Energy spreading thin: why distance costs you
Stand under a streetlamp at night and walk away from it. The light does not vanish — it just gets dimmer, smoothly and predictably, because the same lumens now have to paint a bigger and bigger sphere of darkness. Radio works the same way. A transmitter pours out a fixed number of watts, and those watts spread over the surface of an expanding sphere. Double the distance and the sphere's area quadruples, so the power crossing any one square metre drops to a quarter. This is the radio propagation story in one sentence: nothing is destroyed, the energy is just diluted.
Engineers measure this dilution in decibels because the numbers span an absurd range — a transmitter might emit a watt while the receiver catches a millionth of a billionth of a watt. Working in dB turns multiplication into addition: a link budget becomes a tidy column you can sum on the back of an envelope. The two reference points you will use constantly are dBm (power relative to one milliwatt: 0 dBm = 1 mW, 30 dBm = 1 W) and dBi (antenna gain relative to an ideal isotropic radiator).
Free-space path loss: the formula that prices distance
Let's turn the streetlamp picture into an equation. A transmitter feeding an isotropic antenna with power Pt spreads it over a sphere of area 4πd² at distance d, so the power density is Pt / (4πd²). The receiving antenna grabs energy from an effective area called its aperture, and a fundamental antenna result says that aperture is Ae = Gr·λ²/(4π). Multiply density by aperture and the received power is Pr = Pt·Gt·Gr·λ² / (4πd)². Strip out the antenna gains and what remains — the pure geometry of the channel — is the free-space path loss.
Free-space path loss (FSPL), the channel's pure geometric toll: FSPL = ( 4*pi*d / lambda )^2 (a ratio, >> 1) In decibels, with d in metres and f in hertz: FSPL(dB) = 20*log10(d) + 20*log10(f) - 147.55 The two engineer-friendly forms (note the +/- constant changes with units): FSPL(dB) = 32.45 + 20*log10(d_km) + 20*log10(f_MHz) FSPL(dB) = 92.45 + 20*log10(d_km) + 20*log10(f_GHz) Key behaviour: * +6 dB every time distance DOUBLES (20*log10(2) = 6.02) * +6 dB every time FREQUENCY DOUBLES (same factor) -> higher bands are intrinsically harder to reach with.
Read that formula carefully and a surprise jumps out: higher frequencies lose more, even though the wave travels through empty space exactly the same way. The physics is not that high-frequency waves are 'weaker' — it's that a fixed-gain antenna has a smaller physical aperture at a shorter wavelength, so it scoops up less of the passing wave. This single fact explains why 5 GHz Wi-Fi has shorter range than 2.4 GHz, and why millimetre-wave 5G needs dense small cells. The channel itself is fair; the antennas are what care about wavelength.
The real world bites back: reflection, diffraction, multipath, fading
Outside the textbook sphere, the wave meets matter, and matter does four annoying things to it. Knowing them qualitatively is enough to budget for them — you do not solve Maxwell's equations for every wall, you reserve extra decibels and move on.
- Reflection — the wave bounces off large smooth surfaces (walls, the ground, a metal roof). Each bounce creates a second copy of your signal arriving by a longer path.
- Diffraction — the wave bends around sharp edges (a hilltop, a building corner), letting some energy sneak into the 'shadow' behind an obstacle. This is why you keep a weak signal just around a corner.
- Scattering — rough surfaces and small objects (foliage, rain, rough walls) splash the wave into many weak directions, especially at higher frequencies.
- Multipath — all those copies arrive at slightly different times and phases. Where crests align they reinforce; where a crest meets a trough they cancel. Move half a wavelength (a few cm at Wi-Fi frequencies) and a strong spot can become a dead one.
Sum these up and you get fading: the received power flickers in time and space rather than holding the smooth value FSPL predicts. Slow, large-scale dips as you walk behind a building are *shadow fading* (roughly log-normal, often budgeted as 6–10 dB). Fast, deep nulls from multipath cancellation are *small-scale fading* (Rayleigh when there is no clear line of sight, Rician when there is). Practically, the path loss in a building or city follows Pr ∝ 1/dⁿ with a path-loss exponent n of about 3 to 5, not the free-space n = 2 — distance bites far harder indoors.
The receiver's limit: sensitivity, noise, and the SNR you must clear
A link is only half about how much signal arrives — the other half is how little signal the receiver can survive on. That floor is set by noise. Even a perfect resistor at room temperature hums with thermal noise, and the available noise power in a bandwidth B is kTB. At 290 K this works out to a tidy reference: -174 dBm per hertz of bandwidth. Widen your channel and you let in more noise, exactly 10·log10(B) more.
