Modeling acoustic attenuation of soft tissue with a minimum-phase filter.

Soft biological tissue has been observed to exhibit an acoustic attenuation log-magnitude characteristic which increases as an approximately linear function of frequency. This paper describes the implementation of a finite-impulse-response (FIR) digital filter model for simulating this behavior on a digital computer. To insure that the filter is causal, the minimum-phase constraint is imposed. For minimum-phase filters, the log-magnitude and phase characteristics form a Hilbert Transform pair. The discrete-time Hilbert Transform of the linear log-magnitude characteristic was evaluated to determine the phase of the filter. The inverse Fourier Transform of the resulting real and imaginary components of the frequency transform produces the finite-duration unit-sample response of the digital filter model. Experimental results using plexiglas material, which has a linear-with-frequency loss characteristic, indicate that the minimum-phase model is more accurate than the linear-phase model, resulting in a rms error between predicted and observed time waveforms that is 3 times smaller. The effects of varying the sampling period and the size of the FIR filter are discussed. A FORTRAN program to calculate the minimum-phase unit-sample response from the slope of the log-magnitude characteristic is included in the Appendix.