Download Ultrasound Physics Volume I

Ultrasound Physics & Instrumentation
4th Edition
Volume I
Companion Presentation
Frank R. Miele
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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
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Mathematics: Level 2
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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
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Non-Linear Proportionality
Increase by factor of 4
A proportional relationship between variables in which if one variable
increases by x%, the other variable increases by a different percentage.
y
y = x2
5
4
3
2
1
1
2
3
4
5
Increase by factor of 2
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x
Non-linear Direct Relationships
Fig. 4: y = 3x2 (Pg 43)
Notice the same characteristic parabolic shape of the graph of 3x2 and x2.
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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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Logarithms
Logarithms are a compression scheme which yield a method for dealing
with a very large range of data.
log 10 100 = x
 10x = 100 = 102
x=2
The mathematical approach to solving a logarithm is to go around a circle as
shown above.
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Logarithms
Fig. 6: Log Scale (Pg 51)
Numbers are compressed more and more moving to the right on the graph.
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Linear Scales
Fig. 7: Linear Scale (Pg 52)
Numbers are uniformly distributed along the entire graph.
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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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Properties of Logarithms
log  x  y   log  x   log  y 
x
log    log  x   log  y 
 y
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
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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
4
2
10-1
100
101
-1
0
1
20
5
102
2
0.6
0.3
0.7
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103
3
1.3
Trigonometry and the Unit Circle
Fig. 9: Unit Circle (Pg 53)
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Cosine of 0 Degrees
Fig. 12: Unit Circle
Cosine(0) = 1
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Cosine of 60 Degrees
Fig. 10: Unit Circle (Pg 54)
Cosine(60) = 0.5
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Cosine of 90 Degrees
Fig. 13: Unit Circle (Pg 55)
Cosine(90) = 0
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Trigonometry (Cosine)
y
Cos (  )
0o
1
.7
5
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
-1
210 o
-0.866
225 o
-0.707
240 o
-0.50
270 o
0
-.25
-.25
0
.2
5
.5
.7
5
.-5
.-5
0° = 360°
x
1
-.75
-.75
-1
180°
-1
0
.2
5
Angle (  )
.5
1
90°
270°
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Cosine (Animation)
(Pg 55)
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Sine of 60 Degrees
Fig. 11: Unit Circle (Pg 54)
Sine(60) = 0.866
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Trigonometry (Sine)
Sin (  )
0o
0
.7
5
30 o
0.5
.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
-1
.-5
-.25
0
-.25
-.75
.2
5
.5
.7
5
0° = 360°
x
1
.-5
180°
-1
0
.2
5
Angle (  )
-.75
90°
-1
1
y
270°
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Trigonometry
1
Amplitude
0.8
0.6
cosine
0.4
0.2
0
-0.2
90 180
360
-0.4
sine
-0.6
-0.8
-1
Angle in Degrees
Angle (  )
0o
30 o
45 o
60 o
90 o
120 o
135 o
150 o
180 o
210 o
225 o
240 o
270 o
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Cos (  )
1
0.866
0.707
0.5
0
-0.50
-0.707
-0.866
-1
-0.866
-0.707
-0.50
0
Sin(  )
0
0.5
0.707
0.866
1
0.866
0.707
0.50
0
-0.50
-0.707
-0.866
-1
Graphing the Sine and the Cosine
Fig. 14: Graphical Representation of the Sine and Cosine Versus Angle
(Pg 55)
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Formal Trigonometric Relationships
Fig. 15: Trigonometric Relationship (Pg 56)
adjacent
cos   
hypotenuse
sin   
opposite
hypotenuse
sin   opposite
tan   

cos   adjacent
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Defining Angular Quadrants
Fig. 16: Quadrants of a Circle (Pg 57)
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Positive Cosines
Fig. 17: Cosine is Positive in Quadrant I and Quadrant IV (Pg 57)
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Negative Cosines
Fig. 18: Cosine is Negative in Quadrant II and Quadrant III (Pg 58)
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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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Low Frequency Analog Signal
Fig. 19: Slowly Varying Analog Signal (Pg 66)
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Higher Frequency Analog Signal
Fig. 20: Quickly Varying Analog Signal (Pg 66)
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Analog to Digital Converter (A/D)
Fig. 23: An 8-bit A/D Converter (Pg 67))
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Sampling an Analog Signal
Fig. 21: Graphical Representation of Sampling (Pg 67)
Sampling Clock
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Analog to Digital Conversion
A
m
p
l
i
t
u
d
e
Every time the clock “ticks” the A/D
converter samples the signal and
outputs a digital value representing
the amplitude of the signal.
t2
t3
t4
t5
t6
t7
t9 Time
t8
A/D
“Sampling of a Slowly Varying
Analog Signal”
t1
t2
t3
t4
t5
t6
t7
t8
Analog Input
Signal
t9
Time
Clock
“The Sampling Clock”
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8 bit Digital
Output
t1
Sampled (Digital) Signal
Fig. 24: Graphical Representation of a Digital Signal (Pg 68)
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Reconstructing from a Digital Signal
Fig. 25: Reconstructed Signal (Pg 68)
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Sampling a Higher Frequency Signal
Fig. 26: Sampling a Quickly Varying Analog Signal (Pg 69)
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Digital Signal Representation
Fig. 27: Graphical Representation of the Digital Signal (Pg 69)
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Reconstructing from the Digital Signal
Fig. 28: Reconstructed Signal (Pg 70)
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Original versus Reconstructed Signal
Fig. 29: Reconstructed Versus Original Signal (Pg 70)
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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.
f sampling (minimum)  2  f signal (maximum)
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Determining Nyquist (2 Hz Signal)
Fig. 30: Analog Signal (Pg 72)
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Sampling at 25 Hz
Fig. 31: Sampled Signal (Pg 72)
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Reconstruction (No Aliasing)
Fig. 32: Reconstructed Signal (Pg 72)
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Sampling Too Slowly
Fig. 33: Analog Signal (Pg 73)
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Digital Signal From Sampling at 2 Hz
Fig. 34: Sampled Signal (Pg 73)
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Reconstructed Signal is Aliased
Fig. 35: Reconstructed Signal (Pg 73)
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Reconstructed Signal Not Aliased
Fig. 37: Sampled Signal (Pg 74)
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Sampling at 4 Hz
Fig. 36: Analog Signal (Pg 74)
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Nyquist Limit: Sample Twice as Fast
Fig. 38: Reconstructed Signal (Pg 74)
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Violation of Nyquist: Aliasing
Fig. 39: Aliasing
(Pg 75)
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Wave Addition
Fig. 40: Two In Phase Waves (Pg 76)
+
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Constructive Interference
Fig. 41: Constructive Interference (Pg 77)
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Destructive Interference
Fig. 42: Destructive Interference
(Pg 77)
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Partial Constructive Interference
Fig. 43: Partial Constructive Interference
(Pg 78)
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Addition of Waves
2
1
1
0.75
-0.5
0.5
0
0
0.25
0.5
Constructive Interference
-1
-1.5
-2
Time (sec)
1
0.75
0.5
Sum = 0
1
0.75
-0.25
0.5
0
0
0.25
0.25
Destructive Interference
-0.5
-0.75
-1
Time (sec)
2
1.5
1
1
0.75
-0.5
0.5
0
0.25
0.5
0
Amplitude
Amplitude
Amplitude
1.5
Partial Constructive
Interference
-1
-1.5
-2
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Time (sec)
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