aristotle university of thessaloniki – department of geodesy and surveying a. dermanissignals and...
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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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/1.jpg)
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
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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 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 :
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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 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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/4.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/5.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/6.jpg)
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
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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
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/8.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/9.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/10.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/11.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/12.jpg)
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
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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
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/14.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/15.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/16.jpg)
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
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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 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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/18.jpg)
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
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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 – 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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/20.jpg)
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
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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 – 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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/22.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/23.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/24.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/25.jpg)
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
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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
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
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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
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
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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
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
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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
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
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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
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
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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
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/32.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/33.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/34.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/35.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/36.jpg)
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
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
![Page 37: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/37.jpg)
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)
![Page 38: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/38.jpg)
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)
![Page 39: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/39.jpg)
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)
![Page 40: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/40.jpg)
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)
![Page 41: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/41.jpg)
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|)(|
![Page 42: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/42.jpg)
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
![Page 43: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/43.jpg)
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
![Page 44: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/44.jpg)
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
![Page 45: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/45.jpg)
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
![Page 46: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/46.jpg)
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
![Page 47: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/47.jpg)
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
![Page 48: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/48.jpg)
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
![Page 49: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/49.jpg)
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
![Page 50: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/50.jpg)
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)
![Page 51: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/51.jpg)
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
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/52.jpg)
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
![Page 53: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/53.jpg)
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]
![Page 54: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/54.jpg)
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 [–, +]
![Page 55: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/55.jpg)
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
![Page 56: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/56.jpg)
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
![Page 57: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/57.jpg)
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
![Page 58: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/58.jpg)
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
![Page 59: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/59.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/60.jpg)
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
![Page 61: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/61.jpg)
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
![Page 62: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/62.jpg)
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
![Page 63: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/63.jpg)
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
![Page 64: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/64.jpg)
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
![Page 65: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/65.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/66.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/67.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/68.jpg)
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
![Page 69: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/69.jpg)
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
)(tx )(tyLinput signal output signal
![Page 70: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/70.jpg)
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
![Page 71: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/71.jpg)
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
![Page 72: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/72.jpg)
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
![Page 73: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/73.jpg)
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
![Page 74: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/74.jpg)
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
![Page 75: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/75.jpg)
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
![Page 76: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/76.jpg)
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
![Page 77: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/77.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/78.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/79.jpg)
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
![Page 80: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/80.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/81.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/82.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/83.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/84.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/85.jpg)
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
![Page 86: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/86.jpg)
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
![Page 87: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/87.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/88.jpg)
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
![Page 89: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/89.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/90.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/91.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/92.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/93.jpg)
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
![Page 94: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/94.jpg)
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
![Page 95: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/95.jpg)
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
![Page 96: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/96.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/97.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/98.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/99.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/100.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/101.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/102.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/103.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/104.jpg)
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)
![Page 105: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/105.jpg)
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 106: Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/106.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/107.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/108.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/109.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/110.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/111.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/112.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/113.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/114.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/115.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/116.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/117.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/118.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/119.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/120.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/121.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/122.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/123.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/124.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/125.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/126.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/127.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/128.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/129.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/130.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/131.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/132.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/133.jpg)
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](https://reader030.vdocuments.site/reader030/viewer/2022032702/56649cbb5503460f94982e46/html5/thumbnails/134.jpg)
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