Link Budgets & the Range Equation

A link budget is one accounting sheet that starts at the transmitter's power amplifier and ends at a single number: how many dB of margin the demodulator has left over the ratio it needs to decode. Everything between — antenna gain, spreading loss, atmosphere, feed losses, the receiver's own thermal noise — is a multiplicative factor on power, which is why the whole calculation is done in decibels and reduces to addition. The result decides whether the link exists at all, and it is the same calculation for a satellite transponder, a microwave hop, and a phone talking to a cell tower.

The interesting part is not the arithmetic. It is that a budget which closes with 8 dB of margin on a spreadsheet can be an outage-prone link in the field, because the terms that eat the margin — rain, interference, pointing error, the fact that the demodulator is not the idealised one whose curve you quoted — are exactly the ones that do not appear as a line item unless you put them there.

Why the whole thing is in dB #

Received power in free space is the Friis relation: a product of transmit power, transmit gain, the inverse-square spreading factor, and receive gain. Chained through a real system it becomes a product of a dozen factors spanning twenty orders of magnitude. Take $10\log_{10}$ of the whole thing and every multiplication becomes an addition, every division a subtraction, and the dynamic range collapses into numbers a person can add in their head.

The master equation, in the form worth memorising:

$$\underbrace{\frac{C}{N_0}}_{\text{dBHz}} \;=\; \underbrace{P_T + G_T}_{\text{EIRP (dBW)}} \;-\; \underbrace{L_{\text{fs}} - L_{\text{atm}} - L_{\text{pt}} - L_{\text{pol}}}_{\text{path losses (dB)}} \;+\; \underbrace{\left(\frac{G}{T}\right)_{\!R}}_{\text{dB/K}} \;-\; \underbrace{10\log_{10} k}_{-228.6\ \text{dBW/(K}\cdot\text{Hz)}}$$

$C/N_0$ is carrier power over noise power spectral density, in dB-Hz. It is bandwidth-free, which is what makes it the right intermediate quantity: the propagation physics does not know your symbol rate, and the modem does not care about your antenna. They meet at $C/N_0$.

The constant is the Boltzmann constant, which since the 2019 SI redefinition is exact by definition: $k = 1.380649 \times 10^{-23}$ J/K, so $10\log_{10}k = -228.599$ dBW/(K·Hz). The other constant everyone quotes, $-174$ dBm/Hz, is the same thing evaluated at the reference temperature $T_0 = 290$ K: $kT_0 = 4.0039\times10^{-21}$ W/Hz, so $10\log_{10}(kT_0) + 30 = -173.98$ dBm/Hz. The 290 K itself is a convention inherited from the classical noise-figure definition rather than anything physical — no primary source for it is cited here.

flowchart LR
  PA["PA output Pt dBW"] --> FL["feed loss"]
  FL --> GT["Tx antenna gain"]
  GT --> EIRP["EIRP dBW"]
  EIRP --> FS["free-space loss"]
  FS --> ATM["atmosphere and rain"]
  ATM --> PT["pointing and polarization"]
  PT --> GR["Rx antenna gain"]
  GR --> TS["system noise temperature"]
  TS --> CN0["C/N0 dBHz"]
  CN0 --> ESN0["Es/N0 = C/N0 - 10log Rs"]
  ESN0 --> MAR["margin over required Es/N0"]

Free-space loss is a bookkeeping fiction that happens to be exact #

The spreading term is the one everyone calls "path loss". ITU-R P.525-4 states it two ways — equation (3) as $20\log_{10}(4\pi d/\lambda)$, and equation (4) in practical units:

$$L_{bf} = 32.4 + 20\log_{10} f_{\text{MHz}} + 20\log_{10} d_{\text{km}} \quad \text{dB}$$

The constant is rounded; carrying the speed of light through gives 32.4478. P.525-5 (2024) restates the same equation unchanged. The often-quoted 92.45 constant is the same expression shifted to GHz and km and does not appear in P.525 in that form. The frequency dependence is not a property of space — space does not attenuate — but of the isotropic reference antenna at the receiver, whose effective aperture is $\lambda^2/4\pi$ and therefore shrinks with frequency. Hold both antennas' physical apertures fixed and going up in frequency actually improves the link. This is why Ka-band exists.

All from P.525-4 equation (4); bar length is proportional to the dB value, i.e. logarithmic in the power ratio. Fourteen orders of magnitude between the first and last row is the reason nobody works this in watts.

