Link Budget Analysis | WRL-101

Nothing in this lesson is new. That is the point of it. A link budget is the one calculation that consumes every quantity Modules 5 through 9 have built — the decibel arithmetic of M2-L4, the required SNR per modulation of M5-L2 and M6-L4, the antenna gain of M7-L2, the path loss of M8-L1 and M8-L4, the fading statistics of M8-L3, the noise floor and sensitivity of M9-L1, the interference of M9-L3 — and reduces them to a single column of numbers that either adds up or does not. If you can build one honestly you can answer the only question a radio project ever really asks: will this link work, and how far?

The Budget Is One Line of Addition

Every term in a link budget is a decibel quantity, so the whole calculation is addition and subtraction. That is not a convenience; it is the reason M2-L4 taught logarithms at all, and that lesson already worked a five-term budget ending at −73 dBm. Here is the same line, written properly:

Received Power, in dBm

P_{rx} = P_{tx} + G_{tx} - L_{tx} - PL - L_{misc} + G_{rx} \quad [\text{dBm}]

Gains enter with a plus, losses with a minus, and exactly one term carries a reference letter — the dBm of transmit power — so the answer is in dBm. That is the sanity check M2-L4 recommended, and it catches most sign errors before they propagate. The quantity P_tx + G_tx − L_tx is the EIRP of M8-L1: everything the rest of the link ever sees about the transmitter.

Received power on its own decides nothing. It has to be compared with the weakest signal the receiver can actually use, which M9-L1 named the sensitivity. The difference between the two is the link margin, and it is the number the whole exercise exists to produce:

Link Margin, and the Condition for a Working Link

M_{link} = P_{rx} - S \;\ge\; M_{fade} + M_{shadow} + M_{interf} + M_{impl}

A margin of exactly 0 dB means the link works on average and fails half the time, because half of every random variation is downward. The required margin on the right-hand side is therefore not padding, and not one number: it is a deliberate sum of allowances for the specific things that are known to vary. Getting that sum right is where budgets are won and lost.

Every Term, and Where It Comes From

A budget with a term missing is not conservative; it is wrong in a direction you cannot see. So enumerate the transmit side completely, with realistic values and the lesson each figure comes from. Nothing here is new information — the whole table is a re-reading of Modules 7 and 8 in one column.

Term	Sign	Realistic value	Source and remarks
Transmit power P_tx	+	20 dBm WiFi AP, 23 dBm handset, 46 dBm macro base station	The only absolute term. 20 dBm is 100 mW, 23 dBm is 200 mW, 46 dBm is 40 W (M2-L4)
Feeder and connector loss L_tx	−	0.2–0.5 dB for a short pigtail; 2–4 dB up a mast	Coaxial loss grows with frequency and length. On receive this loss also raises noise figure, which is the classic double-count — see below
Antenna gain G_tx, G_rx	+	0 dBi handset whip, 2 dBi WiFi dipole, 17 dBi sector panel, 38 dBi 1 m dish at 10 GHz	M7-L2. Check the reference: dBi = dBd + 2.15. Arrays buy gain by aperture, not by power (M7-L4)
Polarisation mismatch	−	3 dB linear-to-circular; 20–30 dB fully cross-polarised	M7-L2. GPS and satellite links pay the predictable 3 dB rather than risk the unpredictable 20
Body loss	−	2–4 dB, take 3 dB for a handset at the head	Tissue absorbs and detunes. Absent from fixed links, unavoidable in cellular
Path loss PL	−	80 dB, 100 dB, 150 dB, 205 dB — it spans everything	Free space from M8-L1, log-distance or Hata from M8-L4. Almost always the largest single term
Implementation loss	−	1–3 dB	Phase noise, quantisation, imperfect synchronisation — the gap between a real demodulator and the theory of M5-L2

The receive side is a single number built from four

Everything the receiver contributes collapses into the sensitivity, and M9-L1 already derived it. Four decibel terms, added:

Sensitivity, from M9-L1

S_{\text{dBm}} = -174 + 10\log_{10} B_{\text{Hz}} + \mathrm{NF}_{\text{dB}} + \mathrm{SNR}_{\text{req,dB}}

−174 dBm/Hz is kT₀ at the 290 K reference — physics, and not negotiable. 10 log₁₀B is your channel choice, and it is a two-sided RF bandwidth (M6-L2). NF is your front end, dominated by the first stage through the Friis cascade. SNR_req belongs to the modulation, the code and the target error rate — M5-L2, M6-L4 and M9-L2.

