Lecture 5: Signals – General Characteristics
description
Transcript of Lecture 5: Signals – General Characteristics
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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
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
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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)
c = transmission velocity = velocity of light in vacuum
k = constant, n(t) = noise
ρ = distance transmitter - receiver
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
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
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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)
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
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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
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
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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
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
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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)(
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
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
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
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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
angular frequency :T
f 2
2
(Hertz = cycles / second)
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
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
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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
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
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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)
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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
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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)
(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
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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics
Generalization: Initial epoch t0 0 :
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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics
ΔtΔt0
t0
Τ
t
Generalization: Initial epoch t0 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
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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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
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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
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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
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
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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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/35.jpg)
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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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/36.jpg)
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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
![Page 37: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/37.jpg)
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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics
t
epoch tx(t)
signal at transmitter
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
![Page 38: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/39.jpg)
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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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/40.jpg)
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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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/50.jpg)
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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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/html5/thumbnails/51.jpg)
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A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics
infrared
The electromagnetic spectrum
visible
105 102 3 102 104 106 (μm)
(μm)0.4 0.5 0.6 0.7
visi
ble
refle
cted
ther
mal
mic
row
aves RADIOultravioletΧ raysγ rays
λ
![Page 52: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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: Lecture 5: Signals – General Characteristics](https://reader035.fdocuments.in/reader035/viewer/2022070406/568141e1550346895dadbda9/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