measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

estimating the errors on measured entropy and mutual information abstract information entropy and the related quantity mutual information are used extensively as measures of complexity and to identify nonlinearity in dynamical systems expressions for the probability distribution of entropies and mutual informations calculated from finite amounts of data exist in the literature but the expressions have seldom been used in the field of nonlinear dynamics in this paper formulae for estimating the errors on observed information entropies and mutual informations are derived using the standard error analysis familiar to physicists their validity is demonstrated by numerical experiment for illustration the formulae are then used to evaluate the errors on the time lagged mutual information of the logistic map

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

tsallis conditional mutual information in investigating long range correlation in symbol sequences abstract the presence of long range correlation lrc as opposed to markov chain of a finite order is of primary interest in the study of symbol sequences such as dna among others methods using information theory have been developed for markov chain order estimation in this work we consider the tsallis entropy and define the tsallis conditional mutual information tcmi for markov chain order estimation which is more suitable to identify large orders suggesting effectively lrc the tcmi of order m for a tsallis parameter q is computed for increasing m and a significance test is performed for each m until tcmi is found non significant randomization and parametric significance tests for tcmi are developed for the latter the null distribution of tcmi is approximated with a gamma distribution deriving analytic expressions for its parameters we assess the accuracy of order estimation with the two tests for tcmi and compare them to the respective tests using the shannon entropy extended simulations on markov chains of different orders and structures of the transition probability matrix show that shannon and tsallis tests with parameter q 2 have similar performance for small orders when the problem becomes more demanding for higher markov chain orders and lrcs tsallis tests for q 2 have better approximation to the correct order with the sequence length n finally the tsallis tests are favorably compared to shannon tests to real dna sequences

symbol synchronization for digital signals with multisegment trajectory digital data transition tracking loop the symbol synchronizer presented is an all digital data transition tracking loop which permits simple implementation and voltage control of the loop bandwidth the equivalent loop integration constant is changed by changing the threshold level of a comparator at the output of the loop phase error detector phase error probability distribution function its variance and the average probability of cycle slipping are analyzed analytically for the general case the theory of finite markov chains is used to analyze statistical properties of the tracking error primarily with regard to the phase error distribution and its variance the average probability of the cycle slipping the synchronizer time response and bandwidth as an example numerical results for generalized tamed frequency modulation gtfm with b 0 62 and r 0 36 are presented and discussed

interpolation in digital modems ii implementation and performance for pt i see ibid vol 41 no 3 p 502 208 1993 properties of a specific class of interpolators that are based upon polynomials are discussed several implementations are described one of which is particularly convenient in practical hardware simulations demonstrate that simple interpolators give excellent performance in many cases two point linear interpolation is adequate if better performance is needed classical four point third order polynomials could be used better yet a novel four point interpolating filter with piecewise parabolic impulse response can have performance superior to that of the standard cubic interpolator and still be implemented much more simply the nco based control method presented in part i is shown to be equivalent to a conventional phase locked loop and its operation is verified by simulation

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

markov chain order estimation with conditional mutual information abstract we introduce the conditional mutual information cmi for the estimation of the markov chain order for a markov chain of k symbols we define cmi of order m i c m as the mutual information of two variables in the chain being m time steps apart conditioning on the intermediate variables of the chain we find approximate analytic significance limits based on the estimation bias of cmi and develop a randomization significance test of i c m where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence the significance test is applied for increasing m and the markov chain order is estimated by the last order for which the null hypothesis is rejected we present the appropriateness of cmi testing on monte carlo simulations and compare it to the akaike and bayesian information criteria the maximal fluctuation method peres shields estimator and a likelihood ratio test for increasing orders using divergence the order criterion of cmi testing turns out to be superior for orders larger than one but its effectiveness for large orders depends on data availability in view of the results from the simulations we interpret the estimated orders by the cmi testing and the other criteria on genes and intergenic regions of dna chains

