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

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Page 1: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Lecture 5:Signals – General Characteristics

Signals and Spectral Methodsin Geoinformatics

Page 2: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Signal transmission and processing

τ = n Τ + Δt –Δt0

tt τ

τ

n Τ

Δt0 Δt

Τ

nnT

t

T

tn

cT

c0

0

ρ = c τ

reception t

transmission t τ

ΔΦ = ρ – n λ

Observation :

Page 3: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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

Page 4: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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

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

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

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

Page 6: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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)

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

Page 7: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

τ

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

Page 8: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

kx(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

Page 9: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

k x(t - τ)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

Page 10: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

Monochromatic (sinusoidal) signals

Page 11: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

T = period

x(t)+a

t

a

0 T

Monochromatic (sinusoidal) signals

Page 12: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

T = period

0 1/4 T 1/2 T 3/4 T T

0 1/2 π π 3/2π 2π

0 +1 0 1 0

0 +a 0 a 0

T

t2

t

T

t2sin

)(tx

x(t)+a

t

a

0 T

Monochromatic (sinusoidal) signals

Page 13: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

T = period

0 1/4 T 1/2 T 3/4 T T

0 1/2 π π 3/2π 2π

0 +1 0 1 0

0 +a 0 a 0

T

t2

t

T

t2sin

)(tx

frequency :T

f1

(Hertz = cycles / second)

x(t)+a

t

a

0 T

Monochromatic (sinusoidal) signals

Page 14: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

T = period

0 1/4 T 1/2 T 3/4 T T

0 1/2 π π 3/2π 2π

0 +1 0 1 0

0 +a 0 a 0

T

t2

t

T

t2sin

)(tx

frequency :T

f1

angular frequency :T

f 2

2

(Hertz = cycles / second)

x(t)+a

t

a

0 T

Monochromatic (sinusoidal) signals

Page 15: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

T = period

0 1/4 T 1/2 T 3/4 T T

0 1/2 π π 3/2π 2π

0 +1 0 1 0

0 +a 0 a 0

T

t2

t

T

t2sin

)(tx

frequency :T

f1

angular frequency :T

f 2

2

wavelength :

(Hertz = cycles / second)

cT

c = velocity of light in vacuum

x(t)+a

t

a

0 T

Monochromatic (sinusoidal) signals

Page 16: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

simpler !

Monochromatic signal = periodic signal with sinusoidal from :T

tatx

2sin)(

T = period

0 1/4 T 1/2 T 3/4 T T

0 1/2 π π 3/2π 2π

0 +1 0 1 0

0 +a 0 a 0

T

t2

t

T

t2sin

)(tx

frequency :T

f1

angular frequency :T

f 2

2

tc

atatfaT

tatx

2sin)sin()2sin(

2sin)(

wavelength :

(Hertz = cycles / second)

cT

c = velocity of light in vacuum

Alternative signal descriptions :

x(t)+a

t

a

0 T

Monochromatic (sinusoidal) signals

Page 17: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Signal phase at an instant t :

Signal phase

)(tx

t

Page 18: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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

)(tx

ttt

t

Page 19: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 tT

tt

)(

Tt 0 10

)(tx

ttt

t

(phase = current fraction of the period)

Page 20: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 tT

tt

)(

Tt 0 10

)(tx

ttt

t

(phase = current fraction of the period)

Φ = 0 Φ = 1/4 Φ = 1/2 Φ = 3/4 Φ = 0

Page 21: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 tT

tt

)(

Tt 0 10

= phase angleT

ttt

2)(2)(

20

)(tx

ttt

t

(phase = current fraction of the period)

Φ = 0 Φ = 1/4 Φ = 1/2 Φ = 3/4 Φ = 0

Page 22: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 tT

tt

)(

Tt 0 10

= phase angleT

ttt

2)(2)(

20

)(tx

ttt

t

(phase = current fraction of the period)

(period fraction expressed as an angle)

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

Page 23: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Generalization: Initial epoch t0 0 :

Page 24: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

Generalization: Initial epoch t0 0 :

Page 25: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

n Τ

Generalization: Initial epoch t0 0 : 0)( 00 ttx

T

tt 0

00 )(

T

tt

)(initial phase : current phase :

0)( 00 ttx

Page 26: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

t – t0

n Τ

Generalization: Initial epoch t0 0 : 0)( 00 ttx

T

tt 0

00 )(

T

tt

)(initial phase : current phase :

