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.

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It is known that adaptive Fourier decomposition (AFD) offers efficient rational approximations to functions in the classical Hardy H 2 spaces with significant applications. This study aims at rational approximation in Bergman, and more widely, in weighted Bergman spaces, the functions of which have more singularity than those in the Hardy spaces. Due to lack of an effective inner function theory, direct adaptation of the Hardy-space AFD is not performable. We, however, show that a pre-orthogonal method, being equivalent to AFD in the classical cases, is available for all weighted Bergman spaces. The theory in the Bergman spaces has equal force as AFD in the Hardy spaces. The methodology of approximation is via constructing the rational orthogonal systems of the Bergman type spaces, called Bergman space rational orthogonal (BRO) system, that have the same role as the Takennaka-Malmquist (TM) system in the Hardy spaces. Subsequently, we prove a certain type direct sum decomposition of the Bergman spaces that reveals the orthogonal complement relation between the span of the BRO system and the zero-based invariant spaces. We provide a sequence of examples with different and explicit singularities at the boundary along with a study on the inclusion relations of the weighted Bergman spaces. We finally present illustrative examples for effectiveness of the approximation.

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In this paper we propose a new type of non-negative time-frequency distribution associated with mono-components in both the non-periodic and periodic cases, called transient time-frequency distribution (TTFD), and study its properties. The TTFD of a monocomponent signal can be obtained directly through its analytic instantaneous frequency. The characteristic property of TTFD is its complete concentration along the analytic instantaneous frequency graph. For multi-components there are induced time-frequency distributions called composing transient time-frequency distribution (CTTFD). Each CTTFD is defined as the superposition of the TTFDs of the composing intrinsic mono-components in a suitable mono-components decomposition of the targeted multicomponent. In studying the properties of TTFD and CTTFD the relations between the Fourier frequency and analytic instantaneous frequency are examined.

Abstract-It is accepted knowledge that inner functions and outer functions in complex analysis correspond, respectively, to all-pass filters and signals of minimum phase. The knowledge, however, has not been justified for general inner and outer functions. In digital signal processing the correspondence and related results are based on studies of rational functions. In this paper, based on the recent result on positivity of phase derivatives of inner functions, we establish the theoretical foundation for all-pass filters and signals of minimum phase. We, in particular, deal with infinite Blaschke products and general singular inner functions induced by singular measures. A number of results known for rational functions are generalized to general inner functions. Both the discrete and continuous signals cases are rigorously treated.

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.

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