optimal phase control for equal gain transmission in mimo systems with scalar quantization complexity and algorithms the complexity of the optimal phase control problem in wireless mimo systems with scalar feedback quantization and equal gain transmission is studied the problem is shown to be np hard when the number of receive antennas grows linearly with the number of transmit antennas for the case where the number of receive antennas is constant the problem can be solved in polynomial time an optimal algorithm is explicitly constructed for practical purposes a low complexity algorithm based on local search is presented simulation results show that its performance is nearly optimal

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

quantization methods for equal gain transmission with finite rate feedback we consider the design and analysis of quantizers for equal gain transmission egt systems with finite rate feedback based communication in flat fading multiple input single output miso systems egt is a beamforming technique that maximizes the miso channel capacity when there is an equal power per antenna constraint at the transmitter and requires the feedback of t 1 phase angles when there are t antennas at the transmitter in this paper we contrast two popular approaches for quantizing the phase angles vector quantization vq and scalar quantization sq on the vq side using the capacity loss with respect to egt with perfect channel information at transmitter as performance metric we develop a criterion for designing the beamforming codebook for quantized egt q egt we also propose an iterative algorithm based on the well known generalized lloyd algorithm for computing the beamforming vector codebook on the analytical side we study the performance of q egt and derive closed form expressions for the performance in terms of capacity loss and outage probability in the case of i i d rayleigh flat fading channels on the sq side assuming uniform scalar quantization and i i d rayleigh flat fading channels we derive the high resolution performance of quantized egt and contrast the performance with that of vq we find that although both vq and sq achieve the same rate of convergence to the capacity with perfect feedback as the number of feedback bits b increases there exists a fixed gap between the two

optimal and lattice codes in a mimo system with complete feedback space time coding for a multiple input multiple output mimo system with complete feedback and maximum likelihood ml decoding is considered first a criterion for designing an optimal codebook with complete feedback in the case of high signal to noise ratio snr is given it turns out that designing an optimal codebook with complete feedback in the case of high snr is equivalent to efficiently packing a finite number of points into a dynamic ellipsoid in a finite dimensional real euclidean space second based on this design criterion an optimal tradeoff between diversity and multiplexing is then derived finally practical design using lattice codes is addressed and surprisingly it is shown that the lattice codes can achieve the optimal tradeoff it is also demonstrated that it is not necessary for an optimal lattice code to use all the subchannels generated by the system this is in striking contrast with a general optimal code which generally utilizes all the subchannels

on the design of mimo block fading channels with feedback link capacity constraint in this paper we propose a combined adaptive power control and beamforming framework for optimizing multiple input multiple output mimo link capacity in the presence of feedback link capacity constraint the feedback channel is used to carry channel state information only it is assumed to be noiseless and causal with a feedback capacity constraint in terms of maximum number of feedback bits per fading block we show that the hybrid design could achieve the optimal mimo link capacity and we derive a computationally efficient algorithm to search for the optimal design under a specific average power constraint finally we shall illustrate that a minimum mean square error spatial processor with a successive interference canceller at the receiver could be used to realize the optimal capacity we found that feedback effectively enhances the forward channel capacity for all signal to noise ratio snr values when the number of transmit antennas n sub t is larger than the number of receive antennas n sub r the snr gain with feedback is contributed by focusing transmission power on active eigenchannel and temporal power waterfilling the former factor contributed at most 10log sub 10 n sub t n sub r db snr gain when n sub t n sub r while the latter factor s snr gain is significant only for low snr values

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

bit allocation law for multiantenna channel feedback quantization single user case this paper studies the design and optimization of a limited feedback single user system with multiple antenna transmitter and single antenna receiver the design problem is cast in form of the minimizing the average transmission power at the base station subject to the user s outage probability constraint the optimization is over the user s channel quantization codebook and the transmission power control function at the base station our approach is based on fixing the outage scenarios in advance and transforming the design problem into a robust system design problem we start by showing that uniformly quantizing the channel magnitude in db scale is asymptotically optimal regardless of the magnitude distribution function we derive the optimal uniform in db channel magnitude codebook and combine it with a spatially uniform channel direction codebook to arrive at a product channel quantization codebook we then optimize such a product structure in the asymptotic regime of b where b is the total number of quantization feedback bits the paper shows that for channels in the real space the asymptotically optimal number of direction quantization bits should be m 1 2 times the number of magnitude quantization bits where m is the number of base station antennas we also show that the performance of the designed system approaches the performance of the perfect channel state information system as 2 2b m 1 for complex channels the number of magnitude and direction quantization bits are related by a factor of m 1 and the system performance scales as 2 b m as b

