1. Learning Objectives
By the end of this study guide, the student should be able to:
- Define the space wave (direct wave) and its component paths, and explain its dominance at VHF and above.
- Compute radio-horizon distances using the k = 4/3 effective-earth model, and determine required Fresnel-zone clearances for any link.
- Analyse multipath propagation: two-ray interference, the critical distance, the 1/d⁴ law, Rayleigh and log-normal fading statistics.
- Quantify diffraction loss over knife edges and smooth spherical earth, and account for atmospheric effects: refraction, ducting, tropospheric scatter and rain attenuation.
- Select and apply appropriate path-loss models (free space, two-ray, Okumura–Hata, COST-231, ITU-R P.1546/P.1812) for a given scenario.
- Perform space-wave link budgets including antenna gains (dish formula), thermal noise, required C/N, and fading/rain margins.
- Specify antennas for point-to-point microwave and satellite links, and solve related numerical problems.
2. Introduction
The space wave is the component of a radio wave that travels the direct (line-of-sight) path through the troposphere between transmitting and receiving antennas — aided and abetted by whatever the local environment provides: ground and building reflections, diffraction around obstacles, and occasional atmospheric ducts.
- Dominant from VHF (30 MHz) upward: FM/TV broadcast, cellular, WiFi, microwave backhaul, radar, satellite links, and mm-wave 5G.
- Why not below VHF? Below ~30 MHz the ground wave hugs the surface and the ionosphere returns sky waves; the direct path is swamped by both.
- Engineering character: coverage becomes a problem of geometry (horizon, Fresnel zones), statistics (fading), and weather (rain) — the antenna and the site do most of the talking.
| Band | Frequency | Representative services | Key space-wave issue |
|---|---|---|---|
| VHF | 30–300 MHz | FM radio, TV, airband, 2 m amateur | Horizon + diffraction; modest rain effect |
| UHF | 0.3–3 GHz | TV, cellular, GPS, WiFi | Multipath, clutter; rain still minor |
| SHF (microwave) | 3–30 GHz | Satellite C/Ku/Ka, backhaul, radar, 5G | Rain fade; precise pointing; Fresnel discipline |
| EHF (mm-wave) | 30–300 GHz | 5G mm-wave, automotive radar | LOS only; oxygen/rain absorption; tiny beams |
Why this is core ECE 523 material: every concept of antenna and propagation engineering converges here — aperture and gain, free-space loss, reflection and diffraction, atmospheric refraction, noise and link budgets. Master the space wave and you can budget any terrestrial or satellite link.
3. Line-of-Sight Geometry and Fresnel Zones
3.1 Radio Horizon
Standard atmospheric refraction bends rays downward, equivalent to an Earth of radius k × R with k = 4/3:
Height is coverage: doubling a mast's height adds ≈ 41 % to the horizon radius (d ∝ √h).
3.2 Fresnel Zones — the Volume of LOS
"Line of sight" is not a line but an ellipsoid: secondary waves arriving within the first Fresnel zone add in phase with the direct ray. The engineering rule: keep at least 60 % of the first zone clear.
Maximum (mid-path) r₁max = ½ √( λ d ) (d = total path)
- At 900 MHz, 10 km: r₁max ≈ 29 m — hedges and low buildings genuinely matter.
- At 6 GHz, 30 km: r₁max ≈ 17.3 m — trees on a ridge line can eat a microwave link.
- Obstruction into the zone adds predictable loss (see diffraction, §5).
4. Multipath and Reflection
4.1 Two-Ray Model
Over a flat reflecting surface, the received signal is the sum of the direct ray and the ground-reflected ray (reflection coefficient ≈ −1 at grazing incidence). The result is interference with a striking distance law:
Interpretation −6 dB/octave in free space becomes −12 dB/octave beyond dc
4.2 Multipath Fading Statistics
- Small-scale fading (Rayleigh): many unresolved paths with random phases → deep nulls (20–40 dB) spaced ~λ/2; countered by diversity (space/frequency/polarization), equalization, OFDM.
- Large-scale shadowing (log-normal): buildings/terrain → slow variations over tens of metres with ~8 dB standard deviation; countered by margin and handoff.
- Flat vs frequency-selective fading: if the channel's delay spread ≪ symbol period, all frequencies fade together; otherwise equalization or OFDM subcarriers are required.
Diversity rule of thumb: decorrelate branches by ≥ 10λ (space) or ≥ coherence bandwidth (frequency) so fades do not coincide.
5. Diffraction Loss
When the path is blocked by a ridge, rooftop or building edge, the knife-edge model gives:
Knife-edge loss L(v) ≈ 6.9 + 20 log( √( (v − 0.1)² + 1 ) + v − 0.1 ) dB
Reference points v = 0 (grazing): ≈ 6 dB | v = 1: ≈ 16 dB | v = 2: ≈ 22 dB
- Smooth spherical earth (plains, sea): diffraction loss grows steadily beyond LOS — the service fades gradually rather than cutting off.
- Deliberate use: site links on ridges; use building edges to shield from interference (terrain is a design material).
6. Atmospheric Effects: Refraction, Ducting, Scatter, Rain
- Refraction: the standard atmosphere gives k = 4/3; strong super-refraction (k > 1) extends the horizon; sub-refraction (k < 1) shrinks it.
- Ducting: temperature inversions trap VHF/UHF near the surface → over-water and flat-land interference far beyond normal range; both an opportunity (extended coverage) and a hazard (co-channel interference).
- Tropospheric scatter: common-volume scattering at microwaves enables 100–500 km beyond-LOS links at the price of 30–60 dB scatter loss (huge antennas, high power — a specialist niche).
- Rain attenuation: above ~10 GHz, raindrops absorb and scatter; specific attenuation γR = k Rα (ITU-R P.838), e.g. ≈ 3–4 dB/km at 20 GHz in 25 mm/h rain.
- Other losses: atmospheric absorption peaks at 22 GHz (water vapour) and 60 GHz (oxygen); fog and sandstorms add modest loss at high frequencies.
7. Path-Loss Models
| Model | Applicability | Form | Notes |
|---|---|---|---|
| Free space | LOS, clear path | L = 32.44 + 20log f + 20log d | The anchor; never enough outdoors alone |
| Two-ray | Low antennas, open flat ground | Pr ∝ 1/d⁴ beyond dc | Explains mobile slope of ~40 dB/decade |
| Okumura | Urban UHF (measured curves) | Curves + corrections | Historical basis of cellular planning |
| Hata / COST-231 | 150–1500 / up to 2 GHz | Closed-form urban/suburban/rural | Fast, widely used in tools |
| ITU-R P.1546 / P.1812 | Point-to-area broadcast | Field-strength curves + terrain | Regulatory standard for coverage |
Model selection discipline: match the model to the scenario (height, clutter, frequency), calibrate with measured data, and document the model used in every budget.
8. The Space-Wave Link Budget
Noise N = −174 + 10log B + NF (dBm)
Required C/N ≈ 10 dB (QPSK) · 18 dB (64-QAM) · 24 dB (256-QAM) + implementation loss ~1–3 dB
Fade margin Rayleigh 99.9 % availability ⇒ ~20–30 dB above median; rain per ITU-R P.618 availability targets
- Add scenario losses: clutter, knife-edge diffraction, building penetration (10–20 dB indoors), body/vegetation losses.
- Microwave hops: add rain fade over the full path at the target availability; consider adaptive modulation and ATPC (automatic transmit power control).
- Satellite: add pointing loss (~0.3–1 dB), atmospheric absorption, rain on the slant path, and scintillation margins.
9. Antennas for Space-Wave Links
| Antenna | Typical gain | Pattern | Applications |
|---|---|---|---|
| Dipole / collinear arrays | 0–8 dBi | Omnidirectional azimuth | Broadcast, base-station sectors |
| Yagi–Uda | 7–15 dBi | 20–60° beam | TV/FM reception, point-to-point |
| Panel / sector antennas | 10–20 dBi | 60–120° azimuth | Cellular/WiFi access |
| Parabolic dish | 25–50 dBi | 1–10° beam | Satellite, microwave backhaul |
| Horn / grid / Cassegrain | 10–40 dBi | Various | Microwave, test ranges, feeds |
- Point and match: pointing error ≈ beamwidth costs ~3 dB; polarisation must match the link (linear or circular as specified).
- Feeder discipline: at microwaves, every dB of waveguide/coax loss directly subtracts from margin — mast-head LNA for receivers.
10. Microwave Backhaul and Satellite Links
10.1 Terrestrial Microwave (Backhaul)
- Licensed 6/7/8/11/13/15/18/23/26/32/38/42 GHz; channel widths 7/14/28/56 MHz; capacities to several Gbit/s with 1024–4096 QAM.
- Hop design: path profile → Fresnel clearance → FSPL + rain → fade margin → antenna/feeder → redundancy (1+1, hitless diversity for critical hops).
- Typical availability target 99.99 % (≤ 53 min downtime/year) — drives the rain-margin calculation.
10.2 Satellite Links
| Orbit | Altitude | Path loss (C-band ~6 GHz) | Notes |
|---|---|---|---|
| GEO | 35 786 km | ≈ 200 dB | Fixed pointing; Ku/Ka rain fade; ~250 ms latency |
| MEO | 8 000–20 000 km | ≈ 185–192 dB | GNSS constellations |
| LEO | 500–1 200 km | ≈ 165–172 dB | Steered beams, Doppler, handovers |
11. Worked Examples
Example 1 — Radio horizon
Find the radio horizon for a 60 m tower and a 3 m rooftop.
Example 2 — Fresnel clearance
A 6 GHz, 30 km microwave hop: find the first-Fresnel-zone maximum radius and the required clearance.
Example 3 — Knife-edge loss
A hill 10 m above the LOS line sits mid-path on a 10 km, 1 GHz link. Estimate the loss.
Example 4 — FSPL and received power
FSPL and Pr for 14 GHz, 15 km, 10 W (+40 dBm), 35 dBi dishes each end, 2 dB feeder loss.
Example 5 — Dish sizing
What diameter dish gives 35 dBi at 12 GHz (η = 0.6)? What is its beamwidth?
Example 6 — Rain margin check
The Example 4 link must survive 99.99 % availability in a region where 0.01 % of the year has rain of 10 mm/h at 14 GHz (γR ≈ 2 dB/km). Check the margin.
Example 7 — Two-ray slope check
A base station (30 m) serves a handset (1.5 m) at 900 MHz. Find the critical distance and the slope beyond it.
12. Interactive Calculators
12.1 Radio Horizon (k = 4/3)
12.2 Fresnel-Zone Calculator
12.3 Knife-Edge Diffraction Loss
12.4 Link Budget Calculator
12.5 Dish Gain & Beamwidth
13. Summary of Key Points
- The space wave dominates VHF and above: direct path + reflections + diffraction; the channel is geometry, statistics and weather.
- Radio horizon d ≈ 4.12(√h₁ + √h₂) km with the k = 4/3 effective earth; height is coverage.
- LOS is a volume: r₁ = √(λd₁d₂/(d₁+d₂)), and 60 % of the first Fresnel zone must stay clear.
- Two-ray interference gives the 1/d⁴ law beyond the critical distance; multipath yields Rayleigh (small-scale) and log-normal (shadowing) fading — fight with diversity, equalization, OFDM and margin.
- Knife-edge loss follows the Fresnel parameter v (0 dB at clearance, 6 dB grazing, 16 dB at v = 1); spherical-earth diffraction gives gradual beyond-horizon fade.
- The atmosphere refracts (k factor), ducts (extended interference), scatters (troposcatter links) and rains (γR = kRα above ~10 GHz).
- Choose models to match the scenario: FSPL anchor, two-ray mobile, Hata/COST-231 urban, P.1546/P.1812 regulatory.
- Link budgets: Pr = Pt + Gt + Gr − losses; N = −174 + 10log B + NF; margins for Rayleigh (20–30 dB) and rain (per availability target).
- Antennas: gain ∝ (D/λ)² for dishes, beamwidth ≈ 70°λ/D; pointing and feeder losses matter at microwave.
- Microwave backhaul and satellite links are the same discipline at longer distances — complete the whole budget chain before judging feasibility.
14. Self-Test Quiz (12 Questions)
Attempt each question, then click "Check Answers".
1. The space wave is dominant at:
2. The radio horizon formula d(km) ≈ 4.12(√h₁ + √h₂) uses:
3. The required clearance on a line-of-sight path is:
4. Beyond the critical distance in a two-ray channel, received power falls as:
5. Knife-edge diffraction loss at grazing incidence (v = 0) is approximately:
6. Rain attenuation becomes a first-order design term roughly above:
7. A 60 cm dish at 12 GHz (η = 0.6) has gain of approximately:
8. The thermal noise floor in dBm is:
9. Doubling the bandwidth of a link (with everything else fixed) changes the noise floor by:
10. FSPL at 14 GHz over 15 km is approximately:
11. GEO satellite links overcome ~200 dB path loss primarily via:
12. The standard remedy for flat multipath fading on a microwave hop is:
15. References & Further Reading
- Rappaport, T. S. — Wireless Communications: Principles and Practice, 2nd ed. (two-ray, fading, Hata).
- Stutzman, W. L. & Thiele, G. A. — Antenna Theory and Design (dish gain, patterns).
- Kraus, J. D. & Marhefka, R. J. — Antennas for All Applications.
- ITU-R P.1546 / P.1812 — Point-to-area field-strength prediction methods.
- ITU-R P.838 — Specific attenuation model for rain.
- ITU-R P.618 — Propagation data and prediction methods for Earth–space telecommunication systems.
- Freeman, R. L. — Radio System Design for Telecommunications (microwave link engineering).
Prepared for ECE 523E — Antenna & Radio Wave Propagation, Department of Electrical & Communication Engineering, Masinde Muliro University of Science and Technology.