Identification of minimum-phase-preserving operators on the half-line

Abstract

Minimum phase functions are fundamental in a range of applications, including control theory, communication theory and signal processing. A basic mathematical challenge that arises in the context of geophysical imaging is to understand the structure of linear operators preserving the class of minimum phase functions. The heart of the matter is an inverse problem: to reconstruct an unknown minimum phase preserving operator from its value on a limited set of test functions. This entails, as a preliminary step, ascertaining sets of test functions that determine the operator, as well as the derivation of a corresponding reconstruction scheme. In the present paper we exploit a recent breakthrough in the theory of stable polynomials to solve the stated inverse problem completely. We prove that a minimum phase preserving operator on the half line can be reconstructed from data consisting of its value on precisely two test functions. And we derive an explicit integral representation of the unknown operator in terms of this data. A remarkable corollary of the solution is that if a linear minimum phase preserving operator has rank at least two, then it is necessarily injective.

Constructive solutions to Pólya–Schur problems

“…A minimum phase signal f ∈ L 2 (R + ) is one that maximizes partial energy T 0 |f (t)| 2 dt among all functions having the same power spectrum as f , for all T > 0. Full details of this characterization and its relevance to seismic signal processing are laid out in [4].…”

“…Next we consider the analytic context of the Hardy-Hilbert space H 2 = H 2 (D). See [5,4] for details on how this relates to signal processing and, in particular, to geophysics.…”

“…This in turn can be related to H 2 , but it is no longer shifted outer functions that come into play. Full details of the connection between minimum-phase-preserving operators on L 2 (R + ) and outer-preserving operators on H 2 are given in [4], which is based on an earlier preprint version of the present paper.…”

Hardy space on the polydisk and scattering in layered media

“…the power spectrum of the measured data f * G (τ,R) is simply the power spectrum of the source wavelet f . Assuming further that f is minimum phase (see [7]), it can be recovered from its power spectrum, in turn allowing G (τ,R) to be extracted from the measured data. Theorem 4.1 supports the first of these two assumptions, thereby providing mathematical justification for a longstanding geophysical supposition.…”

“…Further to the three questions, an additional issue motivated the present paper. Recent investigations into minimum phase preserving operators [7] give indirect evidence that Hardy space should somehow be connected to PDEs modeling the propagation of seismic waves-but without showing how. The present paper clarifies this issue by providing a direct link.…”