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

noisy beam alignment techniques for reciprocal mimo channels future multi input multi output mimo wireless communications systems will use beamforming as a first step towards realizing the capacity requirements necessitated by the exponential increase in data demands the focus of this work is on beam alignment for time division duplexing tdd systems for which we propose a number of novel algorithms these algorithms seek to obtain good estimates of the optimal beamformer combiner pair which are the dominant singular vectors of the channel matrix they are motivated by the power method an iterative algorithm to determine eigenvalues and eigenvectors through repeated matrix multiplication in contrast to the basic power method which considers only the most recent iteration and assumes noiseless links the proposed techniques consider information from all the previous iterations of the algorithm and combine them in different ways the first technique sequential least squares method sequentially constructs a least squares estimate of the channel matrix which is then used to calculate the beamformer combiner pair estimate the second technique summed power method aims to mitigate the effect of noise by using a linear combination of the previously tried beams to calculate the next beam providing improved performance in the low snr regime typical for mmwave systems with minimal complexity feedback overhead a third technique least squares initialized summed power method combines the good performance of the first technique at the high snr regime with the low complexity advantage of the second technique by priming the summed power method with initial estimates from the sequential method

performance of quantized equal gain transmission with noisy feedback channels in this paper new results and insights are derived for the performance of multiple input single output systems with beamforming at the transmitter when the channel state information is quantized and sent to the transmitter over a noisy feedback channel it is assumed that there exists a per antenna power constraint at the transmitter hence the equal gain transmission egt beamforming vector is quantized and sent from the receiver to the transmitter the loss in received signal to noise ratio snr relative to perfect beamforming is analytically characterized and it is shown that at high rates the overall distortion can be expressed as the sum of the quantization induced distortion and the channel error induced distortion and that the asymptotic performance depends on the error rate behavior of the noisy feedback channel as the number of codepoints gets large the optimum density of codepoints also known as the point density that minimizes the overall distortion subject to a boundedness constraint is shown to be the same as the point density for a noiseless feedback channel i e the uniform density the binary symmetric channel with random index assignment is a special case of the analysis and it is shown that as the number of quantized bits gets large the distortion approaches the same as that obtained with random beamforming the accuracy of the theoretical expressions obtained are verified through monte carlo simulations

stable embedding of grassmann manifold via gaussian random matrices compressive sensing cs provides a new perspective for data reduction without compromising performance when the signal of interest is sparse or has intrinsically low dimensional structure the theoretical foundation for most of the existing studies on cs is based on the stable embedding i e a distance preserving property of vectors that are sparse or in a union of subspaces via random measurement matrices to the best of our knowledge few existing literatures of cs have clearly discussed the stable embedding of linear subspaces via compressive measurement systems in this paper we explore a volume based stable embedding of multidimensional signals based on grassmann manifold via gaussian random measurement matrices the grassmann manifold is a topological space in which each point is a linear vector subspace and is widely regarded as an ideal model for multidimensional signals generated from linear subspaces in this paper we formulate the linear subspace spanned by multidimensional signal vectors as points on the grassmann manifold and use the volume and the product of sines of principal angles also known as the product of principal sines as the generalized norm and distance measure for the space of grassmann manifold we prove a volume preserving embedding property for points on the grassmann manifold via gaussian random measurement matrices i e the volumes of all parallelotopes from a finite set in grassmann manifold are preserved upon compression this volume preserving embedding property is a multidimensional generalization of the conventional stable embedding properties which only concern the approximate preservation of lengths of vectors in certain unions of subspaces in addition we use the volume preserving embedding property to explore the stable embedding effect on a generalized distance measure of grassmann manifold induced from volume it is proved that the generalized distance measure i e the product of principal sines between different points on the grassmann manifold is well preserved in the compressed domain via gaussian random measurement matrices numerical simulations are also provided for validation

full diversity achieving precoding for asymmetric mimo using partial csit for a block fading n t x n r n t n r mimo channel we propose a precoding scheme that achieves both n tn rth order diversity as well as rate of n t symbols per channel use feeding back b n t 1 bits as partial channel state information to the transmitter csit where 2 b is the number of quantization states available we establish the optimality of the uniform quantizer which achieves minimum loss in coding gain due to quantization of feedback values of our precoding scheme in comparison to the non uniform quantization we also lay down the guidelines for constellation sets with which our precoding scheme can achieve full diversity we also derive the order of complexity involved in computing the precoder matrix and show that it is independent of the size of constellation sets we compare our ber results with that of precoding schemes in the literature utilizing full csit as well as that with partial csit we also investigate the loss in error rate performance due to imperfect channel knowledge at the receiver and present simulation results to verify our claims

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

linear dispersion codes for mimo systems based on frame theory multiple input multiple output mimo wireless communication systems provide high capacity due to the plurality of modes available in the channel existing signaling techniques for mimo systems have focused primarily on multiplexing for high data rate or diversity for high link reliability in this paper we present a new linear dispersion code design for mimo rayleigh fading channels the proposed design bridges the gap between multiplexing and diversity and yields codes that typically perform well both in terms of ergodic capacity as well as error probability this is important because as we show designs performing well from an ergodic capacity point of view do not necessarily perform well from an error probability point of view various techniques are presented for finding codes with good error probability performance monte carlo simulations illustrate performance of some example code designs in terms of ergodic capacity codeword error probability and bit error probability

performance analysis of transmit beamforming for miso systems with imperfect feedback in this paper we analyze the performance of transmit beamforming on multiple antenna rayleigh fading channels with imperfect channel feedback we characterize the feedback imperfections in terms of noisy channel estimation feedback delay and finite rate channel quantization we develop a general framework that is valid for an arbitrary two dimensional linear modulation to capture the aforementioned imperfections and derive the symbol and bit error probability expressions for both m psk and m ary rectangular qam constellations with gray code mapping we show that the proposed analytical formulation is valid for a frequency domain duplexing system with without finite rate channel quantization and a time domain duplexing system we validate the accuracy of the analysis through simulations and assess the relative effects of channel estimation inaccuracy feedback delay and finite rate quantization on the symbol and bit error performances for various constellations

