On Inequalities about Instantaneous Amplitudes (Wavelet analysis and signal processing)

Abstract

For a real signal (a real-valued function) f(t), we consider its analytic signal (mathcal{A}f)(t) = f(t)+i(mathcal{H}f)(t), where (mathcal{H}f)(t) is the Hilbert transform of f(t). Its absolute value A(t) = |(Af)(t)|, which is called instantaneous amplitude, often represents a coarse variation of f(t), and the graph of A(t) looks like an envelope of the graph of |f(t)|. However, for some signals, A(t) changes rather rapidly, and it doesn t look like an envelope of the graph of |f(t)|. We give mathematically rigorous inequalities about hat{A^{2}}( $xi$) (A^{2}(t) = {A(t)}^{2}) which can be considered to explain this difference. We also consider the best possibility of the constants of the inequalities.