0)( 00 ttx

Page 27: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

t – t0

n Τ

Generalization: Initial epoch t0 0 : 0)( 00 ttx

T

tt 0

00 )(

T

tt

)(initial phase : current phase :

0)( 00 ttx

00 ttnTtt

Page 28: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

t – t0

n Τ

Generalization: Initial epoch t0 0 : 0)( 00 ttx

T

tt 0

00 )(

T

tt

)(initial phase : current phase :

TNTTttNTttt 0000

0)( 00 ttx

00 ttnTtt

Page 29: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

t – t0

n Τ

Generalization: Initial epoch t0 0 : 0)( 00 ttx

T

tt 0

00 )(

T

tt

)(initial phase : current phase :

TNTTttNTttt 0000

TtTNtt ])([ 00 Relating time difference to phase difference : mathematical model

for the observationsof phase differences

0)( 00 ttx

00 ttnTtt

Page 30: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

ΔtΔt0

t0

Τ

t

t – t0

n Τ

fTdt

d2

2

Generalization: Initial epoch t0 0 : 0)( 00 ttx

T

tt 0

00 )(

T

tt

)(initial phase : current phase :

TNTTttNTttt 0000

NT

ttt

0

0)( fTdt

d

1

Frequency as the derivative of phase

TTNTtt 00

TtTNtt ])([ 00 Relating time difference to phase difference : mathematical model

for the observationsof phase differences

0)( 00 ttx

00 ttnTtt

Page 31: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

General form of a monochromatic signal :

)(sin)(2sin

)(2sin)( 000

0 tattfaT

ttatx

)(2sin)(2sin

)(2sin 000

0 tattfaT

tta

0000 2)(sin)(sin ttatta

Page 32: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Alternative (usual) form using cosine :

)(cos)(2cos

)(2cos)( 000

0 tattfaT

ttatx

00 0 0cos2 cos2 ( ) cos2 ( )

t ta a f t t a t

T

0000 2)(cos)(cos ttatta

General form of a monochromatic signal :

)(sin)(2sin

)(2sin)( 000

0 tattfaT

ttatx

)(2sin)(2sin

)(2sin 000

0 tattfaT

tta

0000 2)(sin)(sin ttatta

)(tx

t

a

a

0 T

T41

T

T

Page 33: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Alternative (usual) form using cosine :

)(cos)(2cos

)(2cos)( 000

0 tattfaT

ttatx

00 0 0cos2 cos2 ( ) cos2 ( )

t ta a f t t a t

T

0000 2)(cos)(cos ttatta

General form of a monochromatic signal :

)(sin)(2sin

)(2sin)( 000

0 tattfaT

ttatx

)(2sin)(2sin

)(2sin 000

0 tattfaT

tta

0000 2)(sin)(sin ttatta

Θ = phase of a cosine signal

θ = corresponding phase angle

)(tx

t

a

a

0 T

T41

T

T

Page 34: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

2)()(

tt

Alternative (usual) form using cosine :

)(cos)(2cos

)(2cos)( 000

0 tattfaT

ttatx

00 0 0cos2 cos2 ( ) cos2 ( )

t ta a f t t a t

T

0000 2)(cos)(cos ttatta

General form of a monochromatic signal :

)(sin)(2sin

)(2sin)( 000

0 tattfaT

ttatx

)(2sin)(2sin

)(2sin 000

0 tattfaT

tta

0000 2)(sin)(sin ttatta

Θ = phase of a cosine signal

θ = corresponding phase angle

4

1)()( tt

)(tx

t

a

a

0 T

T41

T

T

( 2π )

Page 35: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

2)()(

tt

Alternative (usual) form using cosine :

)(cos)(2cos

)(2cos)( 000

0 tattfaT

ttatx

00 0 0cos2 cos2 ( ) cos2 ( )

t ta a f t t a t

T

0000 2)(cos)(cos ttatta

General form of a monochromatic signal :

)(sin)(2sin

)(2sin)( 000

0 tattfaT

ttatx

)(2sin)(2sin

)(2sin 000

0 tattfaT

tta

0000 2)(sin)(sin ttatta

Θ = phase of a cosine signal

θ = corresponding phase angle

4

1)()( tt

)(tx

t

a

a

0 T

T41

T

T

( 2π )

Usual notation : Θ Φ, θ φ

Page 36: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)

