the beta model maximum likelihood cramer rao bounds and hypothesis testing we study the maximum likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph level the model generalizes the well known beta model for random graphs by replacing the constant model parameters with regression functions cramer rao bounds are derived for special cases of the undirected beta model the directed beta model and the covariate based beta model the corresponding maximum likelihood estimators are compared with the bounds by means of simulations moreover examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance finally the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers

robust mse equalizer design for mimo communication systems in the presence of model uncertainties this paper considers a robust mean square error mse equalizer design problem for multiple input multiple output mimo communication systems with imperfect channel and noise information at the receiver when the channel state information csi and the noise covariance are known exactly at the receiver a minimum mean square error mmse equalizer can be employed to estimate the transmitted signal however in actual systems it is necessary to take into account channel and noise estimation errors we consider here a worst case equalizer design problem where the goal is to find the equalizer minimizing the equalization mse for the least favorable channel model within a neighborhood of the estimated model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and estimated channel models lagrangian optimization is used to convert this min max problem into a convex min min problem over a convex domain which is solved by interchanging the minimization order the robust mse equalizer and associated least favorable channel model can then be obtained by solving numerically a scalar convex minimization problem simulation results are presented to demonstrate the mse and bit error rate ber performance of robust equalizers when applied to the least favorable channel model

demixing can occur in binary hard sphere mixtures with negative nonadditivity a binary fluid mixture of nonadditive hard spheres characterized by a size ratio gamma sigma 2 sigma 1 infinity a demixing transition with a critical consolute point at a packing fraction scaling as eta approximately d2 d is found even for slightly negative nonadditivity if delta 1 8 ln gamma 2 arguments concerning the stability of the demixing with respect to freezing are provided

the probability distribution of the partition function of the random energy model the authors give the expression for both integer and non integer moments of the partition function z of the random energy model in the thermodynamic limit they find that the probability distribution p z can be decomposed into two parts for log z log z finite the distribution is independent of n the size of the system whereas for log z log z positive and of order n the distribution is gaussian these two parts match in the region 1 log z log z n where the distribution is exponential

statistical mechanics of a single particle in a multiscale random potential parisi landscapes in finite dimensional euclidean spaces we construct an n dimensional gaussian landscape with multiscale translation invariant logarithmic correlations and investigate the statistical mechanics of a single particle in this environment in the limit of high dimension n the free energy of the system and overlap function are calculated exactly using the replica trick and parisi s hierarchical ansatz in the thermodynamic limit we recover the most general version of the derrida s generalized random energy model grem the low temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape if the latter consists of k discrete values the system is characterized by a k step replica symmetry breaking solution we argue that our construction is in fact valid in any finite spatial dimensions n 1 we discuss the implications of our results for the singularity spectrum describing multifractality of the associated boltzmann gibbs measure finally we discuss several generalizations and open problems such as the dynamics in such a landscape and the construction of a generalized multifractal random walk

block decision feedback equalization a natural generalization of the conventional decision feedback equalizer dfe based on block processing and maximum a posteriori decisions is presented this block dfe is indexed by two parameters depending on the block length p and the number of decisions q

robust linear channel estimation methods for per subcarrier transmit antenna selection the linear minimum mean squared error mse channel estimator for systems employing per subcarrier transmit antenna selection is developed frequency domain correlations after the selection process are shown to be approximated well using a simple function which makes near optimal channel estimation practically possible however the resultant estimators are not robust to errors in the assumed model in terms of the antenna to subcarrier assignment used at transmission as we motivate both analytically and via simulations this is an issue when the channels for conveying this information to the receiver are severely constrained we present windowed channel estimations and new robust estimation algorithms either individually or in combination as solutions to counter this sensitivity the robust estimators are developed based on the principles of 1 minimising the worst case mse over the set of possible models and 2 minimising the expected mse over the set of possible models the latter estimator is preferred due to the lower implementation complexity and better mse performance we also prove theoretical asymptotic in signal to noise ratio performances of the two estimators used with the proposed correlation model simulation results illustrate the performance gains and improved robustness offered by the developed estimators

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

robust kalman filters for linear time varying systems with stochastic parametric uncertainties we present a robust recursive kalman filtering algorithm that addresses estimation problems that arise in linear time varying systems with stochastic parametric uncertainties the filter has a one step predictor corrector structure and minimizes an upper bound of the mean square estimation error at each step with the minimization reduced to a convex optimization problem based on linear matrix inequalities the algorithm is shown to converge when the system is mean square stable and the state space matrices are time invariant a numerical example consisting of equalizer design for a communication channel demonstrates that our algorithm offers considerable improvement in performance when compared with conventional kalman filtering techniques

binary nonadditive hard sphere mixtures at high dimension at high spatial dimension a suitably scaled classical system of interacting particles truncates at second virial terms a binary mixture of nonadditive hard spheres with sufficiently repulsive interaction between unlike particles decomposes at sufficiently high density into two coexisting phases the region around the critical density behaves classically

hierarchic trees with branching number close to one noiseless kardar parisi zhang equation with additional linear term for imitating two dimensional and three dimensional phase transitions an imitation of two dimensional 2d field theory is formulated by means of a model on the hierarchic tree with branching number close to one with the same potential and the free correlators identical to 2d correlator ones such a model carries on some features of the original model for certain scale invariant theories the renormalization group equation for the free energy is the noiseless kardar parisi zhang equation with an additional linear term

competitive linear estimation under model uncertainties we investigate a linear estimation problem under model uncertainties using a competitive algorithm framework under mean square error mse criteria here the performance of a linear estimator is defined relative to the performance of the linear minimum mse estimator tuned to the underlying unknown system model we then find the linear estimator that minimizes this relative performance measure i e the regret for the worst possible system model two definitions of regret are given first as a difference of mses and second as a ratio of mses we demonstrate that finding the linear estimators that minimize these regret definitions can be cast as a semidefinite programming sdp problem and provide numerical examples

liouville property for groups and manifolds we introduce a method to estimate the entropy of random walks on groups we apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not liouville we also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non trivial poisson boundary if and only if this group is not virtually nilpotent this in particular shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient lie group as another application we get a series of examples of amenable groups such that any finite entropy non degenerate measure on them has non trivial poisson boundary since the groups in question are amenable they do admit measures such that the corresponding random walks have trivial boundary the above shows that such measures on these groups have infinite entropy

poisson furstenberg boundary and growth of groups we study the poisson furstenberg boundary of random walks on permutational wreath products we give a sufficient condition for a group to admit a symmetric measure of finite first moment with non trivial boundary and show that this criterion is useful to establish exponential word growth of groups we construct groups of exponential growth such that all finitely supported not necessarily symmetric possibly degenerate random walks on these groups have trivial boundary this gives a negative answer to a question of kaimanovich and vershik

robust filtering for discrete time systems with saturation and its application to transmultiplexers this paper considers the problem of robust filtering for discrete time linear systems subject to saturation a generalized dynamic filter architecture is proposed and a filter design method is developed our approach incorporates the conventional linear h sub 2 and h sub spl infin filtering as well as a regional l sub 2 gain filtering feature developed specially for the saturation nonlinearity and is applicable to the digital transmultiplexer systems for the purpose of separating filterbank design it turns out that our filter design can be carried out by solving a constrained optimization problem with linear matrix inequality lmi constraints simulations show that the resultant separating filters possess satisfactory reconstruction performance while working in the linear range and less degraded reconstruction performance in the presence of saturation

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

boundedness behavior of the spectral factorization for polynomial data in the wiener algebra spectral factorization is of fundamental importance in many areas of signal processing this paper investigates the boundedness behavior of the spectral factorization mapping in the wiener algebra thereby the focus lies on the factorization of polynomial spectral densities with a finite degree n since such spectra are especially important for practical applications the paper presents a lower and an upper bound on the boundedness behavior which will show that the boundedness constant of the spectral factorization mapping gets worse as the degree n of the spectra increases therewith one obtains independently the known result that the spectral factorization mapping is unbounded on the wiener algebra

rate of convergence in approximating the spectral factor of regular stochastic sequences common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a polynomial and a subsequent factorization of this polynomial it is known that the regularity of the stochastic sequence determines the achievable approximation rate of its spectrum however since the approximative polynomial should be factorized it has to be positive it is shown that this restriction on the approximation polynomial implies a limitation on the approximation rate for linear methods whereas for nonlinear methods the optimal approximation rate can still be achieved this has also consequences for the rate of convergence of the spectral factor which is investigated in the second part there a lower and an upper bound for the error in the spectral factor is derived which shows the dependency on the approximation degree and on the regularity of the stochastic sequence finally if the spectral density is given only on a finite set of sampling points no linear approximation method exists such that the error in the spectral factor can be controlled by the approximation degree

robust mimo equalization for non parametric channel model uncertainty in this paper three mimo robust equalization problems are considered for non parametric classes of channel models defined by weighted h2 or h balls of frequency responses and performance criteria based on h2 variance or h norms of error signals the approach pursued here centers on characterizing the worst case performance of candidate equalizers or upper bounds on it by means of dual lagrangian functionals then for linearly parametrized finite dimensional classes of candidate equalizers the corresponding robust equalization problems are converted into semi definite linear programming problems for which approximate solutions can be effectively computed a simple numerical example is presented involving h2 model uncertainty and error variance performance to illustrate for various levels of uncertainty the changes in the worst case performances of the nominally optimal equalizer and of the one in a specific linear class which minimizes the worst case error variance

on discrete time h sub spl infin fixed lag smoothing the h sub spl infin fixed lag smoothing problem for discrete time linear systems is considered it is well known that such a problem can be reformulated as a filtering problem for a suitable augmented system however in order to check the solvability of the smoothing problem and compute the relevant gains one must solve a riccati equation involving the augmented system matrices which is an approach that may be inefficient especially for long lags in this paper efficient algorithms are worked out that permit checking of solvability and implementation of the smoother relying only on the solution of the h sub spl infin filtering riccati equation in particular this provides a fast method to compute the minimum lag guaranteeing a desired attenuation level

negative fractal dimensions and multifractals abstract a new notion of fractal dimension is defined when it is positive it effectively falls back on known definitions but its motivating virtue is that it can take negative values which measure usefully the degree of emptiness of empty sets the main use concerns random multifractals for which f

robust steady state filtering for systems with deterministic and stochastic uncertainties for uncertain systems containing both deterministic and stochastic uncertainties we consider two problems of optimal filtering the first is the design of a linear time invariant filter that minimizes an upper bound on the mean energy gain between the noise affecting the system and the estimation error the second is the design of a linear time invariant filter that minimizes an upper bound on the asymptotic mean square estimation error when the plant is driven by a white noise we present filtering algorithms that solve each of these problems with the filter parameters determined via convex optimization based on linear matrix inequalities we demonstrate the performance of these robust algorithms on a numerical example consisting of the design of equalizers for a communication channel

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

multifractality and freezing phenomena in random energy landscapes an introduction abstract we start our lectures with introducing and discussing the general notion of multifractality spectrum for random measures on lattices and how it can be probed using moments of that measure then we show that the boltzmann gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet non trivial example of disorder induced multifractal measures the typical values of the multifractality exponents can be extracted from calculating the free energy of the associated statistical mechanics problem to succeed in such a calculation we introduce and discuss in some detail two analytically tractable models for logarithmically correlated potentials the first model uses a special definition of distances between points in space and is based on the idea of multiplicative cascades which originated in theory of turbulent motion it is essentially equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by derrida and spohn 1988 in ref 12 in this way we introduce the notion of the freezing transition which is identified with an abrupt change in the multifractality spectrum second model which allows for explicit analytical evaluation of the free energy is the infinite dimensional version of the problem which can be solved by employing the replica trick in particular the latter version allows one to identify the freezing phenomenon with a mechanism of the replica symmetry breaking rsb and to elucidate its physical meaning the corresponding one step rsb solution turns out to be marginally stable everywhere in the low temperature phase we finish with a short discussion of recent developments and extensions of models with logarithmic correlations in particular in the context of extreme value statistics the first appendix summarizes the standard elementary information about gaussian integrals and related subjects and introduces the notion of the gaussian free field characterized by logarithmic correlations three other appendices provide the detailed exposition of a few technical details underlying the replica analysis of the model discussed in the lectures

on linear h sup spl infin equalization of communication channels as an alternative to existing techniques and algorithms we investigate the merit of the h sup spl infin approach to the linear equalization of communication channels we first give the formulation of all causal h sup spl infin equalizers using the results of and then look at the finite delay ease we compare the risk sensitive h sup spl infin equalizer with the mmse equalizer with respect to both the average and the worst case ber performances and illustrate the improvement due to the use of the h sup spl infin equalizer

l_ 2 e fir smoothers for deterministic discrete time state space signal models in this note a new type of l2 e performance criterion for a fixed lag smoother is introduced which is given by a gain between the energy of the external disturbances during the recent time horizon and the estimation error at the fixed delayed time from the current one by minimizing the maximum value of the l2 e performance criterion the l2 e finite impulse response fir smoother lefs is obtained for a deterministic discrete time state space signal model the lets is designed to meet requirements such as linearity and fir structure with respect to measured inputs and outputs unbiased property in the deterministic sense and independence of any state information simultaneously the proposed lets is represented in a batch form for simplicity the lets is compared with a conventional h infin infinite impulse response iir smoother via simulation

on growth recurrence and the choquet deny theorem for p adic lie groups we first study the growth properties of p adic lie groups and its connection with p adic lie groups of type r and prove that a non type r p adic lie group has compact neighbourhoods of identity having exponential growth this is applied to prove the growth dichotomy for a large class of p adic lie groups which includes p adic algebraic groups we next study p adic lie groups that admit recurrent random walks and prove the natural growth conjecture connecting growth and the existence of recurrent random walks precisely we show that a p adic lie group admits a recurrent random walk if and only if it has polynomial growth of degree at most two we prove this conjecture for some other classes of groups also we also prove the choquet deny theorem for compactly generated p adic lie groups of polynomial growth and also show that polynomial growth is necessary and sufficient for the validity of the choquet deny for all spread out probabilities on zariski connected p adic algebraic groups counter example is also given to show that certain assumptions made in the main results can not be relaxed

on the asymptotic and invariant algebras of random walks on locally compact groups consider a random walk of law on a locally compact second countable groupg let the starting measure be equivalent to the haar measure and denote byq the corresponding markov measure on the space of pathsg we study the relation between the spacesl g a q andl g i q where a and i stand for the asymptotic and invariant algebras respectively we obtain a factorizationl g a q l g i q l c wherec is a cyclic group whose order finite or infinite coincides with the period of the markov shift and is determined by the asymptotic behaviour of the convolution powers n

realizable spatio temporal tomlinson harashima precoders theory and fast computation we derive the optimal realizable tomlinson harashima precoder for frequency selective multiple input multiple output mimo channels with respect to a simplified mean square error mse criterion realizability means that in contrast to other works we do not restrict the internal filters of the precoder to have finite impulse responses firs but nevertheless ensure that the precoder can be operated in real time subject to a finite latency time which can be chosen by the system designer in particular this allows us to consider channels with infinite impulse responses iirs as they occur e g in digital subscriber lines the feedforward filter is located at the transmitter in our system model and an additional scalar gain is employed at the receiver the relocation of the feedforward filter allows us to consider decentralized receivers however it also makes it necessary to impose a transmit power constraint the power constraint often has the effect that the input signals at the receiver have to be rescaled we argue that the inclusion of the additional scalar gain allows us to incorporate the effect of the rescaling operation into the optimization special emphasis is put on the fast computation of the realizable tomlinson harashima precoder via displacement structure theory we propose a fast algorithm which includes a successive construction of a close to optimal ordering of the data streams the complexity of this algorithm is only cubic in the number of channel inputs and quadratic in the latency time the algorithm can also be applied in situations where certain fir constraints have to be fulfilled because we find that the optimal realizable tomlinson harashima precoder involves only fir filters as soon as the channel is fir our results are reproducible

probability measures on groups and some related ideals in group algebras abstract let g be a locally compact group random walks on g some factorization problems in l 1 g and the significance for these of the amenability of g are studied these topics are linked in this paper via ideals in l 1 g of the form l 1 g e where is a probability measure on g cohen s factorization theorem and some ideas from ergodic theory play an important part

norm convergence of random walks on compact hypergroups where ig is the order of g and p is the nth convolut ion power of p they show that this is a natural tool for studying strong uniform times for r a n d o m walks on g and they show in theorem 7 of 1 stated as theorem 5 in 7 chap 4c that if i p ml o as n then s n o as n at a rate that depends on how fast l ip m converges to 0 but does not depend on the size of the group g in particular if j p m is small then s 2n is small

extremes of the discrete two dimensional gaussian free field we consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high or low we then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative for example we compute the width of the largest downward spike of a given length through the prism of these results we find that the extrema of the free field under entropic repulsion minus its mean and those of the unconditioned free field are identical finally when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two dimensional simple random walk on the discrete torus

exact asymptotics of the freezing transition of a logarithmically correlated random energy model we consider a logarithmically correlated random energy model namely a model for directed polymers on a cayley tree which was introduced by derrida and spohn we prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the kpp equation thus translating bramson s work on the kpp equation into a discrete time case we also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point

mimo decision feedback equalization from an h sup spl infin perspective we approach the multiple input multiple output mimo decision feedback equalization dfe problem in digital communications from an h sup spl infin estimation point of view using the standard and simplifying assumption that all previous decisions are correct we obtain an explicit parameterization of all h sup spl infin optimal dfes in particular we show that under the above assumption minimum mean square error mmse dfes are h sup spl infin optimal the h sup spl infin approach also suggests a method for dealing with errors in previous decisions

mimo linear equalization with an h infty criterion in this paper we study the problem of linearly equalizing the multiple input multiple output mimo communications channels from an h infty point of view h infty estimation theory has been recently introduced as a method for designing filters that have acceptable performance in the face of model uncertainty and lack of statistical information on the exogenous signals in this paper we obtain a closed form solution to the square mimo linear h infty equalization problem and parameterize all possible h infty optimal equalizers in particular we show that for minimum phase channels a scaled version of the zero forcing equalizer is h infty optimal the results also indicate an interesting dichotomy between minimum phase and nonminimum phase channels for minimum phase channels the best causal equalizer performs as well as the best noncausal equalizer whereas for nonminimum phase channels causal equalizers cannot reduce the estimation error bounds from their a priori values our analysis also suggests certain remedies in the nonminimum phase case namely allowing for finite delay oversampling or using multiple sensors for example we show that h infty equalization of nonminimum phase channels requires a time delay of at least l units where l is the number of nonminimum phase zeros of the channel

mimo equalization with state space channel models state space models are proposed to represent multiple input multiple output mimo frequency selective wireless channels with the motivation of better model approximation and more robust channel equalization performance when the order of the channel model is lower that of the true channel we develop a simple framework under which the equalizers for state space channel models can be designed using the existing methods for designing equalizers for finite impulse response fir models in particular a mimo minimum mean squared error decision feedback equalizer is developed for state space models when only estimates of the channel are available to the receiver the equalization performance is affected by the channel estimation accuracy because reduced order state space models can provide lower h 2 channel estimation error than reduced length fir models state space based equalizers typically exhibit significantly smaller symbol error rate than fir based ones thus state space channel models can be a more robust choice than fir models in the presence of model order selection error numerical simulation also shows that adaptive state space based receivers have slower but comparable convergence rates compared to fir based adaptive equalizers

entropic repulsion and the maximum of the two dimensional harmonic crystal we consider the lattice version of the free field in two dimensions also called harmonic crystal the main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall the basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure

realizable linear and decision feedback equalizers properties and connections recently there has been renewed interest in the use of infinite impulse response iir linear equalizers les for digital communication channels as a means for both improving performance and blindly initializing decision feedback structures dfes theoretical justification for such an approach is usually given assuming unconstrained filters which are not causal and therefore not implementable in practice we present an analysis of realizable i e causal stable and of finite degree minimum mean square error mmse equalizers for single input multiple output channels both in the le and dfe cases focusing on their structures and filter orders as well as the connections between them the dfe resulting from rearranging the mmse le within a decision feedback loop is given special attention it is shown that although this dfe does not in general coincide with the mmse dfe it still enjoys certain optimality conditions the main tools employed are the wiener theory of minimum variance estimation and kalman filtering theory which show interesting properties of the mmse equalizers not revealed by previous polynomial approaches

anderson localization on the cayley tree multifractal statistics of the transmission at criticality and off criticality in contrast to finite dimensions where disordered systems display multifractal statistics only at criticality the tree geometry induces multifractal statistics for disordered systems also off criticality for the anderson tight binding localization model defined on a tree of branching ratio k 2 with n generations we consider the miller derrida scattering geometry 1994 j stat phys 75 357 where an incoming wire is attached to the root of the tree and where kn outcoming wires are attached to the leaves of the tree in terms of the kn transmission amplitudes tj the total landauer transmission is t j tj 2 so that each channel j is characterized by the weight wj tj 2 t we numerically measure the typical multifractal singularity spectrum f of these weights as a function of the disorder strength w and we obtain the following conclusions for its left termination point w in the delocalized phase w 0 and is associated with a moment index q w 1 at criticality it vanishes wc 0 and is associated with the moment index q wc 1 in the localized phase w wc w 0 is associated with some moment index q w 1 we discuss the similarities with the exact results concerning the multifractal properties of the directed polymer on the cayley tree

product set phenomena for countable groups we develop in this paper novel techniques to analyze local combinatorial structures in product sets of two subsets of a countable group which are large with respect to certain classes of not necessarily invariant means on the group our methods heavily utilize the theory of c algebras and random walks on groups as applications of our methods we extend and quantify a series of recent results by jin bergelson furstenberg weiss beiglbock bergelson fish griesmer and di nasso lupini to general countable groups

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

mmse decision feedback equalizers and coding i equalization results the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure for systems that combine equalization with coded modulation the structure and performance of the mmse dfe are succinctly derived using linear estimation theoretic principles in this first part of this two part paper the front end of the mmse dfe called the mean square whitened matched filter ms wmf is preferable in some ways to a matched filter or a whitened matched filter as a canonical receiver front end in a coded system the feedback filter of the mmse dfe may be implemented in the transmitter using precoding the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels the mmse dfe is biased the optimum unbiased mmse dfe is the mmse dfe with the bias removed removing bias improves error probability but reduces the snr to snr sub mmse dfe u snr sub mmse dfe 1 it is shown that this snr relationship is a particular case of a very general result and that snr sub mmse dfe u gives a more realistic estimate of snr the results are extended to partial response equalization and to equalization with correlated inputs in an appendix

mmse decision feedback equalizers finite length results this paper extends a number of results on the infinite length minimum mean square error decision feedback equalizer mmse dfe reported by cioffi dudevoir eyuboglu and forney see ieee trans commun 1995 to the finite length case cholesky factorization and displacement structure theory are demonstrated to be two powerful analytical tools for analyzing the finite length mmse dfe our objective throughout the paper is to establish finite length analogs of the well known infinite length mmse dfe results similarities and differences between the two cases are examined and delineated finally convergence of our derived finite length results to their well established infinite length counterparts is shown

amenability reiter s condition and liouville property abstract we show that the liouville property and reiter s condition are equivalent for semigroupoids this result applies to semigroups as well as semigroup actions in the special case of measured groupoids and locally compact groupoids our result proves kaimanovich s conjecture of the equivalence of amenability and the liouville property

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

robust minimax mse equalizer designs for mimo wireless communications with time varying channel uncertainties in this paper we propose robust equalizers based on a minimax mean squares error mse scheme for wireless multi input multi output mimo communications subject to time varying channel uncertainties we consider channel uncertainties within a neighborhood of the estimated channel matrix formed by placing a bound on the spectral matrix norm of channel estimation errors first we derive a linear matrix inequality lmi based solution to the targeted problem next we modify the mse cost function of the targeted problem and derive a guaranteed cost based solution which may save computational cost subsequently channel uncertainty is partitioned into finite markov transitioned channel uncertainty states based on the least upper bound of the set of the tightest upper bounds on the matrix spectral norm of channel uncertainty this leads to a multiple model based minimax mse approach on this basis a feasible equalizer can be obtained from a weighted combination of multiple over guaranteed cost based equalizers each of which is designed with respect to a channel uncertain state simulation results show that a weighted combination of a moderate number of multiple over guaranteed cost based equalizers can achieve robust equalization effectively with superior mse and bit error rate ber performance

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

boundary behavior for groups of subexponential growth in this paper we introduce a method for partial description of the poisson boundary for a certain class of groups acting on a segment as an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric infinitely supported measure of finite entropy h this implies that the entropy h of the corresponding random walk is finite and positive as another application we exhibit certain discontinuity for the recurrence property of random walks finally as a corollary of our results we get new estimates from below for the growth function of a certain class of grigorchuk groups in particular we exhibit the first example of a group generated by a finite state automaton such that the growth function is subexponential but grows faster than exp n for any 1 we show that in some of our examples the growth function satisfies exp n ln

functional feynman kac equations for limit lognormal multifractals a novel technique of functional feynman kac equations is developed for the probability distribution of the limit lognormal multifractal process introduced by mandelbrot in statistical models and turbulence m rosenblatt and c van atta eds springer new york 1972 and constructed explicitly by bacry delour and muzy phys rev e 64 026103 2001 the distribution of the process is known to be determined by the complicated stochastic dependence structure of its increments sdsi it is shown that the sdsi has two separate layers of complexity that can be captured in a precise way by a pair of functional feynman kac equations for the laplace transform exact solutions are obtained as power series expansions in the intermittency parameter using a novel intermittency differentiation rule the expansion of the moments gives a new representation of the selberg integral

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

hybrid fm polynomial phase signal modeling parameter estimation and cramer rao bounds parameter estimation for a class of nonstationary signal models is addressed the class contains combination of a polynomial phase signal pps and a frequency modulated fm component of the sinusoidal or hyperbolic type such signals appear in radar and sonar applications involving moving targets with vibrating or rotating components a novel approach is proposed that allows us to decouple estimation of the fm parameters from those of the pps relying on properties of the multilag high order ambiguity function ml haf the accuracy achievable by any unbiased estimator of the hybrid fm pps parameters is investigated by means of the cramer rao lower bounds crlbs both exact and large sample approximate expressions of the bounds are derived and compared with the performance of the proposed methods based on monte carlo simulations

dimensional reduction of integrals of orthogonal invariants the integration of a function of scalar products ofp vectors in rd where d p is reduced to that of a corresponding function of vectors in rp

h sub 2 optimal linear robust sampled data filtering design using polynomial approach a new frequency domain approach to robust multi input multi output mimo linear filter design for sampled data systems is presented the system and noise models are assumed to be represented by polynomial forms that are not perfectly known except that they belong to a certain set the optimal design guarantees that the error variance is kept below an upper bound that is minimized for all admissible uncertainties the design problem is cast in the context of h sub 2 via the polynomial matrix representation of systems with norm bounded unstructured uncertainties the sampled data mix of continuous and discrete time systems is handled by means of a lifting technique however it does not increase the dimensionality or alter the computational cost of the solution the setup adopted allows dealing with several filtering problems a simple deconvolution example illustrates the procedure

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

quantum channels arising from abstract harmonic analysis we present a new application of harmonic analysis to quantum information by constructing intriguing classes of quantum channels stemming from specific representations of multiplier algebras over locally compact groups g beginning with a representation of the measure algebra m g we unify and elaborate on recent counter examples to a conjecture on the structure of fixed point subalgebras in infinite dimensions as well as present an application to the noiseless subsystems method of quantum error correction using a representation of the completely bounded fourier multiplier algebra mcba g we provide a new class of counter examples to the recently solved asymptotic quantum birkhoff conjecture along with a systematic method of producing the examples using a geometric representation of schur maps further properties of our channels including duality quantum capacity and entanglement preservation are discussed along with potential applications to additivity conjectures

joint transmitter receiver optimization for multi input multi output systems with decision feedback the joint transmit receive optimization problem for multiuser communication systems with decision feedback is investigated it is shown that minimization of the geometric mean squared error defined as the determinant of the error covariance matrix leads to a tractable transmitter optimization problem for general multi input multi output decision feedback systems several computational results are included that highlight system performance for a variety of useful transmission scenarios