lecture 5: signals – general characteristics
DESCRIPTION
Signals and Spectral Methods in Geoinformatics. Lecture 5: Signals – General Characteristics. Signal transmission and processing. transmission t - τ. ρ = c τ. t - τ. t. reception t. Τ. Δ t 0. Δ t. n Τ. τ. Observation :. τ = n Τ + Δ t – Δ t 0. ΔΦ = ρ – n λ. - PowerPoint PPT PresentationTRANSCRIPT
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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 :
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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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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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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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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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Monochromatic signal = periodic signal with sinusoidal from :T
tatx
2sin)(
Monochromatic (sinusoidal) signals
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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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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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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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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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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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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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Signal phase at an instant t :
Signal phase
)(tx
t
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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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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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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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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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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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Generalization: Initial epoch t0 0 :
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ΔtΔt0
t0
Τ
t
Generalization: Initial epoch t0 0 :
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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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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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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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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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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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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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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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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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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 )
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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 )
Usual notation : Θ Φ, θ φ
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receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
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t
epoch tx(t)
signal at transmitter
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
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t
epoch tx(t)
signal at transmitter
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
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signal at receiver
y(t) = x(tcρ)
t
epoch t
t
epoch tx(t)
signal at transmitter
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
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signal at receiver
y(t) = x(tcρ)
t
epoch t
t
epoch tx(t)
signal at transmitter
receiver r = ρtransmitter r = 0 Epoch t - Signal traveling in space y(t,r) = x(tcr)
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Energy signals
Energy :
dttxE 2|)(|
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Energy :
Correlation function of two signals x(t) and y(t) :
dttxE 2|)(|
dttytxRxy )()()(
dttytx )()(
Energy signals
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Energy :
Correlation function of two signals x(t) and y(t) :
(Auto)correlation function of a signal :
dttxE 2|)(|
dttytxRxy )()()(
dttytx )()(
dttxtxRxx )()()(
Energy signals
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Energy :
Correlation function of two signals x(t) and y(t) :
(Auto)correlation function of a signal :
Properties
dttxE 2|)(|
dttytxRxy )()()(
dttytx )()(
dttxtxRxx )()()(
)()( yxxy RR )()( xxxx RR ERxx )0(
Energy signals
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Energy :
Correlation function of two signals x(t) and y(t) :
(Auto)correlation function of a signal :
Properties
Applications: GPS, VLBI !
dttxE 2|)(|
dttytxRxy )()()(
dttytx )()(
dttxtxRxx )()()(
)()( yxxy RR )()( xxxx RR ERxx )0(
)(max)0(
xxxx RR
Energy signals
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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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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]
Aristotle University of ThessalonikiAristotle University of 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 [–, +]
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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)()(
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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 :
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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)
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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 )(
Aristotle University of ThessalonikiAristotle University of 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 )(
Aristotle University of ThessalonikiAristotle University of 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 )(
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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
Aristotle University of ThessalonikiAristotle University of 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 :
Aristotle University of ThessalonikiAristotle University of 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