3. requirements1 agenda for understand req activity r1. numbers r2. decibels r3. matrices r4....
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3. Requirements 1
Agenda for understand req activity
1. Numbers2. Decibels3. Matrices4. Transforms5. Statistics6. Software
3. Requirements 2
1. Numbers
Significant digitsPrecisionAccuracy
1. Numbers
3. Requirements 3
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
1. Numbers
3. Requirements 4
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
2510 4 2509.5 to 2510.5
251.0 4 250.95 to 251.05
1. Numbers
3. Requirements 5
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
1. Numbers
3. Requirements 6
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
1. Numbers
3. Requirements 7
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
1. Numbers
3. Requirements 8
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
1. Numbers
3. Requirements 9
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.
1. Numbers
3. Requirements 10
2. Decibels
DefinitionsCommon valuesExamplesAdvantagesDecibels as absolute unitsPowers of 2
2. Decibels
3. Requirements 11
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
2. Decibels
3. Requirements 12
Definitions (2 of 2)
Power ratio
• dB = 10 log10(P2/P1)
• P2/P1=10dB/10
Voltage power
• dB = 10 log10(V2/V1)
• P2/P1=10dB/20
2. Decibels
3. Requirements 13
Common values
dB ratio0 11 1.262 1.63 24 2.55 3.26 47 58 6.39 810 10100 201000 30
2. Decibels
3. Requirements 14
Examples
5000 = 5 x 1000; 7 dB + 30 dB = 37 dB49 dB = 40 dB + 9 dB; 8 x 10,000 = 80,000
2. Decibels
3. Requirements 15
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
2. Decibels
3. Requirements 16
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
2. Decibels
3. Requirements 17
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
2. Decibels
3. Requirements 18
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
2. Decibels
3. Requirements 19
3. Matrices
AdditionSubtractionMultiplicationVector, dot product, & outer productTransposeDeterminant of a 2x2 matrixCofactor and adjoint matricesDeterminantInverse matrixOrthogonal matrix
3. Matrices
3. Requirements 20
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
3. Matrices
3. Requirements 21
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
3. Matrices
3. Requirements 22
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
3. Matrices
3. Requirements 23
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
3. Matrices
3. Requirements 24
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
3. Matrices
3. Requirements 25
Determinant of a 2x2 matrix
2x2 determinant = b11 * b22 - bI2 * b212x2 determinant = b11 * b22 - bI2 * b21
B = 1 -1-2 1
= -1
3. Matrices
3. Requirements 26
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=
3. Matrices
3. Requirements 27
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 row the adjoint matrix = dot product of any row or column in A times
the corresponding row or column in the cofactor matrix
The determinant of A = dot product of any row in A times the corresponding row the adjoint matrix = dot product of any row or column in A times
the corresponding row or column in the cofactor matrix
1 -1 0
=4
2-2-2
= 4
3. Matrices
3. Requirements 28
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
=
3. Matrices
3. Requirements 29
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
=
3. Matrices
3. Requirements 30
4. Transforms
DefinitionExamplesTime-domain solutionFrequency-domain solutionTerms used with frequency responsePower spectrumSinusoidal motionExample -- vibration
4. Transforms
3. Requirements 31
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
4. Transforms
3. Requirements 32
Examples (1 of 3)
English to algebra solution
in algebra
algebra toEnglish
solution in English
problem in English
4. Transforms
3. Requirements 33
Examples (2 of 3)
English tomatrices solution
in matrices
matrices toEnglish
solution in English
problem in English
4. Transforms
3. Requirements 34
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
4. Transforms
3. Requirements 35
Time-domain solution
We typically think in the time domain -- a time input produces a time output
4. Transforms
systemtime
amplitude
time
amplitude
input output
3. Requirements 36
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
4. Transforms
3. Requirements 37
Frequency-domain solution (2 of 2)
4. Transforms
system log frequency
amplitude (dB)
log frequency
magnitude (dB)
input output
log frequency
phase (angle)0
-180
(sinusoids)
3. Requirements 38
Terms used with frequency response
Octave is a range of 2xDecade is a range of 10x
4. 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 decade, power
3. Requirements 39
Power spectrum
A power spectrum is a special form of frequency response in which the ordinate represents power
4. Transforms
g2-Hz (dB)
log frequency
3. Requirements 40
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)
4. Transforms
3. Requirements 41
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
4. Transforms
g2-Hz (dB)
log frequency log frequency log frequency
g2-Hz (dB)amplitude (dB)
input transmissivity-squared output
3. Requirements 42
5. Statistics (1 of 2)
Frequency distributionSample meanSample varianceCEPDensity functionDistribution functionUniformBinomial
5. Statistics
3. Requirements 43
5. Statistics (1 of 2)
NormalPoissonExponential RaleighSamplingCombining error sources
5. Statistics
3. Requirements 44
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
5. Statistics
3. Requirements 45
Sample mean
= xi
An estimate of the population meanExample
= [ 2 x 60 + 4 x61 + 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
5. Statistics
Ni=1
N
3. Requirements 46
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
5. Statistics
N-1i=1
N
3. Requirements 47
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
5. Statistics
3. Requirements 48
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)
x
5. Statistics
3. Requirements 49
Distribution function
Probability that a numerical event x or less occurs
The integral of the density function
F(x)
x
1.0
5. Statistics
3. Requirements 50
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)
5. Statistics
3. Requirements 51
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
5. Statistics
3. Requirements 52
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)]
5. Statistics
3. Requirements 53
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
5. Statistics
3. Requirements 54
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
5. Statistics
3. Requirements 55
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
5. Statistics
3. Requirements 56
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
5. Statistics
3. Requirements 57
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
5. Statistics
3. Requirements 58
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
5. Statistics
3. Requirements 59
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
5. Statistics
3. Requirements 60
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 binomial distributionUsed to describe radial distribution when
uncertainty in x and y are described by normal distributions
5. Statistics
3. Requirements 61
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
5. Statistics
3. Requirements 62
Sampling
A frequent problem is obtaining enough samples to be confident in the answer
5. Statistics
N
M
N>M
3. Requirements 63
Combining error sources (1 of 4)
Variances from multiple error sources can be combined by adding variances
Example
5. Statistics
xorig = standard deviation in original position = 1 mvorig = standard deviation in original velocity = 0.5 m/sT = time between samples = 2 secxcurrent = error in current position
= square root of [(xorig)2 + (vorig * T)2] = sqrt(2)
3. Requirements 64
Combining error sources (2 of 4) 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
5. 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
3. Requirements 65
Combining error sources (3 of 4)
5. 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 0.5 0 1
T = Porig =22
00
0.52
Pcurrent = T * P orig * TT = 1 4 0 1
1 0 4 1
40
00.25
= 164
40.25
3. Requirements 66
Combining error sources (4 of 4)
5. 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
3. Requirements 67
6. Software
MemoryThroughputLanguageDevelopment method
6. Software
3. Requirements 68
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
6. Software
3. Requirements 69
Memory (2 of 3)There shall be 50 percent spare memory capacity
reference 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 percentage6. Software
3. Requirements 70
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
6. Software
3. Requirements 71
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
6. Software
3. Requirements 72
Throughput (2 of 5)There shall be 100 percent 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 denominator6. Software
3. Requirements 73
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 assumption
6. Software
3. Requirements 74
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
6. Software
3. Requirements 75
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
6. Software
3. Requirements 76
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
6. Software
3. Requirements 77
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 used
6. Software
3. Requirements 78
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
6. Software
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