Learning Objectives
By the end of this lecture the student should be able to:
- Define the space wave, identify the frequency bands in which it is the dominant mode, and distinguish it from the ground and sky waves.
- Compute the radio horizon, earth bulge and required antenna heights for a given line-of-sight path.
- Apply the Friis transmission formula and free-space path-loss equation to point-to-point links.
- Analyse the two-ray (plane-earth) model: path difference, phase reversal, break distance and the 1/d⁴ far-field law.
- Assess obstruction effects using knife-edge diffraction and the Fresnel-zone clearance criterion.
- Explain tropospheric refraction, the k-factor and ducting.
- Characterise multipath fading and select appropriate countermeasures (diversity, equalisation).
- Use standard empirical path-loss models (Okumura–Hata) in coverage studies.
Propagation Modes — Where the Space Wave Fits
| Mode | Band | Range |
|---|---|---|
| Ground wave | VLF, LF, MF | ≤ a few hundred km (surface-guided) |
| Sky wave | HF (3–30 MHz) | Intercontinental via ionosphere |
| Space wave | VHF 30–300 MHz, UHF 300 MHz–3 GHz, SHF > 3 GHz | Line of sight (≈ radio horizon) |
Why VHF and above? The ionosphere is transparent to frequencies above ~30 MHz (they penetrate it instead of being refracted back), and ground-wave loss becomes prohibitive — so the wave simply travels straight through the troposphere.
Consequence: coverage of a space-wave link is set by antenna heights and path geometry, not by ionospheric conditions — but it is subject to terrain, buildings, and multipath interference.
Line-of-Sight Geometry & the Radio Horizon
Radio horizon vs optical horizon
- Optical horizon: d = 3.57√h — light rays are not refracted appreciably.
- Radio horizon: d = 4.12√h — the standard atmosphere bends radio rays downward, extending the horizon by ~15%.
- Maximum LOS distance between two antennas: d$_{max}$ = 4.12(√h$_1$ + √h$_2$) km.
Worked micro-example
Two 30 m towers: d$_{max}$ = 4.12(√30 + √30) = 4.12 × 2 × 5.48 ≈ 45 km. A 25 km link is comfortably within LOS, but path clearance (Fresnel criterion, slide 12) still must be verified.
Engineering rule: never design exactly "to the horizon" — atmospheric variability (k-factor swings, ducting) means fading margin and clearance criteria are always added on top of the nominal geometry.
Free-Space Path Loss & the Friis Equation
Friis transmission equation
P$_r$ = P$_t$ G$_t$ G$_r$ ( λ / 4πd )²
Free-space path loss
L$_{FS}$ (dB) = 20log(4πd/λ) = 32.44 + 20log f(MHz) + 20log d(km)
Reading the chart
- Going from 1 km to 10 km costs 20 dB; from 1 GHz to 10 GHz costs another 20 dB.
- Higher frequencies give more antenna gain for a fixed aperture — the extra FSPL is often recovered at the antennas (G ∝ f² for a fixed-dish antenna), which is why microwave links favour high bands.
- "Free-space loss" is not absorption — it is simply the spreading of power over the spherical wavefront (inverse-square law).
Friis applies strictly only in free space — real space-wave links obey it up to the break distance, after which ground reflection takes over (next slides).
Ground Reflection — the Two-Ray Model
Why reflection matters
For terrestrial LOS links the antennas are close to the ground, so the wave reflected from the earth's surface arrives at the receiver with a delay of only a few nanoseconds and a phase reversal (R ≈ −1 at grazing angles). Direct and reflected rays then interfere.
Key relations
Path difference: Δ ≈ 2h$_1$h$_2$/d
Total phase difference: Δφ = (2π/λ)·Δ + π
Total phase difference: Δφ = (2π/λ)·Δ + π
At grazing angles Δφ ≈ π and the two rays cancel almost perfectly — the receiver field is set by the difference of the two nearly-equal phasors, which grows with Δ (i.e. with h₁h₂ and with 1/d):
E ≈ 2E$_0$ sin(2πh$_1$h$_2$ / λd) ≈ E$_0$·(4πh$_1$h$_2$ / λd) (small argument)
The received power therefore falls as 1/d⁴ (field ∝ 1/d²) beyond a break distance — 12 dB per decade worse than free space.
Two-Ray Path Loss & the Break Distance
The break distance
d$_b$ = 4h$_1$h$_2$ / λ
- For d ≪ d$_b$: path difference Δ ≪ λ — the rays cancel almost completely; received power follows free space (1/d²) modulated by interference nulls.
- For d ≫ d$_b$: the phase difference grows with distance and |sin(·)| → its envelope; power falls as (h₁h₂)²/d⁴.
Design formulas
P$_r$ (d ≫ d$_b$) = P$_t$ G$_t$ G$_r$ (h$_1$²h$_2$² / d⁴) [linear]
P$_r$ (dB) = P$_t$ + G$_t$ + G$_r$ + 40log(h$_1$h$_2$ / d²) + constant
P$_r$ (dB) = P$_t$ + G$_t$ + G$_r$ + 40log(h$_1$h$_2$ / d²) + constant
Raising either antenna height extends coverage dramatically (∝ h² in power). This is why cellular towers and microwave masts are tall — and why mobile-phone coverage is height-dominated.
Effect of Antenna Heights and Frequency
Reading the chart
- Doubling antenna height moves the break distance 4× farther and lowers the whole 1/d⁴ branch by 12 dB.
- At a fixed distance, higher frequency moves d$_b$ out too (∝ f) — but the absolute path loss still grows with f² in the near region.
- In the interference region (d < d$_b$) the curves show the classic two-ray nulls — deep fades at distances where Δφ = odd multiples of π. The last null before d$_b$ occurs near 0.7·d$_b$.
The two-ray model explains a classic mobile-network observation: a handset at ground level in a flat rural area loses signal faster (≈ 40 dB/decade) than one in an urban area where scattering breaks up the clean interference pattern (≈ 30–35 dB/decade, Hata urban).
Diffraction — Obstructions on the Path
Knife-edge diffraction
- When the LOS path is blocked by a ridge, roof edge or building corner, the wave still bends around it — diffraction.
- The obstruction is characterised by the dimensionless Fresnel parameter v; losses of 6–20 dB are typical for marginal obstructions.
v = h·√[ 2(d$_1$+d$_2$) / (λ d$_1$ d$_2$) ]
- v < 0: LOS clear (loss ≈ 0, small ripple).
- v = 0: grazing — ≈ 6 dB.
- v > 0: obstruction above LOS — loss rises ~monotonically with v.
Fresnel zones (next slide) give the practical clearance rule: the first Fresnel zone must be kept at least 60% unobstructed to keep diffraction loss small.
Fresnel Zones & the Clearance Criterion
Fresnel zones
The n-th Fresnel zone is the locus of points where the path length TX→P→RX exceeds the LOS path by nλ/2. The first zone carries most of the energy — it must be protected.
F$_n$(s) = √[ n·λ·s(d−s) / d ]
The 0.6 F₁ rule
- Keep the path clear of obstructions to at least 60% of the first-zone radius everywhere along the path.
- Better clearance (≥ 1.0 F₁) is specified for high-availability links; obstacles within the first zone cause partial cancellation (diffraction loss) even though the path is "visually clear".
Worked micro-example: 2 GHz (λ = 0.15 m), d = 20 km. F₁ at mid-path = ½√(λd) ≈ 27 m → keep obstacles below ≈ 16 m along the route. This is why 20 km microwave hops need ~20–30 m towers even with clean LOS.
Tropospheric Refraction, k-Factor & Ducting
Effective earth radius
Because refractive index n decreases with height, rays bend toward the denser (lower) air. The effect is modelled by scaling the earth radius by the k-factor:
R$_e$ = k·R, standard atmosphere: k = 4/3
Variability
- Sub-refraction (k < 4/3, e.g. cool moist air): rays bend less → LOS paths can unexpectedly lose clearance ("earth bulge grows").
- Super-refraction (k > 4/3): extended ranges — coastal/sea-breeze conditions.
- Ducting (k < 0): rays trapped in a layer → anomalous interference over hundreds of km; a major problem for frequency-reuse microwave networks.
Design responses: over-water paths checked with k = 4/3 nominal down to k ≈ 2/3 worst-case; hop lengths and antenna heights chosen against multipath outage objectives (ITU-R P.530).