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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

t

epoch tx(t)

signal at transmitter

receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

t

epoch tx(t)

signal at transmitter

receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

signal at receiver

y(t) = x(tcρ)

t

epoch t

t

epoch tx(t)

signal at transmitter

receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

signal at receiver

y(t) = x(tcρ)

t

epoch t

t

epoch tx(t)

signal at transmitter

receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Energy signals

Energy :

dttxE 2|)(|

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Energy :

Correlation function of two signals x(t) and y(t) :

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Energy :

Correlation function of two signals x(t) and y(t) :

(Auto)correlation function of a signal :

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Energy :

Correlation function of two signals x(t) and y(t) :

(Auto)correlation function of a signal :

Properties

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

)()( yxxy RR )()( xxxx RR ERxx )0(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 !

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

)()( yxxy RR )()( xxxx RR ERxx )0(

)(max)0(

xxxx RR

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 :

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

)()( yxxy RR )()( xxxx RR ERxx )0(

)(max)0(

xxxx RR

deRS i)()(

deSR i)(2

1)(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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 :

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

)()( yxxy RR )()( xxxx RR ERxx )0(

)(max)0(

xxxx RR

deRS i)()(

deSR i)(2

1)(

dSRE )(

2

1)0(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

)()( yxxy RR )()( xxxx RR ERxx )0(

)(max)0(

xxxx RR

deRS i)()(

deSR i)(2

1)(

dSRE )(

2

1)0(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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

dttxE 2|)(|

dttytxRxy )()()(

dttytx )()(

dttxtxRxx )()()(

)()( yxxy RR )()( xxxx RR ERxx )0(

)(max)0(

xxxx RR

deRS i)()(

deSR i)(2

1)(

dSRE )(

2

1)0(

Energy signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

0.20

0.15

0.10

0.05

0

Μλ ( W m2 Ǻ1)

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)

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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

infrared

The electromagnetic spectrum

visible

105 102 3 102 104 106 (μm)

(μm)0.4 0.5 0.6 0.7

visi

ble

refle

cted

ther

mal

mic

row

aves RADIOultravioletΧ raysγ rays

λ

Page 52: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power signals

power for the interval [–Τ /2, Τ /2]

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power signals

power for the interval [–Τ /2, Τ /2]

power for the interval [–, +]

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Total power for the interval [–, +] :

2/

~

2/~

2~ |)(|~

1lim

T

TTdttx

TP

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Total power for the interval [–, +] :

2/

~

2/~

2~ |)(|~

1lim

T

TTdttx

TP

0 TT nTnT (n1)T(n1)T

nTT 2~

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Total power for the interval [–, +] :

2/

~

2/~

2~ |)(|~

1lim

T

TTdttx

TP

0 TT nTnT (n1)T(n1)T

nTT 2~

nT

nTn

nT

nTnTdttx

nTdttx

nT22

2|)(|

2

1lim|)(|

2

1lim

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Total power for the interval [–, +] :

2/

~

2/~

2~ |)(|~

1lim

T

TTdttx

TP

nT

Tn

T

T

Tn

nTndttx

Tdttx

nTdttx

Tdttx

Tn )1(

2

0

20

2)1(

2 |)(|1

|)(|2

1|)(|

1|)(|

1

2

1lim

0 TT nTnT (n1)T(n1)T

nTT 2~

nT

nTn

nT

nTnTdttx

nTdttx

nT22

2|)(|

2

1lim|)(|

2

1lim

Power signals

Page 60: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Total power for the interval [–, +] :

2/

~

2/~

2~ |)(|~

1lim

T

TTdttx

TP

nT

Tn

T

T

Tn

nTndttx

Tdttx

nTdttx

Tdttx

Tn )1(

2

0

20

2)1(

2 |)(|1

|)(|2

1|)(|

1|)(|

1

2

1lim

TTn

Tn

TTTTn

PPnPn

PPPPn

lim22

1lim

2

1lim

0 TT nTnT (n1)T(n1)T

nTT 2~

nT

nTn

nT

nTnTdttx

nTdttx

nT22

2|)(|

2

1lim|)(|

2

1lim

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power of a periodic signal with period Τ

Power for one period Τ :

TT

TT dttx

Tdttx

TP

0

22/

2/

2 |)(|1

|)(|1

Total power for the interval [–, +] :

