Kramers-Kronig relations applied to finite bandwidth data from suspensions of encapsulated microbubbles

In this work, the Kramers-Kronig (K-K) relations are applied to experimental data of resonant nature by limiting the interval of integration to the measurement spectrum. The data are from suspensions of encapsulated microbubbles (Albunex) and have the characteristics of an ultrasonic notch filter. The goal is to test the consistency of this dispersion and attenuation data with the Kramers-Kronig relations in a strict manner, without any parameters from outside the experimental bandwidth entering in to the calculations. In the course of reaching the goal, the artifacts associated with the truncation of the integrals are identified and it is shown how their impacts on the results can be minimized. The problem is first approached analytically by performing the Kramers-Kronig calculations over a restricted spectral band on a specific Hilbert transform pair (Lorentzian curves). The resulting closed-form solutions illustrate the type of artifacts that can occur due to truncation and also show that accurate results can be achieved. Next, both twice-subtracted and lower-order Kramers-Kronig relations are applied directly to the attenuation and dispersion data from the encapsulated microbubbles. Only parameters from within the experimental attenuation coefficient and phase velocity data sets are used. The twice-subtracted K-K relations produced accurate estimates for both the attenuation coefficient and dispersion across all 12 data sets. Lower-order Kramers-Kronig relations also produced good results over the finite spectrum for most of the data. In 2 of the 12 cases, the twice-subtracted relations tracked the data markedly better than the lower-order predictions. These calculations demonstrate that truncation artifacts do not overwhelm the causal link between the phase velocity and the attenuation coefficient for finite bandwidth calculations. This work provides experimental evidence supporting the validity of the subtracted forms of the acoustic K-K relations between the phase velocity and attenuation coefficient.