lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

cooperative transmission for relay networks based on second order statistics of channel state information cooperative transmission in relay networks is considered in which a source transmits to its destination with the help of a set of cooperating nodes the source first transmits locally the cooperating nodes that receive the source signal retransmit a weighted version of it in an amplify and forward af fashion assuming knowledge of the second order statistics of the channel state information beamforming weights are determined so that the signal to noise ratio snr at the destination is maximized subject to two different power constraints i e a total source and relay power constraint and individual relay power constraints for the former constraint the original problem is transformed into a problem of one variable which can be solved via newton s method for the latter constraint this problem is solved completely it is shown that the semidefinite programming sdp relaxation of the original problem always has a rank one solution and hence the original problem is equivalent to finding the rank one solution of the sdp problem an explicit construction of such a rank one solution is also provided numerical results are presented to illustrate the proposed theoretical findings

efficient mesh optimization using the gradient flow of the mean volume the signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron if we consider these up to translation and scaling the resulting quotient space is diffeomorphic to a sphere the mean volume function restricted to this sphere is a quality measure for volume elements we show that the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra hexahedra prisms pyramids and octahedra that is the optimization process converges to regular polyhedra we show that the normalized gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method furthermore we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally

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

optimal linear precoders for mimo wireless correlated channels with nonzero mean in space time coded systems this paper proposes linear precoder designs exploiting statistical channel knowledge at the transmitter in a multiple input multiple output mimo wireless system the paper focuses on channel statistics since obtaining real time channel state information at the transmitter can be difficult due to channel dynamics the considered channel statistics consist of the channel mean and transmit antenna correlation the receiver is assumed to know the instantaneous channel precisely the precoder operates along with a space time block code stbc and aims to minimize the chernoff bound on the pairwise error probability pep between a pair of block codewords averaged over channel fading statistics two pep design criteria are studied minimum distance and average distance the optimal precoder with an orthogonal stbc is established using a convex optimization framework different relaxations then extend the solution to systems with nonorthogonal stbcs in both cases the precoder is a function of both the channel mean and the transmit correlation a linear precoder acts as a combination of a multimode beamformer and an input shaping matrix matching each side to the channel and to the input signal structure respectively both the optimal beam direction and the power of each mode obtained via a dynamic water filling process depend on the signal to noise ratio snr asymptotic analyses of the results reveal that for all stbcs the precoder approaches a single mode beamformer on the dominant right singular vector of the channel mean as the channel k factor increases on the other hand as the snr increases it approaches an equipower multiple mode beamformer matched to the eigenvectors of the transmit correlation design examples and numerical simulation results for both orthogonal and nonorthogonal stbc precoding solutions are provided illustrating the precoding array gain

an analysis of finite difference and finite volume formulations of conservation laws abstract finite difference and finite volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws a new coordinate free formulation of systems of conservation laws is developed which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system the analysis considers general types of equations potential euler and navier stokes three dimensional unsteady flows with time varying grids are described using a single consistent nomenclature for both formulations grid motion due to a non inertial reference frame as well as flow adaptation is covered in comparing the two formulations it is found useful to distinguish between differences in numerical methods and differences in grid definition the former plays a role for non cartesian grids and results in only cosmetic differences in the manner in which geometric terms are handled the differences in grid definition for the two formulations is found to be more important since it affects the manner in which boundary conditions zonal procedures and grid singularities are handled at computational boundaries the proper interpretation of strong and weak conservation law forms for quasi one dimensional and axisymmetric flows is brought out

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

multicell optimization for diversity and interference mitigation we study diversity and interference mitigation with an emphasis on the interaction between the transmitter receiver tx rx pairs in a two user two cell system in which the performance at each rx depends on the actions of the both desired and the interfering txs we will propose two techniques the first scheme is a greedy or competitive algorithm in which each tx rx pair employs an orthogonal space time st block code combined with a multimode beamformer a generalized st code to minimize the error probability to its own rx till an equilibrium is reached the second scheme is a cooperative algorithm also based on a generalized st code in which each tx rx pair minimizes its error probability at its desired rx but cooperates by limiting the amount of interference to the second pair s rx we will model and characterize the equilibrium of the competitive scheme called the nash equlibrium we will then show that by using the cooperative algorithm it is possible to find pareto dominant actions where both tx rx pairs either equal or improve their performance when compared with the ne actions we will analyze the conditions for pareto dominance of the cooperative scheme and finally demonstrate the performance improvements of both schemes over the single tx rx pair generalized st code

