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Transcript of 1 Agenda for design activity r1. Requirements r2. Numbers r3. Decibels r4. Matrices r5. Transforms...
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Agenda for design activity
1. Requirements2. Numbers3. Decibels4. Matrices5. Transforms6. Statistics7. Software
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1. Requirements
Definition of a requirementOccurrence of requirementsGuidelines for a good requirementExamples for each guidelineTools for writing good requirementsNotes
1. Requirements
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Definition of a requirement
Something obligatory or demandedStatement of some needed thing or
characteristic
1. Requirements
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Occurrence of requirements
Writing requirements occurs in both the understand- requirements activity and the design activity
The customer has RAA for requirements in the understand- requirements activity even though the contractor may actually write the requirements
The contractor has RAA for requirements in the design activity
1. Requirements
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Errors in requirements come mainly from incorrect facts (50%), omissions (30%),
inconsistent (15%), ambiguous (2%), misplaced (2%)
Errors in requirements come mainly from incorrect facts (50%), omissions (30%),
inconsistent (15%), ambiguous (2%), misplaced (2%)
Guidelines for a good requirement
NeededCapable of being verifiedFeasible schedule, cost, and
implementationAt correct level in hierarchyCannot be misunderstoodGrammar and spelling correctDoes not duplicate information
1. Requirements
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Example for each guideline
Example 1 -- neededExample 2 -- verificationExample 3 -- feasibleExample 4 -- levelExample 5 -- understandingExample 6 -- duplicationExample 7 -- grammar and spellingExample 8 -- tough requirements
1. Requirements
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Example 1 -- needed
The motor shall weigh less than 10 pounds.The software shall use less than 75 percent of
the computer memory available for software.The MTBF shall be greater than 1000 hours.
1. Requirements
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Example 2 -- verification (1 of 3)
Customer want -- The outside wall shall be a material that requires low maintenance
1. Requirements
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Example 2 -- verification (2 of 3)
First possible rewording -- The outside wall shall be brick. • More verifiable• Limits contractor options• Not a customer requirement
1. Requirements
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Example 2 -- verification (3 of 3)
Second possible rewording -- The outside wall shall be one that requires low maintenance. Low maintenance material is one of the following: brick, stone, concrete, stucco, aluminum, vinyl, or material of similar durability; it is not one of the following: wood, fabric, cardboard, paper or material of similar durability• Uses definition to explain undefined
term
1. Requirements
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Example 3 -- feasible
Not feasible requirement -- The assembly shall be made of pure aluminum having a density of less than 50 pounds per cubic foot
1. Requirements
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Example 4 -- level
Airplane shall be capable carrying up to 2000 pounds Wing airfoil shall be of type Clark Y
airplane
wing
Wing airfoil shall be of type Clark Y
Wing airfoil type is generally a result of design and should appear in the lower product spec
and not in the higher product spec.
Wing airfoil type is generally a result of design and should appear in the lower product spec
and not in the higher product spec.1. Requirements
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Example 5 -- understanding
Avoid imprecise terms such as• Optimize• Maximize• Accommodate• Etc.• Support• Adequate
1. Requirements
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Example 6 -- duplication
Capable of a maximum rate of 100 gpmCapable of a minimum rate of 10 gpmRun BIT while pumping 10 gpm - 100 gpmVs: Run BIT while pumping between min.
and max.
1. Requirements
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Example 7 -- grammar and spelling
The computers is comercial-off-the-shelf items
Incorrect grammar or spelling will divert customer review of the requirements from the technical content
1. Requirements
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Example 8 -- tough requirements
BIT false alarm rate < 3 percentComputer throughput < 75 percent of capacityPerform over all altitudes and speedsConform with all local, state, and national lawsThere shall be no loss of performanceShall be safeThe display shall look the sameTBDs and TBRsStatistics
1. Requirements
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Tools for writing good requirements
Requirements elicitationModelingTrade studies
1. Requirements
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Notes
Perfect requirements can’t always be written
It’s not possible to avoid all calamitiesRequirements and design are similar and
therefore are often confused and placed at the wrong level in the hierarchy
1. Requirements
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2. Numbers
Significant digitsPrecisionAccuracy
2. Numbers
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Significant digits (1 of 5)
The significant digits in a number include the leftmost, non-zero digits to the rightmost digit written.
