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 non-negligible 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

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 "Airy wavelets" 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.Comment: 15 pages, 6 Postscript figure

Estimation de la fréquence instantanée des signaux FM par opérateur d'énergie Psi_B

Psi_B energy operator is an extension of the cross Teager-Kaiser energy operator which is an non-linear energy tracking operator to deal with complex signals and its usefulness for non-stationary signals analysis has been demonstrated. In this letter two new properties of Psi_B are established. The first property is the link between Psi_B and the dynamic signal which is a generalization of the Instantaneous Frequency (IF). The second property obtained for frequency modulated signals is a simple way to estimate the IF. These properties confirm the interest of Psi_B operator to track the non-stationary of a signal. Results of IF estimation in noisy environment of a non-linear FM signal are presented and comparison to Wigner-Ville distribution and Hilbert transform-based method is provided

A new perturbative approach to the adiabatic approximation

A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of "adiabatic" evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well-known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to be not widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.Comment: 9 pages. Added references, discussion of previous results, expanded upon discussion of main result and application of i

Hierarchical Theory of Quantum Adiabatic Evolution

Quantum adiabatic evolution is a dynamical evolution of a quantum system under slow external driving. According to the quantum adiabatic theorem, no transitions occur between non-degenerate instantaneous eigen-energy levels in such a dynamical evolution. However, this is true only when the driving rate is infinitesimally small. For a small nonzero driving rate, there are generally small transition probabilities between the energy levels. We develop a classical mechanics framework to address the small deviations from the quantum adiabatic theorem order by order. A hierarchy of Hamiltonians are constructed iteratively with the zeroth-order Hamiltonian being determined by the original system Hamiltonian. The $k$th-order deviations are governed by a $k$th-order Hamiltonian, which depends on the time derivatives of the adiabatic parameters up to the $k$th-order. Two simple examples, the Landau-Zener model and a spin-1/2 particle in a rotating magnetic field, are used to illustrate our hierarchical theory. Our analysis also exposes a deep, previously unknown connection between classical adiabatic theory and quantum adiabatic theory.Comment: 10 pages, 6 figures, 29 reference

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

Une nouvelle classe d'opérateurs de Teager-Kaiser multidimensionnels basée sur les dérivées directionnelles d'ordre supérieur

This work aims at introducing some energy operators linked to Teager-Kaiser energy operator and its associated higher order versions and expand them to multidimensional signals. These operators are very useful for analyzing oscillatory signals with time-varying amplitude and frequency (AM-FM). We prove that gradient tensors combined with Kronecker powers allow to express these operators by directional derivatives along any n-D vector. In particular, we show that the construction of a large class of non linear operators for AM-FM multidimensional signal demodulation is possible. Also, a new scalar function using the directional derivative along a vector giving the ”sign” of the frequency components is introduced. An application of this model to local n-D AM-FM signal is presented and related demodulation error rates estimates. To show the effectiveness and the robustness of our method in term of envelope and frequency components extraction, results obtained on synthetic and real data are compared to multi-dimensional energy separation algorithm and to our recently introduced n-D operator

Improved Error-Scaling for Adiabatic Quantum State Transfer

We present a technique that dramatically improves the accuracy of adiabatic state transfer for a broad class of realistic Hamiltonians. For some systems, the total error scaling can be quadratically reduced at a fixed maximum transfer rate. These improvements rely only on the judicious choice of the total evolution time. Our technique is error-robust, and hence applicable to existing experiments utilizing adiabatic passage. We give two examples as proofs-of-principle, showing quadratic error reductions for an adiabatic search algorithm and a tunable two-qubit quantum logic gate.Comment: 10 Pages, 4 figures. Comments are welcome. Version substantially revised to generalize results to cases where several derivatives of the Hamiltonian are zero on the boundar

Estimation de l'enveloppe et de la fréquence locales par les opérateurs de Teager-Kaiser en interférométrie en lumière blanche.

In this work, a new method for surface extraction in white light scanning interferometry (WLSI) is introduced. The proposed extraction scheme is based on the Teager-Kaiser energy operator and its extended versions. This non-linear class of operators is helpful to extract the local instantaneous envelope and frequency of any narrow band AM-FM signal. Namely, the combination of the envelope and frequency information, allows effective surface extraction by an iterative re-estimation of the phase in association with a new correlation technique, based on a recent TK crossenergy operator. Through the experiments, it is shown that the proposed method produces substantially effective results in term of surface extraction compared to the peak fringe scanning technique, the five step phase shifting algorithm and the continuous wavelet transform based method. In addition, the results obtained show the robustness of the proposed method to noise and to the fluctuations of the carrier frequency