optimal transmitter eigen beamforming and space time block coding based on channel mean feedback optimal transmitter designs obeying the water filling principle are well documented they are widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information is impossible to be known perfectly at the transmitter in practical wireless systems we design in this paper an optimal multiantenna transmitter based on the knowledge of mean values of the underlying channels our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the correlation matrix of the estimated channel at the transmitter and with proper power loading across beams the optimality pertains to minimizing an upper bound on the symbol error rate which leads to better performance than maximizing the expected signal to noise ratio snr at the receiver coupled with orthogonal space time block codes two directional eigen beamforming emerges as a more attractive choice than conventional one directional beamforming with uniformly improved performance without rate reduction and without essential increase in complexity with multiple receive antennas and reasonably good feedback quality the two directional eigen beamformer is also capable of achieving the best possible performance in a large range of transmit power to noise ratios without a rate penalty

a quasi orthogonal space time block code it has been shown that a complex orthogonal design that provides full diversity and full transmission rate for a space time block code is not possible for more than two antennas previous attempts have been concentrated in generalizing orthogonal designs which provide space time block codes with full diversity and a high transmission rate we design rate one codes which are quasi orthogonal and provide partial diversity the decoder of the proposed codes works with pairs of transmitted symbols instead of single symbols

combining beamforming and orthogonal space time block coding multiple transmit and receive antennas can be used in wireless systems to achieve high data rate communication efficient space time codes have been developed that utilize a large portion of the available capacity these codes are designed under the assumption that the transmitter has no knowledge about the channel in this work on the other hand we consider the case when the transmitter has partial but not perfect knowledge about the channel and how to improve a predetermined code so that this fact is taken into account a performance criterion is derived for a frequency nonselective fading channel and then utilized to optimize a linear transformation of the predetermined code the resulting optimization problem turns out to be convex and can thus be efficiently solved using standard methods in addition a particularly efficient solution method is developed for the special case of independently fading channel coefficients the proposed transmission scheme combines the benefits of conventional beamforming with those given by orthogonal space time block coding simulation results for a narrow band system with multiple transmit antennas and one or more receive antennas demonstrate significant gains over conventional methods in a scenario with nonperfect channel knowledge

improved linear transmit processing for single user and multi user mimo communications systems in this paper we propose a novel linear transmit precoding strategy for multiple input multiple output mimo systems employing improper signal constellations in particular improved zero forcing zf and minimum mean square error mmse precoders are derived based on modified cost functions and are shown to achieve a superior performance without loss of spectrum efficiency compared to the conventional linear and nonlinear precoders the superiority of the proposed precoders over the conventional solutions are verified by both simulation and analytical results the novel approach to precoding design is also applied to the case of an imperfect channel estimate with a known error covariance as well as to the multi user scenario where precoding based on the nullspace of channel transmission matrix is employed to decouple multi user channels in both cases the improved precoding schemes yield significant performance gain compared to the conventional counterparts

application of quasi orthogonal space time block codes in beamforming it is well known that when channel information is available at the transmitter transmit beamforming scheme can be employed to enhance the performance of a multiple antenna system recently jongren et al and zhou giannakis proposed a new performance criterion based on partial channel side information at the transmitter with this criterion an optimal beamforming matrix was constructed for the orthogonal space time block codes however the same method has not been applied to the recently proposed quasi orthogonal space time block codes qstbcs due to the nonorthogonal nature of the quasi orthogonal designs in this paper the issue of combining beamforming with qstbcs is addressed based on our asymptotic analysis we extend the beamforming scheme from jongren et al and construct the beamforming matrices for the quasi orthogonal designs the proposed beamforming scheme accomplishes high transmission rate as well as high order spatial diversity the new qstbc beamformer can be presented as a novel four directional or eight directional eigen beamformer that works for systems with four or more transmit antennas simulations for systems with multiple transmit antennas demonstrate significant performance improvement over several other widely used beamforming methods at various snrs and for channels with different quality of feedback

statistically robust design of linear mimo transceivers the treatment of channel state information csi is critical in the design of mimo systems accurate csi at the transmitter is often not possible or may require high feedback rates herein we consider the robust design of linear mimo transceivers with perfect csi either at the transmitter or at both sides of the link the framework considers the design problem where the imperfect csi consists of the channel mean and covariance matrix or equivalently the channel estimate and the estimation error covariance matrix the robust transceiver design is based on a general cost function of the average mses as well as a design with individual mse based constraints in particular a lower bound of the average mse matrix is explored for the design when only the csi at the transmitter is imperfect under different csi conditions the proposed robust transceivers exhibit a similar structure to the transceiver designs for perfect csi but with a different equivalent channel and or noise covariance matrix

worst case robust mimo transmission with imperfect channel knowledge in this paper we consider robust transmit strategies against the imperfectness of the channel state information at the transmitter csit for multi input multi output mimo communication systems following a worst case deterministic model the actual channel is assumed to be inside an ellipsoid centered at a nominal channel the objective is to maximize the worst case received signal to noise ratio snr or to minimize the worst case chernoff bound of the error probability thus leading to a maximin problem moreover we also consider the qos problem as a complement of the maximin design which minimizes the transmit power consumption and meanwhile keeps the received snr above a given threshold for any channel realization in the ellipsoid it is shown that for a general class of power constraints both the maximin and qos problems can be equivalently transformed into convex problems or even further into semidefinite programs sdps thus efficiently solvable by the numerical methods the most interesting result is that the optimal transmit directions i e the eigenvectors of the transmit covariance are just the right singular vectors of the nominal channel under some mild conditions this result leads to a channel diagonalizing structure as in the cases of perfect csit and statistical csit with mean or covariance feedback and reduces the complicated matrix valued problem to a scalar power allocation problem then we provide the closed form solution to the resulting power allocation problem

robust training sequence design for correlated mimo channel estimation we study how to design a worst case robust training sequence for multiple input multiple output mimo channel estimation we consider mean squared error of channel estimates as the figure of merit which is a function of second order statistics of the mimo channel i e channel covariance matrix in order to optimize training sequences under a total power constraint in practical applications the channel covariance matrix is not known perfectly thus the main aspect of our design is to improve robustness of the training sequences against possible uncertainties in the available channel covariance matrix using a deterministic uncertainty model we formulate a robust training sequence design as a minimax optimization problem where we take such imperfections into account we investigate the robust design problem assuming the general case of an arbitrarily correlated mimo channel and a non empty compact convex uncertainty set we prove that such a problem admits a globally optimal solution by exploiting the convex concave structure of the objective function and propose numerical algorithms to address the robust training design problem we proceed the analysis by considering multiple input single output miso channels and kronecker structured mimo channels along with unitarily invariant uncertainty sets for these scenarios we show that the problem is diagonalized by the eigenvectors of the nominal covariance matrices so that the robust design is significantly simplified from a complex matrix variable problem to a real vector variable power allocation problem for the miso channel we provide closed form solutions for the robust training sequences with the uncertainty sets defined by the spectral norm and nuclear norm

efficient computation of volume in flow predictions an efficient method for calculating cell volumes for time dependent three dimensional flow predictions by finite volume calculations is presented eight arbitrary corner points are considered and the shape face is divided into two planar triangles the volume is then dependent on the orientation of the partitioning in the case of a hexahedron it is noted that any open surface with a boundary that is a closed curve possesses a surface vector independent of the surface shape expressions are defined for the surface vector which is independent of the partitioning surface diagonal used to quantify the volume using a decomposition of the cell volume involving two corners with each the vertex of three diagonals and six corners which are vertices of one diagonal gives portions which are tetrahedra the resultant mesh is can be used for time dependent finite volume calculations one requires less computer time than previous methods

optimal transmitter eigen beamforming and space time block coding based on channel correlations optimal transmitter designs obeying the water filling principle are well documented and widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information csi may be costly or impossible to acquire in rapidly varying wireless environments we develop in this paper statistical water filling approaches for stationary random fading channels these approaches require only knowledge of the channel correlations that do not necessitate frequent updates and can be easily acquired applied to a multiple transmit antenna paradigm our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the channel s correlation matrix and with proper power loading across the beams the optimality pertains to minimizing a tight bound on the symbol error rate the resulting loaded eigen beamforming outperforms not only the equal power allocation across all antennas but also the conventional beamformer that transmits the available power along the strongest direction coupled with orthogonal space time block codes two dimensional 2 d eigen beamforming emerges as a more attractive choice than conventional one dimensional 1 d beamforming with uniformly better performance without rate reduction and without complexity increase

robust mmse precoding in mimo channels with pre fixed receivers in this paper we design robust precoders under the minimum mean square error mmse criterion for different types of channel state information csi in multiple input multiple output mimo channels we consider low complexity pre fixed receivers that may adapt to the channel but are oblivious to the existence of a precoder at the transmitter in particular three types of csi are taken into account i perfect csi ii statistical csi in the form of mean feedback and iii deterministic imperfect csi assuming that the actual channel is within the neighborhood of a nominal channel which leads to the worst case robust design that is the focus of this paper interestingly it is found that under some mild conditions the optimal transmit directions i e the left singular vectors of the precoder are equal to the right singular vectors of the channel the channel mean and the nominal channel for perfect csi statistical csi and the worst case design respectively consequently the matrix valued problems can be simplified to scalar power allocation problems that either admit closed form solutions or can be efficiently solved by the proposed algorithm

optimal joint spatial and temporal power adaptation for space time coded systems with imperfect csi in this correspondence a joint optimization of spatio temporal s t power allocation to minimize the average bit error rate ber subject to an average power constraint is developed for beamforming orthogonal space time block coded multiantenna systems over flat rayleigh fading channels in the presence of imperfect channel state information at the transmitter csit compared to the spatial only s o power allocation methods the optimal s t strategy reduces ber significantly it can be used as a benchmark to evaluate the existing suboptimal s t schemes a necessary and sufficient condition which can determine the number of eigenbeams being used for optimal s t power allocation without computing power allocation is derived this result can greatly simplify the power allocation algorithm numerical results show that the proposed optimal s t algorithm has not only the best ber performance but also the lowest computational complexity in comparison with the existing s t strategies

precoder design for space time coded systems over correlated rayleigh fading channels using convex optimization a class of computationally efficient linear precoders for space time block coded multiple input multiple output wireless systems is derived based on the minimization of the exact symbol error rate ser and its upper bound both correlations at the transmitter and receiver are assumed to be present and only statistical channel state information in the form of the transmit and receive correlation matrices is assumed to be available at the transmitter the convexity of the design based on ser minimization is established and exploited the advantage of the developed technique is its low complexity we also find various relationships of the proposed designs to the existing precoding techniques and derive very simple closed form precoders for special cases such as two or three receive antennas and constant receive correlation the numerical simulations illustrate the excellent ser performance of the proposed precoders

the solution of the two dimensional sine gordon equation using the method of lines the method of lines is used to transform the initial boundary value problem associated with the two dimensional sine gordon equation in two space variables into a second order initial value problem the finite difference methods are developed by replacing the matrix exponential term in a recurrence relation with rational approximants the resulting finite difference methods are analyzed for local truncation error stability and convergence to avoid solving the nonlinear system a predictor corrector scheme using the explicit method as predictor and the implicit as corrector is applied numerical solutions for cases involving the most known from the bibliography line and ring solitons are given

bdma in multicell massive mimo communications power allocation algorithms we investigate power allocation strategies for beam division multiple access transmission in multicell massive multiple input multiple output mimo communications focusing on massive mimo downlink with only statistical channel state information at serving base stations and multiantenna terminals the eigenmatrices of channel transmit covariance matrices become identical and independent of terminals utilizing these eigenmatrices as precoding matrices we consider power allocation of beam domain transmission by treating cochannel user interferences as noises the objective function of achievable ergodic sum rate of the multicell massive mimo downlink reduces to a difference of concave functions we identify the orthogonality conditions for optimal power allocation the results show that the transmit power allocated to the users should be nonoverlapping across beams defined by the channel transmit covariance matrices we present two algorithms to optimize power allocation in the beam space we also prove the convergence of the algorithms and show that the solution obtained by the algorithms satisfies the orthogonality conditions numerical results confirm the efficiency and the improved performance of the proposed algorithms