Satellite link design

The figure below shows the constellation of earth, moon and the two relay satellites in order to get a better understanding the distances between all the involved entities:

Identification of links

In total there are three links (Moon - Satellite1, Satellite1-Satellite2 and Satellite2 - Earth) that need to be considered.

Perfect alignment assumption

Actually the relay satellite no.1 needs to orbit around the L2 point as discussed here. This makes steering of the transmit dish on the moon and the receiver dish of satellite no.2 necessary. However for the following we assume that both satellites remain fixed at their respective locations and that the dishes are perfectly aligned.
For the reception of the signal on earth we make use of NASA's Deep Space Network (DSN) that provides us with three large receive antennas located in the US, Spain and Australia.

Atmospheric Effects

The link from Satellite2 to earth is the only one where we have to deal with atmospheric effects (such as rain attenuation) since the two others are space links.

Identification of free parameters

Since we have the freedom to choose operating frequencies, transmit power and satellite dish antenna sizes (the dish sizes on earth are given since we use the DSN) we encouter a lot of free parameters.
From basic considerations we know that increasing the carrier frequency gives us a benefit in antenna gain for a given dish size. However at the same time the path loss increases also.

Choice of modulation scheme

Before calculating the actual link budget we need to determine the type of modulation we want to use. [4] provides a comparison of different modulation schemes with turbo code error correction and discusses the required SNR for BER of 10^-5 and the spectral efficiency of the different modulation schemes in the following figure:

Since each modulation scheme requires a different bandwidth to transmit a certain bit rate we need to calculate the actual required SNR in order to be able to compare the different modulation schemes:
relative SNR = required SNR + relative increase in noise power (with respect to a reference modulation scheme) - reference SNR

We picked the following modulation schemes as examples:

We took 16-QAM with conv. turbo code at 3bits/s/Hz as a reference and obtain the values in the last column. These values can be interpreted as follows:
For QPSK with 1bit/s/Hz we would need 0.77dB more transmit power or antenna gain in order to compensate for the increased noise power due to increased transmission bandwidth. Likewise for 64-QAM with 4bit/s/Hz we would need 1.55dB additional transmit power or antenna gain. This shows that 16-QAM with 3bit/s/Hz is the optimal transmission scheme. Using 16-QAM with 3bit/s/Hz spectral efficiency means that we will use 14GHz bandwidth to transmit our 42Gbps signal and use a convolutional turbo code with rate 3/4.
Although the actual occupied bandwidth is larger (see here) the effective bandwidth (for SNR calculations) is equal to 14GHz.

Link budget calculation

Taking the required SNR E_b/N₀ = 6dB we can derive the required carrier-to-noise ratio C/N = E_b/N₀ +10*log₁₀(f_b / BW) where f_b = 4bit*14GHz and BW = 14GHz. This yields a required C/N = 12dB which we need to obtain at all receivers.
We created an Excel spreadsheet to calculate link budgets for all three links at different frequencies and account for rain attenuation (data taken from [1] for the DSN complex located in the US) for the Satellite2-earth link.
We used the following values in our link budget calculation:

Parameter	Value
Speed of light c	299,792,458 m/s
Boltzmann constant k	1.3806505E-23
Length of link1 l1	61,500,000 m
Length of link2 l2	439,683,750 m
Length of link3 l3	405,696,000 m
Diameter D of all comm. dishes (except DSN)	5m
Efficiency of all dish antennas	0.9
Diameter D of earth receiver dish (DSN)	34m
Modulation scheme	16-QAM with rate 3/4 turbo code
Effective Bandwidth	14 GHz
Carrier frequency	24.9GHz
System temperature Tsys for sat. receivers	30K
System temperature Tsys for earth receiver	16.2K
Transmit power PT for moon transmitter	3dBW = 2W
Transmit power PT for sat1 transmitter	20dBW = 100W
Transmit power PT for sat2 transmitter	5dBW = 3W
Rain att. (sat2-earth): Region	Region E (Mojave Desert)
Rain att. (sat2-earth): exceedence rate	0.01%
Rain att. (sat2-earth): Rain rate	22mm/h
Rain att. (sat2-earth): effective rain-length	2,000m

From our link budget calculations (including rain attenuation (0.01% exceedence) on the link from Sat.2 to Earth) we get the following results:

Link	Transmit power PT	Received power PR	SNR
Moon-Sat1	3dBW	-99.5..-94dBW	12.8..18.3dB
Sat1-Sat2	20dBW	-99.6..-94.1dBW	12.7..18.2dB
Sat2-Earth	5dBW	-102..-101.2dBW	13.1..14dB

From these results we can make the following conclusions:

• Since we are using a very large bandwidth, attenuation and gains differ and hence we encouter different receive SNRs for different parts of the RF spectrum. In a practical implementation we would need to use an equalizer to compensate for this effect. For simplicity we will omit this device in the discussion of the communication systems.
• The link from Sat1-Sat2 is the most "difficult" one since it has the lowest received SNR (due to longest distance). Our approach works includes a design reserve of 0.7dB.
• Transmit powers have been minimized to just yield enough receive SNR
• Because attenuation and gains differ for different parts of the RF spectrum we could split our data signal into sub-bands and send them sperately to achieve a higher data rate (since we can send at a higher data rate if we have enough receive SNR for a particular subband) For example we could use OFDM to achieve this.

Up to now we have not taken into account the additional noise due to amplifier noise. The broadband low noise amplifier we want to use has a noise figure of 4.5dB which is pretty high. This leads to a decrease of receive SNRs which would mean that we either need to increase the transmit power or user larger dish antennas if we use this amplifier.