performance analysis of an adaptive notch filter with constrained poles and zeros a statistical analysis of the sine wave frequency estimate obtained by the adaptive notch filter anf recently proposed by a nehorai ibid vol assp 33 p 983 96 aug 1985 is presented asymptotically valid expressions for the bias and mean square error of the frequency estimate are provided the analysis verifies the performance exhibited by the anf and clarifies the role of some user variables such as the notch bandwidth and the number of data samples processed

upper and lower conditional probabilities induced by a multivalued mapping abstract given a finitely additive full conditional probability space x f f 0 and a conditional measurable space y g g 0 a multivalued mapping from x to y induces a class of full conditional probabilities on y g g 0 a closed form expression for the lower and upper envelopes and of such class is provided the envelopes can be expressed through a generalized bayesian conditioning rule relying on two linearly ordered classes of possibly unbounded inner and outer measures for every b g 0 b is a normalized totally monotone capacity which is continuous from above if x f f 0 is a countably additive full conditional probability space and f is a algebra moreover the full conditional prevision functional m induced by on the set of f continuous conditional gambles is shown to give rise through to the lower and upper full conditional prevision functionals m and m on the set of g continuous conditional gambles for every b g 0 m b is a totally monotone functional having a choquet integral expression involving finally by considering another conditional measurable space z h h 0 and a multivalued mapping from y to z it is shown that the conditional measures and functionals m m induced by preserve the same properties of and m m

weak approachability in this paper we study repeated games with vector payoffs following blackwell 2 we define weak approachability and its dual property weak excludability we use results from differential games with fixed duration to prove that every set is either weakly approachable or weakly excludable

two algorithms for adaptive retrieval of slowly time varying multiple cisoids in noise two algorithms for tracking parameters of slowly varying multiple complex sine waves cisoids in noise the multiple frequency tracker and the adaptive notch filter are described for high signal to noise ratio snr the properties of the algorithms i e stability noise rejection and tracking speed are studied analytically using a linear filter approximation technique the tradeoff between noise rejection and tracking error for both algorithms is shown to be similar different choices of the design variables are discussed namely i minimal mean square estimation error for random walk modeled frequency variations and ii minimal stationary estimation variance subject to a given tracking delay

subgame perfection in stochastic games with perfect information and recursive payoffs we consider a class of n player stochastic games with the following properties 1 in every state the transitions are controlled by one player 2 the payoffs are equal to zero in every non absorbing state 3 the payoffs are non negative in every absorbing state with respect to the expected average reward we provide a constructive proof that a subgame perfect equilibrium exists in pure strategies for every 0 moreover if all transitions of a game in our class are deterministic then the game has a subgame perfect 0 equilibrium in pure strategies

tracking properties of a gradient based second order adaptive iir notch filter with constrained poles and zeros gradient type adaptive iir notch filters have many attractive merits for various real life applications since they require a small number of computations and yet demonstrate practical performance however it is generally quite difficult to assess their performance analytically their tracking properties in particular have not yet been investigated in this paper the tracking performance of a plain gradient pg algorithm is analyzed in detail for a second order adaptive iir notch filter with constrained poles and zeros which takes a linear chirp signal as its input first two sets of difference equations for the frequency tracking error and mean square error mse are established in the sense of convergence in the mean and convergence in the mean square respectively closed form expressions for the asymptotic tracking error and mse are then derived from these difference equations an optimum step size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking mse it is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size which is an interesting property for a real valued adaptive filtering algorithm extensive simulations are performed to support the analytical findings

spectrum estimation notch filters and music a novel extension of the multiple signal classification music algorithm for the frequency estimation problem is proposed in this work it is shown that the music algorithm is a data dependent optimum notch filter design technique when the input data are passed through this optimum notch filter the signal to noise ratio snr of the output is minimized the zeros of the optimum notch filter are then used to estimate the frequencies in the classical music algorithm this optimization is carried out over the set of all finite impulse response fir filters having a fixed order the framework of this paper allows the use of rational notch filters the fact that the rational filters have sharper spectral characteristics is used to obtain improved estimates particularly at low snr some interesting properties of the proposed algorithm are derived and further perspectives on the implementation aspects are provided the analytical predictions are substantiated using numerical simulation results

a minimal parameter adaptive notch filter with constrained poles and zeros a new algorithm is presented for adaptive notch filtering and parametric spectral estimation of multiple narrow band or sine wave signals in an additive broad band process the algorithm is of recursive prediction error rpe form and uses a special constrained model of infinite impulse response iir with a minimal number of parameters the convergent filter is characterized by highly narrow bandwidth and uniform notches of desired shape for sufficiently large data sets the variances of the sine wave frequency estimates are of the same order of magnitude as the cramer rao bound results from simulations illustrate the performance of the algorithm under a wide range of conditions

subgame perfection in quitting games with perfect information and differential equations we introduce a new approach to studying subgame perfect equilibrium payoffs in stochastic games the differential equations approach we apply our approach to quitting games with perfect information those are sequential games in which at every stage one ofn players is chosen each player is chosen with probability 1 n the chosen playeri decides whether to quit in which case the game terminates and the terminal payoff is some vectora i rn or whether to continue in which case the game continues to the next stage if no player ever quits the payoff is some vectora rn we define a certain differential inclusion prove that it has at least one solution and prove that every vector on a solution of this differential inclusion is a subgame perfect equilibrium payoff

performance analysis of the sign algorithm for a constrained adaptive iir notch filter adaptive infinite impulse response iir notch filters are very attractive in terms of their reasonable performances and low computational requirements generally it is very difficult to assess their performances analytically due to their iir nature this paper analyzes in detail the steady state performance of the sign algorithm sa for a well known adaptive iir notch filter with constrained poles and zeros slow adaptation and gaussianity of the notch filter output are assumed for the sake of analysis two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm steady state estimation error or bias and mean square error mse of the sa are then derived in closed forms a coarse stability bound is also derived for the algorithm theory based comparison between the algorithm and the plain gradient pg algorithm is done in some detail extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations

n monotone exact functionals we study n monotone functionals which constitute a generalisation of n monotone set functions we investigate their relation to the concepts of exactness and natural extension which generalise coherence and natural extension in the behavioural theory of imprecise probabilities we improve upon a number of results in the literature and prove among other things a representation result for exact n monotone functionals in terms of choquet integrals

recursive repeated games with absorbing states we show the existence of stationary limiting average e equilibria e 0 for two person recursive repeated games with absorbing states these are stochastic games where all states but one are absorbing and in the nonabsorbing state all payoffs are equal to zero a state is called absorbing if the probability of a transition to any other state is zero for all available pairs of actions for the purpose of our proof we introduce properness for stationary strategy pairs our result is sharp since it extends neither to the case with more nonabsorbing states nor to the n person case with n 2 moreover it is well known that the result cannot be strengthened to the existence of 0 equilibria and that repeated games with absorbing states generally do not admit stationary e equilibria

repeated games with lack of information on one side we introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by de meyer 1996 a traditional way to study the value of differential games is through discrete time approximations here we follow the opposite approach we identify the limit value of a repeated game in discrete time as the value of a differential game namely we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property the value of the dual differential game exists and is the unique viscosity solution of a first order derivative equation which appears implicitely in de meyer and rosenberg 1999 with a limit condition we identify the solution by translating viscosity properties in the primal

perfect information games with lower semicontinuous payoffs we prove that every multiplayer perfect information game with bounded and lower semicontinuous payoffs admits a subgame perfect e equilibrium in pure strategies this result complements example 3 in solan and vieille solan e n vieille 2003 deterministic multi player dynkin games j math econom 39 911 929 which shows that a subgame perfect e equilibrium in pure strategies need not exist when the payoffs are not lower semicontinuous in addition if the range of payoffs is finite we characterize in the form of a folk theorem the set of all plays and payoffs that are induced by subgame perfect 0 equilibria in pure strategies

conditional belief functions as lower envelopes of conditional probabilities in a finite setting the aim is to provide a characterization of full conditional measures on a finite boolean algebra obtained as lower envelope of the extensions of a full conditional probability defined on another finite boolean algebra such conditional measures are conditional belief functions defined by means of a generalized bayesian conditioning rule relying on a linearly ordered class of belief functions this notion of bayesian conditioning for belief functions is compared with other well known conditioning rules by looking for those conditional measures that can be seen as lower conditional probabilities

factorized all pass based iir adaptive notch filters this paper introduces a new family of infinite impulse response adaptive notch filters that forms multiple notches using a second order factorization of an all pass transfer function the new orthogonal realization is amenable for adaptive filtering to obtain the unknown frequencies of interest two new adaptive filtering algorithms are presented that can achieve fast convergence at low computational cost local convergence analysis for the new algorithms is performed and a detailed discussion of their properties is provided the new all pass based notch realization introduces a different compromise between bias and signal to noise ratio snr when compared with realizations previously reported in the literature specifically it achieves lower bias than other approaches at low snr this property is particularly attractive for the estimation and tracking of multiple sinusoids furthermore the bias can be made arbitrarily small or can be accurately estimated and compensated for extensive computer simulations are provided to illustrate the performance of the proposed adaptive notch filters in terms of bias speed of convergence and tracking capability

three player absorbing games an n player absorbing game is an n player stochastic game where all the states but one are absorbing a state is absorbing if once it is reached the probability to leave it is zero whatever the players play we prove that every three player absorbing game has an undiscounted equilibrium payoff

finitely additive dynamic programming the theory of dynamic programming is formulated using finitely additive probability measures defined on sets of arbitrary cardinality many results from the conventional countably additive theory generalize and the proofs are simpler

tracking characteristics of the constrained iir adaptive notch filter the tracking behavior of a constrained iir notch filter the coefficients of which are estimated using a constant step size gauss newton algorithm is studied using weak convergence theory and the concept of prescaling it is shown that the mean behavior of the adaptive filter coefficients can be studied by using an ordinary differential equation ode approximate and simple closed form results are derived for the tracking behavior of a second order notch filter computer simulations are presented to substantiate the analysis

comments on a robust state observer scheme in this note we point out that the sufficient conditions given in a recent paper to assure convergence of the proposed robust state observer are incomplete

comparative study of four adaptive frequency trackers we study and compare four algorithms for adaptive retrieval of slowly time varying multiple cisoids in noise the adaptive notch filter the multiple frequency tracker the adaptive estimation scheme and the hyperstable adaptive line enhancer the local behavior of the algorithms in a neighborhood of their equilibrium state assuming high signal to noise ratio snr and large data sample for a two cisoid signal is treated in a similar way to the linear filter approximation technique used for a single cisoid case the validity of the results is confirmed by computer simulations

two player nonzero sum stopping games in discrete time we prove that every two player non zero sum stopping game in discrete time admits an equilibrium in randomized strategies for every 0 we use a stochastic variation of ramsey theorem which enables us to reduce the problem to that of studying properties of equilibria in a simple class of stochastic games with finite state space

an adaptive notch filter with improved tracking properties an analysis of the properties of an adaptive notch filter anf applied to time varying frequency tracking is presented starting from the derivation of an expression for anf output power asymptotically optimal values for the pole contraction and forgetting factors are derived for recursive prediction error rpe type anf algorithms based on the obtained results a new anf algorithm that includes adaptation of both pole contraction and forgetting factors is proposed the given experimental results confirm the theoretical conclusions and show that the proposed algorithm is highly efficient in practice

posterior cramer rao bound for adaptive harmonic retrieval the problem of adaptive parameter estimation for a single nonstationary noisy cisoid where the sinusoidal frequency evolves according to gaussian random walks is studied the lower bound on the minimum mean square estimation error which was derived by van trees 1968 is evaluated for the problem it is shown that two estimation methods attain this lower bound

lower previsions induced by filter maps coherent lower previsions are amongst the most prominent uncertainty models within imprecise probability theory we show how they can model information about the joint behaviour of two variables when these are related by means of a filter map a model for the imprecise observation of a random variable by means of a class of filters our construction preserves a number of interesting properties such as n monotonicity and it generalises a number of existing results for multi valued mappings

statistically efficient smoothing algorithm for time varying frequency estimation the problem of extraction elimination of a nonstationary sinusoidal signal from noisy measurements is considered this problem is usually solved using adaptive notch filtering anf algorithms it is shown that the accuracy of frequency estimates can be significantly increased if the results obtained from anf are backward time filtered by an appropriately designed lowpass filter the resulting adaptive notch smoothing ans algorithm can be employed to perform many offline signal processing tasks such as elimination of sinusoidal interference from a prerecorded signal for example we show that when the unknown signal frequency drifts according to the random walk model the optimally tuned ans algorithm is under gaussian assumptions statistically efficient i e it attains the cramer rao type lower smoothing bound which limits accuracy of any whether causal or not frequency estimation scheme

a topological approach to quitting games this paper presents a question of topological dynamics and demonstrates that its affirmation would establish the existence of approximate equilibria in all quitting games with only normal players a quitting game is an undiscounted stochastic game with finitely many players where every player has only two moves to end the game with certainty or to allow the game to continue if nobody ever acts to end the game all players receive payoffs of 0 a player is normal if and only if by quitting alone she receives at least her min max payoff this proof is based on a version of the kohlberg mertens kohlberg e j f mertens 1986 on the strategic stability of equilibria econometrica 54 5 1003 1037 structure theorem designed specifically for quitting games

unbiased estimation of a sinusoid in colored noise via adapted notch filters standard notch models for the estimation of a sinusoid in noise are characterized by two poles on the unit circle and two zeros aligned with the poles these models provide biased estimates of the unknown frequency the bias increasing with the zero pole distance on the other hand keeping the zeros far from the poles has the beneficial effect of making the identification algorithm less sensitive to the adopted initialization in this paper we remove the pole zero alignment constraint the extra degree of freedom thus obtained is used to alleviate the traditional trade off between unbiasedness of the estimates and robustness against poor initializations the proposed technique applies to the case of both white and colored additive noise

funnel filters a new class of filters for frequency estimation of harmonic signals second order notch filters are commonly used for frequency estimation purposes they are characterized by a single design parameter the value of which is selected so as to achieve a suitable compromise between tracking performance and bias variance errors in this paper a new class of filters of fourth order is presented the funnel shaped filters or funnel filters for short these filters enables one to estimate harmonic signals with enhanced tracking capability without any deterioration of the steady state performance

subgame perfection in positive recursive games with perfect information we consider a class of n player stochastic games with the following properties 1 in every state the transitions are controlled by one player 2 the payoffs are equal to zero in every nonabsorbing state 3 the payoffs are nonnegative in every absorbing state we propose a new iterative method to analyze these games with respect to the expected average reward we prove the existence of a subgame perfect e equilibrium in pure strategies for every e 0 moreover if all transitions are deterministic we obtain a subgame perfect 0 equilibrium in pure strategies

closed form unbiased frequency estimation of a noisy sinusoid using notch filters in this note the problem of the frequency estimation of a sinusoid embedded in white noise is considered the approach used herein is the minimization of the sample variance of the output of constrained notch filters fed by the noisy sinusoid in particular this note focuses on closed form expressions of the frequency estimate which can be obtained using notch filters having an all zeros finite impulse response fir structure the results presented in this note are as follows 1 it is shown that the fir notch filters obtained from standard second order infinite impulse response iir filters are inadequate 2 a new second order iir notch filter is proposed which provides an unbiased estimate of the frequency 3 the fir filter obtained from the new iir filter provides a closed form unbiased frequency estimate and 4 the closed form frequency estimate obtained using the new fir notch filter asymptotically converges toward the pisarenko harmonic decomposition estimator and the yule walker estimator

random sets as imprecise random variables given a random set coming from the imprecise observation of a random variable we study how to model the information about the probability distribution of this random variable specifically we investigate whether the information given by the upper and lower probabilities induced by the random set is equivalent to the one given by the class of the probabilities induced by the measurable selections together with sufficient conditions for this we also give examples showing that they are not equivalent in all cases

discounted stochastic games we give an alternative proof to a result of mertens and parthasarathy stating that every n player discounted stochastic game with general setup and with a norm continuous transition has a subgame perfect equilibrium

subgame perfect ϵ equilibria in perfect information games with common preferences at the limit we prove the existence of a pure subgame perfect epsilon equilibrium for every epsilon 0 in multiplayer perfect information games provided that the payoff functions are bounded and exhibit common preferences at the limit if in addition the payoff functions have finite range then there exists a pure subgame perfect 0 equilibrium these results extend and unify recent existence theorems for bounded and semicontinuous payoffs

correlated equilibrium in quitting games a quitting game is a sequential game where each player has two actions to continue or to quit the game continues as long as all players decide to continue the moment at least one player decides to quit the game terminates the terminal payoff depends on the subset of players who quit at the terminating stage if the game continues forever then the payoff for the players is some fixed payoff vector we prove that every quitting game admits a correlated uniform e equilibrium a uniform e equilibrium in an extended game that includes a correlation device that sends one signal to each player before start of play

multiple fully adaptive notch filter design based on allpass sections we develop a canonical adaptive cascade structure iir notch filter to detect and track multiple time varying frequencies in additive white gaussian noise the algorithm uses allpass frequency transformation filters and a truncated gradient simulations indicate that our algorithm is computationally simple converges rapidly and has good frequency resolution

stopping games with randomized strategies we study stopping games in the setup of neveu we prove the existence of a uniform value in a sense defined below by allowing the players to use randomized strategies in constrast with previous work we make no comparison assumption on the payoff processes moreover we prove that the value is the limit of discounted values and we construct e optimal strategies

adaptive comb filtering for harmonic signal enhancement a new algorithm is presented for adaptive comb filtering and parametric spectral estimation of harmonic signals with additive white noise the algorithm is composed of two cascaded parts the first estimates the fundamental frequency and enhances the harmonic component in the input and the second estimates the harmonic amplitudes and phases performance analysis provides new results for the asymptotic cramer rao bound crb on the parameters of harmonic signals with additive white noise results of simulations indicate that the variances of the estimates are of the same order of magnitude as the crb for sufficiently large data sets and illustrate the performance in enhancing noisy artificial periodic signals

on the parametrization and design of an extended kalman filter frequency tracker the problem of estimating the frequency of a harmonic signal embedded in broad band noise is considered the paper focuses on the extended kalman filter frequency tracker which is the application of the extended kalman filter ekf framework to the frequency estimation problem the ekf frequency tracker recently proposed in the literature is characterized by a vector of three design parameters q r spl epsi whose role and tuning is still a controversial and unclear issue in this paper it is shown that a wise parametrization of the extended kalman frequency tracker is characterized by just one parameter the spl epsi must be set to zero to achieve the basic property of unbiasedness in a noise free setting the performances of the tracker are not influenced independently by q and r and what really matters is the ratio spl lambda r q only the proposed simplification of the extended kalman filter frequency tracker allows an easier and more transparent tuning of its tracking behavior

generalized adaptive notch and comb filters for identification of quasi periodically varying systems the problem of identification tracking of quasi periodically varying real valued systems is considered this problem is a generalization to the system case of a classical signal processing task of either elimination or extraction of nonstationary sinusoidal signals buried in noise the solution is based on the exponentially weighted basis function ewbf approach the proposed algorithms are capable of tracking slow changes in system frequencies which means that not only the expansion coefficients in the basis function description of the analyzed system but also the basis functions themselves are adjusted in an adaptive manner first assuming that the system frequencies are known and constant the running basis and fixed basis variants of the ewbf algorithm are derived and their relationship to the classical notch filter with constrained poles and zeros is established next the frequency adaptive versions of both algorithms are obtained using the gradient search and recursive prediction error principles respectively finally the interrelated frequencies case is analyzed and two additional parameter tracking algorithms generalized adaptive comb filters are derived

tracking analysis of an adaptive notch filter with constrained poles and zeros this paper analyzes the asymptotic tracking properties of an adaptive notch filter anf with pole zero constraints for the cancellation or retrieval of multiple time varying sine waves in additive noise the asymptotic mean square error mse is analyzed using the methods of ljung and gunnarsson 1990 when the variations in the underlying frequencies are assumed to be sufficiently small closed form expressions for the mse are derived as functions of the tuning variables of the algorithm the results give insight into the operational properties of the algorithm and are used in order to minimize the mse with respect to the tuning variables computer simulations confirm the validity of the derived results