efficient weighted sum rate maximization with linear precoding achieving the boundary of the capacity region in the multiple input multiple output mimo broadcast channel requires the use of dirty paper coding dpc as practical nearly optimum implementations of dpc are computationally complex purely linear approaches are often used instead however in this case the problem of maximizing a weighted sum rate constitutes a nonconvex and in most cases also a combinatorial optimization problem in this paper we present two heuristic nearly optimum algorithms with reduced computational complexity for this purpose a lower bound for the weighted sum rate under linear zero forcing constraints is used based on this bound both greedy algorithms successively allocate data streams to users in each step the user is determined that is given an additional data stream such that the increase in weighted sum rate becomes maximum thereby the data stream allocations and filters obtained in the previous steps are kept fixed and only the filter corresponding to the additional data stream is optimized the first algorithm determines the receive and transmit filters directly in the downlink the other algorithm operates in the dual uplink from which the downlink transmit and receive filters can be obtained via the general rate duality leading to nonzero forcing in the downlink simulation results reveal marginal performance losses compared to more complex algorithms

a joint source and relay power allocation scheme for a class of mimo relay systems a joint power allocation pa scheme for a class of mimo relay systems is presented in this paper based on the maximization of the capacity or minimization of the mean square error mse two joint pa optimization problems are first formulated since the cost functions thus obtained are in general not convex a tight lower bound of the capacity and a similar upper bound of the mse are derived and employed to modify the two cost functions so as to obtain a convex optimization problem it is confirmed through computer simulations that the proposed pa scheme outperforms the existing method in terms of both the capacity and the mse of mimo relay systems

homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

transmitter optimization for the multi antenna downlink with per antenna power constraints this paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the base station in contrast to the conventional sum power constraint on the transmit antennas this paper adopts a more realistic per antenna power constraint because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier assuming perfect channel knowledge at the transmitter this paper investigates two different transmission schemes under the per antenna power constraint a minimum power beamforming design for downlink channels with a single antenna at each remote user and a capacity achieving transmitter design for downlink channels with multiple antennas at each remote user it is shown that in both cases the per antenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise this generalizes previous uplink downlink duality results and transforms the per antenna transmitter optimization problem into an equivalent minimax optimization problem further it is shown that various notions of uplink downlink duality may be unified under a lagrangian duality framework this new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink per antenna transmitter optimization problem

rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media we establish a logarithmic type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second order pde in strongly mixing media with similar i e logarithmic decorrelation rate the proof consists of two major steps the first which is actually the only place in the paper where probability plays a role establishes the rate for special quadratic data using the methodology developed by the authors and wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media the second is a general argument based on the new notion of viscosity solutions which is introduced in this paper that shows that rates known for quadratic can be extended to general data as an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments the former is algebraic while the latter depends on the particular equation

mitigation of loopback self interference in full duplex mimo relays full duplex relaying is more spectrally efficient than half duplex relaying as only one channel use is needed per two hops however it is crucial to minimize relay self interference to render full duplex feasible for this purpose we analyze a broad range of multiple input multiple output mimo mitigation schemes natural isolation time domain cancellation and spatial suppression cancellation subtracts replicated interference signal from the relay input while suppression reserves spatial dimensions for receive and transmit filtering spatial suppression can be achieved by antenna subset selection null space projection i e receiving and transmitting in orthogonal subspaces or joint transmit and receive beam selection to support more spatial streams by choosing the minimum eigenmodes for overlapping subspaces in addition minimum mean square error mmse filtering can be employed to maintain the desired signal quality which is inherent for cancellation and the combination of time and spatial domain processing may be better than either alone targeting at minimal interference power we solve optimal filters for each scheme in the cases of joint separate and independent design the performance of mitigation schemes is evaluated and compared by simulations the results confirm that self interference can be mitigated effectively also in the presence of imperfect side information