combining beamforming and space time coding using noisy quantized feedback the goal of combining beamforming and spacetime coding is to obtain full diversity order and to provide additional received power array gain compared to conventional space time codes in this work a class of code constellations is proposed called generalized partly orthogonal designs pods and both high rate and low rate feedback information is incorporated with possible feedback errors a binary symmetric channel bsc model characterizes feedback errors two cases are studied first when the bsc bit error probability is known a priori to the transmission ends and second when it is not known exactly based on a minimum pairwise error probability pep design criterion we design a channel optimized vector quantizer covq for feedback information and a precoder matrix codebook to adjust the transmission codewords the attractive property of our combining scheme is that it converges to conventional space time coding with low rate and erroneous feedback and to directional beamforming with high rate and error free feedback this scheme also shows desirable robustness against feedback channel modeling mismatch

binary grassmannian weightbooks for mimo beamforming systems binary weightbooks for multiple input multiple output mimo beamforming systems using quantized feedback are designed based on the grassmannian beamforming criterion it is shown that the grassmannian criterion for binary weightbook design is to maximize the minimum hamming distance of the corresponding block code thus block codes that have large minimum hamming distance can be used for binary weightbook design for mimo beamforming systems using quantized feedback the optimum binary weightbooks when the number of transmit antennas has value from two to six are tabulated

on mimo beamforming systems using quantized feedback we propose the hadamard matrix as a codebook for multiple input multiple output mimo beamforming systems and show that a quantized equal gain transmission qegt scheme using the hadamard codebook has the same symbol error probability performance as a receiver selection diversity combining scheme operating in independent generalized rayleigh fading channels we show that a mimo beamforming system using quantized feedback over rician fading channels can be analyzed by considering a corresponding receiver selection combining scheme operating in single input multiple output simo systems over correlated generalized rician fading channels also we propose a greedy cophasing scheme for mimo beamforming systems using quantized feedback the greedy cophasing scheme provides 3 2 db power gain over an orthogonal space time block code when the number of transmit antennas is four the number of receive antennas is one and the number of feedback bits is three

qrd based precoded mimo ofdm systems with reduced feedback qr decomposition qrd based precoded mimo ofdm systems with reduced feedback are proposed to convert the mimo ofdm channel into layered subchannels qrd m is further combined with either singular value svd or geometric mean decomposition gmd of the time domain channel impulse response matrix as a result the receiver in the proposed systems only needs to feed back information describing one precoding matrix for all carriers simulation results confirm the bit error rate ber and throughput performance superiority of the proposed systems compared to conventional svd per carrier precoding schemes

an upper bound on the ser of transmit beamforming in correlated rayleigh fading we study the symbol error rate ser of maximum ratio transmission transmit antenna selection and codebook based beamforming in correlated rayleigh fading channels assuming maximum ratio combining is performed at the receiver we derive a universal upper bound on the average ser of the three schemes and prove that the bound is asymptotically tight in high signal to noise ratio snr regions however numerical results show that at medium snr the tightness of the bound depends on the condition number of the channel correlation matrix

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

analysis of multiple antenna systems with finite rate feedback using high resolution quantization theory this paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple antenna systems with finite rate feedback inspired by the results of classical high resolution quantization theory the problem of finite rate quantized communication system is formulated as a general fixed rate vector quantization problem with side information available at the encoder or the quantizer but unavailable at the decoder the framework of the quantization problem is sufficiently general to include quantization schemes with general non mean squared distortion functions and constrained source vectors asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the bennett s integral specifically tight lower and upper bounds of the average asymptotic distortion are proposed sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed rate quantization problems the proposed general methodology provides a powerful analytical tool to study a wide range of finite rate feedback systems to illustrate the utility of the framework we consider the analysis of a finite rate feedback multiple input single output miso beamforming system over independent and identically distributed i i d rayleigh flat fading channels numerical and simulation results are presented that further confirm the accuracy of the analytical results

mimo transmit beamforming under uniform elemental power constraint we consider multi input multi output mimo transmit beamforming under the uniform elemental power constraint this is a nonconvex optimization problem and it is usually difficult to find the optimal transmit beamformer first we show that for the multi input single output miso case the optimal solution has a closed form expression then we propose a cyclic algorithm for the mimo case which uses the closed form miso optimal solution iteratively the cyclic algorithm has a low computational complexity and is locally convergent under mild conditions moreover we consider finite rate feedback methods needed for transmit beamforming we propose a simple scalar quantization method as well as a novel vector quantization method for the latter method the codebook is constructed under the uniform elemental power constraint and the method is referred as vq uep we analyze vq uep performance for the miso case specifically we obtain an approximate expression for the average degradation of the receive signal to noise ratio snr caused by vq uep numerical examples are provided to demonstrate the effectiveness of our proposed transmit beamformer designs and the finite rate feedback techniques

space division multiple access with a sum feedback rate constraint on a multiantenna broadcast channel simultaneous transmission to multiple users by joint beamforming and scheduling is capable of achieving high throughput which grows double logarithmically with the number of users the sum rate for channel state information csi feedback however increases linearly with the number of users reducing the effective uplink capacity to address this problem a novel space division multiple access sdma design is proposed where the sum feedback rate is upper bounded by a constant this design consists of algorithms for csi quantization threshold based csi feedback and joint beamforming and scheduling the key feature of the proposed approach is the use of feedback thresholds to select feedback users with large channel gains and small csi quantization errors such that the sum feedback rate constraint is satisfied despite this constraint the proposed sdma design is shown to achieve a sum capacity growth rate close to the optimal one moreover the feedback overflow probability for this design is found to decrease exponentially with the difference between the allowable and the average sum feedback rates numerical results show that the proposed sdma design is capable of attaining higher sum capacities than existing ones even though the sum feedback rate is bounded

achieving the welch bound with difference sets consider a codebook containing n unit norm complex vectors in a k dimensional space in a number of applications the codebook that minimizes the maximal cross correlation amplitude i sub max is often desirable relying on tools from combinatorial number theory we construct analytically optimal codebooks meeting in certain cases the welch lower bound when analytical constructions are not available we develop an efficient numerical search method based on a generalized lloyd algorithm which leads to considerable improvement on the achieved i sub max over existing alternatives we also derive a composite lower bound on the minimum achievable i sub max that is effective for any codebook size n

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

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

convergence of the viscosity solutions for the system of nonlinear elasticity abstract in this paper some special entropy pairs of lax type are constructed for the system of nonlinear elasticity in which the progressive terms are functions of a single variable the necessity estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations the special entropy pairs are used to prove that the family of young measures uniquely determined by the viscosity solution sequence is a family of dirac measures hence a convergence theorem on the viscosity solutions is established by applying the method of compensated compactness

hybrid beamforming in mm wave mimo systems having a finite input alphabet recently there has been significant research effort toward achieving high data rates in the millimeter wave bands by employing large antenna systems these systems are considered to have only a fraction of the rf chains compared with the total number of antennas and employ analog phase shifters to steer the transmit and receive beams in addition to the conventional beamforming bf combining invoked in the baseband domain this scheme which is popularly known as hybrid bf has been extensively studied in the literature to the best of our knowledge all the existing schemes focus on obtaining the bf combining matrices that maximize the system capacity computed using a gaussian input alphabet however this choice of matrices may be suboptimal for practical systems since they employ a finite input alphabet such as quadrature amplitude modulation phase shift keying constellations hence in this paper we consider a hybrid bf combining system operating with a finite input alphabet and optimize the analog as well as digital bf combining matrices by maximizing the mutual information mi this is achieved by an iterative gradient ascent algorithm that exploits the relationship between the minimum mean squared error and the mi furthermore an iterative algorithm is proposed for designing a codebook for the analog and digital bf combining matrices based on a vector quantization approach our simulation results demonstrate that the proposed gradient ascent algorithm achieves an ergodic rate improvement of up to 0 4 bits per channel use bpcu compared with the gaussian input scenario furthermore the gain in the ergodic rate achieved by employing the vector quantization based codebook is about 0 5 bpcu compared with the gaussian input scenario

equal gain transmission in multiple input multiple output wireless systems multiple input multiple output mimo wireless systems are of interest due to their ability to provide substantial gains in capacity and quality the paper proposes equal gain transmission egt to provide diversity advantage in mimo systems experiencing rayleigh fading the applications of egt with selection diversity combining equal gain combining and maximum ratio combining are addressed it is proven that systems using egt with any of these combining schemes achieve full diversity order when transmitting over a memoryless flat fading rayleigh matrix channel with independent entries since in practice full channel knowledge at the transmitter is difficult to realize a quantized version of egt is proposed an algorithm to construct a beamforming vector codebook that guarantees full diversity order is presented monte carlo simulation comparisons with various beamforming and combining systems illustrate the performance as a function of quantization

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

the optimality of transmit beamforming a unified view the optimality of transmit beamforming for a multiple antenna system with partial limited feedback is investigated and a single general necessary and sufficient condition for beamforming to achieve ergodic capacity is derived the condition obtained is universal applicable to all partial limited feedback scenarios in all ergodic fading channel distributions regardless of the number of transmit receive antennas or transmit power using the universal condition we explore the optimality of beamforming for the quantized mean feedback scheme which unifies previous results for the separate cases of mean feedback and quantized feedback numerical results are provided to complement the analysis

systematic design of unitary space time constellations we propose a systematic method for creating constellations of unitary space time signals for multiple antenna communication links unitary space time signals which are orthonormal in time across the antennas have been shown to be well tailored to a rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients the signals can achieve low probability of error by exploiting multiple antenna diversity because the fading coefficients are not known the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximum euclidean distance norm our construction begins with the first signal in the constellation an oblong complex valued matrix whose columns are orthonormal and systematically produces the remaining signals by successively rotating this signal in a high dimensional complex space this construction easily produces large constellations of high dimensional signals we demonstrate its efficacy through examples involving one two and three transmitter antennas

limited feedback for temporally correlated mimo channels with other cell interference limited feedback improves link reliability with a small amount of feedback from the receiver to the transmitter in cellular systems the performance of limited feedback will be degraded in the presence of other cell interference when the base stations have limited or no coordination this paper establishes the degradation in sum rate of users in a cellular system due to uncoordinated other cell interference and feedback delay a goodput metric is defined as the rate when the bits are successfully received at the mobile station and used to derive an upper bound on the performance of limited feedback systems with delay this paper shows that the goodput gained from having delayed limited feedback decreases doubly exponentially as the delay increases the analysis is extended to precoded spatial multiplexing systems where it is shown that the same upper bound can be used to evaluate the decay in the achievable sum rate to reduce the effects of interference zero forcing interference nulling is applied at the receiver where it is shown that the effect of the interference on the achievable sum rate can be suppressed by nulling out the interferer numerical results show that the decay rate of the goodput decreases when the codebook quantization size increases and when the doppler shift in the channel decreases

hybrid mu miso scheduling with limited feedback using hierarchical codebooks the multiuser mimo mu mimo systems tend to have inter user interference in practical systems with limited feedback practical mu mimo systems with limited feedback can be divided into two classes in general one is zero forcing beamforming zfbf and the other is per user unitary rate control pu2rc which is an extended version of random beamforming rbf to improve performance zfbf needs more feedback bits and pu2rc needs a larger number of users we propose a new hybrid mu miso system which has the advantages of the two mu mimo schemes simultaneously and analyze the sum rate performance and the quantization error of the hybrid scheme the analysis shows how the number of feedback bits the number of users and the signal power affect the sum rate the throughput scaling laws are also derived in the high and the medium snr regimes simulation results show that the proposed hybrid mu miso system performs better than either of the two mu mimo systems with the same number of feedback bits another advantage is that it has lower complexity for codeword search since it has a hierarchical codebook structure

existence of the entropy solution for a viscoelastic model in this paper we consider the cauchy problem for a viscoelastic model with relaxationut x 0 f u t 1 f u 0with discontinuous large initial data where 0 0 are constants when the system is nonstrictly hyperbolic under the additional assumptionv0x l the system is reduced to an inhomogeneous scalar balance law by employing the special form of the system itself after introducing a definition of entropy solutions to the system we prove the existence uniqueness and continuous dependence of the global entropy solution for the system when the system is strictly hyperbolic some special entropy pairs of the lax type are constructed in which the progression terms are functions of a single variable and the necessary estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations the special entropy pairs are used to prove the existence of the global entropy solutions for the corresponding cauchy problem by applying the method of compensated compactness

multicarrier beamforming with limited feedback a rate distortion approach this paper studies the optimal use of limited rate feedback of channel state information csi in the case of a wideband multicarrier channel with multiple transmit antennas and a single receive antenna with full knowledge of the csi the receiver should compute the optimal beamforming vectors jointly quantize them and feed them back to the transmitter the achievable forward data rate depends on the rate of the feedback link the optimal tradeoff between the forward and feedback rates is characterized using rate distortion theory in the limit of infinite number of subcarriers with fixed amount of feedback per subcarrier the distortion metric is the difference between the forward rate achieved with limited feedback and the capacity with perfect csi at the transmitter the rate distortion function gives the forward rate as a function of the feedback rate numerical results show that to achieve a target forward rate the required feedback rate can be substantially reduced by joint quantization of the beamformers across subcarriers a simple quantizer amenable to practical implementation is shown to approach the rate distortion bound with near linear computational complexity

rate diversity tradeoff of space time codes with fixed alphabet and optimal constructions for psk modulation we show that for any q spl times m space time code s having a fixed finite signal constellation there is a tradeoff between the transmission rate r and the transmit diversity gain spl nu achieved by the code the tradeoff is characterized by r spl les q spl nu 1 where q is the number of transmit antennas when either binary phase shift keying bpsk or quaternary phase shift keying qpsk is used as the signal constellation a systematic construction is presented to achieve the maximum possible rate for every possible value of transmit diversity gain

well posedness of the global entropy solution to the cauchy problem of a hyperbolic conservation laws with relaxation in this paper we prove that the cauchy problem to a hyperbolic conservation laws with relaxation with singular initial data admits a unique global entropy solution in the sense of definition 1 1 compared with former results in this direction the main ingredient of this paper lies in the fact that it contains a uniqueness result and we do not ask f u to satisfy any convex monotonic conditions and the regularity assumption we imposed on f u is weaker c 2003 elsevier inc all rights reserved

on the expected rate of slowly fading channels with quantized side information we study a multiple layer variable rate system employing quantized feedback to maximize the expected rate over a single input single output slowly fading gaussian channel the transmitter uses partial channel state information which is obtained via an optimized resolution constrained feedback link to adapt the power and to assign code layer rates subject to different power constraints to systematically design the system parameters we develop a simple iterative algorithm that successfully exploits results in the study of parallel broadcast channels we present the necessary and sufficient conditions for single layer coding to be optimal irrespective of the number of code layers that the system can afford unlike in the ergodic case even coarsely quantized feedback is shown to improve the expected rate considerably our results also indicate that with as little as one bit of feedback information the role of multilayer coding reduces significantly

complex codebooks from combinatorial designs codebooks also called signal sets meeting the welch bounds are desirable in code division multiple access cdma systems in 2003 binary codebooks meeting welch s bounds were constructed using difference sets by ding golin and kloslashve recently a generic construction of codebooks with cyclic difference sets meeting welch s bound on the maximum cross correlation amplitude was developed by xia in this correspondence the idea of xia is extended and related constructions of optimal codebooks with both cyclic and noncyclic difference sets are presented these codebooks are optimal in the sense that they also meet this welch bound in addition complex codebooks that almost meet welch s bound on the maximum cross correlation amplitude are also constructed with almost difference sets

transmit equal gain precoding in rayleigh fading channels precoding with limited feedback information can achieve satisfactory performance while the amount of feedback information is kept small in this paper we analyze the theoretical performance of equal gain precoder and find that its performance is at most 1 049 db worse than the optimal precoder no matter how the number of transmit antennas increases moreover we analyze the performance degradation of the equal gain precoder due to scalar quantization theoretically the result shows that 2 3 bits per transmit antenna excluding the first antenna can achieve 0 5 0 25 db performance gap compared with the same scheme without quantization furthermore we found that the equal gain precoder in general can achieve comparable performance with the grassmannian precoder in the same moderate feedback bits simulation results are provided to corroborate the theoretical results

capacity analysis of mimo systems using limited feedback transmit precoding schemes this paper employs a high resolution quantization framework to study the effects of finite rate feedback of the channel state information csi on the performance of multiple input multiple output mimo systems over independently and identically distributed i i d rayleigh flat fading channels the contributions of this paper are twofold first we extend the general distortion analysis of vector quantizers to deal with complex source variables necessary and sufficient conditions that guarantee a concise high resolution distortion analysis in the complex domain is presented second as an application of the proposed complex distortion analysis tight lower bounds on the capacity loss due to the finite rate channel quantization are provided for mimo systems employing a fixed number of equal power spatial beams based on the obtained closed form analytical results it is shown that the system capacity loss decreases exponentially as the ratio of the quantization rate to the total degrees of freedom of the channel state information to be quantized moreover mimo csi quantizers using mismatched codebooks that are only optimized for high signal to noise ratio snr and low snr regimes are also investigated to quantify the penalties incurred by the use of mismatched codebooks in addition the analysis is extended to deal with mimo systems using multi mode spatial multiplexing transmission schemes with finite rate csi feedback finally numerical and simulation results are presented which confirm the tightness of the derived theoretical distortion bounds

robust precoding with limited feedback design based on precoding mse for mu miso systems for the separation of the signals in the vector broadcast channel bc some information about the channel state is necessary at the transmitter in many cases this channel state information csi must be fed back from the receivers to the transmitter we jointly design the channel estimators and the quantizers at the receivers together with the precoder at the transmitter based on a precoder centric criterion i e the minimization of a mean square error mse metric appropriate for the precoder design this is in contrast to our previous works where the quantizer design was based on a csi mse metric i e based on the minimization of the mse between the true channel and the channel recovered by the transmitter using a feedback channel interestingly the estimators resulting from this joint formulation are independent of the used codebook the codebook entries are the employed precoders therefore each receiver feeds back the index of a set of precoders and the intersection of the sets gives the appropriate precoder since the quantizers of the different receivers have to work separately the metric for the computation of the partition cells cannot be expressed as a simple squared error depending on the quantizer output the proposed system based on a joint optimization clearly outperforms previous designs with separate optimization of feedback and precoding

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

transmit selection diversity for unitary precoded multiuser spatial multiplexing systems with linear receivers multiuser spatial multiplexing is a downlink transmission technique that uses linear transmit precoding to multiplex multiple users and precancel interuser interference in such a system the spatial degrees of freedom are used for interference mitigation and generally come at the expense of diversity gain this paper proposes two precoding methods that use extra transmit antennas beyond the minimum required to provide additional degrees of diversity the approach taken is to solve for a unitary transmit precoder under a zero interuser interference constraint that minimizes an upper bound on the symbol error rate ser for each user solutions where all transmit antennas are employed as well as subsets of antennas to reduce analog components are described numerical results confirm a dramatic improvement in terms of ser and mutual information over single user multiple input multiple output mimo methods and static allocation methods for example the proposed techniques achieve a signal to noise ratio snr improvement of 6 10 db at an uncoded ser of 10 3 with only one extra transmit antenna

design and analysis of transmit beamforming based on limited rate feedback this paper deals with design and performance analysis of transmit beamformers for multiple input multiple output mimo systems based on bandwidth limited information that is fed back from the receiver to the transmitter by casting the design of transmit beamforming based on limited rate feedback as an equivalent sphere vector quantization svq problem multiantenna beamformed transmissions through independent and identically distributed i i d rayleigh fading channels are first considered the rate distortion function of the vector source is upper bounded and the operational rate distortion performance achieved by the generalized lloyd s algorithm is lower bounded although different in nature the two bounds yield asymptotically equivalent performance analysis results the average signal to noise ratio snr performance is also quantified finally beamformer codebook designs are studied for correlated rayleigh fading channels and a low complexity codebook design that achieves near optimal performance is derived

