Minimum-phase reflection for a passive acoustic absorber with target absorption spectrum and minimum volume

This paper develops strategies based on the properties of the minimum-phase reflection coefficient to design a passive absorber with minimum volume. It can be demonstrated that, of all the designs achieving the same target absorption spectrum in the one-dimensional reflection problem, the design with minimum-phase reflection requires the smallest total volume. This property is particularly useful for guiding the optimization of acoustic absorbers. We define a cost function involving the complex-valued minimum-phase reflection coefficient, which is uniquely determined from the prescribed absorption spectrum. We also introduce a volume-optimality factor to validate whether a given design achieves minimum-phase reflection. This factor is the volume ratio of a designed system to the corresponding minimum-phase system, thus measuring the volume efficiency of the design. Applying these strategies to absorbers composed of Helmholtz resonators, we show theoretically and numerically that using $N$ resonances yields up to ${2}^{N}$ distinct designs with identical absorption spectra. Specifically, we present designs with single and dual resonators in parallel and series configurations. The minimum-phase design is consistently more compact than the non-minimum-phase designs. These findings validate our design strategies and highlight their importance, as traditional optimization methods for a target absorption spectrum may easily lead to non-minimum-volume designs.