1 Agenda for design activity r1. Requirements r2. Numbers r3. Decibels r4. Matrices r5. Transforms r6. Statistics r7. Software

2 1. Requirements rDefinition of a requirement rOccurrence of requirements rGuidelines for a good requirement rExamples for each guideline rTools for writing good requirements rNotes 1. Requirements

3 Definition of a requirement rSomething obligatory or demanded rStatement of some needed thing or characteristic 1. Requirements

4 Occurrence of requirements rWriting requirements occurs in both the understand- requirements activity and the design activity rThe customer has RAA for requirements in the understand- requirements activity even though the contractor may actually write the requirements rThe 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 rNeeded rCapable of being verified rFeasible schedule, cost, and implementation rAt correct level in hierarchy rCannot be misunderstood rGrammar and spelling correct rDoes not duplicate information 1. Requirements

6 Example for each guideline rExample 1 -- needed rExample 2 -- verification rExample 3 -- feasible rExample 4 -- level rExample 5 -- understanding rExample 6 -- duplication rExample 7 -- grammar and spelling rExample 8 -- tough requirements 1. Requirements

7 Example 1 -- needed rThe motor shall weigh less than 10 pounds. rThe software shall use less than 75 percent of the computer memory available for software. rThe MTBF shall be greater than 1000 hours. 1. Requirements

8 Example 2 -- verification (1 of 3) rCustomer want -- The outside wall shall be a material that requires low maintenance 1. Requirements

9 Example 2 -- verification (2 of 3) rFirst 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) rSecond 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 rNot 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 r Airplane shall be capable carrying up to 2000 pounds r Wing airfoil shall be of type Clark Y airplane wing p 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 rAvoid imprecise terms such as Optimize Maximize Accommodate Etc. Support Adequate 1. Requirements

14 Example 6 -- duplication rCapable of a maximum rate of 100 gpm rCapable of a minimum rate of 10 gpm rRun BIT while pumping 10 gpm gpm rVs: Run BIT while pumping between min. and max. 1. Requirements

15 Example 7 -- grammar and spelling rThe computers is comercial-off-the-shelf items rIncorrect grammar or spelling will divert customer review of the requirements from the technical content 1. Requirements

16 Example 8 -- tough requirements rBIT false alarm rate < 3 percent rComputer throughput < 75 percent of capacity rPerform over all altitudes and speeds rConform with all local, state, and national laws rThere shall be no loss of performance rShall be safe rThe display shall look the same rTBDs and TBRs rStatistics 1. Requirements

17 Tools for writing good requirements rRequirements elicitation rModeling rTrade studies 1. Requirements

18 Notes rPerfect requirements can’t always be written rIt’s not possible to avoid all calamities rRequirements and design are similar and therefore are often confused and placed at the wrong level in the hierarchy 1. Requirements

19 2. Numbers rSignificant digits rPrecision rAccuracy 2. Numbers

20 Significant digits (1 of 5) rThe significant digits in a number include the leftmost, non-zero digits to the rightmost digit written. rFinal answers should be rounded off to the decimal place justified by the data 2. Numbers

21 Significant digits (2 of 5) Examples numberdigitsimplied range to to to x x10 5 to 251.5x x x10 -3 to 2.515x to to Numbers

22 Significant digits (3 of 5) rExample 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 brown eggs for every egg sold. A set of 8000 eggs passes if the number of brown eggs is in the range 2996 to Numbers

23 Significant digits (4 of 5) rThe implied range can be offset by stating an explicit range There shall be 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 brown eggs (±0.1) for every egg sold. A set of 8000 eggs passes only if the number of brown eggs is Numbers

24 Significant digits (5 of 5) r 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 rPrecision refers to the degree to which a number can be expressed. rExamples Computer words The 16-bit signed integer has a normalized precision of Meter readings The ammeter has a range of 10 amps and a precision of 0.01 amp 2. Numbers

26 Accuracy rAccuracy refers to the quality of the number. rExamples 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 rDefinitions rCommon values rExamples rAdvantages rDecibels as absolute units rPowers of 2 3. Decibels

28 Definitions (1 of 2) rThe decibel, named after Alexander Graham Bell, is a logarithmic unit originally used to give power ratios but used today to give other ratios rLogarithm of N The power to which 10 must be raised to equal N n = log 10 (N); N = 10 n 3. Decibels

29 Definitions (2 of 2) rPower ratio dB = 10 log 10 (P 2 /P 1 ) P 2 /P 1 =10 dB/10 rVoltage power dB = 20 log 10 (V 2 /V 1 ) V 2 /V 1 =10 dB/20 3. Decibels

30 Common values dBratio Decibels

31 Examples r5000 = 5 x 1000; 7 dB + 30 dB = 37 dB r49 dB = 40 dB + 9 dB; 8 x 10,000 = 80, Decibels

32 Advantages (1 of 2) rReduces the size of numbers used to express large ratios 2:1 = 3 dB; 100,000,000 = 80 dB rMultiplication in numbers becomes addition in decibels 10*100 =1000; 10 dB + 20 dB = 30 dB rThe 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) rRaising to powers is done by multiplication = 10,000; 2*20dB = 40 dB = 10; 0.5*20dB = 10 dB rCalculations can be done mentally 3. Decibels

34 Decibels as absolute units rdBW = dB relative to 1 watt rdBm = dB relative to 1 milliwatt rdBsm = dB relative to one square meter rdBi = dB relative to an isotropic radiator 3. Decibels

35 Powers of 2 exact valueapproximate value x 1, ,388,6088 x 1,000, ,179,869,18416 x 1,000,000,000 2 xy = 2 y x 10 3x 3. Decibels

36 4. Matrices rAddition rSubtraction rMultiplication rVector, dot product, & outer product rTranspose rDeterminant of a 2x2 matrix rCofactor and adjoint matrices rDeterminant rInverse matrix rOrthogonal matrix 4. Matrices

37 Addition c IJ = a IJ + b IJ A=B= C= C=A+B 4. Matrices

38 Subtraction c IJ = a IJ - b IJ A=B= C= C=A-B 4. Matrices

39 Multiplication c IJ = a I1 * b 1J + a I2 * b 2J + a I3 * b 3J A=B= C= C=A*B 4. Matrices

40 Transpose b IJ = a JI A=B= B=A T 4. Matrices

41 Vector, dot product, & outer product rA vector v is an N x 1 matrix rDot product = inner product = v T x v = a scalar rOuter product = v x v T = N x N matrix 4. Matrices

42 Determinant of a 2x2 matrix 2x2 determinant = b 11 * b 22 - b 12 * b 21 B= = Matrices

43 Cofactor and adjoint matrices A= =B = cofactor = C=B T = adjoint= 4. Matrices

44 Determinant 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 times the 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 times the corresponding row (or column) in the cofactor matrix = = 4 4. Matrices

45 Inverse matrix B = A -1 =adjoint(A)/determinant(A) = = 4. Matrices

46 Orthogonal matrix rAn orthogonal matrix is a matrix whose inverse is equal to its transpose cos  sin  0 -sin  cos  cos  -sin  0 sin  cos  = 4. Matrices

47 5. Transforms rDefinition rExamples rTime-domain solution rFrequency-domain solution rTerms used with frequency response rPower spectrum rSinusoidal motion rExample -- vibration 5. Transforms

48 Definition rTransforms -- a mathematical conversion from one way of thinking to another to make a problem easier to solve transform solution in transform way of thinking inverse transform solution in original way of thinking problem in original way of thinking 5. Transforms

49 Examples (1 of 3) English to algebra solution in algebra algebra to English solution in English problem in English 5. Transforms

50 Examples (2 of 3) English to matrices solution in matrices matrices to English solution in English problem in English 5. Transforms

51 Examples (3 of 3) Fourier transform solution in frequency domain inverse Fourier transform solution in time domain problem in time domain Other transforms Laplace z-transform wavelets 5. Transforms

52 Time-domain solution rWe typically think in the time domain -- a time input produces a time output 5. Transforms system time amplitude time amplitude inputoutput

53 Frequency-domain solution (1 of 2) rHowever, the solution can be expressed in the frequency domain. rA sinusoidal input produces a sinusoidal output rA 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) inputoutput log frequency phase (angle) (sinusoids) log frequency phase (angle) 0 0

55 Terms used with frequency response rOctave is a range of 2x rDecade is a range of 10x 5. Transforms amplitude (dB) power (dB) frequency 6, ,10 Slope = 20 dB/decade, amplitude 6 dB/octave, amplitude 10 dB/decade, power 3 dB/octave, power

56 Power spectrum rA power spectrum is a special form of frequency response in which the ordinate represents power 5. Transforms g 2 -Hz (dB) log frequency

57 Sinusoidal motion rMotion 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 = -  2 sin(  t) x-jerk = -  3 cos(  t) x-yank =  4 sin(  t) 5. Transforms

58 Example -- vibration Output vibration is product of input vibration times the transmissivity-squared at each frequency Output vibration is product of input vibration times the transmissivity-squared at each frequency 5. Transforms g 2 -Hz (dB) log frequency g 2 -Hz (dB)[amplitude (dB)] 2 input transmissivity-squared output

59 6. Statistics (1 of 2) rFrequency distribution rSample mean rSample variance rCEP rDensity function rDistribution function rUniform rBinomial 6. Statistics

60 6. Statistics (1 of 2) rNormal rPoisson rExponential rRaleigh rExcel tools rSampling rCombining error sources 6. Statistics

61 2 Frequency distribution rFrequency distribution -- A histogram or polygon summarizing how raw data can be grouped into classes height (inches) number n = sample size = Statistics

62 Sample mean r  =  x i rAn estimate of the population mean rExample  = [ 2 x x x x x x x x x 68 ] / 39 = 2494/39 = Statistics N i=1 N

63 Sample variance r  2 =  (x i -  ) 2 rAn estimate of the population variance r  = standard deviation rExample  2 = [ 2 x (60 -  ) x (61 -  ) x (62 -  ) x (63 -  ) x (64 -  ) x (65 -  ) x (66 -  ) x (67 -  ) x (68 -  ) 2 ]/(39 - 1] = 183.9/38 = 4.8  = Statistics N-1 i=1 N

64 CEP rCircular error probable is the radius of the circle containing half of the samples r 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 = * sqrt [0.5*(  x 2 +  y 2 )] CEP 6. Statistics

65 Density function rProbability that a discrete event x will occur rNon-negative function whose integral over the entire range of the independent variable is 1 f(x) x 6. Statistics

66 Distribution function rProbability that a numerical event x or less occurs rThe integral of the density function F(x) x Statistics

67 Uniform (1 of 2) rf(x) = 1/(x 2 - x 1 ), x 1  x  x 2 = 0 elsewhere r F(x) = 0, x  x 1 = (x - x 1 ) / (x 2 - x 1 ), x 1  x  x 2 = 1, x > x 2 rMean = (x 2 + x 1 )/2 rStandard deviation = (x 2 - x 1 )/sqrt(12) 6. Statistics

68 Uniform (2 of 2) rExample 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 = Statistics

69 Binomial (1 of 2) rf(x) = n!/[(n-x)!x!]p x (1-p) n-x where p = probability of success on a single trial r Used when all outcomes can be expressed as either successes or failures rMean = np rStandard deviation = sqrt[np(1-p)] 6. Statistics

70 Binomial (2 of 2) rExample 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!)(0.1 2 )(0.9 3 ) = Statistics

71 Normal (1 of 2) rf(x) = 1/[  sqrt(2  )exp[-(x-  ) 2 /(2  2 ) rF(x) = erf[(x-  )/  ] rMean =  rStandard deviation =  rCan be derived from binomial distribution 6. Statistics

72 Normal (2 of 2) rExample 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] = Statistics

73 Poisson (1 of 2) rf(x) = e - x /x!( >0) = average number of times that event occurs per period x = number of time event occurs rMean = rStandard deviation = sqrt( ) rDerived from binomial distribution rUsed to quantify events that occur relatively infrequently but at a regular rate 6. Statistics

74 Poisson (2 of 2) rExample 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! = Statistics

75 Exponential (1 of 2) rF(x) = exp(- x) rF(x) = 1 - exp(- x) rMean = 1/ rStandard deviation = 1/ rUsed in reliability computations where = 1/MTBF 6. Statistics

76 Exponential (2 of 2) rExample 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) = Statistics

77 Raleigh (1 of 2) rf(r) = [1/(2  2 ) * exp[-r 2 /(2  2 )] rF(r) = 1 - exp[-r 2 /(2  2 )] rMean =  sqrt(  /2) rStandard deviation =  sqrt(2) rDerived from normal distribution rUsed to describe radial distribution when uncertainty in x and y are described by normal distributions 6. Statistics

78 Raleigh (2 of 2) rExample 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 2 2 )] = Statistics

79 Excel tools rFunctions COUNT AVERAGE MEDIAN STDDEV BINODIST POISSON rTools Data Analysis Random number generation Histogram 6. Statistics

80 Sampling rA 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) rWhen multiple dimensions are included, covariance matrices can be added rWhen an error source goes through a linear transformation, resulting covariance is expressed as follows 6. Statistics P 1 = covariance of error source 1 P 2 = covariance of error source 2 P = resulting covariance = P 1 + P 2 T = linear transformation T T = transform of linear transformation P orig = covariance of original error source P = T * P * T T

82 Combining error sources (2 of 3) 6. Statistics rExample of propagation of position x orig = standard deviation in original position = 2 m v orig = standard deviation in original velocity = 0.5 m/s T = time between samples = 4 sec x current = error in current position x current = x orig + T * v orig v current = v orig T =P orig = P current = T * P orig * T T = = 8 1 1

83 Combining error sources (3 of 3) 6. Statistics rExample of angular rotation X original = original coordinates X current = current coordinates T = transformation corresponding to angular rotation cos  -sin  sin  cos  T = where  = atan(0.75) P orig = P current = T * P orig * T T = = x’ y’ x y 

84 7. Software rMemory rThroughput rLanguage rDevelopment method 7. Software

85 Memory (1 of 3) rAll general purpose computers shall have 50 percent spare memory capacity rAll digital signal processors (DSPs) shall have 25 percent spare on-chip memory capacity rAll digital signal processors shall have 30 percent spare off-chip memory capacity rAll mass storage units shall have 40 percent spare memory capacity rAll firmware shall have 20 percent spare memory capacity 7. Software

86 Memory (2 of 3) rThere shall be 50 % spare memory capacity referencecapacitymemory-used usagecommonless-common capacity100 Mbytes100 Mbytes memory-used60 Mbytes60 Mbytes spare memory40 Mbytes40 Mbytes percent spare40 percent67 percent pass/failfailpass 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 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 percentage 7. Software

87 Memory (3 of 3) rMemory capacity is most often verified by analysis of load files rMemory capacity is frequently tracked as a technical performance parameter (TPP) rContractors 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) rAll general purpose computers shall have 50 percent spare throughput capacity rAll digital signal processors shall have 25 percent spare throughput capacity rAll firmware shall have 30 percent spare throughput capacity rAll communication channels shall have 40 percent spare throughput capacity rAll communication channels shall have 20 percent spare terminals 7. Software

89 Throughput (2 of 5) rThere shall be 100 % spare throughput capacity referencecapacitythroughput-used usagecommoncommon capacity100 MOPS100 MOPS throughput-used50 MOPS50 MOPS sparethroughput 50 MOPS50 MOPS percent spare50 percent100 percent pass/failfailpass There are two ways of interpreting of spare throughput capacity based on reference used as denominator There are two ways of interpreting of spare throughput capacity based on reference used as denominator 7. Software

90 Throughput (3 of 5) rAvailability 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 7. Software

91 Throughput (4 of 5) rThroughput 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) rNo 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) rRemaining code shall be in Ada Ada is largely a military language and is declining in popularity C++ growing in popularity rLanguage 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 7. Software

95 Development method rSeveral methods are available Structured-analysis-structured-design vs Hatley-Pirba Functional vs object-oriented Classical vs clean-room rGenerally 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