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Agenda for understand req activity

Agenda for understand req activity

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Agenda for understand req activity

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  1. Agenda for understand req activity • 1. Numbers • 2. Decibels • 3. Matrices • 4. Transforms • 5. Statistics • 6. Software

  2. 1. Numbers • Significant digits • Precision • Accuracy 1. Numbers

  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

  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

  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

  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

  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

  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

  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

  10. 2. Decibels • Definitions • Common values • Examples • Advantages • Decibels as absolute units • Powers of 2 2. Decibels

  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

  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

  13. Common values dB ratio 0 1 1 1.26 2 1.6 3 2 4 2.5 5 3.2 6 4 7 5 8 6.3 9 8 10 10 100 20 1000 30 2. Decibels

  14. Examples • 5000 = 5 x 1000; 7 dB + 30 dB = 37 dB • 49 dB = 40 dB + 9 dB; 8 x 10,000 = 80,000 2. Decibels

  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

  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

  17. Decibels as absolute units • dBW = dB relative to 1 watt • dBm = dB relative to 1 milliwatt • dBsm = dB relative to one square meter • dBi = dB relative to an isotropic radiator 2. Decibels

  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 17,179,869,18416 x 1,000,000,000 2xy = 2y x 103x 2. Decibels

  19. 3. Matrices • Addition • Subtraction • Multiplication • Vector, dot product, & outer product • Transpose • Determinant of a 2x2 matrix • Cofactor and adjoint matrices • Determinant • Inverse matrix • Orthogonal matrix 3. Matrices

  20. Addition C=A+B 1 -1 -1 0 4 2 -1 0 1 2 -2 -1 -2 5 -1 1 0 3 1 -1 0 -2 1 -3 2 0 2 A= B= C= cIJ = aIJ + bIJ 3. Matrices

  21. Subtraction C=A-B 1 -1 -1 0 4 2 -1 0 1 0 0 1 -2 -3 -5 3 0 1 1 -1 0 -2 1 -3 2 0 2 A= B= C= cIJ = aIJ - bIJ 3. Matrices

  22. Multiplication C=A-B 1 -1 -1 0 4 2 -1 0 1 1 -5 -3 1 6 1 0 -2 0 1 -1 0 -2 1 -3 2 0 2 A= B= C= cIJ = aI1 * b1J + aI2 * b2J + aI3 * b3J 3. Matrices

  23. Vector, dot product, & outer product • A vector v is an N x 1 matrix • Dot product = inner product = vT x v = a scalar • Outer product = v x vT = N x N matrix 3. Matrices

  24. Transpose B=AT 1 -2 2 -1 1 0 0 -3 2 1 -1 0 -2 1 -3 2 0 2 A= B= bIJ = aJI 3. Matrices

  25. Determinant of a 2x2 matrix 1 -1 -2 1 B = = -1 2x2 determinant = b11 * b22 - bI2 * b21 3. Matrices

  26. Cofactor and adjoint matrices 1 -1 0 -2 1 -3 2 0 2 A= 1 -3 0 2 -2 -3 2 2 -2 1 2 0 2 -2 -2 2 2 -2 3 3 -1 -1 0 0 2 1 0 2 2 1 -1 2 0 B = cofactor = = -1 0 0 -3 1 0 -2 -3 1 -1 -2 1 2 2 3 -2 2 3 -2 -2 -1 C=BT = adjoint= 3. Matrices

  27. Determinant 1 -1 0 -2 1 -3 2 0 2 determinant of A = =4 1 -1 0 2 -2 -2 = 4 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 3. Matrices

  28. Inverse matrix 0.5 0.5 0.75 -0.5 0.5 0.75 -0.5 -0.5 -0.25 B = A-1 =adjoint(A)/determinant(A) = 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 0 0 1 0 0 0 1 = 3. Matrices

  29. Orthogonal matrix • An orthogonal matrix is a matrix whose inverse is equal to its transpose. 1 0 0 0 cos  sin  0 -sin  cos  1 0 0 0 cos  -sin  0 sin  cos  1 0 0 0 1 0 0 0 1 = 3. Matrices

  30. 4. Transforms • Definition • Examples • Time-domain solution • Frequency-domain solution • Terms used with frequency response • Power spectrum • Sinusoidal motion • Example -- vibration 4. Transforms

  31. Definition • Transforms -- a mathematical conversion from one way of thinking to another to make a problem easier to solve problem in original way of thinking solution in original way of thinking transform solution in transform way of thinking inverse transform 4. Transforms

  32. Examples (1 of 3) problem in English solution in English English to algebra solution in algebra algebra to English 4. Transforms

  33. Examples (2 of 3) problem in English solution in English English to matrices solution in matrices matrices to English 4. Transforms

  34. Examples (3 of 3) problem in time domain solution in time domain Fourier transform solution in frequency domain inverse Fourier transform • Other transforms • Laplace • z-transform • wavelets 4. Transforms

  35. Time-domain solution • We typically think in the time domain -- a time input produces a time output input output amplitude amplitude system time time 4. Transforms

  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

  37. Frequency-domain solution (2 of 2) input (sinusoids) output amplitude (dB) magnitude (dB) system log frequency log frequency phase (angle) 0 -180 log frequency 4. Transforms

  38. Terms used with frequency response amplitude (dB) power (dB) • Octave is a range of 2x • Decade is a range of 10x 20,10 • Slope = • 20 dB/decade, amplitude • 6 dB/octave, amplitude • 10 dB decade, power • 3 dB decade, power 6, 3 2 10 frequency 4. Transforms

  39. Power spectrum • A power spectrum is a special form of frequency response in which the ordinate represents power g2-Hz (dB) log frequency 4. Transforms

  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

  41. Example -- vibration transmissivity-squared input output g2-Hz (dB) amplitude (dB) g2-Hz (dB) log frequency log frequency log frequency Output vibration is product of input vibration times the transmissivity-squared at each frequency 4. Transforms

  42. 5. Statistics (1 of 2) • Frequency distribution • Sample mean • Sample variance • CEP • Density function • Distribution function • Uniform • Binomial 5. Statistics

  43. 5. Statistics (1 of 2) • Normal • Poisson • Exponential • Raleigh • Sampling • Combining error sources 5. Statistics

  44. Frequency distribution • Frequency distribution -- A histogram or polygon summarizing how raw data can be grouped into classes number n = sample size = 39 8 6 4 2 2 2 4 5 7 4 6 6 3 2 60 61 62 63 64 65 66 67 68 height (inches) 5. Statistics

  45. Sample mean N i=1 •  =  xi • An estimate of the population mean • Example N  = [ 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

  46. Sample variance N i=1 N-1 • 2=  (xi -  )2 • An estimate of the population variance •  = standard deviation • Example 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

  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 + y2)] CEP 5. Statistics

  48. Density function • Probability that a discrete event x will occur • Non-negative function whose integral over the entire range of the independent variable is 1 f(x) x 5. Statistics

  49. Distribution function • Probability that a numerical event x or less occurs • The integral of the density function F(x) 1.0 x 5. Statistics

  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