Final answers should be rounded off to the decimal place justified by the data
2. Numbers
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Significant digits (2 of 5)
Examples
number digits implied range
251 3 250.5 to 251.5
25.1 3 25.05 to 25.15
0.000251 3 0.0002505 to 0.0002515
251x105 3 250.5x105 to 251.5x105
2.51x10-3 3 2.505x10-3 to 2.515x10-3
2512. 4 2511.5 to 2512.5
251.0 4 250.95 to 251.052. Numbers
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Significant digits (3 of 5)
Example• There shall be 3 brown eggs for every 8
eggs sold. • A set of 8000 eggs passes if the number of
brown eggs is in the range 2500 to 3500
• There shall be 0.375 brown eggs for every egg sold.• A set of 8000 eggs passes if the number of
brown eggs is in the range 2996 to 3004
2. Numbers
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Significant digits (4 of 5)
The implied range can be offset by stating an explicit range• There shall be 0.375 brown eggs (±0.1 of
the set size) for every egg sold.• A set of 8000 eggs passes if the number of
brown eggs is in the range 2200 to 3800
• There shall be 0.375 brown eggs (±0.1) for every egg sold.• A set of 8000 eggs passes only if the
number of brown eggs is 3000
2. Numbers
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Significant digits (5 of 5)
A common problem is to inflate significant digits in making units conversion.• Observers estimated the meteorite had a
mass of 10 kg. This statement implies the mass was in the range of 5 to 15 kg; i.e, a range of 10 kg.
• Observers estimated the meteorite had a mass of 22 lbs. This statement implies a range of 21.5 to 22.5 lb; i.e., a range of 1 pound
2. Numbers
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Precision
Precision refers to the degree to which a number can be expressed.
Examples• Computer words• The 16-bit signed integer has a normalized
precision of 2-15
• Meter readings• The ammeter has a range of 10 amps and a
precision of 0.01 amp
2. Numbers
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Accuracy
Accuracy refers to the quality of the number.
Examples• Computer words• The 16-bit signed integer has a normalized
precision of 2-15, but its normalized accuracy may be only ±2-3
• Meter readings• The ammeter has a range of 10 amps and a
precision of 0.01 amp, but its accuracy may be only ±0.1 amp.
2. Numbers
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3. Decibels
DefinitionsCommon valuesExamplesAdvantagesDecibels as absolute unitsPowers of 2
3. Decibels
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Definitions (1 of 2)
The decibel, named after Alexander Graham Bell, is a logarithmic unit originally used to give power ratios but used today to give other ratios
Logarithm of N• The power to which 10 must be raised to
equal N
• n = log10(N); N = 10n
3. Decibels
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Definitions (2 of 2)
Power ratio
• dB = 10 log10(P2/P1)
• P2/P1=10dB/10
Voltage power
• dB = 20 log10(V2/V1)
• V2/V1=10dB/20
3. Decibels
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Common values
dB ratio0 11 1.262 1.63 24 2.55 3.26 47 58 6.39 810 1020 10030 1000
3. Decibels
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Examples
5000 = 5 x 1000; 7 dB + 30 dB = 37 dB49 dB = 40 dB + 9 dB; 8 x 10,000 = 80,000
3. Decibels
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Advantages (1 of 2)
Reduces the size of numbers used to express large ratios• 2:1 = 3 dB; 100,000,000 = 80 dB
Multiplication in numbers becomes addition in decibels• 10*100 =1000; 10 dB + 20 dB = 30 dB
The reciprocal of a number is the negative of the number of decibels• 100 = 20 dB; 1/100 = -20 dB
3. Decibels
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Advantages (2 of 2)
Raising to powers is done by multiplication• 1002 = 10,000; 2*20dB = 40 dB• 1000.5 = 10; 0.5*20dB = 10 dB
Calculations can be done mentally
3. Decibels
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Decibels as absolute units
dBW = dB relative to 1 wattdBm = dB relative to 1 milliwattdBsm = dB relative to one square
meterdBi = dB relative to an isotropic
radiator
3. Decibels
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Powers of 2
exact value approximate value
20 1 1
24 16 16
210 1024 1 x 1,000
223 8,388,608 8 x 1,000,000
234 17,179,869,184 16 x 1,000,000,000
2xy = 2y x 103x2xy = 2y x 103x
3. Decibels
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4. Matrices
AdditionSubtractionMultiplicationVector, dot product, & outer productTransposeDeterminant of a 2x2 matrixCofactor and adjoint matricesDeterminantInverse matrixOrthogonal matrix
4. Matrices
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Addition
cIJ = aIJ + bIJcIJ = aIJ + bIJ
1 -1 0-2 1 -3 2 0 2
1 -1 -1 0 4 2-1 0 1
A= B=
2 -2 -1 -2 5 -1 1 0 3
C=
C=A+B
4. Matrices
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Subtraction
cIJ = aIJ - bIJcIJ = aIJ - bIJ
1 -1 0-2 1 -3 2 0 2
1 -1 -1 0 4 2-1 0 1
A= B=
0 0 1 -2 -3 -5 3 0 1
C=
C=A-B
4. Matrices
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Multiplication
cIJ = aI1 * b1J + aI2 * b2J + aI3 * b3J cIJ = aI1 * b1J + aI2 * b2J + aI3 * b3J
1 -1 0-2 1 -3 2 0 2
1 -1 -1 0 4 2-1 0 1
A= B=
1 -5 -3 1 6 1 0 -2 0
C=
C=A*B
4. Matrices
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Transpose
bIJ = aJIbIJ = aJI
1 -1 0-2 1 -3 2 0 2
1 -2 2 -1 1 0 0 -3 2
A= B=
B=AT
4. Matrices
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Vector, dot product, & outer product
A vector v is an N x 1 matrixDot product = inner product = vT x v = a
scalarOuter product = v x vT = N x N matrix
4. Matrices
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Determinant of a 2x2 matrix
2x2 determinant = b11 * b22 - b12 * b212x2 determinant = b11 * b22 - b12 * b21
B = 1 -1-2 1
= -1
4. Matrices
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Cofactor and adjoint matrices
1 -1 0-2 1 -3 2 0 2
A=
1 -3 0 2
-1 0 0 2
-1 0 0 -3
-2 -3 2 2
1 0 2 2
1 0-2 -3
-2 1 2 0
1 -1 2 0
1 -1-2 1
2 -2 -22 2 -23 3 -1
=B = cofactor =
2 2 3-2 2 3-2 -2 -1
C=BT = adjoint=
4. Matrices
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Determinant
1 -1 0-2 1 -3 2 0 2
determinant of A =
The determinant of A = dot product of any row in A times the corresponding column of the adjoint matrix =
dot product of any row (or column) in A timesthe corresponding row (or column) in the cofactor matrix
The determinant of A = dot product of any row in A times the corresponding column of the adjoint matrix =
dot product of any row (or column) in A timesthe corresponding row (or column) in the cofactor matrix
1 -1 0
=4
2-2-2
= 4
4. Matrices
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Inverse matrix
B = A-1 =adjoint(A)/determinant(A) = 0.5 0.5 0.75-0.5 0.5 0.75-0.5 -0.5 -0.25
1 -1 0-2 1 -3 2 0 2
0.5 0.5 0.75-0.5 0.5 0.75-0.5 -0.5 -0.25
1 0 00 1 00 0 1
=
4. Matrices
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Orthogonal matrix
An orthogonal matrix is a matrix whose inverse is equal to its transpose.
1 0 00 cos sin 0 -sin cos
1 0 00 cos -sin 0 sin cos
1 0 00 1 00 0 1
=
4. Matrices
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5. Transforms
DefinitionExamplesTime-domain solutionFrequency-domain solutionTerms used with frequency responsePower spectrumSinusoidal motionExample -- vibration
5. Transforms
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Definition
Transforms -- a mathematical conversion from one way of thinking to another to make a problem easier to solve
transformsolution
in transformway of
thinking
inversetransform
solution in original
way of thinking
problem in original
way of thinking
5. Transforms
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Examples (1 of 3)
English to algebra solution
in algebra
algebra toEnglish
solution in English
problem in English
5. Transforms
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Examples (2 of 3)
English tomatrices solution
in matrices
matrices toEnglish
solution in English
problem in English
5. Transforms
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Examples (3 of 3)
Fourier transform
solutionin frequency
domain
inverse Fourier
transform
solution in timedomain
problem in time domain
• Other transforms• Laplace• z-transform• wavelets
5. Transforms
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Time-domain solution
We typically think in the time domain -- a time input produces a time output
5. Transforms
systemtime
amplitude
time
amplitude
input output
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Frequency-domain solution (1 of 2)
However, the solution can be expressed in the frequency domain.
