polarization phase relationships via multiple morse wavelets ii data analysis three examples are provided of the use of multiple generalized morse wavelets for polarization analysis of three component recordings where phase relationships between components of the signal are transient the example three component signals are i synthetic ii the radial transverse and vertical components of an earthquake and iii the radial transverse and normal components of the solar magnetic field measured by the ulysses spacecraft upstream of an interplanetary shock working in the scale time plane a b the multiple wavelet approach detects coherent motion with elliptical polarization in all three cases our graphical presentations fully take into account restrictions imposed by two factors the minimum usable scale and wraparound effects caused by fourier transform computations and consequently a maximum usable scale

domain decomposition schemes with high order accuracy and unconditional stability parallel finite difference schemes with high order accuracy and unconditional stability for solving parabolic equations are presented the schemes are based on domain decomposition method i e interface values between subdomains are computed by the explicit scheme interior values are computed by the implicit scheme the numerical stability and error are derived in the h 1 norm in one dimensional case numerical results of both one and two dimensions examining the stability accuracy and parallelism of the procedure are also presented

generalized morse wavelets this paper examines the class of generalized morse wavelets which are eigenfunction wavelets suitable for use in time varying spectrum estimation via averaging of time scale eigenscalograms generalized morse wavelets of order k the corresponding eigenvalue order depend on a doublet of parameters spl beta spl gamma we extend results derived for the special case spl beta spl gamma 1 and include a proof of the resolution of identity the wavelets are easy to compute using the discrete fourier transform dft and for spl beta spl gamma 2m 2 can be computed exactly a correction of a previously published eigenvalue formula is given this shows that for spl gamma 1 generalized morse wavelets can outperform the hermites in energy concentration contrary to a conclusion based on the spl gamma 1 case for complex signals scalogram analyses must be carried out using both the analytic and anti analytic complex wavelets or odd and even real wavelets whereas for real signals the analytic complex wavelet is sufficient

bivariate instantaneous frequency and bandwidth the generalizations of instantaneous frequency and instantaneous bandwidth to a bivariate signal are derived these are uniquely defined whether the signal is represented as a pair of real valued signals or as one analytic and one anti analytic signal a nonstationary but oscillatory bivariate signal has a natural representation as an ellipse whose properties evolve in time and this representation provides a simple geometric interpretation for the bivariate instantaneous moments the bivariate bandwidth is shown to consist of three terms measuring the degree of instability of the time varying ellipse amplitude modulation with fixed eccentricity eccentricity modulation and orientation modulation or precession an application to the analysis of data from a free drifting oceanographic float is presented and discussed

a high order parallel finite difference algorithm to solve the heat equation on parallel computers a high order parallel finite difference algorithm is presented in this procedure dividing the space domain into several sub domains we calculate the interface values between sub domains by the classical explicit scheme then solve the interior values of sub domains by the forth order compact scheme in parallel the stability bound of the procedure is derived to be 1 63 times that of the classical explicit scheme and the convergence rate is proved to be of order three numerical examples show that this method has much better accuracy than other known methods

analysis of singularities from modulus maxima of complex wavelets complex valued wavelets are normally used to measure instantaneous frequencies while real wavelets are normally used to detect singularities we prove that the wavelet modulus maxima with a complex valued wavelet can detect and characterize singularities this is an extension of the previous wavelet work of mallat and hwang on modulus maxima using a real wavelet with this extension we can simultaneously detect instantaneous frequencies and singularities from the wavelet modulus maxima of a complex valued wavelet some results of singularity detection with the modulus maxima from a real wavelet and an analytic complex valued wavelet are compared we also demonstrate that singularity detection methods can be employed to detect the corners of a planar object

element analysis a wavelet based method for analysing time localized events in noisy time series a method is derived for the quantitative analysis of signals that are composed of superpositions of isolated time localized events here these events are taken to be well represented as rescaled and phase rotated versions of generalized morse wavelets a broad family of continuous analytic functions analysing a signal composed of replicates of such a function using another morse wavelet allows one to directly estimate the properties of events from the values of the wavelet transform at its own maxima the distribution of events in general power law noise is determined in order to establish significance based on an expected false detection rate finally an expression for an event s region of influence within the wavelet transform permits the formation of a criterion for rejecting spurious maxima due to numerical artefacts or other unsuitable events signals can then be reconstructed based on a small number of isolated points on the time scale plane this method termed element analysis is applied to the identification of long lived eddy structures in ocean currents as observed by along track measurements of sea surface elevation from satellite altimetry

explicit implicit conservative domain decomposition procedures for parabolic problems based on block centered finite differences domain decomposition procedures for solving parabolic equations are considered the underlying discretization consists of block centered finite differences in these procedures fluxes at subdomain interfaces are calculated from the solution at the previous time level these fluxes serve as neumann boundary data for implicit block centered discretizations in the subdomains a priori error estimates are presented and numerical results examining the stability accuracy and parallelism of the scheme are presented

on instantaneous amplitude and phase of signals in many questions of signal processing it is important to use the concepts of instantaneous amplitude or phase of signals this is especially the case in communication systems with amplitude or frequency modulation these concepts are often introduced empirically however it is well known that the correct approach for this purpose is to use the concept of analytic signal starting from this point we show some examples of contradictions appearing when using other definitions of instantaneous amplitude or frequency that are commonly admitted this introduces the problem of characterizing pure amplitude modulated or pure phase modulated signals it is especially shown that whereas amplitude modulated signals can be characterized by spectral considerations this is no longer the case for phase modulated signals furthermore signals with constant amplitude have very specific properties which are analyzed in detail some consequences and extensions to random signals are finally discussed

diffusion equations for a markovian jumping process we consider a markovian jumping process which is defined in terms of the jump size distribution and the waiting time distribution with a position dependent frequency in the diffusion limit we assume the power law form for the frequency for small steps we derive the fokker planck equation and show the presence of the normal diffusion subdiffusion and superdiffusion for the l evy distribution of the step size we construct a fractional equation which possesses a variable coefficient and solve it in the diffusion limit then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance we also solve the master equation numerically and demonstrate that there are deviations from the l evy stable distribution for large wave numbers

analysis of modulated multivariate oscillations the concept of a common modulated oscillation spanning multiple time series is formalized a method for the recovery of such a signal from potentially noisy observations is proposed and the time varying bias properties of the recovery method are derived the method an extension of wavelet ridge analysis to the multivariate case identifies the common oscillation by seeking at each point in time a frequency for which a bandpassed version of the signal obtains a local maximum in power the lowest order bias is shown to involve a quantity termed the instantaneous curvature which measures the strength of local quadratic modulation of the signal after demodulation by the common oscillation frequency the bias can be made to be small if the analysis filter or wavelet can be chosen such that the signal s instantaneous curvature changes little over the filter time scale an application is presented to the detection of vortex motions in a set of freely drifting oceanographic instruments tracking the ocean currents

generalized morse wavelets as a superfamily of analytic wavelets the generalized morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets subsuming eight apparently distinct types of analysis filters into a single common form this superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications in addition to a parameter controlling the time domain duration or fourier domain bandwidth the wavelet shape with fixed bandwidth may be modified by varying a second parameter called for integer values of the most symmetric most nearly gaussian and generally most time frequency concentrated member of the superfamily is found to occur for 3 these wavelets known as airy wavelets capture the essential idea of popular morlet wavelet while avoiding its deficiencies they may be recommended as an ideal starting point for general purpose use

conservative domain decomposition procedure with unconditional stability and second order accuracy a new derivation of the conservative domain decomposition procedure for solving the parabolic equation is presented in this procedure fluxes at subdomain interfaces are calculated from the solution at the previous time level then these fluxes serve as neumann boundary data for implicit block centered discretization in the subdomain the unconditional stability and the second order accuracy of solution values as well as fluxes are proved numerical results examining the stability accuracy and parallelism of the procedure are also presented

asymptotic wavelet and gabor analysis extraction of instantaneous frequencies the behavior of the continuous wavelet and gabor coefficients in the asymptotic limit using stationary phase approximations are investigated in particular it is shown how under some additional assumptions these coefficients allow the extraction of some characteristics of the analyzed signal such as frequency and amplitude modulation laws applications to spectral line estimations and matched filtering are briefly discussed

a general approach for obtaining joint representations in signal analysis ii general class mean and local values and bandwidth for pt i see ibid vol 44 no 5 p 1080 90 1996 using the method developed in cohen 1996 part i the concepts of instantaneous frequency and group delay are generalized to arbitrary variables in addition new expressions for mean values and bandwidths are obtained the kernel method is used to define a general class for arbitrary variables as in the time frequency case the general class generates all possible distributions the method is also formulated in terms of the local autocorrelation function

the monogenic wavelet transform this paper extends the 1 d analytic wavelet transform to the 2 d monogenic wavelet transform the transformation requires care in its specification to ensure suitable transform coefficients are calculated and it is constructed so that the wavelet transform may be considered as both local and monogenic this is consistent with defining the transform as a real wavelet transform of a monogenic signal in analogy with the analytic wavelet transform classes of monogenic wavelets are proposed with suitable local properties it is shown that the monogenic wavelet annihilates anti monogenic signals that the monogenic wavelet transform is phase shift covariant and that the transform magnitude is phase shift invariant a simple form for the magnitude and orientation of the isotropic transform coefficients of a unidirectional signal when observed in a rotated frame of reference is derived the monogenic wavelet ridges of local plane waves are given

the homomorphic analytic signal this paper introduces an alternative form of analytic signal that is formed using homomorphic signal processing techniques it may be generated for signals that demonstrate redundancy in their pole zero distributions such as phase signals and real signals the analytic signal is formed by manipulating the positions of the signal s nonminimum phase poles and zeros to create a minimum phase signal a factorization of real signals is presented that demonstrates the properties of the homomorphic analytic signal and a new definition of instantaneous frequency is developed examples are given to verify the theory and comparisons with the linear analytic signal and instantaneous frequency are made

darboux transformations for 1 2 dimensional fokker planck equations with constant diffusion matrix we construct a darboux transformation for 1 2 dimensional fokker planck equations with constant diffusion matrix our transformation is based on the two dimensional supersymmetry formalism for the schrodinger equation the transformed fokker planck equation and its solutions are obtained in explicit form

exact solution of the fokker planck equation for a broad class of diffusion coefficients we consider the langevin equation with a multiplicative noise term that depends on time and space the corresponding fokker planck equation in the stratonovich approach is investigated its exact solution is obtained for an arbitrary multiplicative noise term given by g x t d x t t and the behaviors of probability distributions for some specific functions of d x are analyzed we show that the asymptotic shape of the random walk model and power law decay obtained from other approaches can be reproduced from our solutions by employing two simple functions for g x t in particular for d x approximately x theta 2 the physical solutions for the probability distribution in the ito stratonovich and postpoint discretization approaches can be obtained and analyzed

classical harmonic oscillator with multiplicative noise abstract the first two moments of an underdamped harmonic oscillator were found for additive and multiplicative noises a detailed analysis is performed of the case of a random damping which supplements the case of a random frequency which has received the most study in both cases the second moments are unstable for the sufficient large noise strength the difference appears in the stability conditions of the first moments which can diverge already for white noise for the random damping and only for color noise for the random frequency the difference between these two types of multiplicative noises shows up also in the response to an external periodic force namely white noise influences the first moment only in the case of a random damping however the second moments in both cases show the non monotonic dependence on the noise strength and correlation rate stochastic resonance in linear systems

higher order properties of analytic wavelets the influence of higher order wavelet properties on the analytic wavelet transform behavior is investigated and wavelet functions offering advantageous performance are identified this is accomplished through detailed investigation of the generalized morse wavelets a two parameter family of exactly analytic continuous wavelets the degree of time frequency localization the existence of a mapping between scale and frequency and the bias involved in estimating properties of modulated oscillatory signals are proposed as important considerations wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain as quantified by the third central moments a particular subset of the generalized morse wavelets recognized as deriving from an inhomogeneous airy function emerge as having particularly desirable properties these ldquoairy waveletsrdquo substantially outperform the only approximately analytic morlet wavelets for high time localization special cases of the generalized morse wavelets are examined revealing a broad range of behaviors which can be matched to the characteristics of a signal

on the analytic wavelet transform an exact and general expression for the analytic wavelet transform of a real valued signal is constructed resolving the time dependent effects of nonnegligible amplitude and frequency modulation the analytic signal is first locally represented as a modulated oscillation demodulated by its own instantaneous frequency and then taylor expanded at each point in time the terms in this expansion called the instantaneous modulation functions are time varying functions which quantify at increasingly higher orders the local departures of the signal from a uniform sinusoidal oscillation closed form expressions for these functions are found in terms of bell polynomials and derivatives of the signal s instantaneous frequency and bandwidth the analytic wavelet transform is shown to depend upon the interaction between the signal s instantaneous modulation functions and frequency domain derivatives of the wavelet inducing a hierarchy of departures of the transform away from a perfect representation of the signal the form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal in which case our expressions simplify considerably one may then quantify the time varying bias associated with signal estimation via wavelet ridge analysis and choose wavelets to minimize this bias

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters