Everything in Module 8 so far has been derived. Free-space path loss came out of a sphere’s surface area (M8-L1); the d⁻⁴ two-ray law came out of a phase flip and a path difference (M8-L2); Rayleigh fading came out of summing random phasors (M8-L3). This lesson is where that stops. Nobody derives the path loss of a city from Maxwell’s equations. They measure it, fit a curve, and publish the coefficients — and the resulting models are the ones every network in the world was actually planned with. The shift from physics to empiricism is not a retreat; it is the whole engineering method of this field, and learning to read a fitted model honestly — including its validity limits and its error bars — is the point of the next fifteen minutes.
Consider what an exact prediction of the signal in one apartment would require: the permittivity of every wall, the position of every parked car, the water content of every tree, the pose of every person in the room. A ray tracer given all that would be right, and the data will never exist. So the field made a trade that is worth stating out loud: stop predicting the value, and predict the distribution instead. Measure a few hundred locations, fit a straight line through the decibels against the logarithm of distance, and describe the scatter around that line as a random variable. What you get back is not the signal at your kitchen table. It is the average signal at your distance, plus an honest statement of how wrong that average typically is.
That is the shape of every model in this lesson, and it is the reason coverage is quoted as a percentage. A planner does not promise that your kitchen works; they promise that 95% of locations at the cell edge work. The two halves of the model do the two halves of that job: a deterministic slope sets the average, and a statistical term sets the percentile you design to. Get in the habit of asking, of any propagation number anyone shows you, which of the two it is.
Start with the observation that motivates everything. Plot measured path loss in decibels against log₁₀ of distance, in almost any environment, and the points scatter around a straight line. That is not a coincidence: M8-L1 showed free space loses 20 log₁₀(d), which is a straight line of slope 20 dB per decade, and M8-L2 showed a ground reflection steepens it to 40 dB per decade. Real environments land somewhere in that range and beyond it, so the natural model keeps the straight line and lets its slope be a fitted number:
The exponent n is the whole personality of an environment compressed into one number, and its landmark values are ones you already own. Note what n does to the cost of distance: each doubling costs 10n log₁₀(2) = 3.01n decibels, so n = 2 charges 6 dB per doubling and n = 4 charges 12 dB, exactly the two-ray result from the previous lesson.
| Environment | Typical n | dB per doubling | Where it comes from |
|---|---|---|---|
| Free space | 2.0 | 6.0 | Derived in M8-L1 — the reference case, not a fit |
| Flat ground, two-ray, beyond the breakpoint | 4.0 | 12.0 | Derived in M8-L2 — the direct ray cancelled by its own reflection |
| Urban macrocell, outdoor | 2.7–3.5 | 8.1–10.5 | Fitted; buildings both block and re-radiate, so it lands between the two |
| Obstructed in-building | 4.0–6.0 | 12.0–18.1 | Fitted; every metre of path crosses more material |
| Corridor, tunnel, mine gallery | 1.2–1.8 | 3.6–5.4 | Fitted; the walls guide the wave, so it decays slower than free space |
That last row is worth a pause, because n < 2 looks like it breaks conservation of energy and does not. In free space the power spreads over a sphere whose area grows as d²; in a corridor the walls stop it spreading sideways, so it spreads over something closer to a cylinder, and the geometric dilution is weaker. The corridor is a lossy waveguide. It is also why a tunnel can hold a usable signal for a kilometre while an office floor loses it in thirty metres, and why an in-building design that assumes one exponent for the whole floorplate will be wrong in both directions at once.
Anchor the model with a computed reference instead of a measured one, which is standard practice indoors. Take f = 2.4 GHz and d₀ = 1 m. Using the M8-L1 form of free-space path loss with distance in kilometres and frequency in gigahertz, PL = 92.45 + 20 log₁₀(dkm) + 20 log₁₀(fGHz), one metre is 0.001 km, so:
Compare that with free space over the same 50 m: 40.05 + 20 log₁₀(50) = 40.05 + 33.98 = 74.0 dB. The fitted exponent costs an extra 25.5 dB, which is a factor of 355 in power — and 25.5 dB is roughly the difference between a link that works and a link that does not. This single number is why nobody plans an indoor network with the free-space formula.
