# Satellite Transmission Path

The calculation of noise is more complicated than commonly written. Environmental effects like rain cause not only an attenuation, but also increase the noise temperature, that has a great influence on the final result. Here we shall recapitulate some basics in noise theory and associate it with transmission.

## Parabolic Antenna system

Physical temperatures are named θ and noise temperatures T. The values with index cs describe clear sky. The nominal values (index nom) will be calculated for a specific "nominal antenna noise temperature". These values are given by the component manufacturers and allow qualitative comparison between different stations. Unfortunately there is no standard for it. Satlinker calculates with vacuum, some manufacturers use clear sky condition to describe the system. Some calculate the data in relation to an elevation angle, others use only zenith elevation. These nominal values are not used for link-budget calculation because no operational margins are included such as antenna pointing or rain attenuation. See tooltip help for representative data.

a_{atm} |
dB | Atmospheric absorption caused by humidity and oxygen in the atmosphere. a_{atm} is about 0.2dB at 15GHz, depends on frequency and elevation. |

a_{r,h2o} |
dB/km | Attenuation rate for water vapour, has nothing to do with fog or rain |

a_{r,o2} |
dB/km | Attenuation rate for oxygen |

a_{rain} |
dB | Rain attenuation due to absorption caused by rain drops |

η_{rad,rx} |
dB | Radiation efficiency, attenuation of waveguide components in receiving direction (feed horns, polarizers...) |

B_{n} |
Hz | Noise bandwidth |

d | m | Parabolic antenna diameter |

f | GHz | Frequency, here always in GHz! |

F | 1 | Noise factor |

F_{[dB]} |
dB | Noise figure |

k | W/(K×Hz) | Boltzmann constant 1.38×10^{-23} Joule/Kelvin |

N | W | Noise power in Watt |

φ_{el} |
deg | Elevation angle |

φ_{la} |
deg | Latitude, north+, south- |

ρ | g/m^{3} |
The water vapour density at the surface of location of interest |

R | mm/h | Point rain rate |

s_{0} |
km | Mean 0°C isotherm height, measured from mean sea level to sky |

s_{al} |
km | Altitude above mean sea level of location of interest |

s_{el} |
km | Slant path lenght along the elevation angle from location to s_{0} |

θ | K | Physical temperature in Kelvin, at the location of interest |

θ_{0} |
K | Physical reference temperature 290K by definition in USA and Europe, 293K in Japan, used by component manufacturers |

θ_{atm} |
K | Atmospheric effective or mean radiating temperature |

T | K | Effective noise temperature |

T_{ant} |
K | Antenna noise temperature |

T_{ant,nom} |
K | Antenna noise temperature used by component manufacturers. Might be 40K or another noise temperature, see description |

T_{atm} |
K | Atmospheric noise temperature |

T_{cos} |
K | Cosmic background noise temperature |

T_{gnd} |
K | Ground noise temperature, depends on dish elevation |

T_{lnb} |
K | LNB noise temperature |

## Gain, Attenuation, Transmissivity, Loss

These definitions are important for stated below noise temperature calculations.

Gain: G = P_{out}/P_{in} for P_{out} > P_{in} |
Gain in dB: g = 10 × log G |

Efficiency: η = P_{out}/P_{in} for P_{out} < P_{in} |
Attenuation in dB: a = 10 × log (1/η) |

Fractional transmissivity (Power dissipation) of serial connected resistors R_{1}..R_{i} is defined as power dissipated in R_{i} to total power:

η_{i} = P_{i} / P_{tot} = P_{i} / Σ_{[i=1..i]} (P_{i}) = R_{i} / Σ_{[i=1..i]} (R_{i}) = **10 ^{-ai/10}**

A loss has a unit e.g. [dBWatt], compared to gain or attenuation that have unit [1]:

Power Loss: P_{loss} = P_{in} - P_{out} = P_{in} - P_{in} × η = P_{in} × (1 - η) = **P _{in} × (1 -10^{-a/10})**

### Atmospheric attenuation a_{atm}

The diagram shows the zenith atmospheric attenuation a_{atm,z} (φ_{el} = 90°).

For other elevations: a_{atm} ≈ a_{atm,z} / sin(φ_{el}) , (φ_{el} = 10°...90°)

Simplified atmospheric attenuation model:

Main attenuations of the atmosphere are from water vapour and oxygen for f<57GHz:

Temperature correction factor: x = 1 - 0.01 × (θ - 273.15 - 15)

a_{r,o2} = x × [7.19 × 10^{-3} + 6.09 / (f^{2} + 0.227) + 4.3 / ((f - 57)^{2} + 1.5)] × f^{2} × 10^{-3}

a_{r,h2o} = x×[0.0067 + 3/((f-22.3)^{2}+7.3) + 9/((f-183.3)^{2}+6) + 4.3/((f-323.8)^{2}+10)]×f^{2}×ρ×10^{-4}

The equivalent atmosphere height for oxygen (f < 57GHz) in km is: s_{o2} = 6km

The equivalent atmosphere height for water vapour (f < 350GHz) in km is:

s_{h2o} = 2.2 + 3 / ((f - 22.3)^{2} + 3) + 1 / ((f - 183.3)^{2} + 1) + 1 / ((f - 323.8)^{2} + 1)

The total slant path atmospheric attenuation for φ_{el} > 10°:

a_{atm} = [a_{r,o2} × s_{o2} × e^{(-sal/so2)} + a_{r,h2o} × s_{h2o}] / sin(φ_{el})

### Rain attenuation a_{rain}

The rain attenuation depends on frequency, latitude, elevation and polarization. Here we use the "Simple Attenuation Model" (SAM) from Stutzman and Yen:

Latitude correction factor: s_{la} = 4.8 for |φ_{la}| < 30° and s_{la} = 7.8 - 0.1 × |φ_{la}| for |φ_{la}| > 30°

s_{0} with rain correction: s_{0} = s_{la} for R < 10mm/h and s_{0} = s_{la} + log(R / 10) for R > 10mm/h

The slant path length: s_{el} = (s_{0} - s_{al}) / sin(φ_{el}

Rain attenuation for R < 10mm/h: a_{rain} = x × R^{y} × s_{el}

Rain attenuation for R > 10mm/h:

a_{rain} = x × R^{y} × (1 - exp(-s_{el} × y × z × ln(R / 10) × cos(φ_{el}))) / (y × z × ln(R / 10) × cos(φ_{el}))

x = 4.21 × 10^{-5} × f^{2.49} (for 2.9GHz < f < 54GHz)

x = 4.09 × 10^{-2} × f^{0.699} (for 54GHz < f < 180GHz)

y = 1.41 × f^{-0.0779} (for 8.54GHz < f < 25GHz)

y = 2.63 × f^{-0.272} (for 25GHz < f < 164GHz)

z = 1/14 this is an empirical determined value

For determining a rain rate R, pick a climate zone A to P for the transmitting or receiving location from the CCIR rain climate regions map, then look up the rain rate from table:

%/year |
A |
B |
C |
D |
E |
F |
G |
H |
J |
K |
L |
M |
N |
P |

1.0 |
- | 1 | - | 3 | 1 | 2 | - | - | - | 2 | - | 4 | 5 | 12 |

0.3 |
1 | 2 | 3 | 5 | 3 | 4 | 7 | 4 | 13 | 6 | 7 | 11 | 15 | 34 |

0.1 |
2 | 3 | 5 | 8 | 6 | 8 | 12 | 10 | 20 | 12 | 15 | 22 | 35 | 65 |

0.03 |
5 | 6 | 9 | 13 | 12 | 15 | 20 | 18 | 28 | 23 | 33 | 40 | 65 | 105 |

0.01 |
8 | 12 | 15 | 19 | 22 | 28 | 30 | 32 | 35 | 42 | 60 | 63 | 95 | 145 |

0.003 |
14 | 21 | 26 | 29 | 41 | 54 | 45 | 55 | 45 | 70 | 105 | 95 | 140 | 200 |

0.001 |
22 | 32 | 42 | 42 | 70 | 78 | 65 | 83 | 65 | 100 | 150 | 120 | 180 | 250 |

### Other attenuations

Following attenuations are neglected in the calculations:

**Precipitation:**caused by fog, clouds, sand and dust storms.**Scintillation:**A change in the direction of propagation of a radio wave, caused by refractive index changes in the transmission path. These refractive index variations are the result of temperature, humidity and pressure irregularities called atmospheric turbulence.**Multipath:**is when waves arrive simultaneously at a receiving antenna via several propagation paths and, by interfering with each other, give rise to fading, also called frequency-selective fading. This effect is more significant for decreasing elevation angles due to increase of portion of the propagation path and earth proximity.**Wave front incoherence:**A decrease in effective antenna gain, due to phase decorrelation across the antenna aperture, caused by irregularities in the refractive-index structure. These variations are more significant with large aperture antennas.- The total attenuation is not only a summation, see ITU-R P.618-8: a
_{tot}= a_{atm}+ SQR((a_{rain}+ a_{cloud})^{2}+ a_{scintillation2})

## Noise

Noise has a flat frequency spectrum so the noise power P_{n} per unit bandwidth B_{n} is constant.

**Noise Power Spectral Density:** N_{0} = P_{n} / B_{n} = k × T

**Noise Power:** N = k × T × B_{n} , with B_{n} ≈ 1.12 × B_{3dB} (higher than 3dB bandwidth)

The noise factor indicates the decrease in signal-to-noise ratio (SNR) of a noisy signal (S_{i} plus N_{i}), between input and output of a noisy amplifier, when noise source is θ_{0}. The noisy amplifier is replaced by an ideal amplifier with gain G and an equivalent **Noise Source N _{a,i}** at its input:

**Noise Factor:**F = SNR

_{i}/ SNR

_{o}= (S

_{i}/ N

_{i}) / (G × S

_{i}/ (G × (N

_{i}+ N

_{a,i})) = 1 + N

_{a,i}/ N

_{i}

**Noise Figure:**F

_{[dB]}= 10 × log F

When dealing with satellite sources, the source temperature will generally be much lower than θ_{0}, so the noise figure becomes misleading. In this case, the effective noise temperature of a noise source N_{a,i} is more interesting, because it is independent of any reference.

**Effective Noise Temperature:** T_{a,i} = (F - 1) × θ_{0}

Because of the simple relation between the noise and the resistor temperature, it makes sense to define an effective noise temperature for other noise sources too, even if they are not thermal in origin, e.g. interference from another station. A noise source having a noise temperature T generates a noise power, equal to the thermal noise that would be generated by a resistor at temperature θ. For a perfect absorbing object or so-called blackbody the emanating noise is proportional to the objects' temperature.

Addition of noise temperatures of serial resistors R_{1}..R_{i} depends on fractional transmissivity:

T = Σ_{[i=1..i]} (T_{i} × η_{i}) = Σ_{[i=1..i]} (T_{i} × 10^{-ai/10})

### Antenna noise temperature T_{ant}

**Antenna Noise** consists of ground-, atmospherical-, rain- and cosmic backgound noise. Neglected is galactic noise, that dominates background noise for frequencies below 4GHz and solar noise when tracking near the sun. Note that the increase in noise temperature due to precipitation affects only the antenna temperature of earth station antennas. The satellites' antenna always looks at the hot earth of about θ_{0}.

**Ground noise** increases if the feed over-illuminates the dish and catches backscatter from the warm ground and from reflections e.g. of the feed assembly.

Approximation formula for ground noise temperature: T_{gnd} ≈ 15 + (30/d) + (180/φ_{el}) , [2]

**Cosmic noise:** T_{cos} ≈ 1.7K at 1.7...2.3GHz, 2.5K at 8.4GHz, 2.0K at 32GHz

In general the atmospheric effective temperature θ_{atm} depends upon frequency and attenuation through the physical processes that produce the attenuation. For attenuation values less than 6dB and frequencies below 50GHz it can be approximated, with a weak dependence on frequency (265K in the 10-15GHz range and 270K in the 30GHz range). When surface temperature is known, a first estimation can be also obtained by multiplying the surface temperature by 0.95 in the 20GHz band and by 0.94 for the 30GHz window:

θ_{atm} ≈ 0.95 × θ , [3]

An attenuating atmosphere creates an **Atmospheric Noise Temperature:** T_{atm} = θ_{atm} × (1 - η_{atm}) ≈ 40K

Here we define the **Nominal Antenna Noise Temperature** without ground and cosmic noise, only with an attenuating atmosphere. Manufacturers may use other definitions.

T_{ant,nom} = T_{atm}

T_{ant,cs} = T_{gnd} + θ_{atm} × (1 - η_{atm}) + T_{cos} × η_{atm}

T_{ant} = T_{gnd} + θ_{atm} × (1 - η_{atm} × η_{rain}) + T_{cos} × η_{atm} × η_{rain}

### System noise temperature T_{sys}

consists of LNB-, feed- and antenna noise temperature. The influence of the IRD on the system noise can be neglected, cause noise of first amplifier (LNB) prevails if it has a great amplification.

Radiation efficiency η_{rad,rx} in receiving direction includes all attenuating components of the feed system. It decreases the **Antenna Noise Temperature** ( T_{ant} × η_{rad,rx}) but, unfortunately much more important, it adds a **Feed Noise Temperature** θ × (1 - η_{rad,rx}). Therefor it's very important to keep the feed attenuation low in the receiving path by using short waveguides to the LNB.

**LNB noise temperature** is calculated with θ_{0} by definition: T_{lnb} = θ_{0} × (10^{Flnb[dB]/10} - 1)

T_{sys,nom} = T_{ant,nom} × η_{rad,rx} + θ_{0} × (1 - η_{rad,rx}) + T_{lnb}

T_{sys,cs} = T_{ant,cs} × η_{rad,rx} + θ × (1 - η_{rad,rx}) + T_{lnb}

T_{sys} = T_{ant} × η_{rad,rx} + θ × (1 - η_{rad,rx}) + T_{lnb}

**Downlink degradation** DND caused by rain: DND = a_{rain} + 10 × log(T_{sys} / T_{sys,cs})

### Literature

- Bisante Consortium: GEO Satellite Network Characteristics
- http://photos.imageevent.com/qdf_files/technicalgoodies/training/GT.doc.pdf
- ITU-R P.1322, Radiometric estimation of atmospheric attenuation