Hardy-Sobolev Spaces Decomposition in Signal Analysis

Abstract

Some fundamental formulas and relations in signal analysis are based on amplitude-phase representations (ω) , where the amplitude functions A(t) and B(ω) and the phase functions ϕ(t) and ψ(ω) are assumed to be differentiable. They include the amplitude-phase representations of the first and the second order means of the Fourier frequency and the time, and the equivalence relation 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 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 phase and amplitude derivatives based on 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 using the amplitude-phase representations defined through Hardy-Sobolev spaces decomposition.

A Tighter Uncertainty Principle for Linear Canonical Transform in Terms of Phase Derivative

“…The last two classes of signals can really make the strict inequality hold (see Example 3.7), that is, We provide the Table I in order to compare the existing and proposed results. Remark 3.3: One of the purposes of the series of studies in signal analysis given in [5], [7] and [8], etc., is to establish a theoretical foundation of signal analysis/processing for signals of finite energy, viz., of Lebesgue square integrable functions, those, in particular, are not necessary to be continuous, nor of particular forms. The essence of using Lebesgue integration in the proof is that the integrals eliminate the effect of the possible infinite jumps of the phase derivative induced by discontinuity of signals.…”

“…Because the classical derivatives for signals of finite energy may not exist ( [5], [7]), we adopt the Fourier transform derivatives as follows (also see [8] These formulas have the same forms as those given in the definition. In view of this, when we deal with general signals in Sobolev spaces, since the classical derivatives may not exist, a reasonable replacement of the classical derivative is the Fourier type derivative.…”

“…The assumption of the zero value of the new derivatives at the points is conventional that makes the proofs going. A detailed analysis and comprehensive development of relations of several types of derivatives with applications in signal analysis are given in [5], [7] and [6].…”

“…U NCERTAINTY principle is important in signal analysis ([1]- [4], [7]- [13], [15]). The classical Heisenberg's uncertainty principle provides a lower-bound on the product of spreads of a signal energy in the time and Fourier frequency domains as specified by the inequality (1.1) where and are defined in Definition 2.1.…”

Approximation of functions by higher order Szegö kernels I. Complex variable cases

“…Note that a classical derivative of the phase of f + i H f may not exist. For a class of functions, however, this concept can be defined through boundary limits of the same quantity of the corresponding Hardy space projection inside the disc ( [2,3]). For functions defined on the real line there is a parallel mono-component function theory.…”