Correcting for bounded bandwidth when estimating tissue attenuation from mean frequency downshift

The attenuation of tissue can be estimated utilizing the downshift of the center frequency of a propagating pulse. In general it is assumed that the shape of the emitted pulse can be approximated by a Gaussian function and attenuation is assumed to change linearly with frequency. At this conditions the downshift of the mean frequency of pulse spectrum depends linearly on attenuation coefficient, pulse bandwidth and propagation distance. This is a good approximation for relatively narrowband pulses and small penetration depth. But for short pulses and deep penetration the frequency downshift is large and the ultrasonic pulse is no more Gaussian, thus the previous assumption is no longer correct. The closer is the mean frequency of the pulse to the lower frequency bound of the receiving system the bigger deformation of the pulse spectrum occurs and consequently the attenuation is determined with bigger error. The following paper presents how to correct the experimentally determined mean frequency and to obtain reliable results when investigating tissue attenuation with wideband pulses. We propose a new formula for the dependence between pulse mean frequency, tissue attenuation, pulse bandwidth and traveled distance. The formula was derived from the mean frequency of Gaussian pulse spectrum determined in the limited frequency band. The formula was applied to simulate variation of mean frequency MF of the pulse propagating in the medium with attenuation coefficient corresponding to the attenuation in the tissue mimicking phantom. The MF was also determined (using the correlation estimator of MF and next trend extraction using Single Spectrum Analysis) from the simulated ultrasonic echoes and echoes scattered in the tissue phantom. The corrected nonlinear formula describes well MF variation along the pulse propagation path. The departure from the linear dependence increases with large MF shift, thus it is well pronounced for highly attenuating tissue, the wideband pulses and deep penetration.