bit allocation laws for multiantenna channel feedback quantization multiuser case this paper addresses the optimal design of limited feedback downlink multiuser spatial multiplexing systems a multiple antenna base station is assumed to serve multiple single antenna users who quantize and feed back their channel state information csi through a shared rate limited feedback channel the optimization problem is cast in the form of minimizing the average transmission power at the base station subject to users target signal to interference plus noise ratios sinr and outage probability constraints the goal is to derive the feedback bit allocations among the users and the corresponding channel magnitude and direction quantization codebooks in a high resolution quantization regime toward this end this paper develops an optimization framework using approximate analytical closed form solutions the accuracy of which is then verified by numerical results the results show that for channels in the real space the number of channel direction quantization bits should be m 1 times the number of channel magnitude quantization bits where m is the number of base station antennas moreover users with higher requested quality of service qos i e lower target outage probabilities and higher requested downlink rates i e higher target sinr s should use larger shares of the feedback rate it is also shown that for the target qos parameters to be feasible the total feedback bandwidth should scale logarithmically with the geometric mean of the target sinr values and the geometric mean of the inverse target outage probabilities in particular the minimum required feedback rate is shown to increase if the users target parameters deviate from the corresponding geometric means finally the paper shows that as the total number of feedback bits b increases the performance of the limited feedback system approaches the perfect csi system as 2 b m2

discriminants of brauer algebras in this paper we compute gram determinants associated to all cell modules of brauer algebras fancyscript b n partial theoretically we know when a cell module of fancyscript b n partial is equal to its simple head this gives a solution of this long standing problem

flexible codebook design for limited feedback systems via sequential smooth optimization on the grassmannian manifold grassmannian quantization codebooks play a central role in a number of limited feedback schemes for single and multi user multiple input multiple output mimo communication systems in practice it is often desirable that these codebooks possess additional properties that facilitate their implementation beyond the provision of good quantization performance although some good codebooks exist their design tends to be a rather intricate task the goal of this paper is to suggest a flexible approach to the design of grassmannian codebooks based on sequential smooth optimization on the grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties as one example the proposed approach is used to design rank 2 codebooks that have a nested structure and elements from a phase shift keying psk alphabet in some numerical comparisons codebooks designed using the proposed methods have larger minimum distances than some existing codebooks and provide tangible performance gains when applied to a simple mimo downlink scenario with zero forcing beamforming per user unitary beamforming and rate control pu 2rc and block diagonalization signalling furthermore the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance

a weighted least squares approach to precoding with pilots for mimo ofdm next generation wireless systems are expected to leverage multiple antenna commonly called multiple input multiple output mimo technology transmitting over broadband channels via orthogonal frequency division multiplexing ofdm it is now well known that mimo systems can obtain error rate and capacity improvements by adapting the transmitter to the current channel conditions linear spatial precoding is a popular technique for channel adaptation where the transmitted space time signal for each subcarrier is multiplied by a precoding matrix before transmission spatial precoding is complicated however in ofdm because channel information is often only known for a small number of pilot tones in this correspondence we present a subspace interpolation method for designing spatial precoders for all subcarriers by using the precoders known on pilot tones the nonpilot precoders are solutions to a weighted least squares design on the grassmann manifold simulation results show performance benefits over existing mimo ofdm techniques

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

asymptotic subword complexity of fixed points of group substitutions the subword complexity of fixed points of some types of substitutions was studied by various authors here we introduce a family of substitutions arising from multiplication tables of finite groups and other similar structures and analyze their subword complexity

quantization bounds on grassmann manifolds and applications to mimo communications the grassmann manifold gn p l is the set of all p dimensional planes through the origin in the n dimensional euclidean space ln where l is either r or c this paper considers the quantization problem in which a source in gn p l is quantized through a code in gn q l with p and q not necessarily the same the analysis is based on the volume of a metric ball in gn p l with center in gn q l and our chief result is a closed form expression for the volume of a metric ball of radius at most one this volume formula holds for arbitrary n p q and l while previous results pertained only to some special cases based on this volume formula several bounds are derived for the rate distortion tradeoff assuming that the quantization rate is sufficiently high the lower and upper bounds on the distortion rate function are asymptotically identical and therefore precisely quantify the asymptotic rate distortion tradeoff we also show that random codes are asymptotically optimal in the sense that they achieve the minimum possible distortion in probability as n and the code rate approach infinity linearly finally as an application of the derived results to communication theory we quantify the effect of beamforming matrix selection in multiple antenna communication systems with finite rate channel state feedback

exploiting long term channel correlation in limited feedback sdma through channel phase codebook improving channel information quality at the base station bs is crucial for the optimization of frequency division duplexed fdd multi antenna multiuser downlink systems with limited feedback to this end this paper proposes to estimate a particular representation of channel state information csi at the bs through channel norm feedback and a newly developed channel phase codebook where the long term channel correlation is efficiently exploited to improve performance in particular the channel representation is decomposed into a gain related part and a phase related part with each of them estimated separately more specifically the gain related part is estimated from the channel norm and channel correlation matrix while the phase related part is determined using a channel phase codebook constructed with the generalized lloyd algorithm using the estimated channel representation joint optimization of multiuser precoding and opportunistic scheduling is performed to obtain an sdma transmit scheme computer simulation results confirm the advantage of the proposed scheme over state of the art limited feedback sdma schemes under correlated channel environment

ensemble properties of rvq based limited feedback beamforming codebooks the ensemble properties of random vector quantization rvq codebooks for limited feedback beamforming in multiinput multioutput mimo systems are studied with the metrics of interest being the received ssr snr loss and mutual information loss both relative to a perfect channel state information csi benchmark the simplest case of unskewed codebooks is first studied in the correlated mimo setting and these loss metrics are computed as a function of the number of bits of feedback b transmit antenna dimension nt and spatial correlation in particular it is established that 1 the loss metrics are a product of two components a quantization component and a channel dependent component 2 the quantization component which is also common to analysis of channels with i i d fading decays as b increases at the rate 2 b nt 1 3 the channel dependent component reflects the condition number of the channel further the precise connection between the received ssr snr loss and the squared singular values of the channel is shown to be a schur convex majorization relationship finally the ensemble properties of skewed codebooks that are generated by skewing rvq codebooks with an appropriately designed fixed skewing matrix are studied based on an estimate of the loss expression for skewed codebooks it is established that the structure of a good skewing matrix is critically dependent on the condition numbers of the effective channel product of the true channel and the skewing matrix and the skewing matrix

quasi static multiple antenna fading channels at finite blocklength this paper investigates the maximal achievable rate for a given blocklength and error probability over quasi static multiple input multiple output fading channels with and without channel state information at the transmitter and or the receiver the principal finding is that outage capacity despite being an asymptotic quantity is a sharp proxy for the finite blocklength fundamental limits of slow fading channels specifically the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver at only one of them or at neither of them these results follow from analytically tractable converse and achievability bounds numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the blocklength increases in the example of a particular 1 times 2 single input multiple output rician fading channel the blocklength required to achieve 90 of capacity is about an order of magnitude smaller compared with the blocklength required for an awgn channel with the same capacity for this specific scenario the coding decoding schemes adopted in the lte advanced standard are benchmarked against the finite blocklength achievability and converse bounds

on the geometry and quantization of manifolds of positive semi definite matrices the geometry of different spaces of positive semi definite matrices buffeted by rank and trace constraints is studied in addition to revealing their riemannian structure we derive the normalized volume of a ball over these spaces further we use the leading coefficient from the ball volume expansion to bound the quantization error incurred with finite sized sphere packing codebooks as well as random codebooks to represent sources distributed over general riemannian manifolds

geometry and radiometry invariant matched manifold detection consider a set of deformable objects undergoing geometric and radiometric transformations as a result of the action of these transformations the set of different realizations of each object is generally a manifold in the space of observations assuming the geometric deformations an object undergoes belong to some finite dimensional family it has been shown that the universal manifold embedding ume provides a set of nonlinear operators that universally maps each of the different manifolds where each manifold is generated by the set all of possible appearances of a single object into a distinct linear subspace of an euclidean space in this paper we generalize this framework to the case where the observed object undergoes both an affine geometric transformation and a monotonic radiometric transformation and present a novel framework for the detection and recognition of the deformable objects applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations but is geometry covariant with the affine transformation the set of all possible observations on that object is mapped by the ume into a single linear subspace invariant with respect to both the geometric and radiometric transformations the embedding of the space of observations is independent of the specific observed object hence it is universal the invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations the observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities since each tile is mapped by the radiometry invariant ume to a distinct linear subspace the detection and tracking problems are solved by evaluating distances between linear subspaces classification in this context becomes a problem of determining which labeled subspace in a grassmannian is closest to a subspace in the same grassmannian where the latter has been generated by radiometry invariant ume from an unlabeled observation

spherical cap packing asymptotics and rank extreme detection we study the spherical cap packing problem with a probabilistic approach such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector when the set of unit vectors are themselves independently uniformly distributed we further develop the extreme value distribution limit of the maximal inner product which characterizes its uncertainty around the bound as applications of the above mentioned asymptotic results we derive 1 an asymptotic sharp universal uniform bound on the maximal spurious correlation as well as its uniform convergence in distribution when the explanatory variables are independently gaussian distributed and 2 an asymptotic sharp universal bound on the maximum norm of a low rank elliptically distributed vector as well as related limiting distributions with these results we develop a fast detection method for a low rank structure in high dimensional gaussian data without using the spectrum information

bounds on packings of spheres in the grassmann manifold we derive the gilbert varshamov and hamming bounds for packings of spheres codes in the grassmann manifolds over r and c asymptotic expressions are obtained for the geodesic metric and projection frobenius chordal metric on the manifold

