1. Pegasus Lectures, Inc. COPYRIGHT 2006 Volume I Companion Presentation Frank R. Miele Pegasus Lectures, Inc. Ultrasound Physics & Instrumentation 4 th Edition

2. Pegasus Lectures, Inc. COPYRIGHT 2006 License Agreement This presentation is the sole property of Pegasus Lectures, Inc. No part of this presentation may be copied or used for any purpose other than as part of the partnership program as described in the license agreement. Materials within this presentation may not be used in any part or form outside of the partnership program. Failure to follow the license agreement is a violation of Federal Copyright Law. All Copyright Laws Apply.

3. Pegasus Lectures, Inc. COPYRIGHT 2006 Volume I Outline  Chapter 1: Mathematics  Level 1  Level 2  Chapter 2: Waves  Chapter 3: Attenuation  Chapter 4: Pulsed Wave  Chapter 5: Transducers  Chapter 6: System Operation

4. Pegasus Lectures, Inc. COPYRIGHT 2006 Mathematics: Level 2

5. Pegasus Lectures, Inc. COPYRIGHT 2006 Level 2 Material Level 2 Mathematics delves into concepts of:  Non-linear relationships  Percentage Change  Logarithms  Trigonometry  The Binary System  A/D Conversion  Nyquist Criterion  Constructive and Destructive Interference

6. Pegasus Lectures, Inc. COPYRIGHT 2006 Non-Linear Proportionality A proportional relationship between variables in which if one variable increases by x%, the other variable increases by a different percentage. y = x 2 x y 1345 2 3 4 5 2 1 Increase by factor of 2 Increase by factor of 4

7. Pegasus Lectures, Inc. COPYRIGHT 2006 Non-linear Direct Relationships Fig. 4: y = 3x 2 (Pg 43) Notice the same characteristic parabolic shape of the graph of 3x 2 and x 2.

8. Pegasus Lectures, Inc. COPYRIGHT 2006 Inverse Proportionality Fig. 5: y = 3/x 2 (Pg 44) Notice that in this case small increases in x produce more rapid decreases in y.

9. Pegasus Lectures, Inc. COPYRIGHT 2006 Logarithms Logarithms are a compression scheme which yield a method for dealing with a very large range of data. The mathematical approach to solving a logarithm is to go around a circle as shown above. log 100 = x  10 x = 100 = 10 2  x = 2 10

10. Pegasus Lectures, Inc. COPYRIGHT 2006 Logarithms Fig. 6: Log Scale (Pg 51) Numbers are compressed more and more moving to the right on the graph.

11. Pegasus Lectures, Inc. COPYRIGHT 2006 Linear Scales Fig. 7: Linear Scale (Pg 52) Numbers are uniformly distributed along the entire graph.

12. Pegasus Lectures, Inc. COPYRIGHT 2006 Visualizing Compression by Logarithms Fig. 8: Log Scale and the Log of 2 (Pg 52)  Notice that on a logarithmic graph, the number 2 is farther from 1 than 3 is from 2, and 4 is even closer to 3 than to 2.  Notice that the log of 2 must be greater then 0 and less than 1.  The log of 2 is 0.3.

13. Pegasus Lectures, Inc. COPYRIGHT 2006 Properties of Logarithms log (4) = log (2  2) = log (2) + log (2) = 0.3 + 0.3 = 0.6 log (20) = log (2  10) = log (2) + log (10) = 0.3 + 1.0 = 1.3 log (5) = log (10  2 ) = log (10) - log (2) = 1.0 – 0.3 = 0.7

14. Pegasus Lectures, Inc. COPYRIGHT 2006 Properties of Logarithms log (4) = log (2  2) = log (2) + log (2) = 0.3 + 0.3 = 0.6 log (20) = log (2  10) = log (2) + log (10) = 0.3 + 1.0 = 1.3 log (5) = log (10  2 ) = log (10) - log (2) = 1.0 – 0.3 = 0.7 10 -1 10 0 0 10 1 1 10 2 2 10 3 3 2 5 0.7 20 1.3 4 0.6 0.3

15. Pegasus Lectures, Inc. COPYRIGHT 2006 Trigonometry and the Unit Circle Fig. 9: Unit Circle (Pg 53)

16. Pegasus Lectures, Inc. COPYRIGHT 2006 Cosine of 0 Degrees Fig. 12: Unit Circle Cosine(0  ) = 1

17. Pegasus Lectures, Inc. COPYRIGHT 2006 Cosine of 60 Degrees Fig. 10: Unit Circle (Pg 54) Cosine(60  ) = 0.5

18. Pegasus Lectures, Inc. COPYRIGHT 2006 Cosine of 90 Degrees Fig. 13: Unit Circle (Pg 55) Cosine(90  ) = 0

19. Pegasus Lectures, Inc. COPYRIGHT 2006 Trigonometry (Cosine) y 270° 1 0.5.7 5.2 5.-5-.25 -.75 1 0.5.7 5.2 5.-5 -.25 -.75 x 0° = 360° 90° 180° Angle (  )Cos (  ) 0 o0 o 1 30 o 0.866 45 o 0.707 60 o 0.5 90 o 0 120 o -0.50 135 o -0.707 150 o -0.866 180 o 210 o -0.866 225 o -0.707 240 o -0.50 270 o 0

20. Pegasus Lectures, Inc. COPYRIGHT 2006 Cosine (Animation) (Pg 55)

21. Pegasus Lectures, Inc. COPYRIGHT 2006 Sine of 60 Degrees Fig. 11: Unit Circle (Pg 54) Sine(60  ) = 0.866

22. Pegasus Lectures, Inc. COPYRIGHT 2006 Trigonometry (Sine) Angle (  )Sin (  ) 0 o0 o 0 30 o 0.5 45 o 0.707 60 o 0.866 90 o 1 120 o 0.866 135 o 0.707 150 o 0.5 180 o 0 210 o -0.5 225 o -0.707 240 o -0.866 270 o y 270° 1 0.5.7 5.2 5.-5-.25 -.75 1 0.5.7 5.2 5.-5 -.25 -.75 x 0° = 360° 90° 180°

23. Pegasus Lectures, Inc. COPYRIGHT 2006 Trigonometry Angle (  )Cos (  )Sin(  ) 0 o0 o 10 30 o 0.8660.5 45 o 0.707 60 o 0.50.866 90 o 01 120 o -0.500.866 135 o -0.7070.707 150 o -0.8660.50 180 o 0 210 o -0.866-0.50 225 o -0.707 240 o -0.50-0.866 270 o 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Angle in Degrees 90180360 sine cosine Amplitude

24. Pegasus Lectures, Inc. COPYRIGHT 2006 Graphing the Sine and the Cosine Fig. 14: Graphical Representation of the Sine and Cosine Versus Angle (Pg 55)

25. Pegasus Lectures, Inc. COPYRIGHT 2006 Formal Trigonometric Relationships Fig. 15: Trigonometric Relationship (Pg 56)

26. Pegasus Lectures, Inc. COPYRIGHT 2006 Defining Angular Quadrants Fig. 16: Quadrants of a Circle (Pg 57)

27. Pegasus Lectures, Inc. COPYRIGHT 2006 Positive Cosines Fig. 17: Cosine is Positive in Quadrant I and Quadrant IV (Pg 57)

28. Pegasus Lectures, Inc. COPYRIGHT 2006 Negative Cosines Fig. 18: Cosine is Negative in Quadrant II and Quadrant III (Pg 58)

29. Pegasus Lectures, Inc. COPYRIGHT 2006 Analog Signals and Digital Conversion Signals measured coming from the patient are analog signals. For ease in processing and simplification of electronics, these analog signals are converted to digital signals.  Analog signal are continuous in time.  Digital signals are created by sampling an analog signal at discrete time intervals.  The electronic device used for conversion is referred to as an analog to digital (A/D) converter.  The rate at which the sampling is performed can affect whether the digital signal accurately represents the original analog signal.  Faster signals require faster sampling.

30. Pegasus Lectures, Inc. COPYRIGHT 2006 Low Frequency Analog Signal Fig. 19: Slowly Varying Analog Signal (Pg 66)

31. Pegasus Lectures, Inc. COPYRIGHT 2006 Higher Frequency Analog Signal Fig. 20: Quickly Varying Analog Signal (Pg 66)

32. Pegasus Lectures, Inc. COPYRIGHT 2006 Analog to Digital Converter (A/D) Fig. 23: An 8-bit A/D Converter (Pg 67))

33. Pegasus Lectures, Inc. COPYRIGHT 2006 Sampling an Analog Signal Fig. 21: Graphical Representation of Sampling (Pg 67) Sampling Clock

34. Pegasus Lectures, Inc. COPYRIGHT 2006 Analog to Digital Conversion Analog Input Signal Clock A/D 8 bit Digital Output AmplitudeAmplitude Time t8t8 t7t7 t6t6 t5t5 t4t4 t3t3 t2t2 t1t1 t9t9 “Sampling of a Slowly Varying Analog Signal” “The Sampling Clock” Time t8t8 t7t7 t6t6 t5t5 t4t4 t3t3 t2t2 t1t1 t9t9 Every time the clock “ticks” the A/D converter samples the signal and outputs a digital value representing the amplitude of the signal.