A sinusoidal input produces a sinusoidal output
A series of sinusoidal inputs across the frequency range produces a series of sinusoidal outputs called a frequency response
5. Transforms
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Frequency-domain solution (2 of 2)
5. Transforms
system log frequency
amplitude (dB)
log frequency
magnitude (dB)
input output
log frequency
phase (angle)0
-180
(sinusoids)
log frequency
phase (angle)
0
0
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Terms used with frequency response
Octave is a range of 2xDecade is a range of 10x
5. Transforms
amplitude (dB)power (dB)
frequency
6, 3
2 10
20,10 Slope =• 20 dB/decade, amplitude• 6 dB/octave, amplitude•10 dB/decade, power• 3 dB/octave, power
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Power spectrum
A power spectrum is a special form of frequency response in which the ordinate represents power
5. Transforms
g2-Hz (dB)
log frequency
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Sinusoidal motion
Motion of a point going around a circle in two-dimensional x-y plane produces sinusoidal motion in each dimension• x-displacement = sin(t)• x-velocity = cos(t)• x-acceleration = -2sin(t)• x-jerk = -3cos(t)• x-yank = 4sin(t)
5. Transforms
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Example -- vibration
Output vibration is product of input vibrationtimes the transmissivity-squared at each frequency
Output vibration is product of input vibrationtimes the transmissivity-squared at each frequency
5. Transforms
g2-Hz (dB)
log frequency log frequency log frequency
g2-Hz (dB)[amplitude (dB)] 2
input transmissivity-squared output
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6. Statistics (1 of 2)
Frequency distributionSample meanSample varianceCEPDensity functionDistribution functionUniformBinomial
6. Statistics
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6. Statistics (1 of 2)
NormalPoissonExponential RaleighExcel toolsSamplingCombining error sources
6. Statistics
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2
Frequency distribution
Frequency distribution -- A histogram or polygon summarizing how raw data can be grouped into classes
height (inches)
number
22
4
6
8
4 5 6 67 4 3
n = sample size = 39
2
60 61 62 63 64 65 66 67 68
6. Statistics
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Sample mean
= xi
An estimate of the population meanExample
= [ 2 x 60 + 4 x 61 + 5 x 62 + 7 x 63 + 4 x 64 + 6 x 65 + 6 x 66 + 3 x 67 + 2 x 68 ] / 39 = 2494/39 = 63.9
6. Statistics
Ni=1
N
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Sample variance
2= (xi - )2
An estimate of the population variance = standard deviationExample 2 = [ 2 x (60 - )2 +
4 x (61 - )2 + 5 x (62 - )2 + 7 x (63 - )2 + 4 x (64 - )2 + 6 x (65 - )2 + 6 x (66 - )2 + 3 x (67 - )2 + 2 x (68 - )2 ]/(39 - 1] = 183.9/38 = 4.8 = 2.2 6. Statistics
N-1i=1
N
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CEP
Circular error probable is the radius of the circle containing half of the samples
If samples are normally distributed in the x direction with standard deviation x and normally distribute in the y direction with standard deviation y , then
CEP = 1.1774 * sqrt [0.5*(x2 + y
2)]
CEP
6. Statistics
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Density function
Probability that a discrete event x will occurNon-negative function whose integral over
the entire range of the independent variable is 1
f(x)
x6. Statistics
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Distribution function
Probability that a numerical event x or less occurs
The integral of the density function
F(x)
x
1.0
6. Statistics
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Uniform (1 of 2)
f(x) = 1/(x2 - x1 ), x1 x x2
= 0 elsewhere
F(x) = 0, x x1
= (x - x1 ) / (x2 - x1 ), x1 x x2
= 1, x > x2
Mean = (x2 + x1 )/2
Standard deviation = (x2 - x1 )/sqrt(12)
6. Statistics
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Uniform (2 of 2)
Example• If a set of resistors has a mean of 10,000
and is uniformly distributed between 9,000 and 11,000 , what is the probability the resistance is between 9,900 and 10,100 ?
• F(9900,10100) = 200/2000 = 0.1
6. Statistics
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Binomial (1 of 2)
f(x) = n!/[(n-x)!x!]px (1-p)n-x where p = probability of success on a single trial
Used when all outcomes can be expressed as either successes or failures
Mean = npStandard deviation = sqrt[np(1-p)]
6. Statistics
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Binomial (2 of 2)
Example• 10 percent of a production run of
assemblies are defective. If 5 assemblies are chosen, what is the probability that exactly 2 are defective?
• f(2) = 5!/(3!2!)(0.12)(0.93) = 0.07
6. Statistics
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Normal (1 of 2)
f(x) = 1/[sqrt(2)exp[-(x-)2/(2 2)F(x) = erf[(x-)/] + 0.5Mean = Standard deviation = Can be derived from binomial distribution
6. Statistics
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Normal (2 of 2)
Example• If the mean mass of a set of products is
50 kg and the standard deviation is 5 kg, what is the probability the mass is less than 60 kg?
• F(60) = erf[(60-50)/5] + 0.5 = 0.97
6. Statistics
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Poisson (1 of 2)
f(x) = e-x/x! (>0) = average number of times that event
occurs per period• x = number of time event occurs
Mean = Standard deviation = sqrt()Derived from binomial distributionUsed to quantify events that occur
relatively infrequently but at a regular rate
6. Statistics
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Poisson (2 of 2)
Example• The system generates 5 false alarms per
hour.• What is the probability there will be exactly
3 false alarms in one hour? = 5• x = 3• f(3) = e-5(5)3/3! = 0.14
6. Statistics
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Exponential (1 of 2)
F(x) = exp(- x)F(x) = 1 - exp(- x)Mean = 1/Standard deviation = 1/ Used in reliability computations
where = 1/MTBF
6. Statistics
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Exponential (2 of 2)
Example• If the MTBF of a part is 100 hours, what
is the probability the part will have failed by 150 hours?
• F(150) = 1 - exp(- 150/100) = 0.78
6. Statistics
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Raleigh (1 of 2)
f(r) = [1/(22) * exp[-r2/(2 2)]F(r) = 1 - exp[-r2/(2 2)]Mean = sqrt(/2)Standard deviation = sqrt(2) Derived from normal distributionUsed to describe radial distribution when
uncertainty in x and y are described by normal distributions
6. Statistics
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Raleigh (2 of 2)
Example• If uncertainty in x and y positions are
each described by a normal distribution with zero mean and = 2, what is the probability the position is within a radius of 1.5?
• F(1.5) = 1 - exp[-(1.5)2/(2 x 22)] = 0.25
6. Statistics
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Excel tools
Functions• COUNT• AVERAGE• MEDIAN• STDDEV• BINODIST• POISSON
Tools• Data Analysis• Random number generation• Histogram 6. Statistics
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Sampling
A frequent problem is obtaining enough samples to be confident in the answer
6. Statistics
N
M
N>M
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Combining error sources (1 of 3)
When multiple dimensions are included, covariance matrices can be added
When an error source goes through a linear transformation, resulting covariance is expressed as follows
6. Statistics
P1 = covariance of error source 1P2 = covariance of error source 2P = resulting covariance = P1 + P2
T = linear transformationTT = transform of linear transformationPorig = covariance of original error sourceP = T * P * TT
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Combining error sources (2 of 3)
6. Statistics
Example of propagation of position
xorig = standard deviation in original position = 2 mvorig = standard deviation in original velocity = 0.5 m/sT = time between samples = 4 secxcurrent = error in current position
xcurrent = xorig + T * vorig
vcurrent = vorig
1 4 0 1
T = Porig =22
00
0.52
Pcurrent = T * P orig * TT = 1 4 0 1
1 0 4 1
40
00.25
= 81
10.25
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Combining error sources (3 of 3)
6. Statistics
Example of angular rotation
Xoriginal = original coordinates
Xcurrent = current coordinates
T = transformation corresponding to angular rotation
cos -sin sin cos
T = where = atan(0.75)
Porig =1.64 -0.48-0.48 1.36
Pcurrent = T * P orig * TT = 0.8 -0.60.6 0.8
= 20
01
1.64 -0.48-0.48 1.36
0.8 0.6-0.6 0.8
x’
y’
x
y
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7. Software
MemoryThroughputLanguageDevelopment method
7. Software
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Memory (1 of 3)
All general purpose computers shall have 50 percent spare memory capacity
All digital signal processors (DSPs) shall have 25 percent spare on-chip memory capacity
All digital signal processors shall have 30 percent spare off-chip memory capacity
