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

In this work, the Kramers-Kronig (K-KJ relations are applied to experimenta l data of resonant nature by limiting the interval of integration to the me asurement spectrum. The data are from suspensions of encapsulated microbubb les (Albunex(R)) and have the characteristics of an ultrasonic notch filter , The goal is to test the consistency of this dispersion and attenuation da ta with the Kramers-Kronig relations in a strict manner, without any parame ters from outside the experimental bandwidth entering in to the calculation s. In the course of reaching the goal, the artifacts associated with the tr uncation of the integrals are identified and it is shown how their impacts on the results can be minimized. The problem is first approached analytical ly by performing the Kramers-Kronig calculations over a restricted spectral band on a specific Hilbert transform pair (Lorentzian curves). The resulti ng closed-form solutions illustrate the type of artifacts that can occur du e 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 micro bubbles. Only parameters from within the experimental attenuation coefficie nt and phase velocity data sets are used. The twice-subtracted K-K relation s produced accurate estimates for both the attenuation coefficient and disp ersion across all 12 data sets. Lower-order Kramers-Kronig: relations also produced good results over the finite spectrum fur most of the data, In 2 o f the 12 cases, the twice-subtracted relations tracked the data markedly be tter than the lower-order predictions. These calculations demonstrate that truncation artifacts do not overwhelm the causal link between the phase vel ocity and the attenuation coefficient for finite bandwidth calculations. Th is work provides experimental evidence supporting the validity of the subtr acted forms of the acoustic K-K relations between the phase velocity and at tenuation coefficient, (C) 2000 Acoustical Society of America.