Hardy–Sobolev derivatives of phase and amplitude, and their applications

Abstract

Communicated by I. T. LeongIn time-frequency analysis, there are fundamental formulas expressing the mean and variance of the Fourier frequency of signals, s, originally defined in the Fourier frequency domain, in terms of integrals against the density js.t/j 2 in the time domain. In the literature, the existing formulas are only for smooth signals, for it is the classical derivatives of the phase and amplitude of the signals that are involved. The two representations of the covariance also rely on the classical derivatives and thus are restrictive. In this fundamental study, by introducing a new type of derivatives, called Hardy-Sobolev derivatives, we extend the formulas to signals in the Sobolev space that do not usually have classical derivatives. We also investigate the corresponding formulas for periodic (infinite discrete) and finite discrete signals.

A Tighter Uncertainty Principle for Linear Canonical Transform in Terms of Phase 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].…”

“…To solve this problem, [6]- [8] work with different types of derivatives, viz., Hardy-Sobolev derivative, derivatives as non-tangential boundary limits and Fourier transform derivative, etc., through Hardy decomposition and Fourier transformation. We showed that under certain conditions the various types of derivatives can be unified.…”

Analytic Phase Derivatives, All-Pass Filters and Signals of Minimum Phase

“…We prove in [7] that when the derivatives ϕ (t), s (t) and ρ (t) exist in the classical sense, then the Hardy-Sobolev derivatives ϕ * (t), s * (t) and ρ * (t) coincide with them. A large number of relations for smooth signals are extendable to general functions in the Sobolev space by using Hardy-Sobolev derivatives ( [7]). The frequency spectra or inverse Fourier transforms, respectively for the discrete or continuous allpass filters and signals of minimum phase, are themselves analytic signals.…”

Signal moments for the short‐time Fourier transform associated with Hardy–Sobolev derivatives

“…Now, we recall the definition of HS derivative for a signal (see ). Definition see Let and f ±′ ( t ) be defined by the following non‐tangential limits: where Γ: z → t denote the non‐tangential limits.…”

“…Now, we recall the definition of HS derivative for a signal (see ). Definition see Let and f ±′ ( t ) be defined by the following non‐tangential limits: where Γ: z → t denote the non‐tangential limits. Then the HS derivatives (or ‘HS derivative’) , and of f ( t ), A f ( t ), and ϕ f ( t ) are defined, respectively, by …”