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**Exercises in Medical Ultrasonics (Last Updated ****September 23, 2004****)**

1.
Estimate the
Doppler shift caused by an ambulance as it passes an observer at 30 mph. Assume that the ambulance emits a sound with
frequency of 5 kHz. Let the speed of sound in air be 33,000 cm/s. What is the appropriate Doppler angle for
this problem? Why?

2.
Expand Equation 2
through a

_{}

_{}

_{}

_{}

3.
Use standard
trigonometric identities to show that when _{} is multiplied by _{}, the result can be written in the form of Equation 5:

_{}

4.
Verify that the
phase _{} in Equation 5 is the
same as the phase _{} in _{}.

5.
Use MatLab to
generate and plot the signals on the left hand side of Figure 3. Specifically, generate _{} and _{}, and then multiply them together in the time domain. Then use a low pass filter to obtain the
downmixed Doppler signal. Use _{}, _{}, _{}, and _{}.

6.
Compare the calculated result in 5 to the result of calculating the downmixed
signal directly from Equation 5 and calculating the filtered downmixed signal
directly from the first term on the left hand side of Equation 5.

7.
Use the fast
Fourier transform routine in MatLab to calculate and plot the power spectrum of
the time-domain signals calculated in problem 5. Are the spectra infinitely narrow, as in a
delta function? Why or why not?

8.
Repeat problem 7,
but before taking the Fourier transform zero pad the signal to quadruple the
number of points. How would you describe
the shape of the power spectrum mathematically?

9.
If both the
Fourier transform of a signal and the downmixing of a signal are linear
processes, explain why the power spectrum of a Doppler signal from two
simultaneous scatterers is not equal to the sum of the power spectra of the
individual scatterers.

10. Use MatLab to
generate all of the signals on the left hand side of Figure 5.

11. Use MatLab and the FFT to generate all of the spectra
on the right hand side of Figure 5.

12. Use MatLab to generate the three time-domain signals
in Figure 6.

13. Use MatLab and the FFT to generate all of the spectra
in Figure 6.

14. Show that if the transmitted signal pulse has a
Gaussian shape to it and the receiver gate shape is also Gaussian, then the
dependence of the amplitude of the signal from a particle within the sample
volume on axial location is also Gaussian.

15. Use MatLab to calculate the first moment, mode, and
max frequencies of the Doppler spectrum from a continuous wave downmixed signal
(as calculated in problem 8).

16. Use MatLab to add random noise to a cosine wave
(signal to noise ratio 20 db, i.e. the rms of the noise should be 1/10^{th}
of the rms of the cosine). Plot the
resulting signal.

17. Calculate the first moment, mode and max frequencies
of the signal obtained in 16 above.
Compare these frequencies and explain differences, if any.

18. Create a zero crossing detector in MatLab. Apply this detector to the signal generated
in 16 above and compare the result to the frequencies obtained in 17.

19. For a typical 10 MHz Doppler device used in carotid
diagnosis, with the pulse repetition frequency equal to 32 kHz, calculate the
velocity at which aliasing occurs and the spacing between the multiple sample
volumes.

20. Give mathematical formulations for each of the
parameters _{}, _{}, and _{} in Equation 9 for the following measurement
configuration.

The
velocity profile is Poiseuille, and the seeding is uniform. Let the beam pattern for the probe be
specified as:

Note:
This problem may not be as simple as it appears at first because the coordinate
system for the beam pattern is different from the coordinate system for the
velocity profile. Your best bet is to
rewrite the equation for Poiseuille flow in terms of the coordinate system for
the beam pattern.

21. Use MatLab to calculate and plot the beam pattern for
the far field of a circular transducer element of a 10 MHz device for the
following conditions:

a.
transducer
diameter = 0.5 mm

b.
transducer
diameter = 1 mm

c.
transducer
diameter = 2 mm

22. Derive mathematically the expressions for the spectra
in panels a, b and c of Figure 9.

23. Estimate the
total scattered power from a single red blood cell.

24. Plot scattered power as a function of scattering angle
for a single red blood cell (model the cell as a sphere in this case).

25. Estimate the depth of penetration of a 2 MHz, 5 MHz,
10 MHz and 20 MHz Doppler ultrasound signal.
Assume that depth of penetration is defined as the depth at which the
sound power is 10% of the transmitted power.
Based on this calculation, does the following rule of thumb apply:

a.
Transcraneal
Doppler: 2 MHz

b.
Cardiac Doppler:
5 MHz

c.
Carotid Doppler:
10 MHz

d.
intravascular
Doppler: 20 MHz

26. Estimate the percentage of power that penetrates to 1
cm for the 2 MHz, 5 MHz, 10 MHz and 20 MHz ultrasound devices.

27. Plot the scattering/attenuation function _{} as a function of
range, and show that it has a peak at some value of r. Determine mathematically the value of r at
which the peak occurs.

28. Use Equation 14 to simulate a Doppler signal. Assume that N=10, _{}, and let the spectrum be Gaussian with a standard deviation
of 30 Hz. Simulate 5 seconds worth of
data.

29. Use your PC to play back your simulated signal through
the sound card. Describe the sound
qualitatively.

30. Simulate a pure sine wave at 900 Hz and play it back
through the sound card of your computer.
Describe the difference between this signal and the one generated
through the use of Equation 14.

31. Increase N to 100 in the simulation of problem
28. Describe any qualitative differences
between the two signals in terms of both visual appearance when plotted and
audio quality when played back through your sound card.

32. Use Equation 15 to generate a signal similar to that
of Kitney et al. Compare this visually
to the signal generated in problem 31.
Play it back through the sound card and compare acoustically to the
signal generated in problem 31.

33. Refer to a book on probability and stochastic
processes to determine the relationship between Gaussian, Rayleigh distributed,
and Chi-Square distributed random variables.
Deduce from these relationships that the simulations of equations 14, 16
and 17 are statistically equivalent.

34. Describe several problems that may arise from the
presence of the sharp corners in the pseudorandom signal of Figure 15d.

35. Derive Equation 32 for the relationship between the
first moment frequency and the signal autocorrelation function.

36. Given Doppler shifts from 3 receivers simultaneously,
where the sample volume is at _{}, the transmitter is at _{} and the receivers are
at _{}, _{}, and _{}, derive an expression for the complete velocity vector _{} within the sample
volume.