on some criteria for estimating the order of a markov chain tong 1975 has proposed a procedure for estimating the order of a markov chain based on akaike s information criterion aic in this paper the asymptotic distribution of the aic estimator is derived and it is shown that the estimator is inconsistent as an alternative to the aic procedure the bayesian information criterion bic proposed by schwarz 1978 is shown to be consistent these two procedures yield different estimated orders when applied to specific samples of meteorological observations for parameters based on these meteorological examples the aic and bic procedures are compared by means of simulation for finite samples the results obtained have practical implications concerning whether in the routine fitting of precipitation data it is necessary to consider higher than first order markov chains

nonuniform state space reconstruction and coupling detection we investigate the state space reconstruction from multiple time series derived from continuous and discrete systems and propose a method for building embedding vectors progressively using information measure criteria regarding past current and future states the embedding scheme can be adapted for different purposes such as mixed modeling cross prediction and granger causality in particular we apply this method in order to detect and evaluate information transfer in coupled systems as a practical application we investigate in records of scalp epileptic eeg the information flow across brain areas

the modified cramer rao bound and its application to synchronization problems we introduce the modified cramer rao bound crb which like the true crb is a lower bound to the error variance of any parameter estimator the modified crb proves useful when in addition to the parameter to be estimated the observed data also depend on other unwanted parameters the relationship between the modified and true crb is established and applications are discussed regarding the estimation of carrier frequency offset carrier phase and timing epoch in linearly modulated signals modified crbs for phase and timing estimation have been already discussed in previous works where it is shown that several practical carrier phase and clock recovery circuits do attain such bounds frequency discrimination instead is not so well represented in the literature and a significant contribution of this paper is the calculation of the modified crb for frequency estimation this bound is compared with the performance of some frequency detectors and it is concluded that further work is needed in search of more efficient frequency discrimination methods

sequence estimation and synchronization from nonsynchronized samples the authors study the problem of maximum likelihood sequence estimation and synchronization from samples of the output of a matched filter taken at integer multiples of the symbol rate which is assumed perfectly known by the receiver a general analysis is presented of sampled receivers that handle arbitrary baseband pulse shapes and arbitrary sampling rates it is observed that the optimal processing of the matched filter samples consists of digital interpolation followed by symbol by symbol decoding when sampling is at or above the nyquist rate or viterbi decoding when sampling is below the nyquist rate performance is studied through the cramer rao bound on mean square estimation error and a lower bound on error probability

derivation of gardner s timing error detector from the maximum likelihood principle in a recent paper 1 gardner described a new timing error detector for sampled receivers the detector derives timing information without decision direction from a bpsk or qpsk signal sampled at rate 2 t symbol interval t we will show how this detector can be derived from the maximum likelihood principle and thus give insight into the nature of the algorithm

mutual information functions versus correlation functions this paper studies one application of mutual information to symbolic sequences the mutual information functionm d this function is compared with the more frequently used correlation function d an exact relation betweenm d and d is derived for binary sequences for sequences with more than two symbols no such general relation exists in particular d 0 may or may not lead tom d 0 this linear but not general independence between symbols separated by a distance is studied for ternary sequences also included is the estimation of the finite size effect on calculating mutual information finally the concept of symbolic noise is discussed

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

efficient maximum likelihood based clock and phase estimators for oqpsk signals in this paper we propose an algorithm for joint carrier phase and timing estimation with oqpsk modulations the derivation is based on the maximum likelihood criterion and exploits a very efficient algorithm for the detection of differentially encoded m psk symbols already described in literature though we are mainly interested in measuring the phase and clock parameters estimates of the transmitted symbols are also obtained as by products the resulting scheme has a feedforward structure and provides phase and timing information in a fixed time differently from closed loop architectures it can be implemented in digital form and is particularly suitable for burst mode transmissions its performance is investigated by simulation and the results are compared with cramer rao bounds it turns out that the estimation accuracy is very close to the theoretical limits even with short observation intervals and small values of the excess bandwidth in such conditions the proposed estimators largely outperform other schemes already known in literature their superiority becomes less significant as the signal bandwidth increases

statistical analysis of gene and intergenic dna sequences abstract much of the on going statistical analysis of dna sequences is focused on the estimation of characteristics of coding and non coding regions that would possibly allow discrimination of these regions in the current approach we concentrate specifically on genes and intergenic regions to estimate the level and type of correlation in these regions we apply various statistical methods inspired from nonlinear time series analysis namely the probability distribution of tuplets the mutual information and the identical neighbour fit the methods are suitably modified to work on symbolic sequences and they are first tested for validity on sequences obtained from well known simple deterministic and stochastic models then they are applied to the dna sequence of chromosome 1 of arabidopsis thaliana the results suggest that correlations do exist in the dna sequence but they are weak and that intergenic sequences tend to be more correlated than gene sequences the use of statistical tests with surrogate data establish these findings in a rigorous statistical manner

dna viewed as an out of equilibrium structure the complexity of the primary structure of human dna is explored using methods from nonequilibrium statistical mechanics dynamical systems theory and information theory a collection of statistical analyses is performed on the dna data and the results are compared with sequences derived from different stochastic processes the use of 2 tests shows that dna can not be described as a low order markov chain of order up to r 6 although detailed balance seems to hold at the level of a binary alphabet it fails when all four base pairs are considered suggesting spatial asymmetry and irreversibility furthermore the block entropy does not increase linearly with the block size reflecting the long range nature of the correlations in the human genomic sequences to probe locally the spatial structure of the chain we study the exit distances from a specific symbol the distribution of recurrence distances and the hurst exponent all of which show power law tails and long range characteristics these results suggest that human dna can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the monte carlo rejection sampling method we construct a model dna sequence this method allows us to keep both long and short range statistical characteristics of the native dna data the model sequence presents the same characteristic exponents as the natural dna but fails to capture spatial correlations and point to point details 2014 american physical society

a class of data aided timing recovery schemes this paper develops a class of data aided timing recovery schemes whose structure and operating principle are reminiscent of zero forcing equalizer adaptation the schemes exhibit no pattern dependent jitter and are applicable to a variety of modulation codes and reception techniques they come in both oversampled and baud rate forms and permit maximum likelihood tracking performance whenever the oversampling factor satisfies the sampling theorem subject to this condition even simple schemes can be close to optimum it is shown that various existing schemes belong to the class under study or are closely related to it

symbol synchronization using signal samples and interpolation a method for symbol synchronization using two unsynchronized signal samples per symbol is presented signal values between the sample points are calculated by interpolation and then processed digitally to find a suitable strobe point the interpolated value at the strobe point is used for data detection or input to an equalizer simulations show that low variance timing estimates can be made for pulses with raised cosine spectra degraded by quadratic delay distortion

design of optimal interpolation filter for symbol timing recovery we propose an optimal interpolation filter for symbol timing recovery in a digital receiver where the input analog to digital a d conversion sampling clock is not synchronized to the transmitter symbol clock the optimal filter is designed by minimizing the mean square error mse at the output of the receiver assuming that the correct timing difference between the nonsynchronized input samples and the correct strobe synchronized to the symbol is given the mse minimization procedure results in a system of linear equations which can easily be solved to yield the optimal filter coefficients we also analyze the symbol tracking performance of a symbol synchronizer that employs the proposed optimal interpolation filter although the proposed optimal filter is designed assuming the availability of the correct timing difference we show that it minimizes the timing estimation error variance even when the computed timing difference values deviate from the ideal we also show that the timing estimation is virtually unbiased if the length of the interpolation filter is greater than or equal to 4 simulation results are included to show the performance improvement realizable by employing the optimal interpolation filter

