6.7 Link Budget

Frances Zhu

6. Communications

6.7 Link Budget

_{An Introduction to Satellite Link Budget – Part 1. Video courtesy of YouTube.}

The link budget equation is the fundamental relationship driving the design of the communications architecture. It links together the requirements (data rate, BER) with the main design parameters of both the satellite communications subsystem and the ground segment. This section will describe the components of this equation and how to modify the link budget to close.

These design questions include:

How big should the antennas be?
How much power should we transmit?
How much noise can we accept on the receiver?
What radio frequency should we choose to transmit?

$\tfrac{E_b}{N_0} = \tfrac{P_t G_t G_r L_a L_l \lambda^2}{(4\pi R)^2 k T R_b} > \tfrac{E_b}{N_{0,min}}$

Note that this is a single equation linking 9 parameters, so there are multiple possible combinations of values that can satisfy the requirements:

$P_t$ = transmitter power (Watts)
$L_l$ = transmitter-to-antenna line loss (unitless, Watts/Watts)
$G_t$ = transmit antenna gain (unitless, Watts/Watts)
$L_s$ = space loss (unitless, Watts/Watts)
$L_a$ = transmission path loss (unitless, Watts/Watts)
$G_r$ = receiver antenna gain (unitless, Watts/Watts)
$k$ = Boltzmann’s constant (Joules/Kelvin)
$T_s$ = system noise temperature (Kelvin)

These values can be combined into intermediate expressions.

Power flux density at distance R: $W_t = \tfrac{P_t G_t}{(4\pi R)^2}$
Effect of atmosphere and circuit losses: $W_r = W_t L_a L_l$
Receiver power: $P_r = W_r \cdot A_{eff} = W_r \cdot \tfrac{\lambda^2}{4\pi} G_r$
Energy per bit: $E_b = \tfrac{P_r}{R_b}$
Noise: $N_0 = kT$

Combining all those expressions, we find the link budget equation: $\tfrac{E_b}{N_0} = \tfrac{P_t G_t G_r L_a L_l \lambda^2}{(4\pi R)^2 k T R_b} > \tfrac{E_b}{N_{0,min}}$

The link budget equation is often expressed in the log domain and using the following definitions:

Definition: Equivalent Isotropic Radiated Power: $EIRP \equiv P_t G_t$
Definition: Free space losses $L_s \equiv \tfrac{\lambda^2}{(4\pi R)^2}$
Definition: Gain to Noise temperature $\equiv \tfrac{G_r}{T}$

In this case, all parameters must be in dB $X(dB) ≡ 10 logX$

$\tfrac{E_b}{N_0} (dB) = EIRP (dBW) + L_S (dB) + L_a (dB) + L_l (dB) + \tfrac{G_r}{T} (\tfrac{dB}{K}) − 10 log k − 10 log R_b$

Antenna Gain

“In electromagnetics, an antenna’s power gain or simply gain is a key performance number which combines the antenna‘s directivity and electrical efficiency. In a transmitting antenna, the gain describes how well the antenna converts input power into radio waves headed in a specified direction. In a receiving antenna, the gain describes how well the antenna converts radio waves arriving from a specified direction into electrical power” [Wikipedia]. There are two antennas in the link budget: one in transmission and one in reception.

Directivity and maximum effective aperture diagram. Image by AJAL A.

Regardless, the general expression for antenna gain is:

$G=\eta \tfrac{4 \pi}{\lambda^2} A = \tfrac{\eta 4 \pi}{\lambda^2} D_t D_r$

Where

$\eta$ is the efficiency of an antenna, defined by the power going out of the antenna over the power going into the antenna ${\tfrac{P_o}{P_{in}}}$
$4 \pi$ are the number of steradians in a sphere, which is used for calculating mean radiation regardless of directivity
$\lambda$ is the wavelength
$A$ is the effective aperture area
$D$ is the directivity associated with the transmitter or receiver

The angular resolution achieved by an aperture is:

$\theta = \alpha \tfrac{\lambda}{D}$

where $\alpha = 1.22$ for circular aperture, $\alpha = 1$ for rectangular aperture.

Antenna gain pattern. Image by Electronics 360.

Gains for different shapes of antennas are:

Omnidirectional antenna: $G = 1$ (0 dB)
Parabolic antenna of aperture D:
- Typical values for $\eta =0.55-0.6$
Helical antenna: where L is the length of the antenna
- This gain is obtained when the radius R is equal to $\tfrac{R}{\lambda} = 0.2025 - 0.0079 \tfrac{L}{\lambda} + 0.000515(\tfrac{L}{\lambda})^2$

Various shapes of the patch antenna. Kiruthika, R., and T. Shanmuganantham. “Comparison of different shapes in microstrip patch antenna for X-band applications.” 2016 International Conference on Emerging Technological Trends (ICETT). IEEE, 2016.

Equivalent Isotropic Radiated Power

EIRP is a measurement showing performance at a specific point only (i.e, the measurement at a specific angle (Phi, Theta). Image by Share Tech Note.

Equivalent Isotropic Radiated Power (EIRP) is the primary design parameter on the transmitter side. EIRP is the product of transmitted power and the gain of the transmitting antenna. “Effective isotropic radiated power is the hypothetical power that would have to be radiated by an isotropic antenna to give the same (“equivalent”) signal strength as the actual source antenna in the direction of the antenna’s strongest beam” [Wikipedia]. This means that there is a trade-off between antenna size and transmitted power. We can compensate for a small antenna by transmitting more power and vice-versa, where you want to be in that trade-off depends on cost and other metrics and constraints (e.g. volume constraints). Formally:

$EIRP = P_T - L_C + G_a$

Where $P_T$ is the output power of the transmitter (dBm)

$L_C$ is the cable loss (dB)

Illustration of the definition of equivalent isotropically radiated power (EIRP). The axes have units of signal strength in decibels. R_a is the radiation pattern of a given transmitter driving a directional antenna. It radiates a far-field signal strength of S in its direction of maximum radiation (main lobe) along the z-axis. The green sphere R_iso is the radiation pattern of an ideal isotropic antenna that radiates the same maximum signal strength as the directive antenna does. The transmitter power that would have to be applied to the isotropic antenna to radiate this much power is the EIRP. Image by Chet Vorno.

Free Space Losses

“Free-space path loss is the attenuation of radio energy between the feed points of two antennas that results from the combination of the receiving antenna’s capture area plus the obstacle-free, line-of-sight path through free space (usually air)” [Wikipedia]. Formally:

$L_s \equiv \tfrac{\lambda^2}{(4\pi R)^2}$

Where $R$ is the distance between antennas. “The free-space path loss is the loss factor in this equation that is due to distance and wavelength, or in other words, the ratio of power transmitted to power received assuming the antennas are isotropic and have no directivity” [Wikipedia].

In free space, the intensity of electromagnetic radiation decreases with distance by the inverse square law, because the same amount of power spreads over an area proportional to the square of the distance from the source. CC BY-SA 3.0. Image by Borb.

“The free-space loss increases with the distance between the antennas and decreases with the wavelength of the radio waves due to these factors” [Wikipedia]

Intensity ( $I$ ) – the power density of the radio waves decreases with the square of the distance from the transmitting antenna due to the spreading of the electromagnetic energy in space according to the inverse square law
Antenna capture area ( $A_{eff}$ )– the amount of power the receiving antenna captures from the radiation field is proportional to a factor called the antenna aperture or antenna capture area, which increases with the square of the wavelength. Since this factor is not related to the radio wave path but comes from the receiving antenna, the term “free-space path loss” is a little misleading.

Frequency Selection

Within licensing constraints, the selected radio frequency affects the free space loss, bandwidth, size, antenna gain, cost, and complexity of electronics for you to consider:

The frequencies that are typically used for various space missions are:

S-Band — 2-3 GHz
- Space operation, Earth exploration, Space research
X-Band — 7-8 GHz
- Earth exploration, Space research
Ku-Band — 13-15 GHz
- Space research
- Loss from rain
Ka-Band — 23-28 GHz
- Inter-satellite, Earth exploration

Atmospheric absorption percentages throughout the electromagnetic spectrum. Image courtesy of NASA.

Noise

Noise is any signal that isn’t part of the information sent. Noise can come into the link budget from the original signal, from the system, and from the environment.

Link budgets usually start with the transmitter power and sum all the gains and losses in the system accounting for the propagation losses to find the received power. Then the noise level at the receiver is estimated so we can take the ratio of the signal power to the noise power and work out the performance of the link. Image by Mike Willis.

Signal Noise

Signal noise can come in the form of amplitude noise – an error in the magnitude of a signal and phase noise – an error in the frequency/phase modulation. The communications system receives this signal from the payload and various other subsystems so we will glaze over this.

System Noise

The communications system has noise in its components in the form of passive noise and active noise (amplifiers, mixers, etc…). All real components generate “thermal noise” due to the random motion of atoms. Passive devices’ thermal noise is directly related to the temperature of the device, its bandwidth, and the frequency of operation. Noise is generated by the thermal vibration of bound charges. A moving charge generates an electromagnetic signal. Passive components include resistive loads (power loads) and cables & other such things (like waveguides). The total noise on the receiver (T) has contributions from antenna and receiver:

$T_s = T_A + T_0(F − 1), T_0=290K$

If the line between antenna and receiver is lossy L < 1, it will also contribute noise:

$T_s = T_A + \tfrac{T_0(F − 1)}{L} + \tfrac{T_0 (1-L)}{L}$

where $T_A$ is the antenna noise temperature and it depends on the frequency and on where the antenna is pointing at and $F$ is the noise factor of the receiver. Usually, receivers can be made less noisy on the ground antenna. Noise temperature provides a way of determining how much thermal noise is generated in the receiving system. The physical noise temperature of a device, $T_n$ , results in a noise power of $P_n = K T_n B$ where:

$K$ = Boltzmann’s constant = $1.38 \times 10^{-23} J/K$ ; K in dBW = -228.6 dBW/K

$T_n$ = Noise temperature of source in Kelvin

$B$ = Bandwidth of power measurement device in hertz

Satellite communications systems work with weak signals, so reduce the noise in the receiver as far as possible. Generally, the receiver bandwidth is just large enough to pass the signal. Methods to keep the receiver temperature cool include liquid helium or other thermal solutions.

An example of system noise temperature. Image by Pravin Yalappa Kumbhar.

Environmental Noise

Noise from the space environment can affect the transmitted signal from the galaxy, sun, atmosphere, precipitation, and man-made sources.

“Cosmic noise originates from the stars present in outer space. Distant stars are also very high-temperature bodies and are also termed the sun. The noise generated from the star is similar to that generated by the sun. Cosmic noise is also known as black body noise. Not only the stars but the galaxies and other virtual point sources like quasars and pulsars in outer space produce cosmic noise” [ElectronicsDesk].

The Cosmic Microwave Background temperature fluctuations from the 7-year Wilkinson Microwave Anisotropy Probe data were seen over the full sky. The image is a mollweide projection of the temperature variations over the celestial sphere. The average temperature is 2.725 Kelvin degrees above absolute zero (absolute zero is equivalent to -273.15 ºC or -459 ºF), and the colors represent the tiny temperature fluctuations, as in a weather map. Red regions are warmer and blue regions are colder by about 0.0002 degrees. This map is the ILC (Internal Linear Combination) map, which attempts to subtract out noise from the galaxy and other sources. The technique is of uncertain reliability, especially on smaller scales, so other maps are typically used for detailed scientific analysis. Image courtesy of NASA.

“Solar noise is generated by the sun. As the Sun is a large body with extremely high temperature thus it emits or releases high electrical energy in noise form over a broad frequency range. However, the intensity of the produced noise signal changes timely. This is so because the temperature change of the sun follows 11 years of the life cycle. Hence large electrical disturbances occur after the period of every 11 years. While in other years the noise level is comparatively low” [ElectronicsDesk]. Solar phenomena, like solar flares and coronal mass ejections, can interrupt satellite communications by bursts of radiation that can damage or reset the satellite’s electronics.

NASA’s Solar Dynamics Observatory captured this image of a solar flare, as seen in the bright flash. A loop of solar material, a coronal mass ejection (CME), can also be seen rising up off the right limb of the Sun. Image credit: NASA/SDO/Goddard.

The atmosphere can affect satellite communications in various ways. Rain causes loss, particularly in the Ku band. Lightning creates electromagnetic interference that can also affect signals. Snow affects communications less than rain, due to the difference in density.

Clear atmosphere attenuation of electromagnetic radiation as a function of frequency. Indicated are also the dominant absorption molecules and the Planck law for a 300 K temperature. National Research Council. Assessment of millimeter-wave and terahertz technology for detection and identification of concealed explosives and weapons. National Academies Press, 2007.

Man-made noise comes in many, many forms. Humans transmit radio frequencies that can interfere with incoming spacecraft signals, thus the need for the Federal Communications Commission to regulate the number of waves knocking around in our atmosphere. Potential active threats to satellite communications include anti-satellite ballistic missiles and jamming, which is very rare [Takaya-Umehara].

Atmospheric noise as a function of frequency in the LF, MF, and HF radio spectrum according to CCIR 322. The vertical axis is in decibels above the thermal noise floor. It can be seen that as the frequency drops atmospheric noise dominates other sources. Image by RSGB.

Link Margin

As the communications specialist, you shouldn’t just make sure the link budget closes, you should design the link to have some positive margin with respect to $\tfrac{E_b}{N_{0, min}}$ (e.g. 3dB).

$Margin = (\tfrac{E_b}{N_{0}})_{received [dB]} - (\tfrac{E_b}{N_{0}})_{required [dB]}$

Where $(\tfrac{E_b}{N_{0}})_{required [dB]} = (\tfrac{E_b}{N_{0}})_{theoretical for BER} + \sum Other System Losses_{dB}$

In summary, this section described the various parameters that compose the link budget and discussed methods to either reduce losses or increase the gain. Thankfully, there are many tools available to calculate the exact link budget that keeps track of all of these moving parameters (last section of this chapter) but hopefully, you have general intuition as to how to improve the link budget.

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