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Basic Description of Data

Basic Description of Data. Deterministic Data. X o : Initial displacement k: Spring constant m: mass of body t: time. k. m. m. 0. m. X 0. Deterministic Data: They can be explained by an explicit Mathematical relationship Oceanography: Equilibrium Tides can fall in this category.

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Basic Description of Data

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  1. Basic Description of Data

  2. Deterministic Data X o : Initial displacement k: Spring constant m: mass of body t: time k m m 0 m X0 Deterministic Data: They can be explained by an explicit Mathematical relationship Oceanography:Equilibrium Tides can fall in this category

  3. Sinusoidal Periodic Data T, period X : Amplitude fo : Frequency t : Time Θ : Phase +X t -X Discrete or Line Spectra Example: Theoretical M2 Tide X fo f, frequency

  4. Complex Periodic Data • Periodic data that can be described mathematically by varying functions whose waveform exactly repeats itself at regular intervals. Tp= Period (time) the signal repeats itself

  5. Complex Periodic Data (2) • Periodic data can be described using a Fourier Series: Tp = Period (time) the signal repeats itself fo= 1/To, Frequency (fundamental frequency)

  6. Cartesian vs. Polar Notation Vector F has magnitude |F| and angle θ F (a1,b1) Magnitude |F|=sqrt(a12+b12) Angle θ=atan(b1/a1) Y b1 F θ a1 X

  7. From Polar to Complex Notation Vector F has magnitude |F| and angle θ a=|F| cos(θ) b=|F| sin(θ) |F|=sqrt(a2+b2) θ= atan(b/a) Imag j∙b F F=|F| cos(θ)+j∙|F| sin(θ) F=|F|∙(cos(θ)+j∙sin(θ)) θ a b -b Real -θ Euler’s Relation: cos(θ)+j∙sin(θ)=ej∙θ -j∙b F=|F|∙ej∙θ

  8. Vector Multiplication • θ=ω·t+φ, where ω (=2π/T) is the angular frequency and φ initial phase angle at t=0. • θ= (2π/T )·t+φ, and with f=1/T • θ= 2π·f·t+φ • ejθ=ej(2π·f·t+φ)=ej(2π·f·t)∙ejφ • Unit vector that has orientation φ at t=0 and rotates with period T. ej(2π·f·t) ejφ

  9. Complex Periodic Data (3) • An Alternative Presentation of Periodic data is: To= Period fo = 1/To , Frequency i.e., complex periodic data consist of: Static component , Xo Infinite number of sinusoidal components, called harmonics with amplitudes Xnand phases Θn. The frequencies of the harmonic components are all integral multiples of fo

  10. Complex Periodic Data (4) • Assume a signal of consisting of 3 frequency components, f= 60, 75 and 100Hz. • The highest common divider is 5 (so 5Hz is the fundamental frequency fO) so that: • 5Hz ∙ 12 = 60Hz → fO ∙ 12 = 60Hz → n=12 • 5Hz ∙ 15 = 75Hz → fO ∙ 15 = 75Hz → n=15 • 5Hz ∙ 20 =100Hz → fO ∙ 20 = 100Hz → n=20 Discrete or Line Spectra X15 X12 X20 12∙fo 15∙fo 20∙fo f, frequency

  11. Almost Periodic Data 2πn1fo=2π → fo=1/n1 n1/n2=2/3 n1/n3=2/7 n3/n2=sqrt(50)/3 2πn2fo=3π → fo=3/2n2 2πn3f3=7π → fo=7/2n3 n no integers, but rational numbers (sqrt(50) no integer, sqrt(50)/3 no rational) THERE IS NOT A FUNDAMENTAL FREQUENCY, OR IT IS 0 (T=∞)

  12. Transient no Periodic Data

  13. |X(f)| A/a A Fourier transform will give Continuous spectra, no discrete spectra f aA/(a2+b2) In polar notation f cA f

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