2/

~

2/~

2~ |)(|~

1lim

T

TTdttx

TP

nT

Tn

T

T

Tn

nTndttx

Tdttx

nTdttx

Tdttx

Tn )1(

2

0

20

2)1(

2 |)(|1

|)(|2

1|)(|

1|)(|

1

2

1lim

TTn

Tn

TTTTn

PPnPn

PPPPn

lim22

1lim

2

1lim

The power P of a periodic signal is equal to the power PT for only one period P = PT

0 TT nTnT (n1)T(n1)T

nTT 2~

nT

nTn

nT

nTnTdttx

nTdttx

nT22

2|)(|

2

1lim|)(|

2

1lim

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

2/

2/

)()(1

lim)(T

TTxy dttytx

TR

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Correlation function of two signals x(t) and y(t) :

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

(auto)correlation function of a signal :

2/

2/

)()(1

lim)(T

TTxy dttytx

TR

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Correlation function of two signals x(t) and y(t) :

2/

2/

)()(1

lim)(T

TTxx dttxtx

TR

Power signals

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

(auto)correlation function of a signal :

2/

2/

)()(1

lim)(T

TTxy dttytx

TR

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Correlation function of two signals x(t) and y(t) :

Properties )()( yxxy RR )()( xxxx RR PRxx )0(

2/

2/

)()(1

lim)(T

TTxx dttxtx

TR

Power signals

Page 66: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

(auto)correlation function of a signal :

2/

2/

)()(1

lim)(T

TTxy dttytx

TR

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Correlation function of two signals x(t) and y(t) :

Properties

Εφαρνογές GPS, VLBI !

)()( yxxy RR )()( xxxx RR PRxx )0(

)(max)0(

xxxx RR

2/

2/

)()(1

lim)(T

TTxx dttxtx

TR

Power signals

Page 67: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

(auto)correlation function of a signal :

2/

2/

)()(1

lim)(T

TTxy dttytx

TR

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Correlation function of two signals x(t) and y(t) :

Properties

Εφαρνογές GPS, VLBI !

Power spectral density = Fourier transform of the autocorrelation function :

)()( yxxy RR )()( xxxx RR PRxx )0(

)(max)0(

xxxx RR

deRS i)()(

deSR i)(2

1)(

2/

2/

)()(1

lim)(T

TTxx dttxtx

TR

Power signals

Page 68: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

(auto)correlation function of a signal :

2/

2/

)()(1

lim)(T

TTxy dttytx

TR

Power :

2/

2/

2|)(|1

limT

TTdttx

TP

Correlation function of two signals x(t) and y(t) :

Properties

Εφαρνογές GPS, VLBI !

Power spectral density = Fourier transform of the autocorrelation function :

ισχύς :

)()( yxxy RR )()( xxxx RR PRxx )0(

)(max)0(

xxxx RR

deRS i)()(

deSR i)(2

1)(

dSRP )(

2

1)0(

2/

2/

)()(1

lim)(T

TTxx dttxtx

TR

S(ω) = power (spectral) density_

Power signals

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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

representation of linear system with an integral :

dssxsthtLxty )(),())(()(

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

time translation : )()()(: txtxtxT

representation of linear system with an integral :

dssxsthtLxty )(),())(()(

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

time translation : )()()(: txtxtxT

time invariant system : LTLT )()(: tytxL )()(: tytxL

representation of linear system with an integral :

dssxsthtLxty )(),())(()(

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

time translation : )()()(: txtxtxT

time invariant system : LTLT )()(: tytxL )()(: tytxL

representation of linear system with an integral :

dssxsthtLxty )(),())(()(

Representation of a time invariant linear system with an integral :

dssxsthtLxty )()())(()(

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

time translation : )()()(: txtxtxT

time invariant system : LTLT )()(: tytxL )()(: tytxL

representation of linear system with an integral :

dssxsthtLxty )(),())(()(

Representation of a time invariant linear system with an integral :

dssxsthtLxty )()())(()(

convolution of two functions g(t) and f(t) :

dssfstgtfg )()())((

)(tx )(tyLinput signal output signal

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Linear systems

linear syatem = a mapping Lxy ))(()( tLxty )()(: tytxL

)()()( 22112211 xLaxLaxaxaL linearity :

time translation : )()()(: txtxtxT

time invariant system : LTLT )()(: tytxL )()(: tytxL

representation of linear system with an integral :

dssxsthtLxty )(),())(()(

Representation of a time invariant linear system with an integral :

dssxsthtLxty )()())(()(

convolution of two functions g(t) and f(t) :

dssfstgtfg )()())((

time invariant linear system :

xhLxy

)(tx )(tyLinput signal output signal

Page 78: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Representation of a linear system with an integral :

dssxsthtLxty )(),())(()(

Linear systems

Page 79: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Representation of a linear system with an integral :

for a time-invariant one :

dssxsthtLxty )(),())(()(

),(),( sthsth

Linear systems

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

Linear systems

Page 81: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

Linear systems

Page 82: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

Linear systems

Page 83: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

Linear systems

Page 84: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

Linear systems

Page 85: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

Linear systems

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

(notation simplification)

Linear systems

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

(notation simplification)

Dirac function (impulse):

Linear systems

)(lim)(0

tt

δε(t)

ε

1/ε

Page 88: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

(notation simplification)

Dirac function (impulse):

Linear systems

)(lim)(0

tt

δε(t)

ε

1/εarea = 1

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

(notation simplification)

Dirac function (impulse): )()()(,1)(,00

00)( tfdssstfdss

s

ss

Linear systems

)(lim)(0

tt

δε(t)

ε

1/εarea = 1

Page 90: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

(notation simplification)

Dirac function (impulse): )()()(,1)(,00

00)( tfdssstfdss

s

ss

Linear systems

Page 91: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

h = impulse response function

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

(notation simplification)

Dirac function (impulse): )()()(,1)(,00

00)( tfdssstfdss

s

ss

Linear systems

Page 92: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

h = impulse response function

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

dsssthth )()()(

(notation simplification)

Dirac function (impulse): )()()(,1)(,00

00)( tfdssstfdss

s

ss

Linear systems

Page 93: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

dssxsthsdsxsth )(),()(),(

Representation of a linear system with an integral :

for a time-invariant one :

Proof :

h = impulse response function

dssxsthtLxty )(),())(()(

),(),( sthsth

dssxsthty )(),()(

dssxsthdssxsthtyty )(),()(),()()(

),(),( sthsth

tt),(),( sthsth

),(),( sthsth )()0,(),(),( sthsthsssthsth :hh

dsssthth )()()(

(notation simplification)

Dirac function (impulse): )()()(,1)(,00

00)( tfdssstfdss

s

ss

)(t )(thL

Linear systems

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

Page 97: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

xhy )()()( XHY

Convolution theorem

convolution

Page 98: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

xhy )()()( XHY

Convolution theorem

convolution

Convolution theorem in explicit form :

Page 99: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

xhy )()()( XHY

Convolution theorem

convolution

Convolution theorem in explicit form :

)(|)(|)( XieXX

)(|)(|)( YieYY

)(|)(|)( HieHH

Page 100: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

xhy )()()( XHY

Convolution theorem

convolution

Convolution theorem in explicit form :

|)(||)(|)(| XHY

)()()( XHY

)(|)(|)( XieXX

)(|)(|)( YieYY

)(|)(|)( HieHH

Page 101: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

xhy )()()( XHY

Convolution theorem

convolution

Convolution theorem in explicit form :

|)(||)(|)(| XHY

)()()( XHY

)()()( 21 iXXX

)()()( 21 iYYY

)()()( 21 iHHH

)(|)(|)( XieXX

)(|)(|)( YieYY

)(|)(|)( HieHH

Page 102: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

The convolution theorem

Representation of a time-invariant linear system with an integral :

))(()()())(()( txhdssxsthtLxty

xhLxy convolution

Fourier transforms : ,)()( dtetxX ti

,)()( dtetyY ti

dehH i)()(

xhy )()()( XHY

Convolution theorem

convolution

Convolution theorem in explicit form :

|)(||)(|)(| XHY

)()()( XHY

)()()()()( 22111 XHXHY

)()()()()( 12212 XHXHY

or

)()()( 21 iXXX

)()()( 21 iYYY

)()()( 21 iHHH

)(|)(|)( XieXX

)(|)(|)( YieYY

)(|)(|)( HieHH

Page 103: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Φίλτρα = χρονικά αμετάβλητα γραμμικά συστήματα L με Η(ω) = 0 σε τμήματα συχνοτήτων ω

(= αποκοπή ορισμένων συχνοτήτων)

Page 104: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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)

