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Estimating and Interpreting the Instantaneous Frequency of a Signal  Part 1: Fundamentals
 PROCEEDINGS OF THE IEEE
, 1992
"... The frequency of a sinusoidal signal is a well defined quantity. However, often in practice, signals are not truly sinusoidal, or even aggregates of sinusoidal components. Nonstationary signals in particular do not lend themselves well to decomposition into sinusoidal components. For such signals, t ..."
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Cited by 263 (9 self)
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The frequency of a sinusoidal signal is a well defined quantity. However, often in practice, signals are not truly sinusoidal, or even aggregates of sinusoidal components. Nonstationary signals in particular do not lend themselves well to decomposition into sinusoidal components. For such signals, the notion of frequency loses its effectiveness, and one needs to use a parameter which accounts for the timevarying nature of the process. This need has given rise to the idea of instantaneous frequency. The instantaneous frequency (IF) of a signal is a parameter which is often of significant practical importance. In many situations such as seismic, radar, sonar, communications, and biomedical applications, the IF is a good descriptor of some physical phenomenon. This paper discusses the concept of instantaneous frequency, its definitions, and the correspondence between the various mathematical models formulated for representation of IF. The paper also considers the extent to which the IF corresponds to our intuitive expectation of reality. A historical review of the successive attempts to define the IF is presented. Then the relationships between the IF and the groupdelay, analytic signal, and bandwidthtime (BT) product are explored, as well as the relationship with timefrequency distributions. Finally, the notions of monocomponent and multicomponent signals, and instantaneous bandwidth are discussed. It is shown that all these notions are well described in the context of the theory presented.
Iterated random functions
 SIAM Review
, 1999
"... Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys ..."
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Cited by 223 (2 self)
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Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The
Frequency content of randomly scattered signals
 PART I, WAVE MOTION
, 1990
"... The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equatio ..."
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Cited by 96 (23 self)
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The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equations that characterize multiply scattered signals observed at the surface of the layered medium. The results of extensive numerical simulations are presented, illustrating the scope of the theory. A number of inverse problems for randomly layered media are also formulated where we
Improving Regression Estimation: Averaging Methods for Variance Reduction with Extensions to General Convex Measure Optimization
, 1993
"... ..."
Interdisciplinary application of nonlinear time series methods
 Phys. Reports
, 1998
"... This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situatio ..."
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Cited by 84 (4 self)
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This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.
Strong invariance principles for dependent random variables
 ANNALS PROBA
, 2007
"... We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions. ..."
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Cited by 64 (8 self)
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We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.
Multivariate autoregressive modeling of fmri time series. NeuroImage
, 1477
"... We propose the use of Multivariate Autoregressive (MAR) models of fMRI time series to make inferences about functional integration within the human brain. The method is demonstrated with synthetic and real data showing how such models are able to characterise interregional dependence. We extend lin ..."
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Cited by 55 (10 self)
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We propose the use of Multivariate Autoregressive (MAR) models of fMRI time series to make inferences about functional integration within the human brain. The method is demonstrated with synthetic and real data showing how such models are able to characterise interregional dependence. We extend linear MAR models to accommodate nonlinear interactions to model topdown modulatory processes with bilinear terms. MAR models are time series models and thereby model temporal order within measured brain activity. A further benefit of the MAR approach is that connectivity maps may contain loops, yet exact inference can proceed within a linear framework. Model order selection and parameter estimation are implemented using Bayesian methods. 2 1
Parametric timedomain methods for nonstationary random vibration modelling and analysis A critical survey and comparison
 Mechanical Systems and Signal Processing
"... An overview and comparison of parametric timedomain methods for nonstationary random vibration modelling (identication) and analysis based upon a single vibration signal realization is presented. The considered methods are based upon Timedependent AutoRegressive Moving Average (TARMA) represent ..."
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Cited by 41 (19 self)
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An overview and comparison of parametric timedomain methods for nonstationary random vibration modelling (identication) and analysis based upon a single vibration signal realization is presented. The considered methods are based upon Timedependent AutoRegressive Moving Average (TARMA) representations, and may be classied as unstructured parameter evolution, stochastic parameter evolution, and deterministic parameter evolution. The main methods within each class are presented, and model structure selection is discussed. The methods are compared, via a Monte Carlo study, in terms of achievable model parsimony, prediction accuracy, power spectral density and modal parameter accuracy and tracking, computational simplicity, and ease of use. The results conrm the increased accuracy and performance characteristics of the deterministic, as well as stochastic, parameter evolution methods over those of their unstructured parameter evolution counterparts. 1