35. Pegasus Lectures, Inc. COPYRIGHT 2006 Sampled (Digital) Signal Fig. 24: Graphical Representation of a Digital Signal (Pg 68)

36. Pegasus Lectures, Inc. COPYRIGHT 2006 Reconstructing from a Digital Signal Fig. 25: Reconstructed Signal (Pg 68)

37. Pegasus Lectures, Inc. COPYRIGHT 2006 Sampling a Higher Frequency Signal Fig. 26: Sampling a Quickly Varying Analog Signal (Pg 69)

38. Pegasus Lectures, Inc. COPYRIGHT 2006 Digital Signal Representation Fig. 27: Graphical Representation of the Digital Signal (Pg 69)

39. Pegasus Lectures, Inc. COPYRIGHT 2006 Reconstructing from the Digital Signal Fig. 28: Reconstructed Signal (Pg 70)

40. Pegasus Lectures, Inc. COPYRIGHT 2006 Original versus Reconstructed Signal Fig. 29: Reconstructed Versus Original Signal (Pg 70)

41. Pegasus Lectures, Inc. COPYRIGHT 2006 Nyquist Criterion The Nyquist Criterion states that to avoid aliasing, the sample frequency must be at least twice as fast as the highest frequency in the signal you want to detect.

42. Pegasus Lectures, Inc. COPYRIGHT 2006 Determining Nyquist (2 Hz Signal) Fig. 30: Analog Signal (Pg 72)

43. Pegasus Lectures, Inc. COPYRIGHT 2006 Sampling at 25 Hz Fig. 31: Sampled Signal (Pg 72)

44. Pegasus Lectures, Inc. COPYRIGHT 2006 Reconstruction (No Aliasing) Fig. 32: Reconstructed Signal (Pg 72)

45. Pegasus Lectures, Inc. COPYRIGHT 2006 Sampling Too Slowly Fig. 33: Analog Signal (Pg 73)

46. Pegasus Lectures, Inc. COPYRIGHT 2006 Digital Signal From Sampling at 2 Hz Fig. 34: Sampled Signal (Pg 73)

47. Pegasus Lectures, Inc. COPYRIGHT 2006 Reconstructed Signal is Aliased Fig. 35: Reconstructed Signal (Pg 73)

48. Pegasus Lectures, Inc. COPYRIGHT 2006 Reconstructed Signal Not Aliased Fig. 37: Sampled Signal (Pg 74)

49. Pegasus Lectures, Inc. COPYRIGHT 2006 Sampling at 4 Hz Fig. 36: Analog Signal (Pg 74)

50. Pegasus Lectures, Inc. COPYRIGHT 2006 Nyquist Limit: Sample Twice as Fast Fig. 38: Reconstructed Signal (Pg 74)

51. Pegasus Lectures, Inc. COPYRIGHT 2006 Violation of Nyquist: Aliasing Fig. 39: Aliasing (Pg 75)

52. Pegasus Lectures, Inc. COPYRIGHT 2006 Wave Addition Fig. 40: Two In Phase Waves (Pg 76) +

53. Pegasus Lectures, Inc. COPYRIGHT 2006 Constructive Interference Fig. 41: Constructive Interference (Pg 77)

54. Pegasus Lectures, Inc. COPYRIGHT 2006 Destructive Interference Fig. 42: Destructive Interference (Pg 77)

55. Pegasus Lectures, Inc. COPYRIGHT 2006 Partial Constructive Interference Fig. 43: Partial Constructive Interference (Pg 78)

56. Pegasus Lectures, Inc. COPYRIGHT 2006 Addition of Waves -2 -1.5 -0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 Time (sec) Amplitude -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Time (sec) Amplitude Sum = 0 -2 -1.5 -0.5 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 Time (sec) Amplitude Constructive Interference Destructive Interference Partial Constructive Interference

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