performance study of a near maximum likelihood code aided timing recovery technique in this paper we propose a new code aided ca timing recovery algorithm for various linear constant modulus constellations based on the maximum likelihood ml estimator the first contribution is the derivation of a soft estimator expression of the transmitted symbol instead of its true or hard estimated value which is fed into the timing error detector ted equation the proposed expression includes the log likelihood ratios llrs obtained from a turbo decoder our results show that the proposed ca approach achieves almost as good results as the data aided da approach over a large interval of snr values while achieving a higher spectral efficiency we also derive the corresponding ca cramer rao bounds crb for various modulation orders contrarily to former work we develop here the crb analytical expression for different m psk modulation orders and validate them through comparison to empirical crb obtained by monte carlo iterations the proposed ca estimator realizes an important gain over the nondata aided approach nda and achieves a smaller gap when compared to its relative ca crb especially at moderate snr values where modern systems are constrained to work

the consistency of the bic markov order estimator the bayesian information criterion bic estimates the order of a markov chain with finite alphabet a from observation of a sample path x 1 x 2 x n as that value k k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term a k a 1 2 log n we show that k equals the correct order of the chain eventually almost surely as n oo thereby strengthening earlier consistency results that assumed an apriori bound on the order a key tool is a strong ratio typicality result for markov sample paths we also show that the bayesian estimator or minimum description length estimator of which the bic estimator is regarded as an approximation fails to be consistent for the uniformly distributed i i d process

on necessary and sufficient conditions for chaos to occur in duffing s equation an heuristic approach many different explanations have been offered for the occurrence of chaos in duffing s equation holmes has derived a criterion for the onset of chaos using melnikov theory that is based upon the initial entanglement of the unstable manifold with the stable limit cycle moon offered a criterion based on physical principles by considering how much energy a trajectory had to possess to get out of the potential well of one of the stable limit cycles in this paper conclusive numerical evidence is offered that in order for steady state chaos to occur in this system there must be a near intersection of the stable and unstable steady state limit cycles in the phase space of duffing s equation

a bpsk qpsk timing error detector for sampled receivers a simple algorithm for detection of timing error of a synchronous band limited bpsk or qpsk data stream is proposed the algorithm requires only two samples per symbol for its operation one of the two samples is also used for the symbol decision derivation of the s curve reveals a sinusoidal shape

extended gardner detector for improved symbol timing recovery of m psk signals in a recent paper the modified gardner detector has been proposed for advanced symbol timing recovery of m ary phase shift keying signals compared with the original algorithm the self noise jitter can be considerably reduced for highly bandlimited systems and smaller values of m in this letter a completely different approach is investigated which improves the self noise performance for larger values of m as well

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

a comparison of two types of symbol synchronizers for which self noise is absent for most symbol synchronizers operating on a pam waveform corrupted by noise the timing error consists of a noisedependent term and a data dependent term when the signal to noise ratio snr is high the data dependent term which is often called self noise dominates and the timing error variance becomes independent of snr in this paper we present two types of symbol synchronizers for which self noise is absent the modified maximum likelihood mml synchronizer and the minimum mean square error mmse synchronizer as a consequence of the absence of self noise their timing error variances at high snr are inversely proportional to the snr the performances of these synchronizers are evaluated and compared with the cramer rao bound

bispectral analysis of possessing chaotic motion abstract bispectral analyses of the equations producing a chaotic rossler attractor and the chaotic trajectories of a one mode galerkin approximation of the magnetically buckled beam are presented yielding interesting consequences concerning chaotic system identification for driven oscillators and autonomous systems bicoherence spectra isolate the phase coupling between increasing numbers of triads of fourier modes for a pure period doubling sequence route to chaos for the rossler equations bicoherences from a period doubling intermittency catastrophe route to chaotic motion are also observed for single frequency excitation for the buckled beam although the bicoherence successfully characterizes important aspects of the non linearity of the rossler attractor in the nonchaotic and chaotic regimes where the primary non linearity is quadratic it only partially characterizes the magnetically buckled beam system because the dominant non linearity is cubic

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow