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

Abstract

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.

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

“…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].…”

“…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.…”

New aspects of Beurling–Lax shift invariant subspaces

“…devoted to such construction of bandlimited signals (the band preserving problem) and then apply the results to the phase retrieval problem by the methodology in representing an entire function by Hardamard's formula in an infinite product [12][13][14]. However, almost all the existing constructive results of the band preserving problem are focused on L 2 ðRÞ functions.…”

Adaptive Fourier decompositions and rational approximations, part I: Theory

“…A rigorous definition of analytic phase derivative involves non-tangential boundary limits of holomorphic functions in the relevant domain. 12,5,4 Through a combined effort a large pool of monocomponents is found. Next, one wishes to seek for apropriate decompositions of a signal into mono-components.…”