Signal Strength Variation and Propagation Profiles of UHF ...
Signal Propagation
description
Transcript of Signal Propagation
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Signal Propagation Electro-Magnetic Signal Geometric Approximation ~ Fast Particle Approximation Speed of Light in Vacuum
m/s 299792458c
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1-Way Propagation
Linear Motion of Photon
Fast Motion + Non-Relativistic
000 ttt VXX
c0V
Source
Observer
t = t0
t = t1
photon
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Passive Observables
Arrival Time
Incoming Direction
Received Wavelength
1t
1d
1
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Equation of Light Time within Solar System Departure Time Arrival Time Light Time = Travel Time
Obtain Light Time
RV
S
O01 tt
0t1t
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Derivation of Eq. of Light Time Beginning/End of Photon Motion
2 1 2 1t t x x V Taking the norm
Assumption: Body Motions are known
21R V
tt OS xx ,
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Derivation (contd.)
V c
1 1 2 2
1 1
, ,
,O S
S
t t
R t
x x x x
R R x x
Velocity Expression (Newtonian)
Velocity Expression (Special Relativity)
1S tV c
R
v R
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Solving Eq. of Light Time
Newton Method 0 RVf
'*
fff
''''*
VRVVRRf
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Approximate Solution Initial Guess: Infinite c = Zero Solution First Newton Corrector
Further Correction: General Relativity
111111
1111
*1
,
, ,
0'000
tRV
tR
VcR
RVRf
SSSOSSSO
SSSSO
SO
SO
vvxxvvxxxx
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Light Direction
Aberration: Observer’s Velocity Parallax: Offset of Observer’s Position Periodic: Annual, Diurnal, Monthly, … Correction for Light Time: within Solar
System
RR
VVd
1
1
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Aberration Finiteness of Speed of Light Bradley (1727) Track of Raindrops on Car’s Side Window
c
VV
dvdvd
vdvd
vVvVd
11
1
1
11
11
1
1'
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Annual Aberration Order of Magnitude = Aberration Constant
Angle Expression
"2010km/s 103
km/s 30 45
cvE
sin'cvE
S
E0
’
E1
vE
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Annual Aberration (contd.) Adopting Ecliptic Coordinates Approximate Formula
Mean Longitude of Sun: L Aberration Ellipse
LL
A
A
coscossinsin
1sin
cos22
AA
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Diurnal Aberration Adopting Equatorial Coordinates Approximate Formula
Sidereal Rotation Angle: Geocentric Latitude:
coscos''cossinsincos''
A
A
"3.0106.1m/s103
m/s480' 68
cR EE
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Parallax Offset of Observer’s Position Bessel (1838): 81 Cyg Direction Difference between L&R Eyes
0
01010
010
010
10
10
r
rr
R
dxdxd
xdxd
xxxxRd
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Annual Parallax
Order of Magnitude = Parallax
Angle Expression
0
AU 1r
00 sin Sun E
S
0
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Annual Parallax (contd.) Ecliptic Coordinates Approximate Formula
90°Phase Shift from Aberration Parallactic Ellipse
00
00
sincoscossin
LL
1sin
cos2
0
20
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Diurnal (Geocentric) Parallax Very close objects only: Moon Adopting Equatorial Coordinates Approximate Formula
Geocentric Parallax
sincos''coscossincos''
51 104AU1
sin'
EE R
rR
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Doppler Shift Newtonian Approximation
Outgoing = Red shift Incoming = Blue shift
c
z dvv
10
0
01
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Approximate Doppler Shift Order of Magnitude = Aberration Constant Annual Doppler
Diurnal Doppler
Lz sincos
Θz sincoscos''
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Propagation Delay/Diffractions Vacuum (= Gravitational)
– Wavelength independent Non-Vacuum
– Eminent in Radio wavelength– Intrergalactic, Interstellar, Solar corona– Ionospheric, Tropospheric– Atmospheric
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Wavelength-Dependent Delay
Cancellation by 2 waves measurements– Geodetic VLBI: S-, X-bands– GPS: L1-, L2-bands– Artificial Satellites: Up- and Down-links
Empirical Model– Solar corona, Ionospheric, Tropospheric
2fC
fBAf
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Delay Models Solar Corona (Muhleman and Anderson 19
81)
Tropospheric (Chao 1970)
dsNcf e2CORONA
3.40 6rANe
045.0cot0014.0cos
ns7TROP
zz
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Atmospheric Refraction Variation of Zenith Distance
Saastamoinen (1972)
P: Pressure in hP, PW: Water Vapor Press. T: Temperature in K
zbzaz 3tantan
z
T
PPa W156.0271".16
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Multi-Way Propagation Variation of 1-Way Propagation Series of Light-Time Eq. Ex.: t3, t2, t1, t0
Transponder Delay– Optical: 0– Radio: Constant
Source
Observer
Transponder 1
Transponder 2
t0
t1
t2
t3
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Round Trip Propagation Typical Active Observation Emission/Arrival Times No Need of Target Motion Info Sum of 1-Way Propagations Cancellation of 1-st Order Effects
Observer
Target
t2
t0
t1
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Round Trip Light Time Approximate Mid-Time
Approximate Distance at Mid-Time
2 ,
202
2
120
1tt
cVOtttt
11
202 ,
2ttR
cV
RRttcR
OSSO
SO
SOSO
xx
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Simultaneous Propagation
t2
Almost Simultaneous Arrivals Summed Light Time Eq. Light Time of Mid-Point
Baseline Vector b Mid-Direction k
t1
t0
Observer 1
Observer 2
Source
b
k
212 tt
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Summed Light Time Eq. Approximate Equation
2
210 2
,
cVO
RRc
xxxR
R
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Simult. Propagation (contd.)
t2
Differenced Light Time Eq. Arrival Time Delay
Baseline Vector b Mid-Direction k t1
t0
Observer 1
Observer 2
Source
b
k
12 tt
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Eq. of Interferometric Obs.
1 2
c b k
b x x
Approximate Equation = Equation of VLBI Observation