Atmospheric Absorption & Rain Attenuation (SHF/EHF)
Gaseous absorption
- Resonant absorption by O₂ (~60 GHz) and H₂O vapour (22, 183 GHz) adds γ·d dB to the link loss.
- Below ~10 GHz, gaseous loss (< 0.01 dB/km) is negligible — ignored in the Friis budget.
- The 60 GHz oxygen band (~15 dB/km) both limits and enables systems: short-range 5G/WiGig links reuse spectrum freely since signals cannot travel far.
Rain attenuation
- Above ~10 GHz, rain becomes the dominant weather impairment: γR = k·Rα dB/km, with k, α from ITU-R P.838 and R the rain rate (mm/h).
- Design is availability-based: e.g. a 30 GHz, 5 km hop in heavy rain (50 mm/h) loses ~0.5–1 dB/km → 3–5 dB — covered by a rain-fade margin; deeper fades (0.01% of time) need site diversity or adaptive coding.
Total path loss: L = L$_{FS}$ + γ$_{gas}$·d + γ$_{R}$·d + L$_{cloud/fog}$
Rule of thumb for link planning: below 10 GHz design against multipath; 10–30 GHz add rain margin from local statistics; above 30 GHz rain + absorption dominate the budget.
Multipath Fading in Space-Wave Links
Types of fading
| Type | Cause | Character |
|---|---|---|
| Flat (narrowband) fading | All frequency components fade together | Deep nulls; mitigated by power margin, diversity |
| Frequency-selective fading | Delay spread ≳ symbol period | ISI + notches; mitigated by equalisation, OFDM, RAKE |
| Slow (shadow) fading | Buildings, terrain blocking | Log-normal, σ ≈ 6–10 dB; mitigated by margin, cell planning |
Channel parameters
Delay spread τ$_{rms}$ vs symbol time T$_s$: τ$_{rms}$/T$_s$ ≪ 1 → flat; coherence B$_c$ ≈ 1/(2πτ$_{rms}$)
Small-scale fading statistics are typically Rayleigh (no dominant LOS) or Rician (strong LOS + scattered echoes) — these drive link-margin calculations (e.g. 20 dB margin for ~99% reliability at Rayleigh).
Countermeasures — Diversity & Combining
Diversity principle
Provide the receiver with several statistically independent copies of the signal; the probability that all fade deeply at once is small. Independence is achieved by separating the copies in space, frequency, time, polarisation or angle.
Common techniques
- Space diversity: two antennas ≥ λ/2 apart (base stations use several wavelengths vertically/horizontally).
- Frequency diversity: transmit on two separated carriers (traditional microwave; now less used — spectrum cost).
- Time diversity: interleaving + coding + retransmission (ARQ) — the basis of modern cellular.
- Polarisation diversity: ±45° slant polarisations at the handset.
Combining
Selection combining (pick strongest) · Equal-gain combining · Maximal-ratio combining (MRC) — optimum, gain ≈ SNR sum
Modern form: MIMO — multiple antennas at both ends exploit multipath as parallel spatial channels instead of fighting it.
Empirical Path-Loss Models — Okumura–Hata
Why empirical models?
Two-ray theory assumes a flat, perfectly reflecting earth. Real terrain, clutter and buildings are handled with measurement-based models.
Okumura–Hata (UHF, 150–1500 MHz)
L$_{urban}$ = 69.55 + 26.16log f − 13.82log h$_b$ − a(h$_m$) + (44.9 − 6.55log h$_b$)log d
L$_{suburban}$ = L$_{urban}$ − 2[log(f/28)]² − 5.4
L$_{rural}$ = L$_{urban}$ − 4.78(log f)² + 18.33log f − 40.94
L$_{suburban}$ = L$_{urban}$ − 2[log(f/28)]² − 5.4
L$_{rural}$ = L$_{urban}$ − 4.78(log f)² + 18.33log f − 40.94
f in MHz, d in km, h$_b$ = base-station height, h$_m$ = mobile height; a(h$_m$) is the mobile antenna correction.
Successors: COST-231 Hata (up to 2 GHz), COST-231 Walfisch–Ikegami (urban microcells), and 3GPP/5G statistical models. Modern tools use ray-tracing calibrated against measurements.