### 21.2.2 Subject

### 21.2.3 Velocity Measurement

_{t }(z). Analytical signal g

_{t }(z) was obtained by applying the Hilbert transform to the RF signal r

_{t }(z). Pulse waves with angular frequency ω

_{0}= 2πf

_{0}were transmitted at a time interval of ΔT. Analytical signal of time t and t + ΔT at depth z can be modeled as

where u(z) was the envelope of the analytical signal, f _{s }was the sampling frequency, c _{0} was the sound speed, θ _{0} was the initial phase, and z _{τ }was the true displacement induced by the pulsation. The complex cross-correlation function γ(z _{lag }) at lag z _{lag }was defined as

where N was 1-D cross correlation window of 256 pixels, corresponding to 195 μm depth window with 91 % overlap using hamming window. An index at maximum value of the real part of the Eq. (21.3) corresponded to the index of . In this paper, f _{s }was 1 GHz, and the temporal resolution was enough high to observe the deformation with HFUS of 100 MHz. Additionally, the received RF signal was up-sampled to 4 GHz before implementing the 1-D cross-correlation. Because of these reasons, the estimated displacement was almost equal to the true displacement z _{lag }. The velocity, denoted by v _{t + ΔT/2}(z), of the skin in his forearm between the interval was given as follows:

The acceleration was calculated from the measured velocity. The acceleration was calculated by differentiating the measured velocity as

### 21.2.4 Shear Wave Measurement

where N _{α }was an estimation window of 50 pixels, corresponding to a 50 ms time. A hamming window was used for the estimation. A variable z _{0} was a reference depth, and a variable z was an interest depth. A time lag between the reference depth and the interest depth was expressed as follows:

_{c }was obtained from the power spectrum calculated by applying the Fourier transform to the acceleration signals. Figure 21.3 shows the relationship between the depth z and the time lag. A regression line can be obtained by applying the least square method to the relationship. A slope a

_{1}of the regression line can be obtained by minimizing a least mean square error e as follows: