronoi regions associated with the code points. Furthermore, to compensate for the degradation due to the equal power allocation assumption, we propose a multimode SM transmission strategy wherein the number of data streams is determined based on the average SNR. This approach is shown to allow for effective utilization of the feedback bits resulting in a practical and efficient multiple-input multiple-output system design. Index Terms—Channel capacity, channel information feedback, matrix quantization, multiple antennas, multiple-input multiple-output (MIMO) systems, spatial multiplexing, transmit beamforming. I.

Space-time block codes (STBC) are powerful techniques to achieve full spatial diversity in Rayleigh fading channels with low decoding complexity. When channel state information is available to the transmitter, space-time block codes can be improved through the use of a transmit precoder chosen based in part on the current channel state. In this paper, we investigate the downlink transmission of an Alamouti space-time block coded multiuser wireless system in which complete channel state information is available to the base transceiver station. We propose a novel precoder that, with enough antennas, can effectively precancel co-channel interference at each mobile user thus enabling simple single user space-time block decoding. We show that excess transmit antennas, beyond the minimum required for precancellation, can be used to further improve the diversity performance through careful selection of the precoding matrix. Though the derivation assumes perfect channel state information, we investigate the impact of estimation error by deriving an upper bound on the signal to interference plus noise ratio. Monte Carlo simulations show that our proposed downlink precoder can effectively cancel the interference between mobiles, while still providing good diversity performance. I.

The benefits of using multiple antennas at both the transmitter and the receiver in a wireless system are well established. Multiple-input multiple-output (MIMO) systems enable a growth in transmission rate linear in the minimum of the number of antennas at either end [1][2]. MIMO techniques also enhance link reliability and

Abstract — In this paper, we propose two efficient lowcomplexity quantization methods for multiple-input multipleoutput (MIMO) systems with finite-rate feedback based on proper parameterization of the information to be fed back followed by quantization in the new parameter domain. For a MIMO channel which has multiple orthonormal vectors as channel spatial information, we exploit the geometrical structure of orthonormality while quantizing the spatial information matrix. The parameterization is of two types: one is in terms of a set of unit-norm vectors with different lengths, and the other is in terms of a minimal number of scalar parameters. These parameters are shown to be independent for the i.i.d. flatfading Rayleigh channel, facilitating efficient quantization. In the first scheme, each of the unit-norm vectors is independently quantized with a finite number of bits using an optimal vector quantization (VQ) technique. Bit allocation is needed between the vectors, and the optimum bit allocation depends on the operating SNR of the system. In the second scheme, the scalar parameters are quantized. In slowly time-varying channels, the scalar parameters are also found to be smoothly changing over time, leading to the development of a simple quantization and feedback method using adaptive delta modulation. The results show that the proposed feedback scheme has a channel tracking feature and achieves a capacity very close to perfect feedback with a reasonable feedback rate. Index Terms — Channel information feedback, channel state information, MIMO systems, multiple antennas, parameterization, quantization, transmit beamforming. I.

Abstract — This paper addresses the performance of bitinterleaved coded multiple beamforming (BICMB) with imperfect knowledge of beamforming vectors. Various wireless standards become equivalent to BICMB when they are operated in beamforming mode. In BICMB, the invariance of the precoding matrix under an arbitrary unitary transform widely studied in the literature is not applicable. On the other hand, the optimum precoder and detector are not unique because of invariance under a diagonal unitary transform. We propose an optimal Euclidean distortion measure and a new linear detector. In addition, a new codebook design is proposed via the generalized Lloyd algorithm based on the new distortion measure. We provide simulation results demonstrating the performance improvement achieved with the proposed distortion measure and the linear detector. I.

Transmitter precoding for multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) is an effective way of leveraging the diversity gains afforded by a multiple transmit-multiple receive antenna system in a frequency selective environment. In the limited feedback scenario, optimal precoder representation for narrowband MIMO systems using moderately sized codebooks designed on the Grassmann manifold has been shown to perform remarkably well. In MIMO-OFDM systems precoder matrices have to be designed for all subcarriers and the amount of feedback can get prohibitively large. This is especially true for next generation wireless local area networks and wireless metropolitan area networks which have a large number of subcarriers. In this paper, we present techniques to reduce this feedback requirement and the performance of these algorithms is numerically shown to provide improvement over existing schemes. Index Terms- Closed-loop MIMO systems, Orthogonal frequency division multiplexing,

Abstract. This paper contains a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix. This approach is both powerful and versatile. In cases where experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini– Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing radii that are nearly optimal. 1.