Widen the channel and sensitivity gets worse — 10 log₁₀(20 × 10⁶) = 73.01 dB, so a 20 MHz channel starts 73 dB above the density; a 100 MHz channel starts 80 dB above it, 7 dB worse (M9-L1)
Every decibel of noise figure is a decibel of range budget, which is why the LNA sits at the antenna and not at the bottom of the feeder
Higher-order modulation buys bits with sensitivity — M6-L4 needs 10.5 dB of E_b/N₀ for QPSK at BER 10⁻⁶ and 18.8 dB for 64-QAM, and every one of those decibels comes straight off the allowed path loss
Always quote the bandwidth with the sensitivity. A sensitivity figure without a bandwidth and a required SNR is not a specification, it is a mood

Margins Are Not One Number

This is where budgets go wrong. A spreadsheet with one row labelled “margin: 20 dB” hides which risk it is covering, so it cannot be argued with, checked, or reduced when a measurement campaign earns the right to reduce it. There are four distinct allowances, they answer to four different physical mechanisms, and they should appear as four rows:

Fade margin — against fast, small-scale multipath fading (M8-L3). Its size comes from the Rayleigh tail: 9.5% of the time a Rayleigh channel is more than 10 dB below its own mean, 1.0% more than 20 dB, 0.10% more than 30 dB. So an unprotected narrowband Rayleigh link that may be out 1% of the time needs about 20 dB. Diversity, OFDM across a wide channel, and retransmission all cut that figure, and two independent Rayleigh branches turn the 9.5% figure into roughly 0.9% (M8-L3)
Shadowing margin — against slow, large-scale blockage, which is log-normal with a standard deviation σ of 4–12 dB (M8-L4). The margin is zσ for the reliability you promised: at σ = 8 dB, 90% cell-edge reliability needs 1.28 × 8 = 10.2 dB and 95% needs 1.65 × 8 = 13.2 dB. State whether the reliability is at the cell edge or over the area — they differ by several points
Interference margin — against the fact that your noise floor is not thermal (M9-L3). It is quoted as a rise over thermal: a 3 dB interference margin says the interference power equals the thermal noise power, so the effective floor has doubled and the relevant ratio is an SINR rather than an SNR. In a loaded cellular uplink or a crowded 2.4 GHz band this is not optional
Implementation and ageing margin — against your own hardware. Component tolerance, temperature drift, connector corrosion, a feeder that will be spliced twice in ten years, and the gap between the theoretical sensitivity and the one the chipset actually achieves. Typically 1–3 dB, and unlike the first three it is not random at all: it is a deterministic allowance for imperfect knowledge

Adding margins versus root-sum-square

Margins are sometimes combined by root-sum-square rather than added, and the choice can be worth five decibels — so it needs a rule rather than a preference. The rule follows from what the terms are. Two independent quantities that are both Gaussian in decibels have a sum that is also Gaussian in decibels, with a standard deviation given by the Pythagorean combination, so a single percentile of the combined variable is smaller than the sum of the individual percentiles:

Combining Two Log-Normal Margins

\sigma_{tot} = \sqrt{\sigma_1^2 + \sigma_2^2} \quad\Longrightarrow\quad M = z\,\sigma_{tot} \;\le\; z\sigma_1 + z\sigma_2

Worked: outdoor shadowing at σ₁ = 8 dB combined with building-penetration variability at σ₂ = 6 dB gives √(64 + 36) = 10 dB, so a 90% margin is 1.28 × 10 = 12.8 dB — against 1.28 × 8 + 1.28 × 6 = 17.9 dB if you add the two margins. Root-sum-square saves 5.1 dB, and it is legitimate here because both terms are independent and log-normal.