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

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Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Filters = time-invariant linear systems L with Η(ω) = 0 in parts of the frequency domain

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 107: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 108: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 109: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 110: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 111: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 112: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 113: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 114: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 115: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 116: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 117: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 118: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

Η(ω) = 0 when ω1 < |ω| < ω2

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 119: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

Η(ω) = 0 when ω1 < |ω| < ω2

)(tx L

dssxsthty )()()(

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

Page 120: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals 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| > ω0

Η(ω) = 0 when |ω| < ω1 < ω2

or ω1 < ω2 < |ω|

Η(ω) = 0 when |ω| < ω0

Η(ω) = 0 when ω1 < |ω| < ω2

)(tx L

dssxsthty )()()(

|)(| X |)(| H |)(| X(= removal of some particular frequencies)

L)(X )()()( XHY

Page 121: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H dtieH )(

Page 122: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

dtieH )(

Page 123: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

When Η(ω) = 0 : 0)( Y

dtieH )(

Page 124: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

When Η(ω) = 0 :

When Η(ω) 0 :

0)( Y

dtieH )(

)(])([)( |)(||)(||)(|)()()( YdXXd eYeXeXeXHY tiiti

Page 125: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

dXY tXHY )()(&|)(||)(||)(|

When Η(ω) = 0 :

When Η(ω) 0 :

0)( Y

dtieH )(

)(])([)( |)(||)(||)(|)()()( YdXXd eYeXeXeXHY tiiti

Page 126: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

dXY tXHY )()(&|)(||)(||)(|

Impulse response function of Low Pass ideal filter :

)]([sinc)(

)(sin

2

1)(

2

1)( 0

00d

d

dtititiLPF tt

tt

ttdeedeHth d

When Η(ω) = 0 :

When Η(ω) 0 :

0)( Y

dtieH )(

)(])([)( |)(||)(||)(|)()()( YdXXd eYeXeXeXHY tiiti

Page 127: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

dXY tXHY )()(&|)(||)(||)(|

Impulse response function of Low Pass ideal filter :

)]([sinc)(

)(sin

2

1)(

2

1)( 0

00d

d

dtititiLPF tt

tt

ttdeedeHth d

Casual filters (t = time)

t

dssxsthty )()()( (instesd of )

When Η(ω) = 0 :

When Η(ω) 0 :

0)( Y

dtieH )(

)(])([)( |)(||)(||)(|)()()( YdXXd eYeXeXeXHY tiiti

Page 128: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Ideal filters : when then0)( H

1|)(|&)(|)(|)( )( HteeHH dHtii dH

dXY tXHY )()(&|)(||)(||)(|

Impulse response function of Low Pass ideal filter :

)]([sinc)(

)(sin

2

1)(

2

1)( 0

00d

d

dtititiLPF tt

tt

ttdeedeHth d

Casual filters (t = time)

t

dssxsthty )()()( (instesd of )

When Η(ω) = 0 :

When Η(ω) 0 :

0)( Y

Output y(t) depends only on past ( s t) values s of the input x(s)and not on future values (casuality)

dtieH )(

)(])([)( |)(||)(||)(|)()()( YdXXd eYeXeXeXHY tiiti

Page 129: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Bandwidth

Page 130: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Bandwidth

Low Pass Filter :

0BW

0 0

BW

LPF

Page 131: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Bandwidth

Low Pass Filter :

0BW

0 0

BW

LPF

Band Pass Filter (inside band) :

12 BW

2 1 1 2

BW

BPF

Page 132: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Bandwidth

Low Pass Filter :

0BW

0 0

BW

LPF

Band Pass Filter (inside band) :

12 BW

2 1 1 2

BW

BPF

Low Pass Filter not ideal :

0BW

|)0(||)(|2

10 HH

0 0

BW

|)0(|2

1 H |)0(| H

Page 133: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Bandwidth

Low Pass Filter :

0BW

0 0

BW

LPF

Band Pass Filter (inside band) :

12 BW

2 1 1 2

BW

BPF

Low Pass Filter not ideal :

0BW

|)0(||)(|2

10 HH

0 0

BW

|)0(|2

1 H |)0(| H

1 0 2

BW

|)(| 021 H|)(| 0H

12 BW

|)(||)(||)(| 021

21 HHH|)(|max|)(| 0 HH

Band Pass Filter (inside band) not ideal :

Page 134: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

END