APPENDIX

Analysis of the Ultrasound Field
  An analysis of the ultrasound field was performed for the compound sensor. In the CASS system, the ultrasonic transducers work sequentially; thus, only one transducer is active at any moment. Therefore, it is assumed that there is no interference between the ultrasound fields generated by adjacent transducers. Thus, the analysis of the entire acoustic field of the compound sensor can be simplified to the analysis of the field for a single transducer.

  The ultrasonic transducer acts as a band-pass device with a center frequency, ₀, and bandwidth, D. It can be modeled as a linear time-invariant (LTI) system. Its output can be described as a convolution of an input-signal characteristic and the system-response function in the time domain. The spectrum of the output signal can then be calculated as the product of Fourier transforms of the input signal and the transducer response (32). For an ideal ultrasonic transducer, the spectrum, h(w), can be expressed by the following formula:

[1]

where w₀ is angular frequency, Q is ₀/D. The amplitude coefficient was normalized.

  According to Huygen's principle, the ultrasonic field at any given point in the space above the transducer surface is the sum of contributions from all the Huygen's sources. For a broad-band transducer, the sound pressure at any given point in the ultrasound field is a composite of all frequency components of the transducer's radiating pressure. Therefore, the total sound pressure, consisting of various frequency components, is an integral of the single-frequency component over the transducer bandwidth and the transducer surface (32-34):

[2]

  In Equation 2, r is the density of soft tissue, w is the angular frequency of the sound wave, P₀ is the peak amplitude of pressure on the transducer face, a is the radius of the transducer, s is the radial distance from any point, p, on the transducer surface to the center of transducer, q is the angle between any point, p, on the transducer surface and x axis, c is the speed of sound propagating in soft tissue, r is the distance from observation point to center of transducer, and j is the angle between the transducer axis and the radial vector.

  In the near field, we use a summation equation in place of Equation 2. Then,

[3]

where b(_n) is a spectral density function of the transducer and can be obtained from Equation 1 for various bandwidths, C(r,q,_n) and d(r,q,_n) are the amplitude and the phase of the pressure of the transducer at frequency _n.

  In the calculation, we assumed that the peak amplitude, P₀, of sound pressure at the transducer face is 1, the density of soft tissue is 1, sound velocity, c, is 1,500 m/sec, maximum value of n is 30, and the frequency interval is 0.5 MHz. Simulation results for the ultrasound field of a transducer with three different frequency bandwidths (D/₀=0 percent, D/₀=30 percent, D/₀=90 percent; a=1.5 mm; f_o=7.5 MHz) are shown in Figure 4. Due to the symmetry of the transducer, only the pressure characteristic in the positive XZ plane of the transducer is shown.

  Several sound-field characteristics of the ultrasonic transducer are evident in Figure 10. First, the sound-pressure fluctuation in the near field is smoothed due to the presence of various harmonic components. The broader the bandwidth, the better the near-field homogeneity, indicating that a broad-band transducer is suitable for detecting shallow soft tissue layers in the near field. Second, the peak sound pressure increases more than 50 percent when the bandwidth increases from 30 percent to 90 percent. Thus, the total emitting energy increases with the increase of the bandwidth. Third, the above analysis is based on an ideal ultrasonic transducer with a Gaussian response spectrum with an electrical excitation impulse. The spectrum characteristic of an impulse function, d(t), is assumed to be 1 over the frequency dimension. In the CASS, the excitation source is a square impulse with a 50 ns pulse length and 10 kHz pulse-repetition frequency. The envelope of Fourier series coefficients is a sinc function. Although the frequency bandwidth will be affected by the properties of the emitting power transducer, the bandwidth of the exciting signal can still reach 200 MHz. Therefore, the ideal transducer represents a good approximation of the actual transducer and the preceding sound-field analysis is suitable for our design.

Figure 10. The simulation of the near-field characteristics of an ultrasonic transducer for (a) D/₀=0 percent; (b) D/₀=30 percent; (c) D/₀=90 percent (radius=1.5 mm, ₀=7.5 MHz).

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WANG et al. A Sensor for Biomechanical Analyses