sum power iterative water filling for multi antenna gaussian broadcast channels in this correspondence we consider the problem of maximizing sum rate of a multiple antenna gaussian broadcast channel bc it was recently found that dirty paper coding is capacity achieving for this channel in order to achieve capacity the optimal transmission policy i e the optimal transmit covariance structure given the channel conditions and power constraint must be found however obtaining the optimal transmission policy when employing dirty paper coding is a computationally complex nonconvex problem we use duality to transform this problem into a well structured convex multiple access channel mac problem we exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the mac which can easily be mapped to the optimal bc policies

optimal designs for space time linear precoders and decoders we introduce a new paradigm for the design of transmitter space time coding that we refer to as linear precoding it leads to simple closed form solutions for transmission over frequency selective multiple input multiple output mimo channels which are scalable with respect to the number of antennas size of the coding block and transmit average peak power the scheme operates as a block transmission system in which vectors of symbols are encoded and modulated through a linear mapping operating jointly in the space and time dimension the specific designs target minimization of the symbol mean square error and the approximate maximization of the minimum distance between symbol hypotheses under average and peak power constraints the solutions are shown to convert the mimo channel with memory into a set of parallel flat fading subchannels regardless of the design criterion while appropriate power bits loading on the subchannels is the specific signature of the different designs the proposed designs are compared in terms of various performance measures such as information rate ber and symbol mean square error

minimum ber linear transceivers for mimo channels via primal decomposition this paper considers the employment of linear transceivers for communication through multiple input multiple output mimo channels with channel state information csi at both sides of the link the design of linear mimo transceivers has been studied since the 1970s by optimizing simple measures of the quality of the system such as the trace of the mean square error matrix subject to a power constraint recent results showed how to solve the problem in an optimal way for the family of schur concave and schur convex cost functions in particular when the constellations used on the different transmit dimensions are equal the bit error rate ber averaged over these dimensions happens to be a schur convex function and therefore it can be optimally solved in a more general case however when different constellations are used the average ber is not a schur convex function and the optimal design in terms of minimum ber is an open problem this paper solves the minimum ber problem with arbitrary constellations by first reformulating the problem in convex form and then proposing two solutions one is a heuristic and suboptimal solution which performs remarkably well in practice the other one is the optimal solution obtained by decomposing the convex problem into several subproblems controlled by a master problem a technique borrowed from optimization theory for which extremely simple algorithms exist thus the minimum ber problem can be optimally solved in practice with very simple algorithms

per stream mse based linear transceiver design for mimo interference channels with csi error this paper studies the linear transceiver design for multi stream multiple input multiple output mimo interference channels with imperfect channel state information csi two csi error models are considered the same coding and modulation scheme is assumed for all users and data streams in the first model where the csi error is statistical we minimize the average maximum per stream mean square error mse in the second model where the csi error is norm bounded the worst case maximum per stream mse is optimized since the problems are not jointly convex the alternate optimization approach is adopted to obtain suboptimal transceivers for the statistical csi error a low complexity algorithm utilizing lagrangian duality is proposed which has a semi closed form solution and can be implemented distributively compared with the classical algorithm using the second order cone programming the proposed algorithm significantly reduces the computational complexity without sacrificing the performance for the bounded csi error we present two approaches one uses s procedure to obtain an equivalent problem and the other one solves an approximated problem simulation results demonstrate the robustness of the proposed algorithms and their improved performance over the sum mse and the maximum per user mse based methods the lower complexity of the lagrangian duality based algorithm is also verified numerically

linear precoding via conic optimization for fixed mimo receivers in this paper the problem of designing linear precoders for fixed multiple input multiple output mimo receivers is considered two different design criteria are considered in the first the transmitted power is minimized subject to signal to interference plus noise ratio sinr constraints in the second the worst case sinr is maximized subject to a power constraint it is shown that both problems can be solved using standard conic optimization packages in addition conditions are developed for the optimal precoder for both of these problems and two simple fixed point iterations are proposed to find the solutions that satisfy these conditions the relation to the well known uplink downlink duality in the context of joint transmit beamforming and power control is also explored the proposed precoder design is general and as a special case it solves the transmit rank one beamforming problem simulation results in a multiuser system show that the resulting precoders can significantly outperform existing linear precoders

quantitative stochastic homogenization of elliptic equations in nondivergence form we introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form the main application is an algebraic error estimate asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale assuming a finite range of dependence the results are new even for linear equations the arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the monge ampere equation

linear beamforming and superposition coding with common information for the gaussian mimo broadcast channel a gaussian multiple input multiple output broadcast channel mimo gbc is considered throughout the paper it is assumed that 1 input signals are gaussian and 2 perfect channel state information is available at the transmitter and at the receivers by considering each data stream as a single user the uplink downlink signal to interference plus noise sinr duality is generalized to the mimo case with general cross talk matrix the duality is subsequently applied to finding the solution for the sinr balancing problem the result serves as a tool for characterizing achievable rate regions of different coding strategies next we investigate a superposition coding scheme proposed by cover van der meulen hajek and pursley nicknamed cmhp where there is a common message to both users we consider a mimo broadcast channel with two users each user has two antennas and the transmitter has four antennas assuming one common stream is sent by cmhp coding and successive decoding a lower bound to the cmhp rate region is found behaviors of the cmhp rate region and sumrate are analyzed we find the sumrate gaps between dpc cmhp and mmse at high snr for general 2 user multiple input singleoutput miso gaussian broadcast channel the result suggests when cmhp is beneficial for sumrate

uplink downlink duality via minimax duality the sum capacity of a gaussian vector broadcast channel is the saddle point of a minimax gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint this sum capacity result has been proved using two different methods one based on decision feedback equalization and the other based on a duality between uplink and downlink channels this paper illustrates the connection between the two approaches by establishing that uplink downlink duality is equivalent to lagrangian duality in minimax optimization this minimax lagrangian duality relation allows the optimal transmit covariance and the least favorable noise covariance matrices in a gaussian vector broadcast channel to be characterized in terms of the dual variables in particular it reveals that the least favorable noise is not unique further the new lagrangian interpretation of uplink downlink duality allows the duality relation to be generalized to gaussian vector broadcast channels with arbitrary linear constraints however duality depends critically on the linearity of input constraints duality breaks down when the input constraint is an arbitrary convex constraint this shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink downlink duality representation

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

qos constrained multiuser peer to peer amplify and forward relay beamforming a wireless communication scenario is considered with k single antenna source destination pairs communicating through several half duplex amplify and forward mimo relays where each source is targeting only one destination the relay beamforming matrices are designed in order to minimize the total power transmitted from the relays subject to quality of service constraints on the received signal to interference plus noise ratio at each destination node due to the nonconvexity of this problem several approximations have been used in the literature to find a computationally efficient solution a novel solution technique is developed in which the problem is decomposed into a group of second order cone programs socps parameterized by k phase angles each associated with one of the constraints an iterative algorithm is proposed to search for the phase angles and the relay beamforming matrices sequentially however convergence to the global optimal beamforming matrices cannot be guaranteed two methods for searching for the optimal values of the phase angles are proposed from which the optimal beamforming matrices can be obtained using grid search and bisection and the convergence of these methods to the global optimal solution of the problem is proved numerical simulations are presented showing the superior performance of the proposed algorithms compared to earlier suboptimal approximations at the expense of a moderate increase in the computational complexity

a general robust linear transceiver design for multi hop amplify and forward mimo relaying systems in this paper linear transceiver design for multi hop amplify and forward af multi input multi output mimo relaying systems with gaussian distributed channel estimation errors is investigated commonly used transceiver design criteria including weighted mean square error mse minimization capacity maximization worst mse max mse minimization and weighted sum rate maximization are considered and unified into a single matrix variate optimization problem a general robust design algorithm is proposed to solve the unified problem specifically by exploiting majorization theory and properties of matrix variate functions the optimal structure of the robust transceiver is derived when either the covariance matrix of channel estimation errors seen from the transmitter side or the corresponding covariance matrix seen from the receiver side is proportional to an identity matrix based on the optimal structure the original transceiver design problems are reduced to much simpler problems with only scalar variables whose solutions are readily obtained by an iterative water filling algorithm a number of existing transceiver design algorithms are found to be special cases of the proposed solution the differences between our work and the existing related work are also discussed in detail the performance advantages of the proposed robust designs are demonstrated by simulation results

on the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio temporal media abstract we study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio temporal media from the qualitative and quantitative perspectives under suitable hypotheses solutions to fully nonlinear uniformly parabolic equations in spatio temporal media homogenize almost surely in addition we obtain a logarithmic rate of convergence for this homogenization in measure assuming that the environment is strongly mixing with a prescribed logarithmic rate a general methodology to study the stochastic homogenization and rates of convergence for stochastic homogenization of uniformly elliptic equations was introduced by caffarelli souganidis and wang 1 and caffarelli and souganidis 2 we extend their approach to fully nonlinear uniformly parabolic equations developing a number of new arguments to handle the parabolic structure of the problem

maximization of worst case weighted sum rate for miso downlink systems with imperfect channel knowledge the problem of robust weighted sum rate maximization wsrmax in multicell downlink multi input single output systems is considered we assume that channel state information csi of all users is imperfectly known at the base stations the problem is known to be np hard even in the case of perfect csi we propose optimal and suboptimal but fast converging algorithms for wsrmax problem with csi errors assuming bounded ellipsoidal model for the csi errors we optimize the worst case weighted sum rate the proposed optimal algorithm is based on branch and bound bb technique and it globally solves the worst case wsrmax problem with an optimality certificate as the convergence speed of the bb method can be slow for large networks we also provide a fast but possibly suboptimal algorithm based on alternating optimization technique and sequential convex programming the optimal bb based algorithm can be used to provide performance benchmarks for any suboptimal algorithm numerical results show that the convergence speed of the suboptimal algorithm is fast and it finds a close to optimal solution in only a few iterations

transceiver optimization for multiuser mimo systems we consider the uplink of a multiuser system where the transmitters as well as the receiver are equipped with multiple antennas each user multiplexes its symbols by a linear precoder through its transmit antennas we work with the system wide mean squared error as the performance measure and propose algorithms to find the jointly optimum linear precoders at each transmitter and linear decoders at the receiver we first work with the case where the number of symbols to be transmitted by each user is given we then investigate how the symbol rate should be chosen for each user with optimum transmitters and receivers the convergence analysis of the algorithms is given and numerical evidence that supports the analysis is presented

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

transmission strategies for the mimo mac with mmse receiver average mse optimization and achievable individual mse region in this work we study the multiuser multiple input multiple output mimo multiple access channel mac under the assumption that the base station performs linear multiuser minimum mean square error mmse detection we derive the average normalized mse and the individual mses of users at first we optimize the average normalized mse with respect to the transmit covariance matrices of the users next adaptive power allocation is applied to further minimize the average normalized mse this leads to the general average normalized mse optimization under a sum power constraint we analyze the optimization problems and their connections by their karush kuhn tucker conditions the covariance matrix optimization is solved by iteratively single user average mse optimization in which the interference is treated as noise we develop an iterative algorithm that performs power allocation and covariance matrix optimization in addition to this we study the achievable mse region we show that the mse region is convex for the two user mimo case furthermore we characterize the optimum power allocation among the users with regard to the single user range we show that the user with the maximum singular value of its channel matrix is the first supported user for low snr values the optimal strategy is to have the best user transmitting only in addition we derive the individual mse using single user mmse detectors and study the fulfillment of mse requirements

sum capacity computation for the gaussian vector broadcast channel via dual decomposition a numerical algorithm for the computation of sum capacity for the gaussian vector broadcast channel is proposed the sum capacity computation relies on a duality between the gaussian vector broadcast channel and the sum power constrained gaussian multiple access channel the numerical algorithm is based on a lagrangian dual decomposition technique and it uses a modified iterative water filling approach for the gaussian multiple access channel the algorithm converges to the sum capacity globally and efficiently

regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity we prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form the equation is required to be strictly elliptic but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite d th moment where d is the dimension in the general stationary ergodic framework we show that the equation homogenizes to a deterministic uniformly elliptic equation and we obtain an explicit estimate of the effective ellipticity which is new even in the uniformly elliptic context showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory we prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity which are controlled in the macroscopic limit by the ergodic theorem we show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite p th moment for every p less than d but for which regularity and homogenization break down while these results are new even for linear equations we prove them in the fully nonlinear context in probabilistic terms our homogenization results correspond to quenched invariance principles for controlled diffusion processes in random media including linear diffusions as well as diffusions controlled by one controller or two competing players

maximizing sum rate and minimizing mse on multiuser downlink optimality fast algorithms and equivalence via max min sinr maximizing the minimum weighted signal to interference and noise ratio sinr minimizing the weighted sum mean square error mse and maximizing the weighted sum rate in a multiuser downlink system are three important performance objectives in nonconvex joint transceiver and power optimization where all the users have a total power constraint we show that through connections with the nonlinear perron frobenius theory jointly optimizing power and beamformers in the max min weighted sinr problem can be solved optimally in a distributed fashion then connecting these three performance objectives through the arithmetic geometric mean inequality and nonnegative matrix theory we solve the weighted sum mse minimization and the weighted sum rate maximization in the weak interference regimes using fast algorithms in the general case we first establish optimality conditions to the weighted sum mse minimization and the weighted sum rate maximization problems and provide their further connection to the max min weighted sinr problem we then propose a distributed weighted proportional sinr algorithm that leverages our fast max min weighted sinr algorithm to solve for local optimal solution of the two nonconvex problems and give conditions under which global optimality is achieved numerical results are provided to complement the analysis

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

optimized signaling for mimo interference systems with feedback the system mutual information of a multiple input multiple output mimo system with multiple users which mutually interfere is considered perfect channel state information is assumed to be known to both transmitters and receivers asymptotic performance analysis shows that the system mutual information changes behavior as the interference becomes sufficiently strong in particular beamforming is the optimum signaling for all users when the interference is large we propose several numerical approaches to decide the covariance matrices of the transmitted signals and compare their performance in terms of the system mutual information we model the system as a noncooperative game and perform iterative water filling to find the nash equilibrium distributively a centralized global approach and a distributed iterative approach based on the gradient projection method are also proposed numerical results show that all proposed approaches give better performance than the standard signaling which is optimum for the case without interference both the global and the iterative gradient projection methods are shown to outperform the nash equilibrium significantly

a robust maximin approach for mimo communications with imperfect channel state information based on convex optimization this paper considers a wireless communication system with multiple transmit and receive antennas i e a multiple input multiple output mimo channel the objective is to design the transmitter according to an imperfect channel estimate where the errors are explicitly taken into account to obtain a robust design under the maximin or worst case philosophy the robust transmission scheme is composed of an orthogonal space time block code ostbc whose outputs are transmitted through the eigenmodes of the channel estimate with an appropriate power allocation among them at the receiver the signal is detected assuming a perfect channel knowledge the optimization problem corresponding to the design of the power allocation among the estimated eigenmodes whose goal is the maximization of the signal to noise ratio snr is transformed to a simple convex problem that can be easily solved different sources of errors are considered in the channel estimate such as the gaussian noise from the estimation process and the errors from the quantization of the channel estimate among others for the case of gaussian noise the robust power allocation admits a closed form expression finally the benefits of the proposed design are evaluated and compared with the pure ostbc and nonrobust approaches

zero forcing methods for downlink spatial multiplexing in multiuser mimo channels the use of space division multiple access sdma in the downlink of a multiuser multiple input multiple output mimo wireless communications network can provide a substantial gain in system throughput the challenge in such multiuser systems is designing transmit vectors while considering the co channel interference of other users typical optimization problems of interest include the capacity problem maximizing the sum information rate subject to a power constraint or the power control problem minimizing transmitted power such that a certain quality of service metric for each user is met neither of these problems possess closed form solutions for the general multiuser mimo channel but the imposition of certain constraints can lead to closed form solutions this paper presents two such constrained solutions the first referred to as block diagonalization is a generalization of channel inversion when there are multiple antennas at each receiver it is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high snr the second known as successive optimization is an alternative method for solving the power minimization problem one user at a time and it yields superior results in some e g low snr situations both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined in order to accommodate more general scenarios we also propose a framework for coordinated transmitter receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas while the proposed algorithms are suboptimal they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity

performance analysis of mmse based amplify and forward spatial multiplexing mimo relaying systems in this paper we propose a general framework to quantify the average error probability of the minimum mean squared error based precoding schemes in amplify and forward relay networks where all nodes are equipped with multiple antennas especially we investigate spatial multiplexing schemes which transmit multiple data streams simultaneously due to difficulty in finding an exact expression of the average error rate we exploit the high signal to noise ratio snr based approach which allows a simple and accurate characterization of the performance then we derive new closed form expressions for bit error rate performance of both the optimal source relay joint precoding schemes and the optimal relay only precoding schemes in terms of a coding gain as well as a diversity gain taking a different pathloss in each hop into consideration we evaluate the performance in a generalized environment through our analysis we discuss several interesting observations and provide a helpful guideline for designing mmse based relaying systems monte carlo simulations show that our analytical work accurately predicts numerical results

on fuzzy metric spaces in the present paper the authors define f open sets f closed sets f adherent points f limit points f isolated points f isolated sets f derived sets f closures f interior points f interior f exterior points f exterior f everywhere dense sets f nowhere dense sets and make some characterizations of fuzzy metric spaces

sinr constrained beamforming for a mimo multi user downlink system algorithms and convergence analysis consider a multiple input multiple output mimo downlink multi user channel a well studied problem in such a system is the design of linear beamformers for power minimization with the quality of service qos constraints the most representative algorithms for solving this class of problems are the so called minimum mean square error mmse second order cone programming socp algorithm visotksy and madhow optimum beamforming using transmit antenna arrays proc ieee veh technol conf may 1999 vol 1 pp 851 856 wong zheng and ng convergence analysis of downlink mimo antenna system using second order cone programming proc 62nd ieee veh technol conf sep 2005 pp 492 496 and the uplink downlink duality udd algorithm codreanu tolli juntti and latva aho joint design of tx rx beamformers in mimo downlink channel ieee trans signal process vol 55 no 9 pp 4639 4655 sep 2007 the former is based on alternating optimization of the transmit and receive beamformers while the latter is based on the well known uplink dowlink duality theory despite their wide applicability the convergence to karush kuhn tucker kkt solutions of both algorithms is still open in the literature in this paper we rigorously establish the convergence of these algorithms for qos constrained power minimization qcpm problem with both single stream and multiple streams per user cases key to our analysis is the development and analysis of a new mmse dual algorithm which connects the mmse socp and the udd algorithm our numerical experiments show that 1 all these algorithms can almost always reach points with the same objective value irrespective of initialization and 2 the mmse socp mmse dual algorithm works well while the udd algorithm may fail with an infeasible initialization

sum capacity of gaussian vector broadcast channels this paper characterizes the sum capacity of a class of potentially nondegraded gaussian vector broadcast channels where a single transmitter with multiple transmit terminals sends independent information to multiple receivers coordination is allowed among the transmit terminals but not among the receive terminals the sum capacity is shown to be a saddle point of a gaussian mutual information game where a signal player chooses a transmit covariance matrix to maximize the mutual information and a fictitious noise player chooses a noise correlation to minimize the mutual information the sum capacity is achieved using a precoding strategy for gaussian channels with additive side information noncausally known at the transmitter the optimal precoding structure is shown to correspond to a decision feedback equalizer that decomposes the broadcast channel into a series of single user channels with interference pre subtracted at the transmitter

coordinated beamforming for energy efficient transmission in multicell multiuser systems in this paper we study energy efficient joint power allocation and beamforming for coordinated multicell multiuser downlink systems the considered optimization problem is in a non convex fractional form and hard to tackle we propose to first transform the original problem into an equivalent optimization problem in a parametric subtractive form by which we reach its solution through a two layer optimization scheme the outer layer only involves one dimension search for the energy efficiency parameter which can be addressed using the bi section search the key issue lies in the inner layer where a non fractional sub problem needs to tackle by exploiting the relationship between the user rate and the mean square error we then develop an iterative algorithm to solve it the convergence of this algorithm is proved and the solution is further derived in closed form our analysis also shows that the proposed algorithm can be implemented in parallel with reasonable complexity numerical results illustrate that our algorithm has a fast convergence and achieves near optimal energy efficiency it is also observed that at the low transmit power region our solution almost achieves the optimal sum rate and the optimal energy efficiency simultaneously while at the middle high transmit power region a certain sum rate loss is suffered in order to guarantee the energy efficiency

a closed form power allocation and signal alignment for a diagonalized mimo two way relay channel with linear receivers a novel channel diagonalization scheme for an amplify and forward multiple input multiple output mimo two way relay channel twrc is proposed using generalized singular value decomposition gsvd diagonalization of the mimo twrc is a sub optimal approach that achieves two main purposes reducing the computational complexity for optimizing the linear precoders at each node and reducing the detection complexity at the source nodes by separating the multiple data streams for the given diagonalized structure we first align the entries of the diagonalized channels using a permutation to maximize a lower bound of average achievable sum rate asr and a joint source relay power allocation is then performed to maximize the asr of the aligned twrc the overall problem is divided into two convex subproblems the solutions to which are provided in closed form our analysis for the proposed scheme underlines the benefits of acquiring channel state information simulation results demonstrate that the proposed gsvd based relaying scheme with the signal alignment and closed form power allocation significantly improves the asr while retaining the diagonalized channel structure in addition the proposed scheme achieves the same level of asr with much less computational complexity as compared to the iterative schemes

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

joint design of tx rx beamformers in mimo downlink channel we consider a single cell multiple input multiple output mimo downlink channel where linear transmission and reception strategy is employed the base station bs transmitter is equipped with a scheduler using a simple opportunistic beamforming strategy which associates an intended user for each of the transmitted data streams for the case when the channel of the scheduled users is available at the bs we propose a general method for joint design of the transmit and the receive beamformers according to different optimization criteria including weighted sum rate maximization weighted sum mean square error minimization minimum signal to interference plus noise ratio sinr maximization and sum power minimization under a minimum sinr constraint the proposed method can handle multiple antennas at the bs and at the mobile user with single and or multiple data streams per scheduled user the optimization problems encountered in the beamformer design e g covariance rank constraint are not convex in general therefore the problem of finding the global optimum is intrinsically nontractable however by exploiting the uplink downlink sinr duality we decompose the original optimization problem as a series of simpler optimization problems which can be efficiently solved by using standard convex optimization tools even though each subproblem is optimally solved there is no guarantee that the global optimum has been found due to the nonconvexity of the problem however the simulations show that the algorithms converge fast to a solution which can be a local optimum but is still efficient

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

low complexity iterative algorithm for finding the mimo ofdm broadcast channel sum capacity a novel low complexity and provably convergent algorithm is proposed to find the sum capacity for vector gaussian broadcast channels unlike the recently proposed sum power constraint iterative waterfilling spc iwf algorithms it has lower complexity and requires no additional precautions to ensure the convergence we have proved analytically the convergence with probability one and the computer simulations show that the proposed algorithm converges faster than the earlier variants of spc iwf algorithms we formulate the problem in the context of a multiple input multiple output orthogonal frequency division multiplexing system and discuss the simplifications provided by the block diagonal channel structure

transceiver optimization for block based multiple access through isi channels in this paper we describe a formulation of the minimum mean square error mmse joint transmitter receiver design problem for block based multiple access communication over intersymbol interference isi channels since the direct formulation of this problem turns out to be nonconvex we develop various alternative convex formulations using techniques of linear matrix inequalities lmis and second order cone programming socp in particular we show that the optimal mmse transceiver design problem can be reformulated as a semidefinite program sdp which can be solved using highly efficient interior point methods when the channel matrices are diagonal as in cyclic prefixed multicarrier systems we show that the optimal mmse transceivers can be obtained by subcarrier allocation and optimal power loading to each subcarrier for all the users moreover the optimal subcarrier allocation and power loading can be computed fairly simply in polynomial time by the relative ratios of the magnitudes of the subchannel gains corresponding to all subcarriers we also prove that any two users can share no more than one subcarrier in the optimal mmse transceivers by exploiting this property we design an efficient strongly polynomial time algorithm for the determination of optimal powerloading and subcarrier allocation in the two user case

linear multiuser mimo transceiver design with quality of service and per antenna power constraints joint design of linear multiuser multiple input multiple output mimo transceiver subject to different quality of service qos constraints per user and with per antenna or antenna group power constraints is considered solutions for two linear transceiver optimization problems are proposed i e balancing the weighted signal to interference plus noise ratio sinr per data stream and balancing the weighted rate for each scheduled user with minimum rate requirements per user the proposed joint transceiver optimization algorithms are compared to corresponding optimal nonlinear transmission methods as well as to generalized zero forcing transmission solutions unlike the optimal nonlinear schemes the optimization problems employed in the linear multiuser mimo transceiver design are not convex in general however the proposed algorithms are shown to provide very efficient solutions despite of the fact that the global optimum cannot be guaranteed due to nonconvexity of the problems

gradient based power minimization in mimo broadcast channels with linear precoding we study the problem of minimizing the sum transmit power in mutliple input multiple output mimo downlink channels with linear transceivers if per user quality of service qos constraints expressed in terms of rates have to be fulfilled to find a suboptimal solution of the arising non convex optimization problem we introduce new auxiliary variables representing the di vision of the per user rate constraints into per stream rate targets and we optimize these variables by means of gradient projection steps this new method is combined with alternating updates of the transmit and receive filters furthermore the proposed algorithm ensures that the mean square error mse matrices of all users are diagonal and in the course of the execution of the algorithm it is possible that inactive streams get activated if this leads to a decreased sum transmit power in numerical simulations the new algorithm turns out to be superior to the various existing methods since these methods either lead to a higher sum transmit power than the proposed scheme or have a higher computational complexity or make restrictive assumptions on the system parameters

applications of convex optimization in signal processing and digital communication in the last two decades the mathematical programming community has witnessed some spectacular advances in interior point methods and robust optimization these advances have recently started to significantly impact various fields of applied sciences and engineering where computational efficiency is essential this paper focuses on two such fields digital signal processing and communication in the past the widely used optimization methods in both fields had been the gradient descent or least squares methods both of which are known to suffer from the usual headaches of stepsize selection algorithm initialization and local minima with the recent advances in conic and robust optimization the opportunity is ripe to use the newly developed interior point optimization techniques and highly efficient software tools to help advance the fields of signal processing and digital communication this paper surveys recent successes of applying interior point and robust optimization to solve some core problems in these two fields the successful applications considered in this paper include adaptive filtering robust beamforming design and analysis of multi user communication system channel equalization decoding and detection throughout our emphasis is on how to exploit the hidden convexity convex reformulation of semi infinite constraints analysis of convergence complexity and performance as well as efficient practical implementation

optimization of mimo relays for multipoint to multipoint communications nonrobust and robust designs in this paper we propose algorithms to jointly optimize the multiple multiantenna relays which assist multipoint to multipoint communications in wireless networks assuming that the knowledge of the second order statistics of the channels such as covariance matrices are available the multiple input multiple output mimo relays are designed using two different methods 1 minimize the sum of the powers of the relays while fulfilling the signal to interference plus noise ratio sinr requirements for all destinations and 2 maximize the minimum of the sinrs of all destinations satisfying the transmit power constraint of each mimo relay furthermore considering the fact that the covariance matrices of the channels between the mimo relays and destinations are subject to uncertainty due to feedback and quantization errors the robust versions of the aforementioned methods based on worst case concept are proposed it is shown that the proposed nonrobust as well as robust designs are nonconvex optimization problems but they can be solved accurately and efficiently using the standard semidefinite relaxation and randomization techniques computer simulations verify the improved performance of the robust designs over nonrobust methods