Time-frequency analysis: theory and applications
Time-frequency analysis: theory and applications
Wavelet denoising for signals in quadrature
Integrated Computer-Aided Engineering
Higher-order properties of analytic wavelets
IEEE Transactions on Signal Processing
Bivariate instantaneous frequency and bandwidth
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
Multiridge detection and time-frequency reconstruction
IEEE Transactions on Signal Processing
The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform
IEEE Transactions on Signal Processing
Characterization of signals by the ridges of their wavelettransforms
IEEE Transactions on Signal Processing
The chirplet transform: physical considerations
IEEE Transactions on Signal Processing
Cramer-Rao bounds for wavelet transform-based instantaneous frequency estimates
IEEE Transactions on Signal Processing
On instantaneous amplitude and phase of signals
IEEE Transactions on Signal Processing
The design of approximate Hilbert transform pairs of wavelet bases
IEEE Transactions on Signal Processing
Modified Cohen-Lee time-frequency distributions and instantaneousbandwidth of multicomponent signals
IEEE Transactions on Signal Processing
Analysis of singularities from modulus maxima of complex wavelets
IEEE Transactions on Information Theory
Multidimensional, mapping-based complex wavelet transforms
IEEE Transactions on Image Processing
Wavelet ridge estimation of jointly modulated multivariate oscillations
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
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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.