Real receivers add their own noise on top, quantified by the noise figure (NF) you met in rung 4. The first stage dominates, which is exactly why an LNA sits right behind the antenna: by adding large gain with tiny added noise before anything noisier downstream, it sets the whole chain's noise figure (Friis' formula). The receiver sensitivity — the weakest input it can demodulate — is just the noise floor plus the noise figure plus the SNR the modulation needs:
Receiver noise floor & sensitivity (everything in dB / dBm): Noise floor = -174 dBm/Hz + 10*log10(B) (kTB at 290 K) Sensitivity = Noise floor + NF + SNR_required Worked example -- a 20 MHz Wi-Fi channel, NF = 6 dB, needs SNR = 20 dB: -174 dBm/Hz = -174.0 + 10*log10(20e6) = +73.0 dB (bandwidth) -> -101.0 dBm (thermal floor) + NF = 6 dB -> -95.0 dBm (receiver floor) + SNR_required = 20 dB -> -75.0 dBm = SENSITIVITY Any received signal stronger than -75 dBm demodulates; weaker fails.
Assembling the link budget: a Wi-Fi link, end to end
Now we put transmit, channel, and receive into one column. The link budget is bookkeeping: start with what leaves the transmitter, add every gain (it helps you), subtract every loss (it hurts you), and compare the arriving power against the sensitivity we just computed. The gap between them is the link margin — your safety cushion against the fading and rain you cannot predict exactly.
WI-FI LINK BUDGET -- 5.0 GHz, 100 m line-of-sight outdoor link
TRANSMIT SIDE
Tx power (PA output) +20.0 dBm (100 mW, typical AP)
Tx cable/connector loss -1.0 dB
Tx antenna gain +6.0 dBi
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EIRP (effective radiated power) +25.0 dBm
CHANNEL
FSPL = 92.45 + 20log10(0.1) + 20log10(5)
= 92.45 - 20.0 + 13.98
Free-space path loss -86.4 dB
Misc margin (fading, foliage) -10.0 dB
-------------------------------------------
Total path loss -96.4 dB
RECEIVE SIDE
Rx antenna gain +6.0 dBi
Rx cable loss -1.0 dB
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RECEIVED POWER = +25.0 - 96.4 + 6.0 - 1.0 = -66.4 dBm
COMPARE TO RECEIVER
Sensitivity (from rung 4 calc) -75.0 dBm
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LINK MARGIN = -66.4 - (-75.0) = +8.6 dB --> LINK CLOSES ✓Notice how every earlier rung shows up here. The PA from rung 6 sets the +20 dBm. The antenna gain from the antennas track buys back +6 dB on each side — a directional antenna does not create power, it concentrates the same watts into a narrower beam, which is pure profit on both transmit and receive. The power amplifier could push more watts, but every extra 3 dB only doubles power for 3 dB of margin, whereas a higher-gain antenna or a better LNA is often cheaper margin. And the sensitivity at the bottom is nothing but rung 4's noise figure made into a number.
The same column works for a satellite downlink — only the numbers stretch. A GEO satellite is 36,000 km away, so FSPL at 12 GHz is around 205 dB, a staggering toll. The system survives by stacking gains the Wi-Fi link never needed: a high-power transponder, a tightly focused dish on the satellite, and a large high-gain ground dish (often 30–50 dBi) feeding a cryogenically cooled LNA with a sub-1 dB noise figure. Every term in the budget is the same idea, just turned up to eleven.
Reading the margin: how much cushion is enough?
A positive margin says the link closes *on average*. But averages are cold comfort to a wireless channel that fades, rains, and reflects. The margin is what you spend to buy *reliability* — the probability that the link still works in the bad moments, not just the median one. How much you need depends on how brutal the channel is and how badly a dropout hurts.
- Identify the impairments the average FSPL ignored: shadowing as people and objects move (6–10 dB), small-scale fading (can be 10–30 dB in a deep null), and rain attenuation (severe above 10 GHz — tens of dB for satellite Ka-band).
- Pick a reliability target — 'works 99% of the time' needs less margin than 'works 99.999% of the time'. Each extra nine costs more decibels.
- Set the margin to cover the worst-case sum you are willing to design for. Indoor Wi-Fi often targets 10–20 dB; a deep-space NASA link may run on just 2–3 dB because every decibel is fought for with enormous dishes.
- If you are short, spend where it is cheapest: more antenna gain (both ends, and it's reciprocal), a lower-noise-figure front end, a lower-rate modulation, or — last resort, because watts cost power and heat — more transmit power.