$\epsilon$ -Mono-Component: Its Characterization and Construction

This paper studies the consistency between the analytic amplitude and the physical amplitude for a mono-component of the form s(t)=ρ(t)e^iθ(t). A special class of mono-components, called ε-mono-components, are considered and a parameter ε is introduced to measure the consistency between these two kinds of amplitudes. It is shown that ε controls the number of zerocrossings of s(t) within each monotonic interval of ρ(t), which means that the oscillation of the analytic amplitude ρ(t) is much slower than that of the phase part e^iθ(t) at any instant, provided that ε is sufficiently small. Some sufficient conditions, including the Fourier spectral characterization, for s(t)=ρ(t)e^iθ(t) to be an ε-mono-component are given. Frames and Riesz bases composed of ε-mono-components are constructed. Finally, applications of ε-mono-components to signal decomposition and time-frequency analysis are discussed.