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 an hierarchy of departures of the transform away from a...

• Type: Journal
• Year: 2010
• Where: TIT
• Authors: Jonathan M. Lilly, Sofia C. Olhede
• Keywords: Analytic Wavelet Transform, Education, Instantaneous Modulation Functions, TIT 2010, Wavelets