M8-L3 split the channel’s randomness in two and handed one half to this lesson. Small-scale fading is what you get from moving half a wavelength: multipath phasors re-adding, Rayleigh statistics, deep nulls centimetres apart. Large-scale fading — shadowing — is what you get from moving tens of metres: you are now behind a different building, under a different tree, in a room with a different wall count. Two receivers at exactly the same distance from the same base station can differ by 20 dB for no reason other than what happens to be standing between them and it.
Measurements say that spread is, to a good approximation, Gaussian in decibels — which is to say log-normal in linear power, since decibels are already a logarithm (M2-L4). There is even a plausible reason: a path crosses many independent obstructions, each multiplying the power by its own attenuation factor, and a product of many independent factors becomes a sum of many independent logarithms, which the central limit theorem pushes towards a Gaussian. So:
A zero-mean random term has a brutal consequence that is easy to miss: if you design a cell so that the predicted average path loss at the cell edge exactly equals what your radio can tolerate, then Xσ is positive half the time and half of your cell-edge locations do not work. Fifty per cent coverage is the default outcome of designing to the mean. To do better you must deliberately over-build by some number of decibels, and the Gaussian tells you exactly how many. Ask for the margin M such that the probability of Xσ exceeding M is your outage allowance:
Read the last two lines together, because they contain the practical lesson. The margin is proportional to σ, not to the path loss, so a better-characterised environment is cheaper to cover — the 4 dB you save by measuring instead of guessing buys real cell radius. And going from 90% to 95% reliability at σ = 8 dB costs 3 dB, which is half your transmit power for the last five percentage points. That is the trade that decides how many base stations a city needs.
Cell-edge coverage, not area coverage. The 1.28σ and 1.65σ figures above are the reliability at the cell edge, the worst circle in the cell. Because locations closer in enjoy lower average path loss, the fraction of the whole cell area that works is always higher than the edge figure — typically 90%-edge corresponds to something like 97% of area for n and σ in the ranges used here. Both numbers are quoted in industry, they differ by several points, and a specification that does not say which one it means is not a specification.
The log-distance model has two free parameters and tells you nothing about how to choose them for a city you have not measured. The classic answer comes from Yoshihisa Okumura’s extensive 1968 measurement campaign around Tokyo, which Masaharu Hata reduced in 1980 to a set of closed-form fits. Hata’s formulae are still the reference for sub-2 GHz macrocell coverage, and their standard urban form is one line of arithmetic in four variables — frequency, base-station height, mobile height and distance:
The mobile-height correction a(hm) is a separate small expression. For a small or medium-sized city it is:
Take the canonical GSM-era macrocell — f = 900 MHz, hb = 30 m, hm = 1.5 m, d = 5 km — and evaluate every term. Two logarithms carry the whole calculation: log₁₀(900) = 2.9542 and log₁₀(30) = 1.4771.
Two things in that result are worth more than the total itself. First, free-space path loss over the same 5 km at 900 MHz is 32.45 + 20 log₁₀(5) + 20 log₁₀(900) = 105.5 dB, so Hata charges 45.5 dB more than empty space for the privilege of being in a city. Second, look at that distance slope: 35.22 dB per decade is 10n with n = 3.52. Hata is a log-distance model in disguise, with a height-dependent exponent and a calibrated intercept — which is why the fitted urban n range in the table above brackets it so neatly.
Because the model is a fit to Tokyo measurements, it is only meaningful inside the range those measurements covered. The limits are usually quoted as:
The most common error in practice is extrapolation without saying so: running Hata at 2600 MHz because the spreadsheet accepts the number, or at 200 m because a dense-urban site is 200 m from its neighbour. The formula does not fail loudly. It returns a plausible decibel figure that no measurement supports, and the mistake surfaces months later as a coverage hole. Quote a model’s validity range every time you quote its output, and if you are outside it, either pick a different model or measure.
When European operators needed the same tool for DCS-1800 and UMTS, the COST 231 project re-fitted Hata’s structure to higher-frequency measurements. The result — COST-231 Hata, sometimes called the PCS extension — keeps every term’s shape and changes two coefficients plus a city-type constant:
Alongside it sits a different animal: COST-231 Walfisch–Ikegami. Instead of fitting a curve to distance, it models the actual urban geometry — rooftop diffraction down a street canyon, building separation, street width, the angle between the street and the direction of arrival — using the diffraction machinery of M8-L2 rather than a regression. It is valid from about 800 MHz to 2000 MHz over 20 m to 5 km, it handles base stations below rooftop height that Hata cannot, and it costs you a building database. That trade is the whole spectrum of propagation modelling in miniature: cheap statistical fits at one end, expensive site-specific ray tracing at the other, and Walfisch–Ikegami deliberately in the middle.
Indoor propagation has a different character, and the reason is scale. Outdoors, a 5 km path crosses a statistically similar mess the whole way, so a single fitted slope works. Indoors, a 20 m path crosses a small number of individually identifiable obstacles: this wall, that floor, that lift shaft. With few enough obstacles the averaging that justified a smooth curve stops applying, and the model that works instead is a budget — a distance term plus one explicit decibel entry per partition crossed.
| Partition | Typical loss at 2–2.5 GHz | Note |
|---|---|---|
| Plasterboard / drywall partition | 2–4 dB | The cheap wall; a whole open-plan office may only cost a few of these |
| Interior brick or concrete block wall | 10–15 dB | One of these can undo a doubling of transmit power |
| Reinforced concrete floor / ceiling | 15–20 dB for the first floor | Each additional floor adds only about 4–6 dB more, not another 18 — energy starts arriving round the outside of the building rather than through the slabs |
| Plain glass window | 2–4 dB | Effectively transparent; this is how most outdoor-to-indoor coverage gets in |
| Low-emissivity / metallised coated glass | 25–40 dB | The modern gotcha: the energy-efficiency coating is a thin metal film, so the window becomes a mirror (M8-L2) and the building becomes a Faraday cage |
| Lift shaft, metal-clad plant room, foil-backed insulation | 30 dB and up | Treat as opaque and plan a separate antenna rather than a margin |
Put an access point 20 m away, one floor up, with two drywall partitions and one concrete block wall in between, at 2.4 GHz. Reuse the 40.05 dB reference from earlier and take n = 2.0, since the partitions are now itemised:
Notice the composition of that answer: only 26 dB of it is distance, and 36 dB — more than the distance term — is building fabric. Indoors, geometry is the small term. That is exactly why moving an access point ten metres closer often does nothing while moving it to the other side of one wall fixes everything, and it is why indoor design is a floorplan exercise rather than a radius exercise. For comparison, a single-slope fit with n = 4 over the same 20 m gives 40.05 + 40 × 1.3010 = 92.1 dB — 10 dB more optimistic than the partition budget, because a single exponent cannot know that this particular path happens to cross a floor slab. Neither number is the answer; they are two estimates whose disagreement is itself information.
Assemble the pieces and the professional workflow is short, and every step in it is a hedge against the previous step being a fit rather than a fact:
Step 6 is the one that gets cut when a schedule slips, and it is the only step that produces new information. Everything upstream of it is a prediction from somebody else’s measurements of somebody else’s city. A network that has been drive-tested has a measured n and a measured σ for its own territory, which shrinks the margin it needs and therefore the number of sites it must build — the survey pays for itself in concrete.
What these models are, precisely. Every formula in this lesson is a regression on someone else’s measurements. None of them contains the physics of your building or your city, and none of them predicts the signal at a point — they predict a median and a spread. Used properly that is a powerful instrument: it turns an environment into two numbers you can design against, and it fails in known directions. Used as though it were Maxwell’s equations, it produces confident numbers about places nobody measured. The honest sentence is always the same shape: “this model, in this range, predicts this median, with this σ, at this reliability.”
Module 8 has taken one signal from a transmitter to a receiver and accounted for what the world does to it: spreading loss (M8-L1), the three mechanisms of reflection, diffraction and scattering (M8-L2), the fast statistics of multipath (M8-L3), and now the slow statistics and the fitted models that turn all of it into a coverage radius. What is conspicuously missing is the other side of the comparison. A received power of −95 dBm means nothing on its own; it only means something against the noise floor it competes with, and against the other transmitters using the same frequency. Module 9 supplies exactly that — thermal noise, noise figure, interference, and the full link budget that step 5 above waved at. Module 10 then asks how many users can share the channel you have just learned to characterise, which is where cell reuse turns propagation loss from a problem into the mechanism that makes reuse possible at all.
Module 8 is complete — and the course continues. With this lesson Module 8: Radio Propagation is finished: free-space path loss, the three propagation mechanisms, multipath and fading, and the empirical models and coverage planning above. Next, Module 9 turns to what fights the signal at the far end: thermal noise, bit errors, interference, and the link budget that has to survive all three.