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## Agenda for understand req activity

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**Agenda for understand req activity**• 1. Numbers • 2. Decibels • 3. Matrices • 4. Transforms • 5. Statistics • 6. Software**1. Numbers**• Significant digits • Precision • Accuracy 1. Numbers**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**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**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**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**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**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**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**2. Decibels**• Definitions • Common values • Examples • Advantages • Decibels as absolute units • Powers of 2 2. Decibels**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**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**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**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**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**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**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**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**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**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**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**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**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**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**Determinant of a 2x2 matrix**1 -1 -2 1 B = = -1 2x2 determinant = b11 * b22 - bI2 * b21 3. Matrices**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**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**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**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**4. Transforms**• Definition • Examples • Time-domain solution • Frequency-domain solution • Terms used with frequency response • Power spectrum • Sinusoidal motion • Example -- vibration 4. Transforms**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**Examples (1 of 3)**problem in English solution in English English to algebra solution in algebra algebra to English 4. Transforms**Examples (2 of 3)**problem in English solution in English English to matrices solution in matrices matrices to English 4. Transforms**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**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**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**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**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**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**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**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**5. Statistics (1 of 2)**• Frequency distribution • Sample mean • Sample variance • CEP • Density function • Distribution function • Uniform • Binomial 5. Statistics**5. Statistics (1 of 2)**• Normal • Poisson • Exponential • Raleigh • Sampling • Combining error sources 5. Statistics**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**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**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**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**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**Distribution function**• Probability that a numerical event x or less occurs • The integral of the density function F(x) 1.0 x 5. Statistics**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