# Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals.

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Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Lecture 5: Signals – General Characteristics Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signal transmission and processing τ = n Τ + Δt –Δt 0 t t  τt  τ τ n Τn Τ Δt0Δt0 ΔtΔt Τ ρ = c τ reception t transmission t  τ ΔΦ = ρ – n λ Observation :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) k = constant, n(t) = noise

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) c = transmission velocity = velocity of light in vacuum k = constant, n(t) = noise ρ = distance transmitter - receiver Signal traveling time: τ = ρ / c

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics x(t)x(t) t τ t x(t - τ) Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) c = transmission velocity = velocity of light in vacuum k = constant, n(t) = noise ρ = distance transmitter - receiver Signal traveling time: τ = ρ / c

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) x(t)x(t) t c = transmission velocity = velocity of light in vacuum The function g(t) = f(t – τ) obtains at instant t the value which f had at the instance t – τ, at a time period τ before = delay of τ = transposition by τ of the function graph to the right (= future) k = constant, n(t) = noise ρ = distance transmitter - receiver τ t x(t - τ) Signal traveling time: τ = ρ / c

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ x(t)x(t) t t x(t - τ) Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) c = transmission velocity = velocity of light in vacuum k = constant, n(t) = noise ρ = distance transmitter - receiver Signal traveling time: τ = ρ / c

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics k x(t)x(t) t t x(t - τ) Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) c = transmission velocity = velocity of light in vacuum k = constant, n(t) = noise ρ = distance transmitter - receiver Signal traveling time: τ = ρ / c

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics k x(t - τ) x(t)x(t) t t Noise n(t) =external high frequency interference (atmosphere, electonic parts of transmitter and receiver) + n(t) Signal transmission and reception Signal at transmitter: x(t) Signal at receiver: y(t) = k x(t - τ) + n(t) c = transmission velocity = velocity of light in vacuum k = constant, n(t) = noise ρ = distance transmitter - receiver Signal traveling time: τ = ρ / c

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Monochromatic signal = periodic signal with sinusoidal from : Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Monochromatic signal = periodic signal with sinusoidal from : T = period x(t)x(t) +a t aa 0T Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Monochromatic signal = periodic signal with sinusoidal from : T = period 0 1/4 T1/4 T 1/2 T1/2 T 3/4 T3/4 TT 0 1 / 2 ππ 3/2π3/2π2π 0+10 11 0 0+a+a0  a a 0 x(t)x(t) +a t aa 0T Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Monochromatic signal = periodic signal with sinusoidal from : T = period 0 1/4 T1/4 T 1/2 T1/2 T 3/4 T3/4 TT 0 1 / 2 ππ 3/2π3/2π2π 0+10 11 0 0+a+a0  a a 0 frequency : (Hertz = cycles / second) x(t)x(t) +a t aa 0T Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Monochromatic signal = periodic signal with sinusoidal from : T = period 0 1/4 T1/4 T 1/2 T1/2 T 3/4 T3/4 TT 0 1 / 2 ππ 3/2π3/2π2π 0+10 11 0 0+a+a0  a a 0 frequency : angular frequency : (Hertz = cycles / second) x(t)x(t) +a t aa 0T Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Monochromatic signal = periodic signal with sinusoidal from : T = period 0 1/4 T1/4 T 1/2 T1/2 T 3/4 T3/4 TT 0 1 / 2 ππ 3/2π3/2π2π 0+10 11 0 0+a+a0  a a 0 frequency : angular frequency : wavelength : (Hertz = cycles / second) c = velocity of light in vacuum x(t)x(t) +a t aa 0T Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics simpler ! Monochromatic signal = periodic signal with sinusoidal from : T = period 0 1/4 T1/4 T 1/2 T1/2 T 3/4 T3/4 TT 0 1 / 2 ππ 3/2π3/2π2π 0+10 11 0 0+a+a0  a a 0 frequency : angular frequency : wavelength : (Hertz = cycles / second) c = velocity of light in vacuum Alternative signal descriptions : x(t)x(t) +a t aa 0T Monochromatic (sinusoidal) signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signal phase at an instant t : Signal phase

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t – Δt = immediately preceding instance with x(t – Δt ) = 0 and x(t – Δt + ε ) > 0 (= beginning of current cycle) Signal phase at an instant t : Signal phase

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t – Δt = immediately preceding instance with x(t – Δt ) = 0 and x(t – Δt + ε ) > 0 (= beginning of current cycle) Signal phase at an instant t : Signal phase = phase at instant t (phase = current fraction of the period)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t – Δt = immediately preceding instance with x(t – Δt ) = 0 and x(t – Δt + ε ) > 0 (= beginning of current cycle) Signal phase at an instant t : Signal phase = phase at instant t (phase = current fraction of the period) Φ = 0Φ = 0Φ = 1/4Φ = 1/2Φ = 3/4Φ = 0Φ = 0