space time autocoding prior treatments of space time communications in rayleigh flat fading generally assume that channel coding covers either one fading interval in which case there is a nonzero outage capacity or multiple fading intervals in which case there is a nonzero shannon capacity however we establish conditions under which channel codes span only one fading interval and yet are arbitrarily reliable in short space time signals are their own channel codes we call this phenomenon space time autocoding and the accompanying capacity the space time autocapacity let an m transmitter antenna n receiver antenna rayleigh flat fading channel be characterized by an m spl times n matrix of independent propagation coefficients distributed as zero mean unit variance complex gaussian random variables this propagation matrix is unknown to the transmitter it remains constant during a t symbol coherence interval and there is a fixed total transmit power let the coherence interval and number of transmitter antennas be related as t spl beta m for some constant spl beta a t spl times m matrix valued signal associated with r spl middot t bits of information for some rate r is transmitted during the t symbol coherence interval then there is a positive space time autocapacity c sub a such that for all r c sub a the block probability of error goes to zero as the pair t m spl rarr spl infin such that t m spl beta the autocoding effect occurs whether or not the propagation matrix is known to the receiver and c sub a nlog 1 spl rho in either case independently of spl beta where spl rho is the expected signal to noise ratio snr at each receiver antenna lower bounds on the cutoff rate derived from random unitary space time signals suggest that the autocoding effect manifests itself for relatively small values of t and m for example within a single coherence interval of duration t 16 for m 7 transmitter antennas and n 4 receiver antennas and an 18 db expected snr a total of 80 bits corresponding to rate r 5 can theoretically be transmitted with a block probability of error less than 10 sup 9 all without any training or knowledge of the propagation matrix

from random matrix theory to coding theory volume of a metric ball in unitary group volume estimates of metric balls in manifolds find diverse applications in information and coding theory in this paper new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory the first result is an integral representation of the exact volume which involves a toeplitz determinant of bessel functions a simple but accurate limiting volume formula is then obtained by invoking szeg s strong limit theorem for large toeplitz matrices the derived asymptotic volume formula enables analytical evaluation of some coding theoretic bounds of unitary codes in particular the gilbert varshamov lower bound and the hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived moreover bounds on the scaling law of code rate are found finally a closed form bound on the diversity sum relevant to unitary space time codes is obtained which was only computed numerically in the literature

correction to bounds on packings of spheres in the grassmann manifold in the above titled paper ibid vol 48 no 9 pp 2450 2454 sep 02 corrections were made to haar measurements in various equations

reduced feedback mimo ofdm precoding and antenna selection 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

design and analysis of mimo spatial multiplexing systems with quantized feedback 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

mimo transceivers with decision feedback and bit loading theory and optimization this paper considers mimo transceivers with linear precoders and decision feedback equalizers dfes with bit allocation at the transmitter zero forcing zf is assumed considered first is the minimization of transmitted power for a given total bit rate and a specified set of error probabilities for the symbol streams the precoder and dfe matrices are optimized jointly with bit allocation it is shown that the generalized triangular decomposition gtd introduced by jiang li and hager offers an optimal family of solutions the optimal linear transceiver which has a linear equalizer rather than a dfe with optimal bit allocation is a member of this family this shows formally that under optimal bit allocation linear and dfe transceivers achieve the same minimum power the dfe transceiver using the geometric mean decomposition gmd is another member of this optimal family and is such that optimal bit allocation yields identical bits for all symbol streams no bit allocation is necessary when the specified error probabilities are identical for all streams the qr based system used in vblast is yet another member of the optimal family and is particularly well suited when limited feedback is allowed from receiver to transmitter two other optimization problems are then considered a minimization of power for specified set of bit rates and error probabilities the qos problem and b maximization of bit rate for fixed set of error probabilities and power it is shown in both cases that the gtd yields an optimal family of solutions

sphere packing bounds in the grassmann and stiefel manifolds applying the riemann geometric machinery of volume estimates in terms of curvature bounds for the minimal distance of packings codes in the grassmann and stiefel manifolds will be derived and analyzed in the context of space time block codes this leads to a monotonically increasing minimal distance lower bound as a function of the block length this advocates large block lengths for the code design

ber criterion and codebook construction for finite rate precoded spatial multiplexing with linear receivers precoded spatial multiplexing systems with rate limited feedback have been studied recently based on various precoder selection criteria instead of those based on indirect performance indicators we in this paper propose a new criterion directly based on the exact bit error rate ber that is applicable to systems with linear receivers and rectangular square quadrature amplitude modulation constellations the ber criterion outperforms any other alternative in terms of optimizing the ber performance for an uncoded system with linear receivers we then develop a precoder codebook construction method based on the generalized lloyd algorithm from the vector quantization literature this construction is not directly based on the ber criterion hence it is suboptimal in the ber sense however relative to those currently available our newfound codebooks improve considerably various minimum distances between any pair of codewords of the codebook finally we analyze the ber optimal precoder in the asymptotic case with infinite rate feedback that amounts to perfect channel knowledge at the transmitter the infinite rate optimal precoder based on the ber criterion is drastically different from the counterparts with other criteria and it leads to a benchmark performance for finite rate precoded spatial multiplexing systems we observe from numerical results that the ber performance of finite rate feedback with suboptimal codebooks approaches quickly the benchmark performance of infinite rate feedback this suggests that i the number of feedback bits in practical systems need not be large and ii the room for performance improvement via further codebook optimization shrinks quickly as the codebook size increases

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold