design and implementation of two dimensional fully recursive digital filters the two dimensional 2 d fully recursive filter was recently introduced by the authors as a more general 2 d recursive filter in this paper we present a design algorithm and an implementation method for this class of filters the design algorithm incorporates a novel stability test we discuss the boundary condition problem which occurs in the implementation of fully recursive filters we propose a direct fully recursive implementation method which has been verified in software several design examples and comparisons are provided

algebraic approach to system identification a variety of identification procedures exists for estimating the parameters of an autoregressive moving average arma process from noise free excitation and noise contaminated response data in this paper an identification procedure is proposed for the more realistic situation in which both the excitation and response are contaminated by white noises the method is based upon the null space characterization of an associated data matrix some of the more important algebraic properties possessed by this data matrix are first established in the ideal noise free data case in particular it is found that an overordering of the arma model will not impair the identification of the underlying system in the more realistic noise contaminated data case an approximation of the data matrix s null space is affected by using an eigenvalue eigenvector decomposition by incorporating this null space approximation the deleterious effects of the noise are significantly reduced thereby giving rise to improved modeling performance this improvement is demonstrated by means of a standard example in which the proposed identification method is shown to produce a better modeling behavior than does the classical least squares method the corresponding iterative generalized least squares method and a commonly employed instrumental variable method

chebyshev approximations for the direct design of 2 d quarter plane iir filters the development of various direct design techniques for inherently stable causal recursive two dimensional 2 d digital filters is described and illustrated by examples these techniques center around the development of 2 d z domain functions for use as arguments in the chebyshev function and provide for the design of low pass bandpass and high pass filters the specifications that can be prescribed are passband ripple filter order and normalized passband position parameters in addition to being inherently stable the filters possess an approximately flat group delay in most of the filter s passband a feature necessary for most applications

limitedness theorem on finite automata with distance functions an algebraic proof a distance function on a finite automaton m is defined by assigning to each transition a distance of nonnegative integer value m is said to be limited in distance if there is a nonnegative integer k such that for each accepted string w in the language of m there is an accepting path p for w on which the sum of distances is bounded by k the limitedness theorem 6 9 on finite automata with distance functions states that it is decidable if an arbitrary finite automaton with a distance function is limited in distance in this paper we give an algebraic proof of the theorem and derive from it an exponential time decision algorithm in addition we prove that the decision problem is pspace hard

on exit times of multivariate random walk with some applications to finance abstract this paper analyzes multivariate delayed random walk processes and multivariate poisson processes and presents some applications to the stock market when trading with stock options or having a stock portfolio including ones by mutual funds we are interested in various parameters such as exit times and stock exit price values when the stock drops or when it exceeds a specified price or even one trade ahead of the time when an exit would occur we obtain closed form functionals of the named random parameters we also present an example of a random walk embedded in a multivariate poisson process

continuity of closest rank p approximations to matrices in signal processing the singular value decomposition and rank characterization of matrices play prominent roles the mapping which associates with any complex m n matrix x its closest rank p approximation x p need not be continuous when the pth and the p 1 st singular values of x are equal this mapping maps in fact a matrix to a set of matrices furthermore an example is given to show that large errors in computing x p can be expected when p is sufficiently close to p 1 it is finally shown that this mapping is closed in the sense of zangwill the property of closedness is an essential assumption of a global convergence proof for algorithms involving this mapping e g see 1

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

signal enhancement a composite property mapping algorithm a signal enhancement algorithm is developed that seeks to recover a signal from noise contaminated distorted measurements made on that signal this object is achieved by utilizing a set of properties which the signal is known or is hypothesized as possessing the measured signal is modified to the smallest degree necessary to sequentially possess each of the individual properties conditions for the algorithm s convergence are established in which the primary requirement is that a composite property mapping be closed this is a relatively unrestricted condition in comparison to that required of most existing signal enhancement algorithms

level crossings of an oscillating marked random walk this paper deals with a class of real valued random walk processes observed over random epochs of time that forms a delayed renewal process the present model does not restrict this class to a merely monotone random walk which is easier to analyze and find explicit form functional the objective is to find the first passage of the process exiting a rectangular set and registering the value of the process at this time thus generalizing past models where either the observed process was monotone or the first passage time reduced to the moment of the first drop the joint transformation of the named random characteristics of the process are derived in a closed form the paper concludes with examples including numerical examples demonstrating the use of the results as well as practical applications to finance

multilayers in a modulated stochastic game we are concerned with an antagonistic stochastic game between two players a and b which finds applications in economics and warfare the actions of the players are manifested by a series of strikes of random magnitudes at random times exerted by each player against his opponent each of the assaults inflicts a random damage to enemy s vital areas in contrast with traditional games in our setting each player can endure multiple strikes before perishing predicting the ruin time exit of player a along with the total amount of casualties to both players at the exit is a main objective of this work in contrast to the time sensitive analysis earlier developed to refine the information on the game we insert auxiliary control levels which both players will cross in due game before the ruin of a this gives a and also b an additional opportunity to reevaluate his strategy and change the course of the game we formalize such a game and also allow the real time information about the game to be randomly delayed the delayed exit time cumulative casualties to both players and prior crossings are all obtained in a closed form joint functional

estimation of linear parametric models using inverse filter criteria and higher order statistics considers the problem of estimating the parameters of a stable scalar arma p q signal model causal or noncausal minimum phase or mixed phase driven by an i i d non gaussian sequence the driving noise sequence is not observed the wiggins donoho 1978 1991 class of inverse filter criteria for estimation of model parameters are analyzed and extended these criteria have been considered in the past only for moving average inverse filters these criteria are extended to general arma inverses computer simulation examples are presented to illustrate the proposed approaches

recognizable closures and submonoids of free partially commutative monoids abstract the main result of this paper is the presentation of two sufficient conditions for the closure x of x and the closure x of x respectively in any free partially commutative monoid for any regular language x to be recognizable several previously obtained sufficient conditions for x to be recognizable can be seen to imply one of those two conditions

design of 1 d and 2 d iir eigenfilters an effective method is proposed for the design of recursive digital filters in the time spatial domain the method is based on the computation of an eigenvalue and its corresponding eigenvector of an appropriate real symmetric and positive definite matrix derived from the objective error function of the impulse response in the least squares sense the method can be used to design 1d iir digital filters as well as 2d iir digital filters the method is easy and the performance is comparable to those of the existing methods one of the main advantages of the proposed method is that the solution is obtained directly without the need for iteration

two dimensional iir filter design with magnitude and phase error criteria two dimensional iir filters are designed using an error criterion consisting of the weighted sum of magnitude phase and stability errors thereby extending the spectral factorization based method to complex approximation the weights are adjusted to effectively constrain stability while minimizing a weighted sum of frequency domain magnitude and phase errors we use a derivative free variant of the marquardt optimization algorithm augmented by a one dimensional search we present examples of approximation of both linear and nonlinear phase ideal transfer functions and conduct comparisons across a range of numerator and denominator orders from fir to all pole while keeping the total number of coefficients approximately constant we find that the designed iir filters offer better magnitude response in the case of linear phase ideal functions even considering coefficient reflection symmetry but of course the fir filter has the better zero phase error when the ideal transfer function has nonlinear phase and hence no coefficient reflection symmetry we find that the designed iir filter can perform better than the fir filter with regard to both magnitude and phase error when magnitude error is weighted more heavily than phase error

rational relations and rational series a rational relation is a rational subset of the direct product of two free monoids r a b consider r as a function of a into the family of subsets of b by posing for all u a r u v b u v r assume r u is a finite set for all u a we study how the cardinality of r u behaves as the length of u tends to infinity and we show that there exists an infinite hierachy of growth functions

on the degree of ambiguity of finite automata we investigate the ambiguity behavior of finite automata in connection with their inner structure we show that the degree of ambiguity of a finitely ambiguous nondeterministic finite automaton nfa with n states is at most 5n2 nn there is a simple criterion which characterizes the infinite degree of ambiguity of an nfa and which is decidable in polynomial time the degree of growth of the ambiguity of an nfa is computable in polynomial time starting from the first result we discuss the maximal finite degree of ambiguity of an nfa with n states and we present subclasses of nfas where this quantity is of order 2 n

on the equivalence and containment problems for unambiguous regular expressions regular grammars and finite automata the known proofs that the equivalence and containment problems for regular expressions regular grammars and nondeterministic finite automata are pspace complete sm depend upon consideration of highly unambiguous expressions grammars and automata here we prove that such dependence is inherent deterministic polynomial time algorithms are presented for the equivalence and containment problems for unambiguous regular expressions unambiguous regular grammars and unambiguous finite automata the algorithms are then extended to ambiguity bounded by a fixed k our algorithms depend upon several elementary observations on the solutions of systems of homogeneous linear difference equations with constant coefficients and their relationship with the number of derivations of strings of a given length n by a regular grammar

an efficient approach for the synthesis of 2 d recursive fan filters using 1 d prototypes an elliptic approximation based design approach is proposed for obtaining 2 d recursive fan filters the 1 d elliptic filter is reduced to a cascade parallel combination of all pass sections and is then used as a prototype for fan filter synthesis resulting in final realization of 2 d transfer functions using allpass filters it is shown that the synthesis procedure not only gives a filter that has far fewer coefficients but also enjoys a very low computational complexity

layers of noncooperative games abstract we model and analyze classes of antagonistic stochastic games of two players the actions of the players are formalized by marked point processes recording the cumulative damage to the players at any moment of time the processes evolve until one of the processes crosses its fixed preassigned threshold of tolerance once the threshold is reached or exceeded at some point of the time exit time the associated player is ruined both stochastic processes are being observed by a third party point stochastic process over which the information regarding the status of both players is obtained we succeed in these goals by arriving at closed form joint functionals of the named elements and processes furthermore we also look into the game more closely by introducing an intermediate threshold see a layer which a losing player is to cross prior to his ruin in order to analyze the game more scrupulously and see what makes the player lose the game

phase equalization of one and two dimensional recursive filers an effective optimization algorithm is proposed to design multidimensional recursive phase equalizers the optimization algorithm uses the modified version of the rosenbrock 1960 pattern search method to minimize the maximum group delay ripple this numerical method eliminates the increased complexity that derivative methods show when multiple variables are to be found the criteria used to guarantee the stability of the equalizer are based on the decarlo strintzis theorem by applying this theorem the stability test procedure in m dimensions is reduced to m one dimensional stability test routines using this stabilization technique and the rosenbrock pattern search method for optimization the intensity of computations is considerably reduced in higher dimensions

algebraic approach to two dimensional recursive digital filter synthesis an effective algorithm is developed for synthesizing two dimensional recursive digital filters which approximate prescribed ideal frequency response specifications the algorithm is based on an algebraic approach that uses the eigenvalue eigenvector decomposition of the ideal filter s excitation response matrix in conjunction with a recently developed signal enhancement method this results in a recursive filter which has a unit impulse that closely approximates the ideal filter s unit impulse response illustrative examples and comparisons to an existing technique are included

range of fluctuation of brownian motion on a complete riemannian manifold we investigate the escape rate of the brownian motion w x t on a complete noncompact riemannian manifold assuming that the manifold has at most polynomial volume growth and that its ricci curvature is bounded below we prove that dist w x t x ct log t for all large t with probability 1 on the other hand if the ricci curvature is nonnegative and the volume growth is at least polynomial of the order n 2 then formula math again for all large t with probability 1 where e 0

relating the type of ambiguity of finite automata to the succinctness of their representation we consider the problem of how the size of a nondeterministic finite automaton nfa representing a regular language depends on the degree of ambiguity of the nfa we obtain results for the unary and bounded inputs and partial results for the unrestricted inputs one of the main results of this paper shows that for unrestricted inputs deterministic unambiguous and nondeterministic machines form a hierarchy with respect to the number of states solving an open problem of stearns and hunt we also propose a new approach to the study of the succinctness of representation through regularity preserving closure properties and obtain some results in this direction

deciding equivalence of finite tree automata it is shown that for every constant m it can be decided in polynomial time whether or not two m ambiguous finite tree automata are equivalent in general inequivalence for finite tree automata is dexptime complete with respect to logspace reductions and pspace complete with respect to logspace reductions if the automata in question are supposed to accept only finite languages for finite tree automata with weights in a field r a polynomial time algorithm is presented for deciding ambiguity equivalence provided r operations and r tests for 0 can be performed in constant time this result is used to construct an algorithm deciding ambiguity inequivalence of finite tree automata in randomized polynomial time finally for every constant m it is shown that it can be decided in polynomial time whether or not a given finite tree automaton is m ambiguous

on the decidability of some problems about rational subsets of free partially commutative monoids let i a b be a partially commutative alphabet such that two letters commute if one of them belongs to a and the other one belongs to b let m a b denote the free partially commutative monoid generated by i we consider the following six problems for rational given by regular expressions subsets x y of m q1 x y o q2 x y di x y q4 x m l25 m x finite x is recognizable it was proved by choffrut see 2 that all these problems are undecidable if card a 1 and card b 1 and they are decidable if card a card b 1 card u denotes the cardinality of u it was conjectured see 2 p 79 that these problems are decidable in the remaining cases where card a 1 and card b 1 in this paper we show that if card a 1 and card b 1 then the problem 01 is decidable and problems q2 06 are undecidable our paper is an application of results concerning reversal bounded nondeterministic multicounter machines and nondeterministic general sequential machines

a design method for two dimensional recursive digital filters a method for designing two dimensional symmetric half plane recursive digital filters is presented a filter is first designed as a parametrized family of one dimensional filters a simple approximation is then used to find a rational stable two dimensional filter some advantages and disadvantages of the method are discussed and several examples are given

on the finite valuedness problem for sequential machines abstract for a set a a 1 a m of n x n matrices with nonnegative integer entries let p a be the set of all finite products of matrices in a we show that there is a square space algorithm to decide given a whether or not p a is finite as a corollary we show that there is an exponential space algorithm to decide given a nondeterministic mealy sequential machine m with accepting states whether there exists an integer d 0 such that on every input the number of accepting computations of m producing distinct outputs is at most d if d exists the smallest such d can be computed in space exponential in size m log d the space bound reduces to polynomial for the analogous problem of ambiguity of nondeterministic finite acceptors

time sensitive analysis of independent and stationary increment processes we study the behavior of independent and stationary increments jump processes as they approach fixed thresholds the exact crossing time is unavailable because the real time information about successive jumps is unknown instead the underlying process a t is observed only upon a third party independent point process n the observed time series a n presents crude delayed data the crossing is first observed upon one of the observations denoted we develop and further explore a new technique to revive the real time paths of a t for all t belonging to an interval before the pre crossing observation 0 1 or between the observations just before and just after the crossing 1 as a joint laplace stieltjes transform and probability generating function of a t a 1 a 1 and joint probability distributions are obtained from the transforms in a tractable form and they are applied to modeling of stochastic networks under cyber attacks by accurately predicting their crash

string matching problems over free partially commutative monoids abstract this paper studies two string matching problems over free partially commutative monoids we analyze these two problems in detail and present two efficient polynomial time algorithms for solving them

the parallel complexity of finite state automata problems the goal of this paper is to study the exact complexity of several important problems concerning finite state automata and to classify the degrees of ambiguity of nondeterministic finite state automata our results are as follows 1 minimization of deterministic finite automata is nc 1 complete for nl 2 testing whether the degree of ambiguity of a nondeterministic finite automaton is exponential or polynomial or bounded is nc 1 complete for nl 3 checking whether a given nondeterministic finite automaton is unambiguous or k ambiguous is nc 1 complete for nl where k is some fixed constant 4 the bounded nonuniversality problem for nondeterministic finite automata which is the problem of deciding whether l m s n s n for a given nondeterministic finite automaton m and a unary integer n is log space complete for np 5 the bounded nonuniversality problem for unambiguous finite automata is in det the class of problems nc 1 reducible to computing the determinants of integer matrices and for deterministic finite automata it is nc 1 complete for nl 6 the inequivalence problems for unambiguous and k ambiguous finite automata are both in det where k is some fixed constant

the fully recursive filter a general 2 d recursive filter the conventional 2 d recursive filter can be thought of as performing a linear combination of 1 d filterings on past and present input and output data rows all of these component 1 d filters are fir except for that of the current output row in a fully recursive filter all the component 1 d filters are iir

symmetry exploitation in the design and implementation of recursive 2 d rectangularly sampled digital filters it is well known that exploiting the symmetries existing in the frequency response of 2 d filters results in a reduction in design and implementation complexities in this paper we explore how to exploit a general class of symmetries in the design and implementation of recursive two dimensional 2 d filters three types of filters are being studied namely causal factorizable noncausal and unfactorizable noncausal the capability of each of these filters to exhibit symmetries is discussed a method is then proposed to obtain the required form of the filter transfer function for exhibiting the different types of symmetries examples are solved both to illustrate the proposed method and to compare the performance of these filter types for the same implementation complexity

probabilistic estimation of the number of prefixes of a trace abstract we present a generating function technique to evaluate the number of strings of a given length recognized by a particular kind of finite state automaton using the method we determine some asymptotic estimations of the number of prefixes in free partially commutative monoids more precisely we prove that for every concurrent alphabet c assuming equiprobable strings all of length n the average number of prefixes of a trace of length n is n k o n k 1 and its variance is o n 2 k 1 where k is the number of components of the dependency graph c c and is a rational constant depending only on c these combinatorial results allow to determine the probabilistic behaviour of some algorithms for problems on trace languages including the membership problems for regular and context free trace languages

design of 2 d recursive digital filters using nonsymmetric half plane allpass filters a novel structure using recursive nonsymmetric half plane nshp digital allpass filters dafs is presented for designing 2 d recursive digital filters first several important properties of 2 d recursive dafs with nshp support regions for filter coefficients are investigated the stability of the 2 d recursive nshp dafs is guaranteed by using a spectral factorization based algorithm then the important characteristics regarding the proposed novel structure are discussed the design problem of 2 d recursive digital filters using the novel structure is considered we formulate the problem by forming an objective function consisting of the weighted sum of magnitude group delay and stability related errors a design technique using a trust region newton conjugate gradient method in conjunction with the analytic derivatives of the objective function is presented to efficiently solve the resulting optimization problem the novelty of the presented 2 d structure is that it possesses the advantage of better performance in designing a variety of 2 d recursive digital filters over existing 2 d filter structures finally several design examples are provided for conducting illustration and comparison