Abstract
The causality condition, which states that the response of a passive system cannot preceed its cause, places mathematical constraints on the complex Fourier transform of the response. For minimum-phase-shift systems, these constraints take the form of Hilbert transform relations between the magnitude and phase of the Fourier transform. These relations have historically had a profound impact on the design of electrical circuits, as evidenced in Bode’s text Network Analysis and Feedback Amplifier Design. The present work presents new applications in the area of structural acoustics which show how the mechanical impedance of a passive fluid-loaded structure can sometimes be inferred from the magnitude of its acoustic reflection coefficient. The approach is demonstrated on a one-dimensional problem in which an incident acoustic wave reflects from a passive structure. An equivalent form of the Hilbert transform, known as the Wiener-Lee transform, is used for the numerical calculations. If the reflection coefficient is minimum-phase-shift, then its phase can be uniquely computed from its magnitude. In this work, it is shown that the minimum-phase-shift property of the reflection coefficient actually depends on the real part of the mechanical impedance of the structure. Therefore, in an inversion problem where one attempts to infer the mechanical impedance given the magnitude of reflection, additional assumptions about the structure must be made in order to arrive at a unique answer. In cases where one attempts to design a passive structure with a specified reflection coefficient magnitude, the causality condition yields a mechanical impedance which corresponds to a minimum-phase-shift reflection coefficient.
