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Lecture 5: Signals – General Characteristics

Signals and Spectral Methods in Geoinformatics. Lecture 5: Signals – General Characteristics. Signal transmission and processing. transmission t - τ. ρ = c τ. t - τ. t. reception t. Τ. Δ t 0. Δ t. n Τ. τ. Observation :. τ = n Τ + Δ t – Δ t 0. ΔΦ = ρ – n λ.

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Lecture 5: Signals – General Characteristics

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  1. SignalsandSpectral Methods in Geoinformatics Lecture 5: Signals – General Characteristics

  2. Signal transmission and processing transmissiont-τ ρ = c τ t-τ t receptiont Τ Δt0 Δt nΤ τ Observation: τ = nΤ + Δt –Δt0 ΔΦ = ρ –nλ

  3. Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) k = constant, n(t) =noise

  4. Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) Signal traveling time: τ = ρ / c ρ = distancetransmitter - receiver c = transmission velocity = velocity of light in vacuum k = constant, n(t) =noise

  5. x(t - τ) x(t) τ t t Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) Signal traveling time: τ = ρ / c ρ = distancetransmitter - receiver c = transmission velocity = velocity of light in vacuum k = constant, n(t) =noise

  6. x(t - τ) x(t) τ t t Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) Signal traveling time: τ = ρ / c ρ = distancetransmitter - receiver c = transmission velocity = velocity of light in vacuum k = constant, n(t) =noise The functiong(t) = f(t – τ)obtains at instanttthe value whichfhad at the instancet – τ, at a time periodτbefore = delay ofτ = transposition by τ of the function graph to the right (= future)

  7. x(t) t Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) Signal traveling time: τ = ρ / c ρ = distancetransmitter - receiver c = transmission velocity = velocity of light in vacuum k = constant, n(t) =noise x(t - τ) τ t

  8. x(t) t Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) Signal traveling time: τ = ρ / c ρ = distancetransmitter - receiver c = transmission velocity = velocity of light in vacuum k = constant, n(t) =noise k x(t - τ) t

  9. x(t) t Signal transmission and reception Signal at transmitter:x(t) Signal at receiver:y(t) = kx(t - τ) + n(t) Signal traveling time: τ = ρ / c ρ = distancetransmitter - receiver c = transmission velocity = velocity of light in vacuum k = constant, n(t) =noise kx(t - τ) + n(t) t Noisen(t) =external high frequency interference (atmosphere, electonic parts of transmitter and receiver)

  10. Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from :

  11. x(t) +a t 0 T -a Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from : T = period

  12. x(t) +a t 0 T -a Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from : T = period

  13. x(t) +a t 0 T -a Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from : T = period frequency : (Hertz =cycles / second)

  14. x(t) +a t 0 T -a Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from : T = period frequency : (Hertz =cycles / second) angular frequency :

  15. x(t) +a t 0 T -a Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from : T = period frequency : (Hertz =cycles / second) angular frequency : wavelength : c = velocity of light in vacuum

  16. x(t) +a t 0 T -a Monochromatic (sinusoidal) signals Monochromatic signal = periodic signal with sinusoidal from : T = period frequency : (Hertz =cycles / second) angular frequency : wavelength : c = velocity of light in vacuum Alternative signal descriptions: simpler !

  17. Signal phase Signal phase at an instant t :

  18. Signal phase Signal phase at an instant t : t – Δt = immediately preceding instance with x(t – Δt) = 0andx(t – Δt+ ε) > 0 (= beginning of current cycle)

  19. Signal phase Signal phase at an instant t : t – Δt = immediately preceding instance with x(t – Δt) = 0andx(t – Δt+ ε) > 0 (= beginning of current cycle) = phase at instantt (phase = current fraction of the period)

  20. Φ=0 Φ=1/4 Φ=1/2 Φ=3/4 Φ=0 Signal phase Signal phase at an instant t : t – Δt = immediately preceding instance with x(t – Δt) = 0andx(t – Δt+ ε) > 0 (= beginning of current cycle) = phase at instantt (phase = current fraction of the period)

  21. Φ=0 Φ=1/4 Φ=1/2 Φ=3/4 Φ=0 Signal phase Signal phase at an instant t : t – Δt = immediately preceding instance with x(t – Δt) = 0andx(t – Δt+ ε) > 0 (= beginning of current cycle) = phase at instantt (phase = current fraction of the period) = phase angle

  22. Φ=0 Φ=1/4 Φ=1/2 Φ=3/4 Φ=0 φ=0 φ=π/4 φ=π/2 φ=3π/4 φ=0 Signal phase Signal phase at an instant t : t – Δt = immediately preceding instance with x(t – Δt) = 0andx(t – Δt+ ε) > 0 (= beginning of current cycle) = phase at instantt (phase = current fraction of the period) = phase angle (period fraction expressed asan angle)

  23. Generalization:Initial epoch t0 0 :

  24. t0 t Τ Δt0 Δt Generalization:Initial epoch t0 0 :

  25. t0 t Τ Δt0 Δt Generalization:Initial epoch t0 0 : nΤ current phase : initial phase :

  26. t0 t Τ Δt0 Δt Generalization:Initial epoch t0 0 : nΤ t – t0 current phase : initial phase :

  27. t0 t Τ Δt0 Δt Generalization:Initial epoch t0 0 : nΤ t – t0 current phase : initial phase :

  28. t0 t Τ Δt0 Δt Generalization:Initial epoch t0 0 : nΤ t – t0 current phase : initial phase :

  29. t0 t Τ Δt0 Δt Relating time difference to phase difference: mathematical model for the observations of phase differences Generalization:Initial epoch t0 0 : nΤ t – t0 current phase : initial phase :

  30. t0 t Τ Δt0 Δt Relating time difference to phase difference: mathematical model for the observations of phase differences Generalization:Initial epoch t0 0 : nΤ t – t0 current phase : initial phase : Frequency as the derivative of phase

  31. General form of a monochromatic signal:

  32. General form of a monochromatic signal: Alternative (usual) formusing cosine:

  33. General form of a monochromatic signal: Alternative (usual) formusing cosine: Θ = phase of a cosinesignal θ =corresponding phase angle

  34. General form of a monochromatic signal: Alternative (usual) formusing cosine: Θ = phase of a cosinesignal θ =corresponding phase angle ( 2π )

  35. General form of a monochromatic signal: Alternative (usual) formusing cosine: Θ = phase of a cosinesignal θ =corresponding phase angle ( 2π ) Usualnotation : Θ  Φ, θ  φ

  36. Epocht- Signal traveling in space y(t,r) = x(t-cr) transmitter r = 0 receiver r = ρ

  37. Epocht- Signal traveling in space y(t,r) = x(t-cr) transmitter r = 0 receiver r = ρ epocht x(t) t signal at transmitter

  38. Epocht- Signal traveling in space y(t,r) = x(t-cr) transmitter r = 0 receiver r = ρ epocht x(t) t signal at transmitter

  39. Epocht- Signal traveling in space y(t,r) = x(t-cr) transmitter r = 0 receiver r = ρ epocht epocht x(t) y(t) = x(t-cρ) t t signal at transmitter signal at receiver

  40. Epocht- Signal traveling in space y(t,r) = x(t-cr) transmitter r = 0 receiver r = ρ epocht epocht x(t) y(t) = x(t-cρ) t t signal at transmitter signal at receiver

  41. Energy signals Energy :

  42. Energy signals Energy : Correlation function of two signalsx(t)andy(t):

  43. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal:

  44. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal: Properties

  45. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal: Properties Applications:GPS, VLBI !

  46. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal: Properties Applications:GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function :

  47. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal: Properties Applications:GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : Energy:

  48. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal: Properties Applications:GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : S(ω) =energy (spectral) density Energy:

  49. Energy signals Energy : Correlation function of two signalsx(t)andy(t): (Auto)correlation functionof a signal: Properties Applications:GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : S(ω) =energy (spectral) density Energy: Example: x(t) = solar radiation on earth surface, S(ω)  S(λ) = chromatic spectrum

  50. Energy spectral density of the solar electromagnetic radiation Μλ( W m-2Ǻ-1) 0.20 Black body radiationat 6000 Κ Radiation above the atmosphere 0.15 Radiation on the surface of the earth 0.10 0.05 ορατό 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 wavelengthλ (μm) (energy per wavelength unit arriving on a surface with unit area within a unit of time)

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