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t – Δt = immediately preceding instance with x(t – Δt ) = 0 and x(t – Δt + ε ) > 0 (= beginning of current cycle) Signal phase at an instant t : Signal phase = phase at instant t = phase angle (phase = current fraction of the period) Φ = 0Φ = 0Φ = 1/4Φ = 1/2Φ = 3/4Φ = 0Φ = 0

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t – Δt = immediately preceding instance with x(t – Δt ) = 0 and x(t – Δt + ε ) > 0 (= beginning of current cycle) Signal phase at an instant t : Signal phase = phase at instant t = phase angle (phase = current fraction of the period) (period fraction expressed as an angle) Φ = 0Φ = 0Φ = 1/4Φ = 1/2Φ = 3/4Φ = 0Φ = 0 φ = 0φ = 0φ = π/4φ = π/2φ = 3π/4φ = 0φ = 0

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Generalization: Initial epoch t 0  0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t Generalization: Initial epoch t 0  0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t n Τn Τ Generalization: Initial epoch t 0  0 : initial phase : current phase :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t t – t 0 n Τn Τ Generalization: Initial epoch t 0  0 : initial phase : current phase :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t t – t 0 n Τn Τ Generalization: Initial epoch t 0  0 : initial phase : current phase :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t t – t 0 n Τn Τ Generalization: Initial epoch t 0  0 : initial phase : current phase :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t t – t 0 n Τn Τ Generalization: Initial epoch t 0  0 : initial phase : current phase : Relating time difference to phase difference : mathematical model for the observations of phase differences

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics ΔtΔtΔt0Δt0 t0t0 Τ t t – t 0 n Τn Τ Generalization: Initial epoch t 0  0 : initial phase : current phase : Frequency as the derivative of phase Relating time difference to phase difference : mathematical model for the observations of phase differences

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics General form of a monochromatic signal :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Alternative (usual) form using cosine : General form of a monochromatic signal :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Alternative (usual) form using cosine : General form of a monochromatic signal : Θ = phase of a cosine signal θ = corresponding phase angle

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Alternative (usual) form using cosine : General form of a monochromatic signal : Θ = phase of a cosine signal θ = corresponding phase angle (  2π  )

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Alternative (usual) form using cosine : General form of a monochromatic signal : Θ = phase of a cosine signal θ = corresponding phase angle (  2π  ) Usual notation : Θ  Φ, θ  φ

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics receiver r = ρ transmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(t  cr)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t epoch t x(t)x(t) signal at transmitter receiver r = ρ transmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(t  cr)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics t epoch t x(t)x(t) signal at transmitter receiver r = ρ transmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(t  cr)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics signal at receiver y(t) = x(t  cρ) t epoch t t x(t)x(t) signal at transmitter receiver r = ρ transmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(t  cr)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics signal at receiver y(t) = x(t  cρ) t epoch t t x(t)x(t) signal at transmitter receiver r = ρ transmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(t  cr)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy signals Energy :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Properties Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Properties Applications: GPS, VLBI ! Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Properties Applications: GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Properties Applications: GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : Energy : Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Properties Applications: GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : Energy : S(ω) = energy (spectral) density Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Energy : Correlation function of two signals x(t) and y(t) : (Auto)correlation function of a signal : Properties Applications: GPS, VLBI ! Energy spectral density = Fourier transform of autocorrelation function : Energy : S(ω) = energy (spectral) density Example : x(t) = solar radiation on earth surface, S(ω)  S(λ) = chromatic spectrum Energy signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics 0.40.81.21.62.02.42.83.2 0.20 0.15 0.10 0.05 0 Μ λ ( W m  2 Ǻ  1 ) wavelength λ (μm) Black body radiation at 6000 Κ Radiation above the atmosphere Radiation on the surface of the earth Energy spectral density of the solar electromagnetic radiation ορατό (energy per wavelength unit arriving on a surface with unit area within a unit of time)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics infrared The electromagnetic spectrum visible 10  5 10  2 310 2 10 4 10 6 (μm) 0.40.50.60.7 visible reflected thermal microwaves RADIOultravioletΧ raysγ rays λ

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power signals power for the interval [–Τ /2, Τ /2]

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power signals power for the interval [–Τ /2, Τ /2] power for the interval [– , +  ]

