Hardy-Sobolev Spaces Decomposition in Signal Analysis

Some fundamental formulas and relations in signal analysis are based on the amplitude-phase representations s(t) = A(t)e(i phi(t)) and (s) over cap(omega) = B(omega)e(i psi(omega)), where the amplitude functions A(t) and B(omega) and the phase functions phi(t) and psi(omega) are assumed to be differentiable. They include the amplitude-phase representations of the first and second order means of the Fourier frequency and time, and the equivalence between two forms of the covariance. A proof of the uncertainty principle is also based on the amplitude-phase representations. In general, however, signals of finite energy do not necessarily have differentiable amplitude-phase representations. The study presented in this paper extends the classical formulas and relations to general signals of finite energy. Under the formulation of the phase and amplitude derivatives based on the Hardy-Sobolev spaces decomposition the extended formulas reveal new features, and contribute to the foundations of time-frequency analysis. The established theory is based on the equivalent classes of the L-2 space but not on particular representations of the classes. We also give a proof of the uncertainty principle by using the amplitude-phase representations defined through the Hardy-Sobolev spaces decomposition.