Wireless 101
M08 · L04
Module 8 · Lesson 4

Where Physics Gives Up

Nobody derives a city from Maxwell’s equations. They measure it, fit a line, and publish the coefficients — and every network on Earth was planned that way. This is how to read a fitted model honestly.

01 / 11
Wireless 101
M08 · L04
The log-distance model

A Straight Line in Decibels

Plot measured loss against log distance and the points scatter about a straight line. So keep the line, and let its slope be a fitted number.

Log-distance path loss
PL(d) = PL(d_0) + 10\,n\log_{10}\!\big(\tfrac{d}{d_0}\big) + X_\sigma

An anchor at a close-in d₀, a slope 10n dB per decade, and a random term for everything the line cannot know.

02 / 11
Wireless 101
M08 · L04
The exponent is the environment

One Number for a Whole City

  • n = 2 — free space, 6 dB per doubling (M8-L1)
  • n = 4 — two-ray ground, 12 dB per doubling (M8-L2)
  • n = 2.7–3.5 — urban outdoor, fitted
  • n = 4–6 — obstructed in-building
  • n < 2 — corridors and tunnels: the walls guide the wave

Each doubling of distance costs 3.01n dB, so n is the price list.

03 / 11
Wireless 101
M08 · L04
2.4 GHz, d₀ = 1 m, n = 3.5

Fifty Metres Indoors

PL(1 m) = 92.45 − 60 + 7.60 = 40.05 dB. Then 35 × log₁₀(50) = 35 × 1.699 = 59.46 dB, so PL(50 m) = 99.5 dB — against 74.0 dB in free space.

PL at 1 m
40.05 dB
PL at 50 m
99.5 dB
Vs free space
+25.5 dB
04 / 11
Wireless 101
M08 · L04
Shadow fading — the term M8-L3 deferred

Design to the Mean, Cover Half

Log-normal shadowing and its margin
X_\sigma \sim \mathcal{N}(0,\sigma^2)\ \text{[dB]} \;\Rightarrow\; M = z\,\sigma

Xσ is Gaussian in dB, σ ≈ 4–12. At σ = 8 dB, 90% edge coverage needs 1.28 × 8 = 10.2 dB of margin and 95% needs 1.65 × 8 = 13.2 dB.

05 / 11
Wireless 101
M08 · L04
Try it — 130 dB allowed path loss, 2.4 GHz

How Big Is the Cell?

Fixed budget, fixed 40.05 dB anchor at 1 m. Move n, σ and the reliability target.

Exp n n 3.5
Sigma σ 8.0 dB
Target 90% edge
z 1.28 margin 10.2 dB usable 79.7 dB radius 189 m
06 / 11
Wireless 101
M08 · L04
Hata — 900 MHz, hb 30 m, hm 1.5 m, 5 km

Every Term, Evaluated

  • 69.55 constant, then 26.16 × 2.9542 = +77.28
  • −13.82 × 1.4771 = −20.41 for mast height
  • a(hm) = 3.8245 − 3.8086 = 0.016 dB, negligible at 1.5 m
  • Slope 44.9 − 9.675 = 35.22 dB/decade, i.e. n = 3.52
  • 35.22 × 0.6990 = +24.62 → total 151.0 dB (free space: 105.5)

Valid 150–1500 MHz, hb 30–200 m, d 1–20 km. Outside that it lies quietly.

07 / 11
Wireless 101
M08 · L04
Indoors — 20 m, one floor, three walls

Geometry Is the Small Term

Itemise the building instead of steepening n: 40.05 + 20 × 1.3010 = 66.07 dB of distance, plus 6 + 12 + 18 = 36 dB of fabric → 102.1 dB. Coated low-E glass alone is 25–40 dB.

Distance part
26.0 dB
Partitions
36 dB
Total
102.1 dB
08 / 11
Wireless 101
M08 · L04
Coverage planning

Predict, Then Go and Measure

  • Name the environment — that choice dominates the answer
  • Pick a model valid there, in its units and range
  • Predict the median loss at the intended cell edge
  • Add zσ for your reliability target, edge or area
  • Check the link budget (M9-L4), then drive-test and re-fit

These are statistical fits, not physics: they predict an average, and you design for a percentile.

09 / 11
Wireless 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

10 / 11
Wireless 101
M08 · L04
Recap

What you learned

  • PL = PL(d₀) + 10n log₁₀(d/d₀) + Xσ
  • n runs from below 2 in a tunnel to 6 in a building
  • σ = 8 dB → 10.2 dB for 90%, 13.2 dB for 95%
  • Hata at 900 MHz, 5 km: 151.0 dB, slope n = 3.52
  • Indoors add partitions, not exponent: 102.1 dB
Module 8 complete — up next
Module 9: Noise →
All lessons Read article
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