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Total power for the interval [– , +  ] : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Total power for the interval [– , +  ] : (n  1)T  (n  1)T Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Total power for the interval [– , +  ] : (n  1)T  (n  1)T Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Total power for the interval [– , +  ] : (n  1)T  (n  1)T Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Total power for the interval [– , +  ] : (n  1)T  (n  1)T Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power of a periodic signal with period Τ Power for one period Τ : Total power for the interval [– , +  ] : The power P of a periodic signal is equal to the power P T for only one period P = P T (n  1)T  (n  1)T Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Power : Correlation function of two signals x(t) and y(t) : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (auto)correlation function of a signal : Power : Correlation function of two signals x(t) and y(t) : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (auto)correlation function of a signal : Power : Correlation function of two signals x(t) and y(t) : Properties Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (auto)correlation function of a signal : Power : Correlation function of two signals x(t) and y(t) : Properties Εφαρνογές GPS, VLBI ! Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (auto)correlation function of a signal : Power : Correlation function of two signals x(t) and y(t) : Properties Εφαρνογές GPS, VLBI ! Power spectral density = Fourier transform of the autocorrelation function : Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics (auto)correlation function of a signal : Power : Correlation function of two signals x(t) and y(t) : Properties Εφαρνογές GPS, VLBI ! Power spectral density = Fourier transform of the autocorrelation function : ισχύς : S(ω) = power (spectral) density _ Power signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : representation of linear system with an integral : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : time translation : representation of linear system with an integral : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : time translation : time invariant system : representation of linear system with an integral : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : time translation : time invariant system : representation of linear system with an integral : Representation of a time invariant linear system with an integral : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : time translation : time invariant system : representation of linear system with an integral : Representation of a time invariant linear system with an integral : convolution of two functions g(t) and f(t) : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Linear systems linear syatem = a mapping linearity : time translation : time invariant system : representation of linear system with an integral : Representation of a time invariant linear system with an integral : convolution of two functions g(t) and f(t) : time invariant linear system : input signaloutput signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : (notation simplification) Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : (notation simplification) Dirac function (impulse): Linear systems δε(t)δε(t) ε 1/ε

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : (notation simplification) Dirac function (impulse): Linear systems δε(t)δε(t) ε 1/ε area = 1

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : (notation simplification) Dirac function (impulse): Linear systems δε(t)δε(t) ε 1/ε area = 1

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : (notation simplification) Dirac function (impulse): Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : h = impulse response function (notation simplification) Dirac function (impulse): Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : h = impulse response function (notation simplification) Dirac function (impulse): Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Representation of a linear system with an integral : for a time-invariant one : Proof : h = impulse response function (notation simplification) Dirac function (impulse): Linear systems

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms : Convolution theorem convolution

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms : Convolution theorem convolution Convolution theorem in explicit form :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms : Convolution theorem convolution Convolution theorem in explicit form :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms : Convolution theorem convolution Convolution theorem in explicit form :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms : Convolution theorem convolution Convolution theorem in explicit form :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The convolution theorem Representation of a time-invariant linear system with an integral : convolution Fourier transforms : Convolution theorem convolution Convolution theorem in explicit form : or

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Φίλτρα = χρονικά αμετάβλητα γραμμικά συστήματα L με Η(ω) = 0 σε τμήματα συχνοτήτων ω (= αποκοπή ορισμένων συχνοτήτων)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : Η(ω) = 0 when |ω| > ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : Η(ω) = 0 when |ω| > ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : Η(ω) = 0 when |ω| > ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : BPF = Band Pass Filter (outside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : BPF = Band Pass Filter (outside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 Η(ω) = 0 when ω 1 < |ω| < ω 2 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : BPF = Band Pass Filter (outside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 Η(ω) = 0 when ω 1 < |ω| < ω 2 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain LPF = Low Pass Filter : HPF = High Pass Filter : BPF = Band Pass Filter (inside band) : BPF = Band Pass Filter (outside band) : Η(ω) = 0 when |ω| > ω 0 Η(ω) = 0 when| ω| < ω 1 < ω 2 or ω 1 < ω 2 < | ω| Η(ω) = 0 when |ω| < ω 0 Η(ω) = 0 when ω 1 < |ω| < ω 2 (= removal of some particular frequencies)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then When Η(ω) = 0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then When Η(ω) = 0 : When Η(ω)  0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then When Η(ω) = 0 : When Η(ω)  0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then Impulse response function of Low Pass ideal filter : When Η(ω) = 0 : When Η(ω)  0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then Impulse response function of Low Pass ideal filter : Casual filters ( t = time) (instesd of ) When Η(ω) = 0 : When Η(ω)  0 :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Ideal filters : when then Impulse response function of Low Pass ideal filter : Casual filters ( t = time) (instesd of ) When Η(ω) = 0 : When Η(ω)  0 : Output y(t) depends only on past (   s  t ) values s of the input x(s) and not on future values (casuality)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Bandwidth

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Bandwidth Low Pass Filter :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Bandwidth Low Pass Filter :Band Pass Filter (inside band) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Bandwidth Low Pass Filter :Band Pass Filter (inside band) : Low Pass Filter not ideal :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Bandwidth Low Pass Filter :Band Pass Filter (inside band) : Low Pass Filter not ideal :Band Pass Filter (inside band) not ideal :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics END

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