Add, rather than combine, in three cases. When the terms are not both Gaussian-in-dB — fast Rayleigh fading is not, so a Rayleigh fade margin and a log-normal shadowing margin are properly added, not root-sum-squared. When they are correlated, because the Pythagorean form assumes independence and correlation pushes the answer back towards the plain sum. And when a term is deterministic rather than statistical: an implementation or ageing allowance is not a percentile of anything, so it simply adds. The honest summary is that root-sum-square is a specific tool for combining independent log-normal variabilities, not a general discount on caution.

Worked Example: A 2.4 GHz WiFi Link

An office access point at 2.4 GHz on a 20 MHz channel, talking to a handheld client that wants 64-QAM. Start with the sensitivity, using M9-L1’s formula and a client noise figure of 6 dB, and the 20 dB of SNR that M9-L1 attributed to 64-QAM at rate 3/4:

Bandwidth term: 10 log₁₀(20 × 10⁶) = 10 log₁₀(2) + 70 = 3.01 + 70 = 73.01 dB
Thermal floor: −174 + 73.01 = −100.99 dBm, the −101 dBm of M9-L1’s table
Receiver floor: −100.99 + 6 = −94.99 dBm
Sensitivity: −94.99 + 20 = −74.99, i.e. −75.0 dBm
Cross-check against M9-L1, which ran the same case with NF = 5 dB and got −76 dBm: the two answers differ by exactly 1 dB, which is exactly the difference in noise figure ✓

Now the full budget. Every term from the table above that applies, in one column:

Line	Value	Running total	Note
AP transmit power	+20.0 dBm	`20.0`	100 mW, a typical 2.4 GHz AP per chain
Connector and pigtail loss	−0.5 dB	`19.5`	Short internal cable
AP antenna gain	+2.0 dBi	`21.5`	EIRP = 21.5 dBm (M8-L1)
Path loss	−PL	`21.5 − PL`	The unknown we are solving for
Client antenna gain	+2.0 dBi	`23.5 − PL`	Small dipole, M7-L2
Body loss	−3.0 dB	`20.5 − PL`	Hand-held device
Implementation loss	−2.0 dB	`18.5 − PL`	Real demodulator versus theory
Received power	—	`18.5 − PL dBm`	Exactly one reference letter survives ✓
Fade margin	8.0 dB	—	OFDM spreads a symbol across 20 MHz, far wider than the coherence bandwidth of an office, so this link earns frequency diversity and does not need the 20 dB a flat-faded Rayleigh link would (M8-L3)
Shadowing margin	5.1 dB	—	1.28 × 4 dB, indoor σ at 90% reliability (M8-L4)
Interference margin	3.0 dB	—	2.4 GHz is shared with everything (M9-L3)
Total required margin	16.1 dB	—	8.0 + 5.1 + 3.0, added because they are not all log-normal

Solve it. The link works when received power exceeds sensitivity by the required margin, so 18.5 − PL ≥ −75.0 + 16.1 = −58.9 dBm, which gives an allowed path loss of 18.5 + 58.9 = 77.4 dB. Turn that into a distance with M8-L4’s log-distance model, anchored at PL(1 m) = 40.05 dB for 2.4 GHz and using n = 3.0 for an office floor:

77.4 = 40.05 + 10 × 3.0 × log₁₀(d), so 30 log₁₀(d) = 37.35
log₁₀(d) = 1.245, so d = 10^1.245 = 17.6 m
Sanity check the arithmetic backwards: 40.05 + 30 × 1.245 = 40.05 + 37.35 = 77.4 dB ✓

Seventeen metres. That is the honest 64-QAM range of a 2.4 GHz access point in an office with these margins, and it is much less than the range people quote for WiFi — because the quoted range is the range of the slowest modulation, not the fastest. Redo the last three lines with the lowest rate instead, which needs about 5 dB of SNR rather than 20: sensitivity becomes −174 + 73.01 + 6 + 5 = −90.0 dBm, allowed path loss becomes 18.5 + 90.0 − 16.1 = 92.4 dB, and the range becomes 10^{(92.4−40.05)/30} = 10^1.745 = 55.6 m. Fifteen decibels of relaxed SNR bought a factor of 10^15/30 = 3.16 in distance, and 55.6/17.6 = 3.16 ✓. One access point, one physical environment, two ranges differing by more than three times, and the only thing that changed was how many bits per symbol you asked for.

The small losses are not small. Drop the 0.5 dB connector, the 3 dB body loss and the 2 dB implementation loss — the three lines a hurried budget omits — and the allowed path loss rises from 77.4 dB to 20 + 2 + 2 + 75.0 − 16.1 = 82.9 dB, giving a range of 10^{(82.9−40.05)/30} = 26.8 m. Those 5.5 forgotten decibels are 10^5.5/30 = 1.53 in distance, so the tidy budget predicts 52% more range than the honest one. At n = 3 a decibel is worth 7.7% of your range; the same 5.5 dB under free-space spreading, n = 2, would be worth 1.9 times.

Worked Example: A 900 MHz Cellular Cell, and Why the Uplink Loses

Now the case where the budget has to be done twice, because a cellular link is not symmetric. Take a 900 MHz macrocell with a 30 m mast, a 17 dBi sector panel, 2 dB of feeder, and a handset on the other end. The base station transmits at 46 dBm and the handset at 23 dBm — a 23 dB advantage to the downlink — and the base station also has a better antenna and a better noise figure. The interesting question is whether those advantages cancel. They do not, and the direction of the answer decides how many base stations a city needs. Both budgets below use a 1.08 MHz allocation, so 10 log₁₀(1.08 × 10⁶) = 60.33 dB, and a required SNR of 0 dB for QPSK with rate-1/3 coding.

The uplink: handset to base station

Base-station sensitivity: −174 + 60.33 + NF 2.0 + SNR 0.0 = −111.7 dBm — the 2 dB noise figure is specified at the radio input, below the feeder
Is that 0 dB of SNR plausible? SNR = E_b/N₀ + 10 log₁₀(η) from M9-L1, and QPSK at rate 1/3 has η = 2 × 1/3 = 0.667 bit/s/Hz, so 10 log₁₀(0.667) = −1.76 dB and the implied E_b/N₀ is 0 − (−1.76) = +1.76 dB. Uncoded QPSK needs 6.79 dB at BER 10⁻³ (M9-L2), so this asks for 6.79 − 1.76 = 5.03 dB of coding gain — comfortably inside the 5–8 dB classical and 10–11 dB turbo range M9-L2 quotes ✓
Handset power +23.0 dBm, handset antenna 0 dBi, body loss −3.0 dB, so 20.0 dBm leaves the user
Base-station antenna +17.0 dBi and feeder loss −2.0 dB, so received power = 20.0 − PL + 17.0 − 2.0 = 35.0 − PL dBm
Margins: fade 10.0 dB (two-branch receive diversity turns a 10 dB fade into a ~1% event, M8-L3), shadowing 1.28 × 8 = 10.2 dB at 90% cell edge (M8-L4), interference 3.0 dB (M9-L3) — total 23.2 dB
Allowed path loss: 35.0 + 111.7 − 23.2 = 123.5 dB

The downlink: base station to handset

Handset sensitivity: −174 + 60.33 + NF 7.0 + SNR 0.0 = −106.7 dBm — 5 dB worse than the base station, which is the whole of what the better front end is worth
Base-station power +46.0 dBm, feeder −2.0 dB, antenna +17.0 dBi, so EIRP = 61.0 dBm
Handset antenna 0 dBi and body loss −3.0 dB, so received power = 61.0 − PL − 3.0 = 58.0 − PL dBm
Same 23.2 dB of margin, because it describes the channel and the band, not the direction of travel
Allowed path loss: 58.0 + 106.7 − 23.2 = 141.5 dB

There is the asymmetry, and it is large: 141.5 − 123.5 = 18.0 dB. And it reconciles exactly, which is the check that the two budgets were built consistently. The downlink starts 46 − 23 = 23 dB ahead on transmit power. The base station’s superior noise figure gives 5 dB of that back to the uplink, since its sensitivity is −111.7 dBm against the handset’s −106.7 dBm. Every other line — the 17 dBi panel, the 2 dB feeder, the 3 dB body loss, all three margins — appears identically in both directions and therefore cancels. So 23 − 5 = 18 dB ✓, and the uplink is the binding constraint even though the base station is better at everything except output power. A handset runs on a battery; a base station runs on the grid, and no amount of tower engineering repeals that.