G/T: why gain and noise temperature are one number #

The receive side of the budget never needs antenna gain and noise temperature separately — they always appear as a ratio, because $C/N$ is (gain × signal) over (Boltzmann × temperature × bandwidth). So the receiving station is characterised by a single figure of merit:

$T_{\text{sys}}$ is not the physical temperature of anything. It is the temperature a resistor would need for its Johnson noise to equal all the noise the system actually produces: sky brightness collected by the antenna, ground spillover into the sidelobes, the ohmic loss of everything ahead of the first amplifier, and the amplifier's own noise. For a lossy element at physical temperature $T_p$ with loss factor $L$ ahead of a receiver of noise temperature $T_{\text{rx}}$:

$$T_{\text{sys}} = T_{\text{ant}} + (L-1)\,T_p + L\,T_{\text{rx}}, \qquad T_{\text{rx}} = T_0\left(10^{NF/10} - 1\right)$$

Work it for the receive earth stations in the ETSI DSNG example — 65% aperture efficiency, 0.3 dB coupling loss, 0.5 dB pointing loss, 1.2 dB noise figure, at 11.75 GHz:

$T_{\text{sys}} = 30 + (1.0715-1)(290) + 1.0715 \times 92.3 = 149.6$ K, taking a clear-sky antenna temperature of 30 K (sky brightness plus ground spillover). The 1.2 dB noise figure alone contributes 92 K — more than half the total. Reproducing all three of the standard's stated figures to within a decibel is the useful check that the decomposition is right; the residual is the standard rounding down to whole dB.

From C/N0 to a decision: Es/N0, Eb/N0, and what the modem actually demands #

$C/N_0$ becomes a decodability statement only after you divide by a rate:

$$\frac{E_s}{N_0} = \frac{C}{N_0} - 10\log_{10} R_s, \qquad \frac{E_b}{N_0} = \frac{E_s}{N_0} - 10\log_{10}\eta$$

where $R_s$ is the symbol rate and $\eta$ the spectral efficiency in information bits per symbol — modulation order times code rate, minus framing overhead. DVB-S2 states its requirement as $E_s/N_0$ and notes the conversion in exactly this form.

The required $E_s/N_0$ is a property of the modulation and the code, and it is tabulated. EN 302 307-1 Table 13 gives it for the AWGN channel with 50 LDPC decoding iterations, perfect carrier and symbol synchronisation, no phase noise, at quasi-error-free (PER < 10⁻⁷ before the demultiplexer):

η and ideal $E_s/N_0$ are quoted from EN 302 307-1 Table 13 (normal FECFRAME, no pilots); $E_b/N_0$ and the Shannon column $10\log_{10}(2^\eta - 1)$ are computed from them. A modern LDPC-coded waveform sits 1.5–2 dB from capacity over most of its range, widening to under 3 dB at the densest constellation — which is why "just use a better code" is no longer an available answer when a budget refuses to close.

Worked budget 1: Ku-band GEO, uplink and downlink #

The scenario is ETSI's own DSNG example (TR 102 376-1 clause 7.2.5): a 36 MHz transponder at 0°E carrying four equal 7.2 Mbaud carriers in 9 MHz slots, DVB-S2 QPSK 3/4, transponder driven to 8 dB total input back-off / 3.4 dB total output back-off. Transmit terminal is a 1.2 m DSNG flyaway with a 250 W TWT (63 dBW EIRP at 3 dB OBO); receive station is 4.5 m, G/T 30 dB/K. Satellite figures are the beam-edge values: saturated EIRP 42 dBW, G/T −0.5 dB/K, uplink 14.25 GHz, downlink 11.75 GHz, atmospheric absorption 0.3 dB up and 0.2 dB down.

Geometry is the one thing the standard leaves to you. Put the earth station at Madrid, 40.42°N 3.70°W, looking at 0°E: central angle 40.56°, elevation 43.1°, slant range 37 548 km.

The two hops are in series, and noise adds in linear power, not in dB. That is the one place where the dB trick fails and you must convert back:

$$\left(\frac{C}{N_0}\right)^{-1}_{\text{tot}} = \left(\frac{C}{N_0}\right)^{-1}_{\text{up}} + \left(\frac{C}{N_0}\right)^{-1}_{\text{down}} + \left(\frac{C}{I_0}\right)^{-1}_{\text{IM}}$$

Combining the two thermal terms gives $C/N_0 = 81.65$ dB-Hz and $E_s/N_0 = 13.08$ dB — 2.18 dB worse than the weaker hop on its own. That penalty is structural: two equal hops cost 3 dB, and a hop 10 dB stronger than the other costs 0.4 dB. It is the reason the ETSI note about the feeder link in a broadband system reads "dimensioned in order not to contribute significantly to the overall link budget": you buy a big gateway antenna specifically so the uplink term drops out of that sum.

Now subtract what the demodulator wants:

Three defensible numbers for "the margin", 2.9 dB apart, differing only in which real-world terms you admitted. Across the terminal/station combinations the standard lists, against the practical 4.9 dB requirement:

Read the rows and columns against each other: going from a 4.5 m to an 8.1 m receive station buys only 1.4 dB with the 1.2 m terminal, because at that point the uplink is the weak hop and the combination equation refuses to reward you for improving the strong one. That saturation is the single most useful thing the two-hop equation tells you, and it is invisible if you budget each direction separately.

Worked budget 2: a 5G NR uplink, where nothing is shaped like a satellite budget #

The terrestrial version of the same calculation looks different for three reasons: the receiver is specified by a conformance floor rather than a G/T, propagation is a fitted statistical model rather than $1/d^2$, and the dominant impairment is other cells rather than thermal noise.

Take the ITU-R M.2412-0 Dense Urban-eMBB Configuration A parameter set: 4 GHz carrier, base station at 25 m, UE at 1.5 m, BS noise figure 5 dB, BS antenna element gain 8 dBi, UE antenna element gain 0 dBi, thermal noise −174 dBm/Hz, inter-site distance 200 m. UE transmit power is 23 dBm — power class 3, the default in 3GPP TS 38.101-1.

The receiver floor. 3GPP does not publish a base-station G/T; it publishes a reference sensitivity that a conforming radio must meet. TS 38.104 Table 7.2.2-1 gives a Wide Area BS with 30 kHz subcarrier spacing $P_{\text{REFSENS}} = -95.6$ dBm, measured on reference channel G-FR1-A1-5: 51 resource blocks of QPSK at code rate 1/3, throughput ≥ 95% of the channel's maximum. 51 RB × 12 subcarriers × 30 kHz = 18.36 MHz, so the thermal floor is $-174 + 10\log_{10}(18.36\times10^6) = -101.36$ dBm.

Three unrelated bandwidths, the same 5.8 dB of excess — which is the whole receiver budget folded into one specification number: noise figure plus the SNR a QPSK-1/3 channel needs plus implementation margin. With the 5 dB noise figure M.2412 assumes for a macro base station, that leaves about 0.8 dB of required SNR. The satellite budget spends four lines getting to the same place.

The budget. Receive antenna gain is the one term neither standard fixes; take a 16-element panel of the 8 dBi elements M.2412 specifies, combining coherently for $8 + 10\log_{10}16 = 20.0$ dBi, and 0.5 dB of jumper loss — a conventional macro sector, and a design input rather than a cited value.

Invert the UMa NLOS model from TR 38.901 Table 7.4.1-1, $\text{PL} = 13.54 + 39.08\log_{10}d_{3D} + 20\log_{10}f_c - 0.6(h_{UT}-1.5)$ with $f_c$ in GHz and $d_{3D}$ in metres:

Against M.2412's 200 m inter-site distance — a cell radius of roughly 115 m — the uplink closes with more than 20 dB to spare. The link budget is not what sizes a dense urban network. Capacity and inter-cell interference are; the budget only becomes binding indoors and at the rural end.

Add TR 38.901's low-loss O2I model at 4 GHz: $\text{PL}_{tw} = 5 - 10\log_{10}(0.3 \cdot 10^{-L_{\text{glass}}/10} + 0.7 \cdot 10^{-L_{\text{concrete}}/10})$ with $L_{\text{glass}} = 2 + 0.2f$ and $L_{\text{concrete}} = 5 + 4f$ gives 12.9 dB through the wall, plus $0.5\,d_{2D\text{-in}}$ inside — 6.25 dB at a 12.5 m depth. That 19.1 dB takes the 95% range from 424 m to 137 m, and the whole comfortable margin is gone. Substituting the high-loss model (IRR-coated glass, $L_{\text{IRR}} = 25.4 + 0.11f$) makes it worse still. Indoor coverage, not outdoor range, is what a 4 GHz macro budget is really about.

Run the same arithmetic rurally — 700 MHz, RMa NLOS, $\sigma_{SF} = 8$ dB, 25 RB at 15 kHz so $P_{\text{REFSENS}} = -101.7$ dBm — and the maximum allowable path loss rises to 144.2 dB, giving 2.4 km at 95% against M.2412's 1732 m inter-site distance. Rural is where the budget binds and where every dB of antenna gain gets spent.

Fade margin and availability #

Everything above was clear sky. A link is specified not by whether it works but by what fraction of the year it works, and that percentage converts directly into decibels of margin you must buy up front.

Above about 10 GHz, rain is the term that sets those minutes. ITU-R P.838-3 gives specific attenuation as a power law $\gamma_R = k R^{\alpha}$ dB/km, with $k$ and $\alpha$ tabulated per frequency and polarisation and combined for circular polarisation at path elevation $\theta$ via $k = [k_H + k_V + (k_H-k_V)\cos^2\theta\cos 2\tau]/2$ (with $\tau = 45°$ the cosine term vanishes):

Specific attenuation in dB/km, circular polarisation, 43° elevation; coefficients from P.838-3 Table 5. Note what the α column does: below about 25 GHz the exponent is above 1, so attenuation grows faster than rain rate; above it, slower. Multiply by an effective path length of a few kilometres and Ku-band moderate rain is a decibel or two, while Ka-band uplink in the same storm is 15–25 dB. This is why the jump from Ku to Ka is a systems decision, not a frequency change.

The full P.618-14 §2.2.1.1 procedure turns a site's 0.01% point rainfall rate into $A_{0.01}$ through rain height, horizontal reduction and vertical adjustment factors. What matters for budgeting is its final step, equation (8), which extrapolates to any other availability. For an earth station at latitude ≥ 36° it reduces to

$$A_p = A_{0.01}\left(\frac{p}{0.01}\right)^{-\left(0.655 + 0.033\ln p - 0.045\ln A_{0.01}\right)}$$

Computed from P.618-14 equation (8). The shape is the point: the last decade of availability is where the decibels live. Going from 99.9% to 99.99% on the 6 dB path costs 4.1 dB; going from 99.99% to 99.999% costs another 7.3 dB. Each additional nine costs more than the one before it, so "five nines" on a Ka-band link is not a slightly more expensive version of "four nines" — it is a different spacecraft.

Rain charges you twice #

The trap is that rain attenuation is not just attenuation. An absorbing medium also emits, so the same rain cell that takes signal away fills the antenna's beam with thermal noise. P.618-14 equation (69) gives it directly:

$$T_{\text{sky}} = T_{mr}\left(1 - 10^{-A/10}\right) + 2.7 \times 10^{-A/10} \ \text{K}$$

with $T_{mr} = 275$ K recommended in the absence of local data. Apply it to the 4.5 m earth station above ($T_{\text{rx}} = 92$ K from a 1.2 dB noise figure, 0.3 dB coupling loss, 15 K of ground spillover):

$T_{\text{sky}}$ from P.618-14 eq. (69) with $T_{mr} = 275$ K; the rest computed from the noise-temperature decomposition above.

Terrestrial fade margin is a different distribution #

The satellite side buys margin against a time distribution (P.618's percentage-of-year). The terrestrial side buys it against a spatial distribution: log-normal shadow fading with $\sigma_{SF}$ from TR 38.901 Table 7.4.1-1 — 4 dB for UMa LOS, 6 dB for UMa NLOS, 8 dB for RMa NLOS, 7.8 dB for the optional UMa NLOS model. The margin for a target location probability $P$ is $z_{1-P}\,\sigma_{SF}$: 1.28σ for 90%, 1.64σ for 95%. On the rural link that is 13.2 dB of the budget — more than the entire base-station antenna gain. Fast multipath fading is a third distribution again, handled by diversity and interleaving rather than by static margin; see fading and diversity.

Closing on paper versus closing in the field #

Adaptive coding and modulation is the modern answer to the whole margin problem: rather than sizing the link for the worst 0.01% and wasting the dB for the other 99.99%, DVB-S2 lets the MODCOD track the measured $E_s/N_0$, trading throughput for robustness minute by minute. A link that would be QPSK 1/4 at all times becomes 16APSK most of the year and QPSK 1/4 only during storms. The budget then produces a distribution of throughput rather than a single rate, and the availability specification applies to the lowest MODCOD rather than to the whole link.

What still separates a spreadsheet that closes from a link that works:

The range equation itself — the balance between radiated power, spreading, and the noise floor that decides whether a link exists — is developed with the propagation and noise physics behind it in the KPI radio-systems course; the physics chapters there have not aged, though its cellular and satellite system descriptions are a 2012 snapshot. For the terms that feed the budget, see radio propagation, noise sources and antennas; for what the budget's output is spent on, digital modulation and channel coding.