All mass storage units shall have 40 percent spare memory capacity
All firmware shall have 20 percent spare memory capacity
7. Software
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Memory (2 of 3)
There shall be 50 % spare memory capacityreference capacity memory-used
usage common less-common
capacity 100 Mbytes 100 Mbytes
memory-used 60 Mbytes 60 Mbytes
spare memory 40 Mbytes 40 Mbytes
percent spare 40 percent 67 percent
pass/fail fail passThere are at least two ways of interpreting the meaning of spare memory capacity based on the reference used
as the denominator in computing the percentage
There are at least two ways of interpreting the meaning of spare memory capacity based on the reference used
as the denominator in computing the percentage7. Software
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Memory (3 of 3)
Memory capacity is most often verified by analysis of load files
Memory capacity is frequently tracked as a technical performance parameter (TPP)
Contractors don’t like to consider that firmware is software because firmware is often not developed using software development methodology and firmware is not as likely to grow in the future
Memory is often verified by analysis, and firmware is often not considered to be software
Memory is often verified by analysis, and firmware is often not considered to be software
7. Software
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Throughput (1 of 5)
All general purpose computers shall have 50 percent spare throughput capacity
All digital signal processors shall have 25 percent spare throughput capacity
All firmware shall have 30 percent spare throughput capacity
All communication channels shall have 40 percent spare throughput capacity
All communication channels shall have 20 percent spare terminals
7. Software
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Throughput (2 of 5)
There shall be 100 % spare throughput capacity
reference capacity throughput-used
usage common common
capacity 100 MOPS 100 MOPS
throughput-used 50 MOPS 50 MOPS
spare throughput 50 MOPS 50 MOPS
percent spare 50 percent 100 percent
pass/fail fail passThere are two ways of interpreting of spare throughput
capacity based on reference used as denominatorThere are two ways of interpreting of spare throughput
capacity based on reference used as denominator7. Software
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Throughput (3 of 5)
Availability of spare throughput• Available at the highest-priority-
application level -- most common• Available at the lowest-priority-application
level -- common• Available in proportion to the times spent
by each segment of the application -- not common
Assuming the spare throughput is available at the highest-priority-application level is
the most common assumption
Assuming the spare throughput is available at the highest-priority-application level is
the most common assumption7. Software
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Throughput (4 of 5)
Throughput capacity is most often verified by test• Analysis -- not common• Time event simulation -- not common• Execution monitor -- common but
requires instrumentation code and hardware
7. Software
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Throughput (5 of 5)
• Execution of a code segment that uses at least the number of spare throughput instructions required -- not common but avoids instrumentation
Instrumenting the software to monitor runtime or inserting a code segment that uses at least the
spare throughput are two methods of verifying throughput
Instrumenting the software to monitor runtime or inserting a code segment that uses at least the
spare throughput are two methods of verifying throughput
7. Software
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Language (1 of 2)
No more than 15 percent of the code shall be in assembly language.• Useful for device drivers and for speed• Not as easily maintained
7. Software
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Language (2 of 2)
Remaining code shall be in Ada• Ada is largely a military language and is
declining in popularity• C++ growing in popularity
Language is verified by analysis of code
C++ is becoming the most popular programming language but assembly language may still need
to be used
C++ is becoming the most popular programming language but assembly language may still need
to be used7. Software
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Development method
Several methods are available• Structured-analysis-structured-design
vs Hatley-Pirba• Functional vs object-oriented• Classical vs clean-room
Generally a statement of work issue and not a requirement although customer prefers a proven, low-risk approach
Customer does not usually specify the development method
Customer does not usually specify the development method
7. Software