Turn both into a cell radius with M8-L4’s Hata fit at 900 MHz, h_b = 30 m, h_m = 1.5 m. That lesson computed 151.0 dB at 5 km with a distance slope of 44.9 − 6.55 log₁₀(30) = 35.22 dB per decade, so the model rearranges to PL(d) = 126.4 + 35.22 log₁₀(d_km). Check that against the source: 126.4 + 35.22 × log₁₀(5) = 126.4 + 24.62 = 151.0 dB ✓.

Uplink: log₁₀(d) = (123.5 − 126.4)/35.22 = −0.082, so d = 0.83 km
Downlink: log₁₀(d) = (141.5 − 126.4)/35.22 = 0.429, so d = 2.68 km
Radius ratio: 10^18.0/35.22 = 3.24, and 2.68/0.83 = 3.23 ✓
Area ratio: 3.24² = 10.5 — sizing the cell on the downlink would leave about 90% of its area unable to answer
Honesty check on the model: M8-L4 states that Hata is valid from 1 km, so the 0.83 km answer sits just outside its range. Quote it as “about 1 km, at the edge of Hata’s validity” rather than as 830 m

That last bullet is not pedantry. A model that is asked a question outside its range does not fail loudly; it returns a confident number, exactly as M8-L4 warned. And the 10.5× area ratio is the most useful single insight in this lesson: it is why cellular systems have uplink power control, why they hand narrow allocations to distant users, and why the coverage figure a network commits to is always an uplink figure.

Plausibility check on that sensitivity. −111.7 dBm looks aggressive until you notice it is for 1.08 MHz. Sensitivity scales with the bandwidth term, so widen the same radio to a 9 MHz occupied allocation and it degrades by 10 log₁₀(9/1.08) = 9.2 dB, to −102.5 dBm — the order of figure quoted for wide-area base stations. Both numbers describe the same hardware. This is why a sensitivity without a bandwidth cannot be compared with anything, and why a narrow uplink allocation is a real tool for extending cell edge: fewer hertz means less noise.

Reading and Building the Spreadsheet

A link budget in practice is a two-column sheet: a name and a signed decibel value, with a running total beside it and a source note for every row. Three conventions and three checks make the difference between a sheet that can be defended and a sheet that merely sums.

Sign convention, stated at the top. Gains positive, losses negative, and every loss entered as a negative number rather than as a positive number you remember to subtract. A budget that mixes the two habits will eventually add a loss
One row, one physical mechanism. Never a row called “miscellaneous”, and never one called “margin”. Four margin rows with four names can each be challenged; one lump cannot
A source column. Measured, datasheet, standard, or assumed — and if assumed, by whom. This is the column that lets a drive test replace a guess two months later and shrink the margin honestly
Check 1: the double-counted loss. Feeder loss ahead of the LNA can be booked as a 2 dB budget line with the radio’s own NF of 2 dB, or deleted from the budget and folded into an antenna-referred NF of 2 + 2 = 4 dB (M9-L1’s Friis corollary). Both are correct. Doing both charges the same 2 dB twice, and it is the single commonest error in real sheets
Check 2: the double-counted margin. If your required ratio is an SINR that already contains an interference allowance, do not also carry an interference margin row. Likewise, if the required SNR was taken from a faded channel measurement, the fade margin is already inside it. Ask of every margin: what varies, and is that variation priced anywhere else on this sheet?
Check 3: does the answer match reality? The budget said 17.6 m for indoor 64-QAM, which is the right order for an office; if it had said 2 km, the exponent or the margins are wrong. And sanity-check the sensitivity: a datasheet claiming −95 dBm for 64-QAM in 20 MHz implies NF + SNR_req = −95 + 101 = 6 dB in total, which is impossible when the modulation alone needs about 20 dB. A budget is a model, and the only thing that validates it is a measurement

One habit is worth more than all of these. Do the budget twice, once forwards to find the received power at a stated distance, and once backwards to find the distance at a stated margin. If the two disagree you have a sign error, and you will find it in about a minute.

What Modules 5 to 9 Built

This lesson closes an arc of twenty lessons, and it is worth naming what they add up to, because the link budget is precisely the place where they meet. Read the WiFi table again and notice that every row is somebody else’s lesson:

Module 5 turned bits into symbols and gave the error probability its shape — the SNR_req term
Module 6 priced bits against bandwidth, power and delay, and set what SNR_req must be for a chosen spectral efficiency and code
Module 7 supplied both gain terms, and the reminder that gain is geometry rather than amplification
Module 8 supplied the path-loss term and, more importantly, the statistics that make two of the margins computable rather than invented
Module 9 supplied the sensitivity from first principles, the meaning of an error rate, the interference margin — and this lesson, which spends all of it

And notice what the budget cannot do. Every calculation on this page describes one link, alone in its band, using the whole channel. Real systems put many users in the same spectrum at once, split the channel into thousands of narrow subcarriers, and use several antennas at both ends to send parallel streams through the same multipath that this lesson treated purely as a hazard. Those three ideas — multiple access, OFDM and MIMO — are Module 10 and beyond, and each of them changes the budget rather than replacing it: it acquires new rows, and the fading that cost you 8 dB here becomes something a system can exploit.

Key Takeaways

P_rx = P_tx + G_tx − L_tx − PL − L_misc + G_rx, all in dB, with exactly one reference letter surviving into the answer (M2-L4). Link margin = P_rx − sensitivity, and it must exceed the sum of the required margins.
Sensitivity = −174 + 10 log₁₀B + NF + SNR_req (M9-L1). For 20 MHz, NF 6 dB and 20 dB for 64-QAM: −174 + 73.01 + 6 + 20 = −75.0 dBm, exactly 1 dB from M9-L1’s NF-5 answer of −76 dBm.
Four margins, four mechanisms: fade (Rayleigh, ~20 dB for 1% outage unprotected, M8-L3), shadowing (zσ, 1.28 × 8 = 10.2 dB at 90% edge, M8-L4), interference (3 dB means I = N, M9-L3), and implementation or ageing (1–3 dB). A single lumped “margin” row hides which risk you covered.
Root-sum-square only for independent log-normal terms: σ = √(8² + 6²) = 10 dB gives 12.8 dB at 90%, against 17.9 dB added — a 5.1 dB difference. Rayleigh fading and deterministic allowances are added, not combined.
WiFi budget: 20 − 0.5 + 2 + 2 − 3 − 2 = 18.5 dBm before path loss; margins 8 + 5.1 + 3 = 16.1 dB; allowed PL = 18.5 + 75.0 − 16.1 = 77.4 dB; at PL(1 m) = 40.05 dB and n = 3 that is 17.6 m for 64-QAM and 55.6 m at the lowest rate.
Cellular is uplink-limited: allowed PL is 123.5 dB up and 141.5 dB down, an 18.0 dB gap that resolves exactly as 23 dB of transmit power minus 5 dB of noise figure. In Hata terms that is 0.83 km against 2.68 km — a 3.24× radius and a 10.5× area ratio.
The two classic sheet errors are double-counting a feeder loss that is already inside the noise figure, and double-counting a margin already inside a required SINR. Do the budget forwards and backwards, and check the answer against something real.

Module 9 is complete — and so is the arc. Thermal noise and SNR (M9-L1), bit error rate (M9-L2), sources of interference (M9-L3) and this link budget close the module, and with it Modules 5 through 9. Next, Module 10 asks how many users share one channel at once — by frequency, by time, by code, and by space.

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