1

Agenda for design activity

1. Requirements2. Numbers3. Decibels4. Matrices5. Transforms6. Statistics7. Software

2

1. Requirements

Definition of a requirementOccurrence of requirementsGuidelines for a good requirementExamples for each guidelineTools for writing good requirementsNotes

1. Requirements

3

Definition of a requirement

Something obligatory or demandedStatement of some needed thing or

characteristic

1. Requirements

4

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

5

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

6

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

7

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

8

Example 2 -- verification (1 of 3)

Customer want -- The outside wall shall be a material that requires low maintenance

1. Requirements

9

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

10

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

11

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

12

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

13

Example 5 -- understanding

Avoid imprecise terms such as• Optimize• Maximize• Accommodate• Etc.• Support• Adequate

1. Requirements

14

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

15

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

16

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

17

Tools for writing good requirements

Requirements elicitationModelingTrade studies

1. Requirements

18

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

19

2. Numbers

Significant digitsPrecisionAccuracy

2. Numbers

20

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

21

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

22

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

24

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

25

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

26

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

27

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

31

Examples

5000 = 5 x 1000; 7 dB + 30 dB = 37 dB49 dB = 40 dB + 9 dB; 8 x 10,000 = 80,000

3. Decibels

32

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

33

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

34

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

36

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

38

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

39

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

40

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

41

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

43

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

-

- -

-

44

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

45

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

46

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

48

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

50

Examples (2 of 3)

English tomatrices solution

in matrices

matrices toEnglish

solution in English

problem in English

5. Transforms

51

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

52

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

53

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

54

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

55

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

56

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

57

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

58

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

59

6. Statistics (1 of 2)

Frequency distributionSample meanSample varianceCEPDensity functionDistribution functionUniformBinomial

6. Statistics

60

6. Statistics (1 of 2)

NormalPoissonExponential RaleighExcel toolsSamplingCombining error sources

6. Statistics

61

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

62

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

63

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

64

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

65

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

66

Distribution function

Probability that a numerical event x or less occurs

The integral of the density function

F(x)

x

1.0

6. Statistics

67

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

68

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

69

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

70

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

71

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

72

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

73

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

74

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

75

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

76

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

77

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

78

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

79

Excel tools

Functions• COUNT• AVERAGE• MEDIAN• STDDEV• BINODIST• POISSON

Tools• Data Analysis• Random number generation• Histogram 6. Statistics

80

Sampling

A frequent problem is obtaining enough samples to be confident in the answer

6. Statistics

N

M

N>M

81

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

82

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

83

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

84

7. Software

MemoryThroughputLanguageDevelopment method

7. Software

85

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

86

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

87

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

88

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

89

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

90

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

91

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

92

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

93

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

94

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

95

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