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

the cauchy problem for the systems tz x zy 0 in this paper we solve the cauchy problem for the systems tz xz 0 where z u iv c and 1 2 these systems are nonstrictly hyperbolic possessing an isolated umbilic point at z 0 we use the vanishing viscosity method with the help of the theory of compensated compactness uniform bounds in l for the solutions of the viscous systems are not available but such bounds can be found in l2 this makes necessary the use of the generalized young measures and improvements in known techniques of the compensated compactness theory applied to conservation laws

channel adaptive quantization for limited feedback mimo beamforming systems multiple input multiple output mimo wireless systems can achieve significant diversity and array gain by using transmit beamforming and receive combining techniques in the absence of full channel knowledge at the transmitter the transmit beamforming vector can be quantized at the receiver and sent to the transmitter using a low rate feedback channel in the literature quantization algorithms for the beamforming vector are designed and optimized for a particular channel distribution commonly the uncorrelated rayleigh distribution when the channel is not uncorrelated rayleigh however these quantization strategies result in a degradation of the receive signal to noise ratio snr in this paper switched codebook quantization is proposed where the codebook is dynamically chosen based on the channel distribution the codebook adaptation enables the quantization to exploit the spatial and temporal correlation inherent in the channel the convergence properties of the codebook selection algorithm are studied assuming a block stationary model for the channel in the case of a nonstationary channel it is shown using simulations that the selected codebook tracks the distribution of the channel resulting in improvements in snr simulation results show that in the case of correlated channels the snr performance of the link can be significantly improved by adaptation compared with nonadaptive quantization strategies designed for uncorrelated rayleigh fading channels

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

interpolation based transmit beamforming for mimo ofdm with limited feedback transmit beamforming and receive combining are simple methods for exploiting spatial diversity in multiple input multiple output orthogonal frequency division multiplexing mimo ofdm system optimal beamforming requires channel state information in the form of the beamforming vectors for each ofdm subcarrier this paper proposes a limited feedback architecture that combines beamforming vector quantization and smart vector interpolation in the proposed system the receiver sends a fraction of information about the optimal beamforming vectors to the transmitter and the transmitter computes the beamforming vectors for all subcarriers through interpolation a new spherical interpolator is developed that exploits parameters for phase rotation to satisfy the phase invariance and unit norm properties of the transmitted beamforming vectors the beamforming vectors and phase parameters are quantized at the receiver and the quantized information is provided to the transmitter the proposed quantization system provides only a moderate increase in complexity versus over comparable approaches numerical simulations show that the proposed scheme performs better than existing diversity techniques with the same feedback data rate

transmit beamforming in multiple antenna systems with finite rate feedback a vq based approach this paper investigates quantization methods for feeding back the channel information through a low rate feedback channel in the context of multiple input single output miso systems we propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization vq design algorithm the criterion is based on maximizing the mean squared weighted inner product mswip between the optimum and the quantized beamforming vector the performance of systems with quantized beamforming is analyzed for the independent fading case this requires finding the density of the squared inner product between the optimum and the quantized beamforming vector which is obtained by considering a simple approximation of the quantization cell the approximate density function is used to lower bound the capacity loss due to quantization the outage probability and the bit error probability the resulting expressions provide insight into the dependence of the performance of transmit beamforming miso systems on the number of transmit antennas and feedback rate computer simulations support the analytical results and indicate that the lower bounds are quite tight

beam selection gain versus antenna selection gain we consider beam selection using a fixed beamforming network fbn at a base station with m array antennas in our setting a butler matrix is deployed at the rf stage to form m beams and then the best beam is selected for transmission we introduce some properties of the noncentral chi square distribution and prove the resulting properties of the beam selection gain verifying that beam selection is superior to antenna selection in rician channels with any k factors furthermore we find asymptotically tight stochastic bounds of the beam selection gain which yield approximate closed form expressions of the expected selection gain and the ergodic capacity beam selection has the order of growth of the ergodic capacity logm regardless of user location in contrast to log logm for antenna selection

zero relaxation and dissipation limits for hyperbolic conservation laws we are interested in hyperbolic systems of conservation laws with relaxation and dissipation particularly the zero relaxation limit such a limit is of interest in several physical situations including gas flow near thermo equilibrium kinetic theory with small mean free path and viscoelasticity with vanishing memory in this article we study hyperbolic systems of two conservation laws with relaxation for the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit for the marginally stable case where the equilibrium speed may equal one of the frozen speeds we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation our analysis includes the construction of suitable entropy pairs to derive energy estimates we need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems the theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit 1993 john wiley sons inc

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

selection rules for topology change it is shown that there are restrictions on the possible changes of topology of space sections of the universe if this topology change takes place in a compact region which has a lorentzian metric and spinor structure in particular it is impossible to create a single wormhole or attach a single handle to a spacetime but it is kinematically possible to create such wormholes in pairs another way of saying this is that there is a 2 invariant for a closed oriented 3 manifold which determines whether can be the spacelike boundary of a compact manifoldm which admits a lorentzian metric and a spinor structure we evaluate this invariant in terms of the homology groups of and find that it is the mod2 kervaire semi characteristic

novel transmit beamforming schemes for time selective fading multiantenna systems transmit beamforming has been widely adopted for wireless systems with multiple transmit antennas for a block fading channel the grassmannian beamformer has been shown to provide very good performance for finite rate feedback however the original grassmannian beamformer does not take the time domain correlation of the channel fading into consideration in this paper based on a first order autoregressive ar1 dynamic fading model we develop two new classes of beamforming algorithms that exploit the interframe correlations in the channel fading we first introduce an algorithm based on a standard predictive vector quantization pvq approach and the resulting pvq beamformer accomplishes superior power delivery at the receiver however the error performance of the pvq beamformer is not satisfactory at high signal to noise ratios and it also has a high implementation complexity to resolve these issues we then develop a novel successive beamforming sbf algorithm the new sbf scheme uses the knowledge of the previous fading blocks to aid the beamforming codebook design of the current fading block the beamforming codebook is constructed based on the successive partition of the surface of a spherical cap the new sbf scheme accomplishes nearly the same performance as that of the pvq beamformer and it has a much simpler implementation through numerical simulations we demonstrate that the proposed beamformers outperform the other previously proposed beamformers at various fading scenarios

grassmannian beamforming for multiple input multiple output wireless systems transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple input multiple output mimo wireless systems unfortunately optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector both are hard to realize in this article a quantized maximum signal to noise ratio snr beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter by using the distribution of the optimal beamforming vector in independent and identically distributed rayleigh fading matrix channels the codebook design problem is solved and related to the problem of grassmannian line packing the proposed design criterion is flexible enough to allow for side constraints on the codebook vectors bounds on the codebook size are derived to guarantee full diversity order results on the density of grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or snr loss monte carlo simulations are presented that compare the probability of error for different quantization strategies

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

space time trellis codes based on channel phase feedback space time coding stc has been proposed recently for multiple antenna wireless communication systems most of the proposed stc schemes use the assumption that either no channel state information or the channel mean covariance information is available at the transmitter in this paper we propose a new stc scheme for a closed loop transmission system where quantized channel phase information is available at the transmitter a new performance criterion is derived for the quasi static fading channel this design criterion is then used to construct a new class of space time trellis codes sttcs the proposed code construction is based on the concatenation of a standard multiple trellis coded modulation outer code with an inner code the inner code is selected from a series of inner codes using the channel phase feedback the series of inner codes are constructed based on the systematic set partitioning of several classes of space time signal designs simulation results show significant performance improvement over the other sttcs in the literature in addition the proposed coding scheme enjoys low peak to average power ratio simple decoding and power efficient low cost implementation

gaussian source coding with spherical codes a fixed rate shape gain quantizer for the memoryless gaussian source is proposed the shape quantizer is constructed from wrapped spherical codes that map a sphere packing in spl ropf sup k 1 onto a sphere in spl ropf sup k and the gain codebook is a globally optimal scalar quantizer a wrapped leech lattice shape quantizer is used to demonstrate a signal to quantization noise ratio within 1 db of the distortion rate function for rates above 1 bit per sample and an improvement over existing techniques of similar complexity an asymptotic analysis of the tradeoff between gain quantization and shape quantization is also given

new constructions of codebooks nearly meeting the welch bound with equality an n k codebook c is a set of n unit norm complex vectors in bbck optimal codebooks meeting the welch bound with equality are desirable in a number of areas however it is very difficult to construct such optimal codebooks there have been a number of attempts to construct codebooks nearly meeting the welch bound with equality i e the maximal cross correlation amplitude imax c is slightly higher than the welch bound equality but asymptotically achieves it for large enough n in this paper using difference sets and the product of abelian groups we propose new constructions of codebooks nearly meeting the welch bound with equality our methods yield many codebooks with new parameters in some cases our constructions are comparable to known constructions

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

generalized asymptotically optimal codebooks abstract recently luo et al proposed two new codebooks based on the operations of some given sets which are asymptotically optimal in the paper we present new construction of codebooks we determine maximal cross correlation amplitude of the constructed codebooks the constructed codebooks are near optimal to the welch bound this new codebooks in the paper generalize luo et al s constructions the parameters of constructed codebooks are nimble and new under some circumstances

design guidelines for limited feedback in the spatially correlated broadcast channel a linear beamformer design for the broadcast channel where the base station is equipped with n t antennas and signals to m 2 users each with a single antenna and where the vector channels are spatially correlated is considered this problem is of relevance in wireless standardization efforts where two users are simultaneously scheduled instead of the theoretically feasible n t user scheduling to minimize signaling overhead the users are assumed to have perfect channel state information csi whereas the base station has statistical information of the channels in the first part of this work the role of the relevance of feedback is studied via the quantification of the gap in ergodic sum rate between the perfect csi and statistics only extremes in this direction the importance of orthogonality of the dominant eigenmodes of the two users in maximizing the gap is established in the second part of this work in scenarios with a large gap limited feedback beamforming codebooks and a codeword selection metric are designed for the low rate feedback setting the main contribution here is the proposal of a generalized eigenvector codebook and feedback of the codeword index that maximizes an estimate of the signal to interference and noise ratio textsf sinr of each user extensions of this scheme are also proposed for the general n t antenna case with m users where m le n t it is shown via numerical studies that this scheme leads to significant performance improvement across a large family of channels over schemes such as grassmannian random vector quantization single user codebooks with the channel projection metric for codeword selection

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

on beamforming with finite rate feedback in multiple antenna systems we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations assuming that the transmitter uses the quantized information for beamforming we derive a universal lower bound on the outage probability for any finite set of beamformers the universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel using the bound it is shown that finite information systems approach the perfect information case as t 1 2 sup b t 1 where b is the number of feedback bits and t is the number of transmit antennas the geometrical bounding technique used in the proof of the lower bound also leads to a design criterion for good beamformers whose outage performance approaches the lower bound the design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook and is equivalent to the problem of designing unitary space time codes under certain conditions finally we show that good beamformers are good packings of two dimensional subspaces in a 2t dimensional real grassmannian manifold with chordal distance as the metric