the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

new exact solutions of the broer kaup equations in 2 1 dimensional spaces abstract with the aid of maple several new kinds of exact solutions for the broer kaup equations in 2 1 dimensional spaces are obtained by using a new ansatz this approach can also be applied to other nonlinear evolution equations

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

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

transitions and kinematics of reaction convection fronts in a cell population model abstract here we consider cell population dynamics in which there is a simultaneous proliferation and maturation the mathematical model of this process is formulated as a nonlinear first order partial differential equation for the cell density u t x in which there is retardation delay in the temporal t variable thus we consider a transient reaction convection equation in which the cell density is convected with maturation velocity r for localizedd initial perturbations the equation has positive and negative traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution and negative fronts correspond to the reversed case three classes of fronts are found according to the strength of the convection velocity i for strong convection r 1 the fronts are simple translations of the initial data regardless of delay strength ii for weak convection r 1 two types of fronts exist a reaction convection fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state and b convection fronts arise if the localized initial perturbation acts at zero maturation on a finite amplitude state for weak convection cases ii a and ii b a further classification arises according to whether the magnitude of the temporal delay is larger or smaller than a critical value b it is found that the critical delay b corresponds to the hopf bifircation of the reaction equation that is obtained in the absence of convection r 0 for delays larger than b the convective and reaction convection fronts are oscillatory in addition all the reaction convection fronts reverse their direction of motion undergo a positive to negative transition and display non uniform kinematics simulation results are validated and interpreted using solutions for the unretarded equations and by local stability analysis

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

adaptive quantization on a grassmann manifold for limited feedback beamforming systems in this paper we examine delay limited adaptive quantization on the grassmann manifold of 1 dimensional subspaces in n dimensional space due to strict delay limits vector quantization over multiple time instants cannot be applied to exploit the temporal correlation of the source signal instead a vector predictive quantizer is proposed that combines prediction and differential quantization algorithms to achieve an efficient quantization of the correlated grassmannian source the proposed predictor is based on adaptive finite impulse response filters to adjust to the temporal statistics of the source signal it is shown that the prediction error in the tangent space associated with the grassmann manifold behaves approximately gaussian provided its norm is sufficiently small the proposed quantization algorithm is applied to channel state information quantization in multi user beamforming wireless communication systems large throughput gains are demonstrated in comparison to memoryless quantization due to reduced multi user interference

mimo systems with quantized covariance feedback a unified approach is developed for the study of multi input multi output mimo systems under fast fading with quantized covariance feedback in such systems the receiver computes using perfect channel state information csi the covariance matrices to be adopted by the transmitter corresponding to the current channel realization and feeds back a quantized version of this information to the transmitter using finite say nf bits per channel realization our analysis is based on a geometric framework we developed in a companion paper for manifolds of positive semi definite covariance matrices with various trace and rank constraints that analysis applies to mimo systems in a unified manner to optimal as well as various suboptimal precoding methods for obtaining input covariances these include for example the capacity achieving spatio temporal water filling strategy the incorporation of just spatial water filling and reduced rank beamforming with power allocation in both cases as well as mimo systems with antenna selection for a given system strategy including the one that achieves capacity the gap between the communication rates in the perfect and quantized covariance cases is shown to be o 2 nf n where n is the dimension of the particular pn manifold used for quantization this dimension depends on the precoding strategy adopted and strongly determines how quickly the communication rate with quantized covariance feedback approaches that with perfect csi at the transmitter csit our results show how the choice of covariance quantization manifold can be tailored to the available feedback rate in mimo system design

linear multiuser receivers in random environments we study the signal to interference sir performance of linear multiuser receivers in random environments where signals from the users arrive in random directions such a random environment may arise in a ds cdma system with random signature sequences or in a system with antenna diversity where the randomness is due to channel fading assuming that such random directions can be tracked by the receiver the resulting sir performance is a function of the directions and therefore also random we study the asymptotic distribution of this random performance in the regime where both the number of users k and the number of degrees of freedom n in the system are large but keeping their ratio fixed our results show that for both the decorrelator and the minimum mean square error mmse receiver the variance of the sir distribution decreases like 1 n and the sir distribution is asymptotically gaussian we compute closed form expressions for the asymptotic means and variances for both receivers simulation results are presented to verify the accuracy of the asymptotic results for finite sized systems

game theoretic opportunistic spectrum sharing strategy selection for cognitive mimo multiple access channels this paper studies opportunistic spectrum sharing strategy i e quantized precoding strategy selection for cognitive multiple input multiple output multiple access channels with limited feedback under interference power constraint and maximum transmission stream number constraint we put forward a game theoretic framework to model the precoding strategy selection behaviors of the secondary users under the specified constraints first we prove the formulated discrete game is a potential game which possesses at least one feasible pure strategy nash equilibrium the feasibility and optimality of the nash equilibrium are also analyzed then we prove that the solution to the sum rate maximization problem constitutes a feasible pure strategy nash equilibrium of our formulated game furthermore we design two algorithms the iterative precoding strategy selection algorithm based on the best response rule is designed to attain a feasible nash equilibrium the modified algorithm is designed to improve the sum rate performance simulation results show that our designed algorithms can achieve optimal or near optimal sum rate performance with low complexity

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

sinh gordon equation painleve property and backlund transformation abstract the painleve property for partial differential equation proposed by weiss et al is studied for the sinh gordon equation we performed a singular point analysis to show that the painleve property is a valuable analytical test for integrability and we found in a remarkably straightforward manner the backlund transformation of the sinh gordon equation

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

selection and rate adaptation schemes for mimo multiple access channels with low rate channel feedback in this paper two selection schemes are proposed for coded transmission over multiple input multiple output mimo multiple access channels mac to yield a much higher diversity multiplexing gain tradeoff dmt performance these schemes require a channel feedback but at an extremely low rate the first scheme is based on user selection and can be easily implemented in the existing mimo mac systems upper bounds on the minimal computational complexity required by sphere decoders to decode dmt optimal codes for this scheme as well as for mimo mac without feedback are given it is shown that this scheme can offer both a much larger dmt and an exponential reduction on decoding complexity compared with the latter the second scheme selects jointly the users and their transmit antennas it requires an additional design of rate assignments for performance optimization a very general framework on the design of optimal rate assignments is thus provided it is shown that this scheme can yield dmt performances far superior to the optimal mimo mac dmt without channel feedback the simulation results confirm that in some cases this scheme can provide an astonishing snr gain of 14 64 db at outage probability 10 6 compared with the optimal mimo mac coding schemes without feedback

qam and psk codebooks for limited feedback mimo beamforming this paper considers the problem of beamforming in multiple input multiple output mimo wireless systems assuming perfect channel state information at the receiver the choice of the beamforming vector is made possible through a noiseless limited rate feedback of one or more bits per coefficient to the transmitter this paper proposes the use of beamforming codebooks based on quadrature amplitude modulation qam and phase shift keying psk constellations which essentially eliminates the need for storage of the codebook we show that such codebooks perform arbitrarily close to the perfect feedback case as the constellation size increases and that full diversity order is achieved we demonstrate an equivalence between the beamforming codebook search problem with that of noncoherent sequence detection based on this we propose fast beamforming vector search algorithms monte carlo simulations are presented to show that the performance is comparable to the best known codebooks and that the search complexity can be reduced by several orders of magnitude

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

noncoherent trellis coded quantization a practical limited feedback technique for massive mimo systems accurate channel state information csi is essential for attaining beamforming gains in single user su multiple input multiple output mimo and multiplexing gains in multi user mu mimo wireless communication systems state of the art limited feedback schemes which rely on pre defined codebooks for channel quantization are only appropriate for a small number of transmit antennas and low feedback overhead in order to scale informed transmitter schemes to emerging massive mimo systems with a large number of transmit antennas at the base station one common approach is to employ time division duplexing tdd and to exploit the implicit feedback obtained from channel reciprocity however most existing cellular deployments are based on frequency division duplexing fdd hence it is of great interest to explore backwards compatible massive mimo upgrades of such systems for a fixed feedback rate per antenna the number of codewords for quantizing the channel grows exponentially with the number of antennas hence generating feedback based on look up from a standard vector quantized codebook does not scale in this paper we propose noncoherent trellis coded quantization ntcq whose encoding complexity scales linearly with the number of antennas the approach exploits the duality between source encoding in a grassmannian manifold for finding a vector in the codebook which maximizes beamforming gain and noncoherent sequence detection for maximum likelihood decoding subject to uncertainty in the channel gain furthermore since noncoherent detection can be realized near optimally using a bank of coherent detectors we obtain a low complexity implementation of ntcq encoding using an off the shelf viterbi algorithm applied to standard trellis coded quantization we also develop advanced ntcq schemes which utilize various channel properties such as temporal spatial correlations monte carlo simulation results show the proposed ntcq and its extensions can achieve near optimal performance with moderate complexity and feedback overhead

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

a nonlinear age and maturity structured model of population dynamics ii chaos instability hypercyclicity and chaos are investigated in a nonlinear model of age and maturity structured population dynamics it is demonstrated that the behavior of solutions depends on the viewpoint of the observer viewed in the direction of age structure the population stabilizes to a regular distribution viewed in the direction of maturity structure there is the possibility of chaotic behavior

information capacity of a random signature multiple input multiple output channel a closed form expression for information capacity of the random signature multiple input multiple output channel is given a direct calculation of the capacity is provided additional to the proof involving the capacity of minimum mean square error systems provided by verdu and shamai see ieee trans inform theory vol 45 p 622 40 1999

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

signature optimization for cdma with limited feedback we study the performance of joint signature receiver optimization for direct sequence code division multiple access ds cdma with limited feedback the receiver for a particular user selects the signature from a signature codebook and relays the corresponding b index bits to the transmitter over a noiseless channel we study the performance of a random vector quantization rvq scheme in which the codebook entries are independent and isotropically distributed assuming the interfering signatures are independent and have independent and identically distributed i i d elements we evaluate the received signal to interference plus noise ratio sinr in the large system limit as the number of users processing gain and feedback bits b all tend to infinity with fixed ratios this sinr is evaluated for both the matched filter and linear minimum mean squared error mmse receivers furthermore we show that this large system sinr is the maximum that can be achieved over any sequence of codebooks numerical results show that with the mmse receiver one feedback bit per signature coefficient achieves close to single user performance we also consider a less complex and suboptimal reduced rank signature optimization scheme in which the user s signature is constrained to lie in a lower dimensional subspace the optimal subspace coefficients are scalar quantized and relayed to the transmitter the large system performance of the quantized reduced rank scheme can be approximated and numerical results show that it performs in the vicinity of the rvq bound finally we extend our analysis to the scenario in which a subset of users optimize their signatures in the presence of random interference

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

generalized solitary and periodic wave solutions to a 2 1 dimensional zakharov kuznetsov equation abstract in this paper the exp function method is employed to the zakharov kuznetsov equation as a 2 1 dimensional model for nonlinear rossby waves the observation of solitary wave solutions and periodic wave solutions constructed from the exponential function solutions reveal that our approach is very effective and convenient the obtained results may be useful for better understanding the properties of two dimensional coherent structures such as atmospheric blocking events

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

asymptotic capacity of multicarrier transmission with frequency selective fading and limited feedback we study the capacity of multicarrier transmission through a slow frequency selective fading channel with limited feedback which specifies channel state information our results are asymptotic in the number of subchannels we first assume independent and identically distributed i i d subchannel gains and show that for a large class of fading distributions a uniform power distribution over an optimized subset of subchannels or on off power allocation gives the same asymptotic growth in capacity as optimal water filling e g with rayleigh fading furthermore the growth in data rate can be achieved with a feedback rate as if the number of active subchannels is bounded the capacity grows only as with the feedback rate of we then consider correlated subchannels modeled as a markov process and study the savings in feedback assuming a fixed ratio of coherence bandwidth to the total bandwidth the ratio between minimum feedback rates with correlated and i i d subchannels converges to zero with e g as for rayleigh fading subchannels satisfying a first order autoregressive process we also show that adaptive modulation or rate control schemes in which the rate on each subchannel is selected from a quantized set achieves the same asymptotic growth rates in capacity and required feedback finally our results are extended to cellular uplink and downlink channel models

global stability of a partial differential equation with distributed delay due to cellular replication in this paper we investigate a nonlinear partial differential equation arising from a model of cellular proliferation this model describes the production of blood cells in the bone marrow it is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population

asymptotic behavior for a cellular replication and maturation model abstract we consider a nonlocal first order partial differential equation with time delay that models simultaneous cell replication and maturation processes we establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the equation we then analyze the asymptotic behavior of the solution via upper lower solution technique

signature quantization in fading cdma with limited feedback in this work we analyze the performance of a signature quantization scheme for reverse link direct sequence ds code division multiple access cdma assuming perfect estimates of the channel and interference covariance the receiver selects the signature that minimizes interference power or maximizes signal to interference plus noise ratio sinr for a desired user from a signature codebook the codebook index corresponding to the optimal signature is then relayed to the user with a finite number of bits via a feedback channel here we are interested in the performance of a random vector quantization rvq codebook which contains independent isotropically distributed vectors assuming arbitrary transmit power allocation we consider additive white gaussian noise awgn channel first with no fading and subsequently with multipath fading we derive the corresponding sinr in a large system limit at the output of matched filter and linear minimum mean squared error mmse receiver numerical examples show that the derived large system results give a good approximation to the performance of finite size system and that the mmse receiver achieves close to a single user performance with only one feedback bit per signature element

degrees of freedom of the network mimo channel with distributed csi in this paper we discuss the joint precoding with finite rate feedback in the so called network multiple input multiple output mimo where the txs share the knowledge of the data symbols to be transmitted we introduce a distributed channel state information dcsi model where each tx has its own local estimate of the overall multiuser mimo channel and must make a precoding decision solely based on the available local csi we refer to this channel as the dcsi mimo channel and the precoding problem as distributed precoding we extend to the dcsi setting the work from jindal in 2006 for the conventional mimo broadcast channel bc in which the number of degrees of freedom dofs achieved by zero forcing zf was derived as a function of the scaling in the logarithm of the signal to noise ratio of the number of quantizing bits particularly we show the seemingly pessimistic result that the number of dofs at each user is limited by the worst csi across all users and across all txs this is in contrast to the conventional mimo bc where the number of dofs at one user is solely dependent on the quality of the estimation of his own feedback consequently we provide precoding schemes improving on the achieved number of dofs for the two user case the derived novel precoder achieves a number of dofs limited by the best csi accuracy across the txs instead of the worst with conventional zf we also advocate the use of hierarchical quantization of the csi for which we show that considerable gains are possible finally we use the previous analysis to derive the dofs optimal allocation of the feedback bits to the various txs under a constraint on the size of the aggregate feedback in the network in the case where conventional zf is used

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

handbook of the normal distribution genesis an historical background basic properties expansions and algorithms characterizations sampling distributions limit theorems and expansions normal approximations to distributions order statistics from normal samples the bivariate normal distribution bivariate normal sampling distributions point estimation statistical intervals

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

mimo systems with limited rate differential feedback in slowly varying channels in this paper an adaptive limited feedback linear precoding technique for temporally correlated multiple input multiple output mimo channels is proposed where the receiver has perfect channel knowledge but the transmitter only receives a quantized channel direction to perform adaptation to the time correlation structure we employ a differential feedback where the amount of the perturbation added to the previous precoder is determined by the statistics of the directional variation based on random matrix quantization analysis we develop a spherical cap codebook approach where the cap is centered at the previous precoder and the radius of the cap is determined proportional to the identified directional variation if the channel is highly correlated in time it is shown that the proposed differential feedback scheme achieves significant throughput improvement in the large codebook size regime the rest of the paper is devoted to developing a systematic spherical cap codebook generation method the developed approach employs a feedback scheme that uses a differential rotation of the previously used precoder our codebook adaptation is based on generating a perturbation in euclidean space and projecting the perturbation onto the unitary space simulation results show that the proposed adaptation scheme accurately tracks the channel using only a small rate of feedback

irregular wakes in reaction diffusion waves abstract reaction diffusion equations have proved to be highly successful models for a wide range of biological and chemical systems but chaotic solutions have been very rarely documented we present a new mechanism for generating apparently chaotic spatiotemporal irregularity in such systems by analysing in detail the bifurcation structure of a particular set of reaction diffusion equations on an infinite one dimensional domain with particular initial conditions we show that possible solutions include travelling fronts which leave behind either regular or irregular spatiotemporal oscillations using a combination of analytical and numerical analysis we show that the irregular behaviour arises from the instability of oscillations induced by the passage of the front finally we discuss the generality of this mechanism as a way in which spatiotemporal irregularities can arise naturally in reaction diffusion systems

capacity of a multiple antenna fading channel with a quantized precoding matrix given a multiple input multiple output mimo channel feedback from the receiver can be used to specify a transmit precoding matrix which selectively activates the strongest channel modes here we analyze the performance of random vector quantization rvq in which the precoding matrix is selected from a random codebook containing independent isotropically distributed entries we assume that channel elements are independent and identically distributed i i d and known to the receiver which relays the optimal rate maximizing precoder codebook index to the transmitter using b bits we first derive the large system capacity of beamforming rank one precoding matrix as a function of b where large system refers to the limit as b and the number of transmit and receive antennas all go to infinity with fixed ratios rvq for beamforming is asymptotically optimal i e no other quantization scheme can achieve a larger asymptotic rate we subsequently consider a precoding matrix with arbitrary rank and approximate the asymptotic rvq performance with optimal and linear receivers matched filter and minimum mean squared error mmse numerical examples show that these approximations accurately predict the performance of finite size systems of interest given a target spectral efficiency numerical examples show that the amount of feedback required by the linear mmse receiver is only slightly more than that required by the optimal receiver whereas the matched filter can require significantly more feedback

a nonlinear age and maturity structured model of population dynamics i basic theory a nonlinear model of age and maturity structured population dynamics is analyzed the population is structured by age and maturity of the individuals and the nonlinearity in the equations corresponds to density dependent limitation of population growth the existence and asymptotic behavior of solutions are studied

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

on the amount of downlink training in correlated massive mimo channels in this paper we study the effect of downlink dl training on the achievable sum rate of multiuser massive mimo channels with spatial correlations and derive sufficient conditions of the dl channel estimation error covariance matrices that maintain the full multiplexing gain at high data signal to noise ratios snrs given the derived conditions a simple asymptotic upper bound on the average sum rate loss due to channel estimation is obtained we derive in closed form training sequences of limited duration that satisfy these conditions the training duration is variable and increases with the data snr while the sequences lie in a subspace spanned by a variable number of user spatial covariance matrices eigenvectors we additionally study the problem of sequence codebook design and find solutions to this problem for uniform linear and rectangular arrays using asymptotic results for the aforementioned training structure the designed codebooks are observed numerically to be near optimal for a moderate number of base station antennas due to their ability to identify a sufficient limited number of channel directions to train the proposed solutions can substantially reduce dl training overheads while providing achievable rates that are comparable with the rates achieved with perfect channel state information

numerical integration of a mathematical model of hematopoietic stem cell dynamics a mathematical model of hematopoiesis describing the dynamics of stem cell population is investigated this model is represented by a system of two nonlinear age structured partial differential equations describing the dynamics of resting and proliferating hematopoietic stem cells it differs from previous attempts to model the hematopoietic system dynamics by taking into account cell age dependence of coefficients that prevents a usual reduction of this system to an unstructured delay differential system we prove the existence and uniqueness of a solution to our problem and we investigate the existence of stationary solutions a numerical scheme adapted to the problem is presented we show the effectiveness of this numerical technique in the simulation of the dynamics of the solution numerical simulations show that long period oscillations can be obtained in this model corresponding to a destabilization of the system these oscillations can be related to observations of some periodical hematological diseases such as chronic myelogenous leukemia

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

design and analysis of mimo spatial multiplexing systems with quantized feedback this paper investigates the problem of transmit beamforming in multiple antenna spatial multiplexing sm systems employing a finite rate feedback channel assuming a fixed number of spatial channels and equal power allocation we propose a new criterion for designing the codebook of beamforming matrices that is based on minimizing an approximation to the capacity loss resulting from the limited rate in the feedback channel using the criterion we develop an iterative design algorithm that converges to a locally optimum codebook under the independent identically distributed channel and high signal to noise ratio snr assumption the effect on channel capacity of the finite bit representation of the beamforming matrix is analyzed central to this analysis is the complex multivariate beta distribution and tractable approximations to the voronoi regions associated with the code points furthermore to compensate for the degradation due to the equal power allocation assumption we propose a multimode sm transmission strategy wherein the number of data streams is determined based on the average snr this approach is shown to allow for effective utilization of the feedback bits resulting in a practical and efficient multiple input multiple output system design

predictive coding of bit loading for time correlated mimo channels with a decision feedback receiver in this paper we consider variable rate transmission over a slowly varying multiple input multiple output mimo channel with a decision feedback receiver the transmission rate is adapted to the channel by dynamically assigning bits to the subchannels of the mimo system predictive quantization is used for the feedback of bit loading to take advantage of the time correlation inherited from the temporally correlated channel due to the use of decision feedback at the receiver the bit loading is related to the cholesky decomposition of the channel gram matrix assuming the channel is modeled by a slowly varying gauss markov process we show that the nested submatrices generated during the process of cholesky decomposition can be updated as time evolves based on the update we derive the optimal predictor of the next bit loading for predictive quantization furthermore we derive the statistics of the prediction error which are then exploited to design the quantizer to achieve a smaller quantization error simulations are given to demonstrate that the proposed predictive quantization gives a good approximation of the desired transmission rate with a low feedback rate

on transmit beamforming for miso ofdm channels with finite rate feedback with finite rate feedback we propose two feedback methods for transmit beamforming in a point to point miso ofdm channel for the first method a receiver with perfect channel information quantizes and feeds back the optimal transmit beamforming vectors of a few selected subcarriers which are equally spaced based on those quantized vectors the transmitter applies either constant linear or higher order interpolation with the remaining beamforming vectors with constant interpolation we derive the approximate sum achievable rate and the optimal cluster size that maximizes the approximate rate for linear interpolation we derive a closed form expression for the phase rotation by utilizing the correlation between ofdm subcarriers we also propose a higher order interpolation that requires more than two quantized vectors to interpolate transmit beamformers and is based on existing channel estimation methods numerical results show that interpolation with the optimized cluster size can perform significantly better than that with an arbitrary cluster size for the second proposed method a channel impulse response is quantized with a uniform scalar quantizer with channel quantization we also derive the approximate sum achievable rate we show that switching between the two methods for different feedback rate requirements can perform better than the existing schemes

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

connection aspects of nonlinear lattice equations in 1 1 dimensions considers two nonlinear lattice equations and studies some geometrical features the author demonstrates that two nonlinear partial differential equations are deduced from the fact that the sl 2 r connection has zero curvature

differential feedback of geometrical mean decomposition precoder for time correlated mimo systems for a time correlated channel we consider the differential feedback of the geometrical mean decomposition gmd precoder which is known to be optimal for a number of criteria when the channel varies slowly we can expect the optimal gmd precoders of consecutive channel uses to be close we consider the feedback of the so called differential precoder and show that it lies in a neighborhood of the identity matrix using matrix perturbation theory furthermore the radius of the neighborhood is shown to be proportional to a time correlation parameter such a characterization allows for subsequent efficient quantization of the differential precoder we show how to design codewords for the prescribed radius by perturbing the identity matrix and applying qr decomposition on the perturbed matrix simulations are given to demonstrate that with a small feedback rate the performance of the proposed differential gmd comes close to the case when perfect channel state information is available to the transmitter for slowly time varying channels like those in indoor and microcellular environments

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

propagation of population pulses and fronts in a cell replication problem non locality and dependence on the initial function abstract here we study the solution types in a simplified equation arising from a cell population dynamics model formulated as a non linear first order partial differential equation for the cell density u t x in the presence of non local maturational x interactions this can be considered as a reaction convection equation in which the cell density is convected with maturation velocity r the non local action affects only the reaction birth rate an initial function representing a pulse of small amplitude on an almost extinguished population may be dissipated may produce multiple pulses or may remain intact as it travels driven by reaction and convection depending on the magnitude of the non local action alternately for localized initial perturbations the equation also has positive traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution for weak convection r 1 which is the only case treated here reaction driven fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state a classification arises according to whether the magnitude of the maturation delay is larger or smaller than a critical value and scaling arguments are used to determine the critical delay for delays larger than critical the reaction driven fronts are oscillatory and correspond to the case for which the correlation length for non local action is of the same order as the characteristic length of maturation population gradients that yield oscillatory birth rates simulation results are validated and interpreted using scaling arguments and solutions for the local equations

energy efficient spectrum sharing strategy selection for cognitive mimo interference channels in this paper we propose a promising discrete game theoretic framework for distributed energy efficient discrete spectrum sharing strategy selection i e joint discrete power control and multimode precoding strategy selection with limited feedback for cognitive mimo interference channels given the competitive nature the secondary users are assumed to be selfish and noncooperative each of whom attempts to maximize its individual energy efficiency under a minimum data rate constraint and an interference power constraint a mechanism for shutting down links is proposed to reduce interference and save energy a payoff function is designed to guarantee the feasibility of the pure strategy nash equilibrium with no need to know the infeasible strategy profiles a spectrum sharing strategy profile is said to be feasible if the stated constraints are satisfied otherwise the spectrum sharing strategy profile is said to be infeasible i e they may not satisfy the interference power constraint and minimum data rate constraint in advance we then investigate the existence and the feasibility of the pure strategy nash equilibrium and further devise a pricing based distributed algorithm for spectrum strategy selection the proposed algorithm is proved to converge to a feasible pure strategy nash equilibrium under specific conditions moreover by studying the relationship between our proposed game and the social optimum we find that the pricing mechanism can result in pareto improvement and lead to better convergence for the proposed distributed algorithm numerical results show that our designed algorithm significantly outperforms the random selection algorithm and the pricing mechanism has a dramatic effect in improving the system performance

delay limited and ergodic capacities of mimo channels with limited feedback we consider a fixed data rate slow fading mimo channel with a long term power constraint p at the transmitter a relevant performance limit is the delay limited capacity which is the largest data rate at which the outage probability is zero it is well known that if both the transmitter and the receiver have full channel state information csi and if either of them has multiple antennas the delay limited capacity is non zero and grows logarithmically with p achieving even a positive delay limited capacity however becomes a difficult task when the csi at the transmitter csit is imperfect in this context the standard partial csit model where the transmitter has a fixed finite bit of quantized csi feedback for each channel state results in zero delay limited capacity we show that by using a variable length feedback scheme that utilizes a different number of feedback bits for different channel states a non zero delay limited capacity can be achieved if the feedback rate is greater than 1 bit per channel state moreover we show that the delay limited capacity loss due to finite rate feedback decays at least inverse linearly with respect to the feedback rate we also discuss the applications to ergodic mimo channels