When reading this post, also consider the comment made by Alex Pettit at the bottom of the post.
The link equation can be written as:
P_rx = P_tx + G_tx - L_tx - L_prop + G_rx - L_rx
where:
P_rx = Received power (dBm)
P_tx = Transmitter power (dBm)
G_tx = Transmitting antenna gain (dBi)
G_rx = Receiving antenna gain (dBi)
L_tx = Transmitter cable/hardware losses (dB)
L_rx = Receiver cable/hardware losses (dB)
L_prop = Propagation losses (dB)
In free space, the propagation loss is often replaced by the Free Space Path Loss (FSPL):
FSPL(dB) = 32.44 + 20 log10(d_km) + 20 log10(f_MHz)
where:
d_km = distance in kilometres
f_MHz = frequency in MHz
so the equation becomes:
P_rx = P_tx + G_tx + G_rx - L_tx - L_rx - FSPL
Astronomical applications
Radio astronomers use essentially the same link budget ideas, although the “transmitter” is usually a natural cosmic source.
Radio astronomy receiving equation
For an astronomical source:
P_rx = S A_eff B
where:
P_rx = received power (W)
S = source flux density (W m^-2 Hz^-1)
A_eff = effective collecting area (m^2)
B = receiver bandwidth (Hz)
Flux density is commonly measured in janskys:
1 Jy = 10^-26 W m^-2 Hz^-1
Effective area of a dish
The effective collecting area is:
A_eff = eta pi D^2 / 4
where:
eta = aperture efficiency
D = dish diameter (m)
For example, a 1.2 m dish with 60% efficiency has:
A_eff = 0.60 x pi x (1.2)^2 / 4
= 0.68 m^2
Antenna gain from dish diameter
Dish gain can be estimated from:
G = eta (pi D / lambda)^2
or in dBi:
G_dBi = 10 log10[ eta (pi D / lambda)^2 ]
where:
lambda = wavelength (m)
At the hydrogen line:
f = 1420.40575 MHz
lambda = 0.211 m
A 1.2 m dish with eta = 0.60 gives approximately:
G_dBi ~ 22.8 dBi
System noise temperature
Radio astronomers often replace losses and noise figures with temperature:
P_noise = k T_sys B
where:
k = 1.38 x 10^-23 J K^-1
T_sys = system noise temperature (K)
B = bandwidth (Hz)
Typical values are:
Professional observatories: 15-30 K
Good amateur systems: 40-100 K
Basic amateur systems: 100-300 K
Radiometer equation
The sensitivity of a radio telescope improves with integration time:
Delta_T = T_sys / sqrt(B tau)
where:
Delta_T = rms noise fluctuation (K)
tau = integration time (s)
B = bandwidth (Hz)
This equation explains why faint signals from the Milky Way, hydrogen line emission, Jupiter bursts, or masers become detectable by averaging for longer periods.
Signal-to-noise ratio
Combining these ideas gives:
SNR = P_signal / P_noise
or, for radio astronomy,
SNR = (S A_eff sqrt(B tau)) / (k T_sys)
Thus, improving an astronomical “link budget” means:
Increase dish diameter.
Increase aperture efficiency.
Reduce T_sys.
Use a wider bandwidth (if appropriate).
Integrate for longer.
Minimise cable and feed losses.
Observe when terrestrial interference is low.
Although developed for telecommunications, the link equation underpins everything from satellite communications to detecting the 21 cm hydrogen line, observing Jupiter’s radio bursts, and measuring weak cosmic signals with amateur radio telescopes.
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In considering the above, also review the comment below made by Alex Pettit:
The problem with this communication link budget equation, and its radio astronomy derivative above, is that they disregard Ground Noise Spillover which is a significant performance factor in radio astronomy signal reception.
I have never calculated the Tsys for any of my antennas as the value assumes ideal electrical and mechanical fabrication is employed.
Measurement is difficult. The classic “S7” Calibration Standard is too small a Reference Source for wide beam width antennas …
I have used Dec+40deg @ RA20:30Hrs for my comparisons and hardware optimization.
Their H-line signal level over Cold_Sky Noise performance is usually quite high.
They of course lack the angular resolution of larger aperture systems.
Regards,
Alex Pettit