Mathematics – Statistics Theory
Scientific paper
2007-11-24
Lilly, J. M., and S. C. Olhede (2010). On the analytic wavelet transform. IEEE Transactions on Information Theory, 56 (8), 413
Mathematics
Statistics Theory
Scientific paper
10.1109/TIT.2010.2050935
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.
Lilly Jonathan M.
Olhede Sofia C.
No associations
LandOfFree
On the Analytic Wavelet Transform does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the Analytic Wavelet Transform, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Analytic Wavelet Transform will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-562953