1. Volume I Companion Presentation Frank R. Miele Pegasus Lectures, Inc.
Ultrasound Physics & Instrumentation 4th Edition
Volume I
Companion Presentation
Frank R. Miele
Pegasus Lectures, Inc.
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2. License Agreement Pegasus Lectures, Inc. All Copyright Laws Apply.
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3. Volume I Outline Pegasus Lectures, Inc. Chapter 1: Mathematics Level 1
Chapter 2: Waves
Chapter 3: Attenuation
Chapter 4: Pulsed Wave
Chapter 5: Transducers
Chapter 6: System Operation
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4. Mathematics: Level 2
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5. Level 2 Material Pegasus Lectures, Inc.
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
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6. 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 = x2 x y 1 3 4 5 2
Increase by factor of 2
Increase by factor of 4
The contrast between linear and non-linear growth should be stressed. Non-linear relationships express situations where very small changes in one parameter can result in significant changes in the related parameter. Most of the world behaves non-linearly such as growth of plants, spread of disease, perceived brightness with distance, etc…
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7. Non-linear Direct Relationships
Fig. 4: y = 3x2 (Pg 43)
Notice the same characteristic parabolic shape of the graph of 3x2 and x2.
Notice that the constant term of 3 effects that overall value, but not the rate of growth. This fact can be seen by comparing this graph with the graph of the previous slide (y = x2). Notice that both graphs have the same characteristic shape. In both cases, if x is increased by a factor of 4, y increases by a factor of 16. The constant factor of three affects the absolute value but not the relative value. This is why both equations would lead us to stating that y is proportional to x squared.
You should also note that how the data is graphed can lead to misperceptions. Note that in the previous slide, the y axis increases by ones (from zero to five) whereas this graph increases by tens (from zero to 90). As a result, the previous graph artificially “appears” to express a greater rate of change than this graph.
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8. Inverse Proportionality
Fig. 5: y = 3/x2 (Pg 44)
Notice that in this case small increases in x produce more rapid decreases in y.
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9. Logarithms Pegasus Lectures, Inc.
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 = x  10x = 100 =  x = 2 10
The value of logarithms becomes clear in Chapter 6 when we discuss the need to compress the enormous dynamic range so that real-time imaging can be achieved. Without a means by which to compress the extraordinary range of signals that return from the body, we would need to use some type of technique in which we would have to view multiple images at varying greyscale levels.
Also notice that the base used in this example is 10. It is clearly possible to use any base one desires, but since we generally use a counting system based on factors of 10 (decimal), the only logarithm the students are expected to know is base 10. However, the concept is exactly the same for all other bases.
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10. Logarithms Pegasus Lectures, Inc. Fig. 6: Log Scale (Pg 51)
Numbers are compressed more and more moving to the right on the graph.
Using the above scale, it is easy to see that the log (base 10) of a power of 10 is simply the exponent. In other words, since 100 can be written as 102, the log of 100 is 2. It is helpful to point out the fact that reciprocals are clearly demonstrated by this graph (since the log of 100 is 2, the log of 1/100 is
-2).
It is also valuable to point out that as the we move farther up the graph, there is more and more compression (numbers become closer and closer together). In other words, logarithms are a non-linear method of displaying (graphing) data.
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11. Linear Scales Pegasus Lectures, Inc. Fig. 7: Linear Scale (Pg 52)
Numbers are uniformly distributed along the entire graph.
In contrast to the log scale, a “distance” on the linear scale always represents an equal value. If one inch of a linear graph equals the value 7, then two inches equals the value of 14.
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12. 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.
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13. Properties of Logarithms
log (4) = log (2  2) = log (2) + log (2) = = 0.6
log (20) = log (2  10) = log (2) + log (10) = = 1.3
log (5) = log (10 2 ) = log (10) - log (2) = 1.0 – 0.3 = 0.7
Logarithms convert the operation of multiplication into addition. Not surprisingly, logarithms also convert division into subtraction (inverse operations). This fact is useful to solve logarithms of numbers that are not powers of 10.
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14. Properties of Logarithms
log (4) = log (2  2) = log (2) + log (2) = = 0.6
log (20) = log (2  10) = log (2) + log (10) = = 1.3
log (5) = log (10 2 ) = log (10) - log (2) = 1.0 – 0.3 = 0.7
10-1 -1
100
101 1
102 2
103 3 5
0.7 20
1.3 4
0.6
0.3
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15. Trigonometry and the Unit Circle
Fig. 9: Unit Circle (Pg 53)
The use of the unit circle for trigonometry is extremely helpful. It is referred to as the unit circle since the length of the radius is 1, or 1 unit. This radius makes life easy since all cosine and sine values are referenced to the value of 1, making it easy to express percentages.
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16. Cosine of 0 Degrees Pegasus Lectures, Inc. Fig. 12: Unit Circle
It is important that the student learns the convention of how angles are measured. The angle is always measured relative to the x-axis. Notice that zero degrees aligns with the x-axis.
The cosine is the projection of the intersection of the angle with the unit circle toward the x-axis. At zero degrees, the projection is already on the x-axis so the cosine is 1, or 100%.
Cosine(0) = 1
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17. Cosine of 60 Degrees Pegasus Lectures, Inc.
Fig. 10: Unit Circle (Pg 54)
Cosine(60) = 0.5
Again notice that the angle is measured relative to the x-axis. (60 degrees is 2/3 of the arc from 0 degrees and 90 degrees). When projecting to the x-axis, notice that the “shadow” produced on the x-axis is only 0.5, or 50%.
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18. Cosine of 90 Degrees Pegasus Lectures, Inc.
Fig. 13: Unit Circle (Pg 55)
At 90 degrees, the projection of the angle with the circle back to the x-axis has a length of zero on the x-axis, so the cosine of 90 degrees is 0.
Cosine(90) = 0
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19. Trigonometry (Cosine)
270° 1 -1 .5
.75
.25
.-5
-.25
-.75 x
0° = 360°
90°
180°
Angle (  )
Cos (  )
0 o 1
30 o
0.866
45 o
0.707
60 o
0.5
90 o
120 o
-0.50
135 o
-0.707
150 o
-0.866
This slide is intended to show the students the cosine of the “basic” angles. Knowing the basic angles is useful since it provides a reference for estimating the cosine of “non-basic” angles. Notice that the basic angles are all relatively easily visualized on the unit circle. In this slide, the basic angles between 0 and 90 degrees (inclusively) are shown, but the cosine values are also given in the table for angles greater than 90 degrees. From this table it should be evident that 45 degrees is basically the “same” angle as 135 degrees, 225 degrees, and 315 degrees (not shown), just as 30 degrees is the same basic angle as 150 degrees, 210 degrees, and 330 degrees.
Important points to note:
The angle is always measured with respect to the x-axis
The cosine is the projection of the intersection of the angle with the unit circle toward the x-axis.
The cosine decreases as the angle increases from 0 degrees to 90 degrees.
The cosine is positive for angles between 0 degrees and 90 degrees and between 270 degrees and 360 degrees
The cosine is negative for angles between 90 degrees and 270 degrees.
180 o -1
210 o
-0.866
225 o
-0.707
240 o
-0.50
270 o
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20. Cosine (Animation)
(Pg 55)
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21. Sine of 60 Degrees Pegasus Lectures, Inc. Fig. 11: Unit Circle (Pg 54)
Like the cosine, the sine of an angle can be determined from intersecting the angle with the unit circle but then projecting towards and y-axis. Whereas the cosine is the projection to the x-axis, the sine is the projection to the y-axis. Because of the obvious circular symmetry, the sine of 60 degrees is the same as the cosine of 30 degrees.
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22. Trigonometry (Sine) Pegasus Lectures, Inc. y 270° 1 -1 .5 .75 .25 .-5.5
.75
.25
.-5
-.25
-.75 x
0° = 360°
90°
180°
Angle (  )
Sin (  )
0 o
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
As was done for the cosine, the basic angles are included together on one slide as well as within a table.
Some important points to notice:
Notice that the angles are still specified relative to the x-axis. (Angles are always measured with respect to the x-axis)
The sine of 45 degrees is the same as the cosine of 45 degrees (45 degrees is halfway between the x-axis and y-axis – so the projection to the x-axis (cosine) and the projection to the y-axis (sine) must be equal.
The sine if positive between 0 degree and 180 degrees.
The sine is negative between 180 degrees and 360 degrees.
180 o
210 o
-0.5
225 o
-0.707
240 o
-0.866
270 o -1
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23. Trigonometry Pegasus Lectures, Inc. Angle (  ) Cos (  ) Sin(  ) 0 o1
30 o
0.866
0.5
45 o
0.707
60 o
90 o
120 o
-0.50
135 o
-0.707
150 o
-0.866
0.50
180 o -1
210 o
225 o
240 o
270 o 1
0.8
0.6
cosine
Amplitude
0.4
0.2 90
180
360
-0.2
-0.4
sine
-0.6
-0.8
Notice also that each time around the circle, the values repeat. Hence, when graphed, the graph shows periodicity (a characteristic of a wave). -1
Angle in Degrees
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24. Graphing the Sine and the Cosine
Fig. 14: Graphical Representation of the Sine and Cosine Versus Angle
(Pg 55)
Notice that there is a 90 degree phase shift between the sine and the cosine. This fact should not be surprising since the cosine is the projection to the x-axis and the sine is the projection to the y-axis. Since there are 90 degrees between the x-axis and the y-axis, there is a 90 degree phase shift between the sine and the cosine.
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25. Formal Trigonometric Relationships
Fig. 15: Trigonometric Relationship (Pg 56)
These slides are added for completeness. Most math classes teach the trigonometric functions such as the sine and the cosine (tangent, secant, cosecant and cotangent) in terms of ratios of the opposite, adjacent, and hypotenuse sides of a right triangle. This slide is therefore intended to give the students a means by which to associate what they might have learned elsewhere with the approach illustrated through use of the unit circle.
Points to notice:
The “adjacent” side corresponds to the value on the x-axis
The “opposite” side corresponds to the value on the y-axis
Since the circle has a radius of 1 (unit circle), the hypotenuse = 1, so the adjacent divided by the hypotenuse (cosine) is simply the adjacent value. (similarly the opposite divided by the hypotenuse (the sine) is imply the opposite side.
Please note that if your students have never learned trigonometry in this formal way, then this slide can be ignored for them. From the ultrasound standpoint – it is critical that understand sines and cosines, but it is not critical that they know the formal definition as described here.
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26. Defining Angular Quadrants
Fig. 16: Quadrants of a Circle (Pg 57)
Often on the boards, the angles are referred to in terms of quadrants. Obviously the term quadrant refers to the fact that a “quadrant” represents a quarter of a circle, or 90 degrees. Since angles are measured counter-clockwise, the quadrants are also numbered counter-clockwise, as depicted in the figure above.
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27. Positive Cosines Pegasus Lectures, Inc.
Fig. 17: Cosine is Positive in Quadrant I and Quadrant IV (Pg 57)
Understanding for what angles the cosine is positive and for what angles the cosine is negative is very important to be able to correctly identify flow direction (as will be seen in Chapter 7 on Doppler). Since the cosine is the projection towards the x-axis, the cosine is positive when the projection is on the positive portion of the x-axis, which occurs when the angle is between 0 and 90 degrees and when the angle is between 270 and 360 degrees.
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28. Negative Cosines Pegasus Lectures, Inc.
Fig. 18: Cosine is Negative in Quadrant II and Quadrant III (Pg 58)
Similar to the last slide, the cosine is negative when the projection onto the x-axis is onto the negative portion of the x-axis which occurs when angles are between 90 degrees and 270 degrees.
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29. 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.
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30. Low Frequency Analog Signal
Fig. 19: Slowly Varying Analog Signal (Pg 66)
Before demonstrating the effects of sampling, it is useful to show a few analog signals of varying frequency. This first signal varies very slowly with time, and hence, is considered to be a low frequency signal.
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31. Higher Frequency Analog Signal
Fig. 20: Quickly Varying Analog Signal (Pg 66)
In comparison to the last slide, notice how much faster the signal is varying with respect to time. This signal constitutes a higher frequency signal.
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32. Analog to Digital Converter (A/D)
Fig. 23: An 8-bit A/D Converter (Pg 67))
The analog signal is input into the A/D converter. A clock is attached to the clock line which sets the conversion (sampling) rate. Every time the clock “ticks” the signal is sampled and output as a digital signal (in binary format). The smallest division of the A/D converter output is called a bit. The more bits, the higher the output dynamic range. Imagine if a converter only had one bit, there would only be two possible signal level outputs (on or off) or 1 and 0. Having more bits yields more combinations, thereby allowing the ability to specify more signal levels (higher dynamic range).
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33. Sampling an Analog Signal
Fig. 21: Graphical Representation of Sampling (Pg 67)
Sampling Clock
Notice that each time the clock “ticks” there is a “sample” taken of the analog signal. Between the clock ticks, there is no sampling and hence no record of how the signal is changing during these intervals.
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34. Analog to Digital Conversion
Amplitude
Time t8 t7 t6 t5 t4 t3 t2 t1 t9
Every time the clock “ticks” the A/D converter samples the signal and outputs a digital value representing the amplitude of the signal.
Analog Input Signal
Clock
A/D
8 bit Digital Output
“Sampling of a Slowly Varying Analog Signal”
“The Sampling Clock”
Time t8 t7 t6 t5 t4 t3 t2 t1 t9
This slide combines the pieces of the previous slides into one to show that the dynamic range of the digital signal is limited by the number of bits of the A/D converter.
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35. Sampled (Digital) Signal
Fig. 24: Graphical Representation of a Digital Signal (Pg 68)
This slide graphically shows the digital signal and the fact that between each sample there is no information retained. In cases like this in which the signal is varying slowly, it is evident that not much information was lost ( as seen on the next slide).
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36. Reconstructing from a Digital Signal
Fig. 25: Reconstructed Signal (Pg 68)
Ultimately the samples at discrete time intervals are reconnected to create the “reconstructed” signal. In this case, the reconstructed signal looks very similar to the original signal and hence accurately represents the original analog signal.
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37. Sampling a Higher Frequency Signal
Fig. 26: Sampling a Quickly Varying Analog Signal (Pg 69)
This slide mimics the earlier slide showing how sampling of an analog signal occurs. This time, the analog signal is higher frequency (the signal changes faster with respect to time). Notice that the sampling rate is the same in this slide as it was with the lower frequency analog signal. This point will be important to understand the upcoming slides.
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38. Digital Signal Representation
Fig. 27: Graphical Representation of the Digital Signal (Pg 69)
This slide represents the digital signal that resulted from sampling the analog signal. Note (as was the case with the previous lower frequency signal) that there is no information about what occurred between sampling times.
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39. Reconstructing from the Digital Signal
Fig. 28: Reconstructed Signal (Pg 70)
This slide shows the results of when the digital signal is reconstructed.
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40. Original versus Reconstructed Signal
Fig. 29: Reconstructed Versus Original Signal (Pg 70)
Now we can clearly see the effect of sampling a higher frequency signal too slowly. Notice how the reconstructed signal has the basic trend of the original analog signal, but does not faithfully reproduce the quick variations (higher frequency components) of the signal. Note that the “poor” reproduction occurred even though the sample rate was the same as it was for the slower analog signal. Clearly, higher frequency signals require faster sampling for adequate reconstruction. This fact will lead to a desire to specify how fast a signal needs to be sampled to produce faithful reconstructions of the analog signal after sampling. This criterion is referred to as the Nyquist Criterion as shown on the next slide.
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41. Nyquist Criterion Pegasus Lectures, Inc.
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.
Nyquist is valuable because it expresses the basic fact that what you see is not always reality. It is a good idea to have students answer some basic questions about the sampling rate and highest detectable frequency at this point (making sure to ask the question in both the forward and reciprocal direction) For example:
If the sampling frequency is 20 kHz, what is the highest frequency signal that can be detected according to the Nyquist Criterion?
(Answer 10 kHz)
If the signal has a frequency of 20 kHz, what is the lowest sample rate possible for accurate detection based on the Nyquist Criterion
(Answer 40 kHz)
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42. Determining Nyquist (2 Hz Signal)
Fig. 30: Analog Signal (Pg 72)
This image is the first of three consecutive images that need to be considered together.
This first image shows a 2 Hz signal (2 cycles per second).
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43. Sampling at 25 Hz Pegasus Lectures, Inc.
Fig. 31: Sampled Signal (Pg 72)
This second image shows the resulting digital signal from sampling the 2 Hz signal at 25 Hz.
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44. Reconstruction (No Aliasing)
Fig. 32: Reconstructed Signal (Pg 72)
This third slide shows the reconstructed signal by “connecting” the points of the digital signal. Notice that the correct 2 Hz signal is reconstructed. This was anticipated by the fact that the Nyquist criterion was satisfied. (A sampling rate of 25 Hz is clearly greater than 2 times 2 Hz).
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45. Sampling Too Slowly Pegasus Lectures, Inc.
Fig. 33: Analog Signal (Pg 73)
In this case, the sample rate for the 2 Hz signal is only 2 Hz (which is lower than the 4 Hz dictated by Nyquist).
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46. Digital Signal From Sampling at 2 Hz
Fig. 34: Sampled Signal (Pg 73)
Notice that the there are only two sample points in the digital signal.
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47. Reconstructed Signal is Aliased
Fig. 35: Reconstructed Signal (Pg 73)
In this slide, the reconstructed signal is demonstrated. In marked contrast to the previous example, notice that Nyquist is not satisfied and the reconstructed signal frequency does not match the original analog signal (Aliasing). The reconstructed signal frequency that results is actually 0 Hz (clearly incorrect).
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48. Reconstructed Signal Not Aliased
Fig. 37: Sampled Signal (Pg 74)
This example shows what happens when the 2 Hz signal is sampled at exactly the Nyquist limit of 4 Hz.
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49. Sampling at 4 Hz Pegasus Lectures, Inc. Fig. 36: Analog Signal (Pg 74)
This slide graphically demonstrates the digital signal that results from sampling the 2 Hz signal at 4 Hz.
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50. Nyquist Limit: Sample Twice as Fast
Fig. 38: Reconstructed Signal (Pg 74)
Notice now, that the reconstructed signal frequency does match the original signal frequency of 2 Hz.
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51. Violation of Nyquist: Aliasing
Fig. 39: Aliasing (Pg 75)
The sampling of this 9 Hz signal clearly does not meet the Nyquist criterion. Notice that the resulting reconstructed frequency is 2 Hz, clearly incongruous with the original 9 Hz signal.
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52. Wave Addition Pegasus Lectures, Inc. +
Fig. 40: Two In Phase Waves (Pg 76) +
“ In phase” implies that both waves reach a maxima and a minima at the same time. Notice that if these two waves were overlaid, they would appear as one wave.
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53. Constructive Interference
Fig. 41: Constructive Interference (Pg 77)
If two or more in phase waves are created in the same physical region, the energy of the waves combines to construct one larger wave. In this case with two waves, the resultant wave has an amplitude twice as big as the original individual waves. Constructive interference is an important concept to understand how phased array transducers behave and are capable of steering and focusing.
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54. Destructive Interference
Fig. 42: Destructive Interference (Pg 77)
In contrast to the previous case, these two waves are completely out of phase (180 degree phase shift). Two waves are considered purely out of phase when the maxima of one wave aligns with the minima of the other wave. The result is wave cancellation as shown referred to as destructive interference. Destructive interference is also very important in phased array operation. Ideally, to make a narrow beam, there would be pure constructive interference in a very narrow region along the direction of the beam and pure destructive interference everywhere else.
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55. Partial Constructive Interference
Fig. 43: Partial Constructive Interference (Pg 78)
When two waves are not purely in phase or out of phase, partial constructive interference occurs. With partial constructive interference the waves add up only partially building a “bigger” wave, but not completely as the sum of the two waves. Most of the time, waves from elements from phased array transducers add partially constructively.
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56. Addition of Waves Pegasus Lectures, Inc. Constructive Interference2
1.5 1
Amplitude
0.5
Constructive Interference
-0.5
0.25
0.5
0.75 1 -1
-1.5 -2
Time (sec) 1
0.75
0.5
Sum = 0
Amplitude
0.25
Destructive Interference
0.25
0.5 1
-0.25
0.75
-0.5
-0.75
This slides recapitulates the previous three slides so that all three cases can be seen simultaneously. -1
Time (sec) 2
1.5 1
Partial Constructive Interference
Amplitude
0.5
0.25
0.5
0.75 1
-0.5 -1
-1.5 -2
Time (sec)
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57. Add Title Pegasus Lectures, Inc. Blank Slide:
This blank slide is here to help facilitate adding new content. If you would like to add material to this presentation, copy this slide and place in the correct location.
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