interference alignment with finite extensions in partially connected networks in this paper we investigate the interference alignment ia for a class of partially connected multi input multi output heterogeneous networks which consist of l fully connected pico cells j partially connected pico cells and one macro cell in particular we investigate the feasibility of ia conditions with no channel extensions t independent channel extensions and t dependent channel extensions respectively for the single beam case we present three theorems to characterize some necessary conditions which l j t and the channel diversity order overline l must jointly satisfy when ia conditions are feasible these necessary conditions also implicitly impose the constraints on the upper bounds of achievable degrees of freedom with different channel extensions

interference alignment with asymmetric complex signaling settling the host madsen nosratinia conjecture it has been conjectured by ho madsen and nosratinia that complex gaussian interference channels with constant channel coefficients have only one degree of freedom regardless of the number of users while several examples are known of constant channels that achieve more than 1 degree of freedom these special cases only span a subset of measure zero in other words for almost all channel coefficient values it is not known if more than 1 degree of freedom is achievable in this paper we settle the host madsen nosratinia conjecture in the negative we show that at least 1 2 degrees of freedom are achievable for all values of complex channel coefficients except for a subset of measure zero for the class of linear beamforming and interference alignment schemes considered in this paper it is also shown that 1 2 is the maximum number of degrees of freedom achievable on the complex gaussian 3 user interference channel with constant channel coefficients for almost all values of channel coefficients to establish the achievability of 1 2 degrees of freedom we use the novel idea of asymmetric complex signaling i e the inputs are chosen to be complex but not circularly symmetric it is shown that unlike gaussian point to point multiple access and broadcast channels where circularly symmetric complex gaussian inputs are optimal for interference channels optimal inputs are in general asymmetric with asymmetric complex signaling we also show that the 2 user complex gaussian x channel with constant channel coefficients achieves the outer bound of 4 3 degrees of freedom i e the assumption of time variations frequency selectivity used in prior work to establish the same result is not needed

generalized interference alignment part i theoretical framework interference alignment ia has attracted enormous research interest as it achieves optimal capacity scaling with respect to signal to noise ratio on interference networks ia has also recently emerged as an effective tool in engineering interference for secrecy protection on wireless wiretap networks however despite the numerous works dedicated to ia two of its fundamental issues i e feasibility conditions and transceiver design are not completely addressed in the literature in this two part paper a generalized interference alignment gia technique is proposed to enhance the ia s capability in secrecy protection a theoretical framework is established to analyze the two fundamental issues of gia in part i and then the performance of gia in large scale stochastic networks is characterized to illustrate how gia benefits secrecy protection in part ii the theoretical framework for gia adopts methodologies from algebraic geometry determines the necessary and sufficient feasibility conditions of gia and generates a set of algorithms for solving the gia problem this framework sets up a foundation for the development and implementation of gia

generalized l sub 2 synthesis introduces a general framework for control design which can be used to solve several open problems in robust and optimal control these include systems subject to independently norm bounded disturbances robust stability for systems subject to element by element bounded structured uncertainty certain classes of robust performance problems and systems subject to deterministic white noise disturbances this is achieved by combining concepts such as h infinity optimization linear matrix inequalities and integral quadratic constraints necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

rate region boundary of the siso z interference channel with improper signaling this paper provides a complete characterization of the boundary of an achievable rate region called the pareto boundary of the single antenna z interference channel z ic when interference is treated as noise and users transmit complex gaussian signals that are allowed to be improper by considering the augmented complex formulation we derive a necessary and sufficient condition for improper signaling to be optimal this condition is stated as a threshold on the interference channel coefficient which is a function of the interfered user rate and which allows insightful interpretations into the behavior of the achievable rates in terms of the circularity coefficient i e degree of impropriety furthermore the optimal circularity coefficient is provided in closed form the simplicity of the obtained characterization permits interesting insights into when and how improper signaling outperforms proper signaling in the single antenna z ic we also provide an in depth discussion on the optimal strategies and the properties of the pareto boundary

minimization of csi feedback dimension for interference alignment in mimo interference multicast networks it is well known that interference alignment ia can achieve substantial theoretical gains in multiple input multiple output mimo networks however the conventional works usually assume the full channel state information csi is available at the transmitter side which would create a overwhelming csi feedback overhead for practical frequency division duplexing fdd wireless systems to implement ia in practice it is highly desirable to reduce the amount of required csi feedback overhead in this paper we consider ia in mimo interference multicast networks under partial csi feedback and we attempt to minimize the csi feedback cost subject to ia feasibility constraints with a given degree of freedom dof requirements first we propose a csi feedback profile to embrace two important csi feedback reduction strategies and we use the metric of csi feedback dimension to quantify the associated csi feedback cost we then formulate the ia conditions under partial csi feedback in mimo interference multicast networks and we derive new ia feasibility conditions under the proposed partial csi feedback framework based on these results we consider the csi feedback dimension minimization subject to the ia feasibility constraints with a given dof requirements in mimo interference multicast networks which is formulated as a combinatorial optimization problem based on the specific problem structure we derive an asymptotically optimal solution for a category of network topologies and we further obtain closed form tradeoff results between dofs and the csi feedback cost for mimo interference multicast networks

necessary and sufficient conditions for optimal offers in electricity markets in this paper we consider the optimal policy for a generator offering power into a wholesale electricity market operating under a pool arrangement anderson and philpott math oper res 27 2002 pp 82 100 recently discussed necessary conditions for an optimal offer curve when there is uncertainty in the demand and in the behavior of other participants in the market they show that the objective function in these circumstances can be expressed as a line integral along the offer curve of a profit function integrated with respect to a market distribution function in this paper we prove the existence of an optimal offer stack and we extend the analysis of math oper res 27 2002 pp 82 100 to include necessary conditions of a higher order in the presence of horizontal and or vertical sections in an offer curve finally we establish sufficient conditions for an offer curve to be locally optimal

feasibility of interference alignment for the mimo interference channel we study vector space interference alignment for the multiple input multiple output interference channel with no time or frequency diversity and no symbol extensions we prove both necessary and sufficient conditions for alignment in particular we characterize the feasibility of alignment for the symmetric three user channel where all users transmit along d dimensions all transmitters have m antennas and all receivers have n antennas as well as feasibility of alignment for the fully symmetric m n channel with an arbitrary number of users an implication of our results is that the total degrees of freedom available in a k user interference channel using only spatial diversity from the multiple antennas is at most 2 this is in sharp contrast to the k 2 degrees of freedom shown to be possible by cadambe and jafar with arbitrarily large time or frequency diversity moving beyond the question of feasibility we additionally discuss computation of the number of solutions using schubert calculus in cases where there are a finite number of solutions

interference alignment with partial csi feedback in mimo cellular networks interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks however most existing ia designs require full channel state information csi at the transmitters which could lead to significant csi signaling overhead there are two techniques namely csi quantization and csi feedback filtering that reduce csi feedback overhead in this paper we consider ia processing with csi feedback filtering in mimo cellular networks we introduce a novel metric namely the feedback dimension to quantify the first order csi feedback cost associated with the csi feedback filtering the csi feedback filtering poses several important challenges in ia processing first there is a hidden partial csi knowledge constraint in ia precoder design which cannot be handled using conventional ia design methodology furthermore existing results on the feasibility conditions of ia cannot be applied due to the partial csi knowledge finally it is very challenging to find out how much csi feedback is actually needed to support ia processing we shall address the above challenges and propose a new ia feasibility condition under partial csit knowledge in mimo cellular networks based on this we consider the csi feedback dimension minimization subject to the degrees of freedom requirements and further propose an asymptotically optimal solution and derive closed form trade off results between the csi feedback cost and ia performance in mimo cellular networks

communication over mimo x channels interference alignment decomposition and performance analysis in a multiple antenna system with two transmitters and two receivers a scenario of data communication known as the x channel is studied in which each receiver receives data from both transmitters in this scenario it is assumed that each transmitter is unaware of the other transmitter s data noncooperative scenario this system can be considered as a combination of two broadcast channels from the transmitters points of view and two multiple access channels from the receivers points of view taking advantage of both perspectives two signaling schemes for such a scenario are developed in these schemes some linear filters are employed at the transmitters and at the receivers which decompose the system into either two noninterfering multiple antenna broadcast subchannels or two noninterfering multiple antenna multiple access subchannels the main objective in the design of the filters is to exploit the structure of the channel matrices to achieve the highest multiplexing gain mg it is shown that the proposed noncooperative signaling schemes outperform other known noncooperative schemes in terms of the achievable mg in particular it is shown that in some specific cases the achieved mg is the same as the mg of the system if full cooperation is provided either between the transmitters or between the receivers in the second part of the paper it is shown that by using mixed design schemes rather than decomposition schemes and taking the statistical properties of the interference terms into account the power offset of the system can be improved the power offset represents the horizontal shift in the curve of the sum rate versus the total power in decibels

degrees of freedom of mimo cellular networks decomposition and linear beamforming design this paper investigates the symmetric degrees of freedom dof of multiple input multiple output mimo cellular networks with g cells and k users per cell having n antennas at each base station and m antennas at each user in particular we investigate techniques for achievability that are based on either decomposition with asymptotic interference alignment or linear beamforming schemes and show that there are distinct regimes of g k m n where one outperforms the other we first note that both one sided and two sided decomposition with asymptotic interference alignment achieve the same dof we then establish specific antenna configurations under which the dof achieved using decomposition based schemes is optimal by deriving a set of outer bounds on the symmetric dof using these results we completely characterize the optimal dof of any g cell network with single antenna users for linear beamforming schemes we first focus on small networks and propose a structured approach to linear beamforming based on a notion called packing ratios packing ratio describes the interference footprint or shadow cast by a set of transmit beamformers and enables us to identify the underlying structures for aligning interference such a structured beamforming design can be shown to achieve the optimal spatially normalized dof sdof of two cell two user cell network and the two cell three user cell network for larger networks we develop an unstructured approach to linear interference alignment where transmit beamformers are designed to satisfy conditions for interference alignment without explicitly identifying the underlying structures for interference alignment the main numerical insight of this paper is that such an approach appears to be capable of achieving the optimal sdof for mimo cellular networks in regimes where linear beamforming dominates asymptotic decomposition and a significant portion of sdof elsewhere remarkably polynomial identity test appears to play a key role in identifying the boundary of the achievable sdof region in the former case

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

robust controller synthesis for uncertain time varying systems in this paper we develop and present new convex synthesis conditions for robust performance in linear time varying systems subject to time varying perturbations in particular these results are exact for sensitivity minimization in the presence of multiplicative perturbations the methods apply to a number of additional robust performance problems and are always both necessary and sufficient when the system plant is periodic otherwise the conditions provided are always sufficient and a controller can be directly constructed when this condition holds

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

optimal offer construction in electricity markets in this paper we study strategies for generators making offers into electricity markets in circumstances where both the demand for electricity and the behaviour of competing generators is unknown but can be represented by a probability distribution given this probability distribution we derive necessary optimality conditions for a broad class of supply offer curves we show how these conditions can be used to construct an optimal solution for a simple example we also consider the case in which a generator is restricted in the number of prices at which power can be offered

a new approach for analysis and synthesis of time varying systems in this paper new techniques are developed for the analysis of linear time varying ltv systems these lead to a formally simple treatment of robust control problems for ltv systems allowing methods more usually restricted to time invariant systems to be employed in the time varying case as an illustration of this methodology the so called h sub spl infin synthesis problem is solved for ltv systems

subspace alignment chains and the degrees of freedom of the three user mimo interference channel we show that the three user mt mr multiple input multiple output mimo interference channel where each transmitter is equipped with mt antennas and each receiver is equipped with mr antennas has m d m n min m 2 1 n 2 1 degrees of freedom dof normalized by time frequency and space dimensions where min mt mr n max mt mr m n m while the dof outer bound of d m n is established for every mt mr value the achievability of d m n dof is established in general subject to a normalization with respect to spatial extensions i e the scaling of the number of antennas at all nodes in particular we show that qd m n dof are achievable for the three user qmt qmr mimo interference channel for some positive integer q which may be seen as a spatial extension factor q is the scaling factor needed to make the value qd m n an integer given spatial extensions the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency in the absence of spatial extensions it is shown through examples how essentially the same interference alignment scheme may be applied over time extensions over either constant or time varying channels the central new insight to emerge from this paper is the notion of subspace alignment chains as the dof bottlenecks the subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the dof outer bound as well as in the construction of the optimal interference alignment schemes the dof value d m n is a piecewise linear function of m n with either m or n being the bottleneck within each linear segment whereas the other value contains some redundancy i e it can be reduced without reducing the dof the corner points of these piecewise linear segments correspond to two sets a 1 2 2 3 3 4 and b 1 3 3 5 5 7 the set a contains all those values of m n and only those values of m n for which there is redundancy in both m and n i e either can be reduced without reducing the dof the set b contains all those values of m n and only those values of m n for which there is no redundancy in either m or n i e neither can be reduced without reducing the dof because a and b represent settings with maximum and minimum redundancy essentially they are the basis for the dof outer bounds and inner bounds respectively our results settle the question of feasibility of linear interference alignment introduced previously by cenk et al for the three user mt mr mimo interference channel completely for all values of mt mr in particular we show that the linear interference alignment problem mt mr d 3 as defined in previous paper by cenk et al is feasible if and only if d d m n with the exception of the values m n b and only with that exception we show that for every m n value there are proper systems as defined by cenk et al that are not feasible evidently the redundancy contained in all other values of m n manifests itself as superfluous variables that are not discounted in the definition of proper systems thus creating a discrepancy between proper and feasible systems our results show that m n a are the only values for which there is no dof benefit of joint processing among co located antennas at the transmitters or receivers this may also be seen as a consequence of the maximum redundancy in the m n a settings

an iteratively weighted mmse approach to distributed sum utility maximization for a mimo interfering broadcast channel consider the multiple input multiple output mimo interfering broadcast channel whereby multiple base stations in a cellular network simultaneously transmit signals to a group of users in their own cells while causing interference to each other the basic problem is to design linear beamformers that can maximize the system throughput in this paper we propose a linear transceiver design algorithm for weighted sum rate maximization that is based on iterative minimization of weighted mean square error mse the proposed algorithm only needs local channel knowledge and converges to a stationary point of the weighted sum rate maximization problem furthermore the algorithm and its convergence can be extended to a general class of sum utility maximization problem the effectiveness of the proposed algorithm is validated by numerical experiments

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

ergodic interference alignment this paper develops a new communication strategy ergodic interference alignment for the k user interference channel with time varying fading at any particular time each receiver will see a superposition of the transmitted signals plus noise the standard approach to such a scenario results in each transmitter receiver pair achieving a rate proportional to 1 k its interference free ergodic capacity however given two well chosen time indices the channel coefficients from interfering users can be made to exactly cancel by adding up these two observations each receiver can obtain its desired signal without any interference if the channel gains have independent uniform phases this technique allows each user to achieve at least 1 2 its interference free ergodic capacity at any signal to noise ratio prior interference alignment techniques were only able to attain this performance as the signal to noise ratio tended to infinity extensions are given for the case where each receiver wants a message from more than one transmitter as well as the x channel case with two receivers where each transmitter has an independent message for each receiver finally it is shown how to generalize this strategy beyond gaussian channel models for a class of finite field interference channels this approach yields the ergodic capacity region

the approximate capacity of the many to one and one to many gaussian interference channels recently etkin tse and wang found the capacity region of the two user gaussian interference channel to within 1 bit s hz a natural goal is to apply this approach to the gaussian interference channel with an arbitrary number of users we make progress towards this goal by finding the capacity region of the many to one and one to many gaussian interference channels to within a constant number of bits the result makes use of a deterministic model to provide insight into the gaussian channel the deterministic model makes explicit the dimension of signal level a central theme emerges the use of lattice codes for alignment of interfering signals on the signal level

interference alignment with asymmetric complex signaling on mimo x channels it is known that with conventional spatial interference alignment i e linear interference alignment without symbol extension the achievable degrees of freedom dof of mimo x channels is no more than lfloor d outer rfloor where d outer denotes the outer bound of dof of mimo x channels in this paper we propose a spatial interference alignment and zero forcing framework in combination with asymmetric complex signaling which can achieve the dof of 1 2 lfloor 2d outer rfloor for arbitrary number of antennas on each terminal note that 1 2 lfloor 2d outer rfloor is equal to or larger than lfloor d outer rfloor the result shows that the technique of asymmetric complex signaling which was originally proposed for single antenna systems can be useful for mimo channels in particular when symbol extensions are not allowed

linear interference alignment in full duplex mimo networks with imperfect csi in this paper we consider a system where full duplex fd base stations bss communicate with half duplex hd downlink dl and uplink ul users in a multi user multi cell network where all nodes are equipped with multiple antennas the introduction of fd bss offers potential to increase spectral efficiency however it also causes a surge in the number of interference links compared with the hd network counterpart here we apply linear interference alignment ia to manage interference in this network under imperfect channel state information csi first we characterize the performance losses incurred with respect to the achievable sum rate and degrees of freedom dofs results show that the general trend in performance loss is mainly determined by how the error scales with the signal to noise ratio snr in particular full ul and dl dof can be achieved even under imperfect csi when the channel error is at least inversely proportional to snr moreover in such cases the sum rate loss is always finite and either goes to zero or is upper bounded by a derived value second we design two linear ia algorithms applicable to the system under consideration these are based on minimizing the mean square error mmse and maximizing the signal to interference plus noise ratio and consider statistical knowledge of the csi error for added robustness the proposed algorithms follow specific design principles that distribute the different interference components amongst the various beamformers and result in unitary receivers and precoders in addition we show that under certain conditions both designs result in identical beamforming solutions even though the mmse algorithm has lower computational complexity finally we also derive the proper condition for ia feasibility in the multi cell system under consideration

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

finding supply function equilibria with asymmetric firms firms compete in supply functions when they offer a schedule of prices and quantities into a market for example this occurs in many wholesale electricity markets we study the equilibrium behaviour when firms differ both with regard to their costs and their capacities we characterize strong equilibrium solutions in which given the other players supply functions optimal profits are achieved for every demand realisation if the demand can be low enough for it to be met economically with supply from just one firm then the supply function equilibria are ordered in a natural way we consider equilibria in which for the highest levels of demand all but one of the firms have reached their capacity limit we show that there can be at most one supply function equilibrium with this property we also propose a new numerical method to find asymmetric supply function equilibria using piecewise linear approximations and a discretization of the demand distribution we show that this approach has good theoretical convergence behaviour finally we present numerical results from an implementation using gams to demonstrate that the approach is effective in practice

degrees of freedom for the mimo interference channel in this correspondence we show that the exact number of spatial degrees of freedom dof for a two user nondegenerate full rank channel matrices multiple input multiple output mimo gaussian interference channel with m1 m2 antennas at transmitters 1 2 and n1 n2 antennas at the corresponding receivers and perfect channel knowledge at all transmitters and receivers is min m1 m2 n1 m2 max m1 n2 max m2 n1 a constructive achievability proof shows that zero forcing is sufficient to achieve all the available dof on the two user mimo interference channel we also show through an example of a share and transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available dof are not increased

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

tangent operators sensitivity expressions and optimal design of non linear elastica in contact with applications to beams the tangent operator and design sensitivity expressions for non linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method the sensitivity computations are then combined with a numerical programming package to create an optimal design environment to exemplify the optimal design environment a beam s contact surface is contoured to minimize bending stress this work combines the research on the sensitivity analysis of non linear elastic bodies subject to constraints finite strain non linear elastic beam analysis and contact analyses the analysis is valid for any smooth contact surface and specialized for the case in which the surface is represented by a cubic spline the direct differentiation method is utilized to perform the sensitivity analysis in an example problem the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve

on the number of interference alignment solutions for the k user mimo channel with constant coefficients in this paper we study the number of different interference alignment ia solutions in a k user multiple input multiple output mimo interference channel when the alignment is performed via beamforming and no symbol extensions are allowed we focus on the case where the number of ia equations matches the number of variables in this situation the number of ia solutions is finite and constant for any channel realization out of a zero measure set and as we prove in this paper it is given by an integral formula that can be numerically approximated using monte carlo integration methods more precisely the number of alignment solutions is the scaled average of the determinant of a certain hermitian matrix related to the geometry of the problem interestingly while the value of this determinant at an arbitrary point can be used to check the feasibility of the ia problem its average properly scaled gives the number of solutions for single beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented nonetheless our results can be applied to arbitrary interference mimo networks with any number of users antennas and streams per user

on the existence of supply function equilibria supply function equilibria are used in the analysis of divisible good auctions with a large number of identical objects to be sold or bought an important example occurs in wholesale electricity markets despite the substantial literature on supply function equilibria the existence of a pure strategy nash equilibria for a uniform price auction in asymmetric cases has not been established in a general setting in this paper we prove the existence of a supply function equilibrium for a duopoly with asymmetric firms having convex non decreasing marginal costs with decreasing concave demand subject to an additive demand shock provided that the second derivative of the demand function is small enough and not increasing the proof is constructive and also gives insight into the structure of the equilibrium solutions

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

degrees of freedom of mimo x networks spatial scale invariance and one sided decomposability we show that an m n user mimo x network with a antennas at each node has a mn m n 1 degrees of freedom dof thus resolving in this case a discrepancy between the spatial scale invariance conjecture scaling the number of antennas at each node by a constant factor will scale the total dof by the same factor and a decomposability property of overconstrained wireless networks while the best previously known general dof outer bound is consistent with the spatial invariance conjecture the best previously known general dof inner bound inspired by the k user mimo interference channel was based on the decomposition of every transmitter and receiver into multiple single antenna nodes transforming the network into an am an user siso x network while such a decomposition is dof optimal for the k user mimo interference channel a gap remained between the best inner and outer bounds for the mimo x channel here we close this gap with the new insight that the mimo x network is only one sided decomposable i e either all the transmitters or all the receivers but not both can be decomposed by splitting multiple antenna nodes into multiple single antenna nodes without loss of dof the result is extended to simo and miso x networks as well and in each case the dof results satisfy the spatial scale invariance property

on the feasibility of linear interference alignment for mimo interference broadcast channels with constant coefficients in this paper we analyze the feasibility of linear interference alignment ia for multi input multi output mimo interference broadcast channel mimo ibc with constant coefficients we pose and prove the necessary conditions of linear ia feasibility for general mimo ibc except for the proper condition we find another necessary condition to ensure a kind of irreducible interference to be eliminated we then prove the necessary and sufficient conditions for a special class of mimo ibc where the numbers of antennas are divisible by the number of data streams per user since finding an invertible jacobian matrix is crucial for the sufficiency proof we first analyze the impact of sparse structure and repeated structure of the jacobian matrix considering that for the mimo ibc the sub matrices of the jacobian matrix corresponding to the transmit and receive matrices have different repeated structure we find an invertible jacobian matrix by constructing the two sub matrices separately we show that for the mimo ibc where each user has one desired data stream a proper system is feasible for symmetric mimo ibc we provide proper but infeasible region of antenna configurations by analyzing the difference between the necessary conditions and the sufficient conditions of linear ia feasibility

on the degrees of freedom achievable through interference alignment in a mimo interference channel consider a k user flat fading mimo interference channel where the kth transmitter or receiver is equipped with mk respectively nk antennas if an exponential in k number of generic channel extensions are used either across time or frequency cadambe and jafar 1 showed that the total achievable degrees of freedom dof can be maximized via interference alignment resulting in a total dof that grows linearly with a even if mk and nk are bounded in this work we consider the case where no channel extension is allowed and establish a general condition that must be satisfied by any degrees of freedom tuple d1 d2 dk achievable through linear interference alignment for a symmetric system with mk m nk n dk d for all k this condition implies that the total achievable dof cannot grow linearly with k and is in fact no more than k m n k 1 we also show that this bound is tight when the number of antennas at each transceiver is divisible by d the number of data streams per user

a feasibility test for linear interference alignment in mimo channels with constant coefficients in this paper we consider the feasibility of linear interference alignment ia for multiple input multiple output mimo channels with constant coefficients for any number of users antennas and streams per user and propose a polynomial time test for this problem combining algebraic geometry techniques with differential topology ones we first prove a result that generalizes those previously published on this topic in particular we consider the input set complex projective space of mimo interference channels the output set precoder and decoder grassmannians and the solution set channels decoders and precoders satisfying the ia polynomial equations not only as algebraic sets but also as smooth compact manifolds using this mathematical framework we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective if that mapping is not surjective then the solution variety projects into the input space in a singular way and the projection is a zero measure set this result naturally yields a simple feasibility test which amounts to checking the rank of a matrix we also provide an exact arithmetic version of the test which proves that testing the feasibility of ia for generic mimo channels belongs to the bounded error probabilistic polynomial complexity class

csi feedback reduction for mimo interference alignment interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks most of the existing ia designs require full channel state information csi at the transmitters which induces a huge csi signaling cost hence it is desirable to improve the feedback efficiency for ia and in this paper we propose a novel ia scheme with a significantly reduced csi feedback to quantify the csi feedback cost we introduce a novel metric namely the feedback dimension this metric serves as a first order measurement of csi feedback overhead due to the partial csi feedback constraint conventional ia schemes can not be applied and hence we develop a novel ia precoder decorrelator design and establish new ia feasibility conditions via dynamic feedback profile design the proposed ia scheme can also achieve a flexible tradeoff between the degree of freedom dof requirements for data streams the antenna resources and the csi feedback cost we show by analysis and simulations that the proposed scheme achieves substantial reductions of csi feedback overhead under the same dof requirement in mimo interference networks

on the achievable degrees of freedom of two cell multiuser mimo interference networks in this paper we study the sum degrees of freedom dof of an uplink two cell multiuser mimo interference network with asymmetric number of users in the cells the achievable dof is devised based on a two dimensional space time spreading code framework with linear precoding decoding design and finite channel extension the derivation of the achievable dof is shown related to a rank minimization problem which corresponds to the minimization of the dimension of the interference subspace the problem is solved by the proposed grouping algorithm ga based on aligning interfering signals into a low dimensional subspace as a group and attaining the minimum number of groups the achievable sum dof derived based on the proposed ga is shown to be greater than prior arts and achieves the theoretic upper bound in several cases we also give a closed form expression of the max imum achievable sum dof when there is the maximum number of admissible users in the considered finite diversity environment

isotone equilibrium in games of incomplete information an isotone pure strategy equilibrium exists in any game of incomplete information in which each player s action set is a finite sublattice of multidimensional euclidean space types are multidimensional and atomless and each player s interim expected payoff function satisfies two nonprimitive conditions whenever others adopt isotone pure strategies i single crossing in own action and type and ii quasi supermodularity in own action conditions i ii are satisfied in supermodular and log supermodular games given affiliated types and in games with independent types in which each player s ex post payoff satisfies supermodularity in own action and nondecreasing differences in own action and type this result is applied to provide the first proof of pure strategy equilibrium existence in the uniform price auction when bidders have multi unit demand nonprivate values and independent types

robust linear time varying control for trajectory tracking computation and an experimental application this paper provides computational and experimental verification of the use of recently developed linear time varying ltv robust control synthesis methods for controlling systems along prespecified trajectories the linearization of system dynamics along trajectories yield ltv systems and we compute and design robust controllers with respect to such linearizations the techniques employed require the computation of potentially large linear matrix inequalities and complexity reduction using new model reduction for ltv systems is also demonstrated