Wireless Notes
Learn shadowing log-normal fading with distribution model, standard deviation values, shadow margin calculation, coverage probability, correlation distance, and network planning implications for engineering students.
Shadowing (or shadow fading) occurs when large obstacles (buildings, hills, groups of trees) come in the signal path and randomly attenuate the signal.
🏢 What is Shadowing?
Shadowing consists of slow, random variations in the signal caused by large obstacles. When you move and go behind a building, the signal drops suddenly – this is shadowing.
| ╲ | |
|---|---|
| ╲ ← Path loss model (predicted) | |
| ╲╲ | |
| ╲╲ ╱╲ ← Actual signal (with shadowing) | |
| ╲╲╱ ╲ | |
| ╲ ╲╱╲ | |
| ╲ ╲ ╱╲ | |
| ╲╲ ╲╱ ╲╲ | |
| ╲╲ ╲╲ | |
| ╲╲ ╱╲ ╲╲ |
Shadowing vs Small-Scale Fading:
| Property | Shadowing | Small-Scale Fading |
|---|---|---|
| Scale | 10s-100s of meters | Centimeters (λ/2) |
| Speed | Slow variations | Fast fluctuations |
| Cause | Large obstacles | Multipath interference |
| Distribution | Log-Normal (dB: Gaussian) | Rayleigh/Rician |
| Also called | Large-scale fading | Fast fading |
| Variation | 4-12 dB (std dev) | Up to 30-40 dB |
📊 Log-Normal Distribution
An interesting property of shadowing is that signal variations measured in dB follow a Gaussian (Normal) distribution. That is why it is called Log-Normal fading.
| │ In dB | Shadowing ~ N(0, σ²) (Gaussian) │ |
| │ PDF (in dB) | │ |
| │ Where | │ |
PDF Shape:
f(X)↑
│ ╱╲
│ ╱ ╲
│ ╱ ╲
│ ╱ ╲
│ ╱ ╲
│ ╱ ╲
│ ╱ ╲
│────╱──────────────╲────── X (dB)
│ -2σ 0 (mean) +2σ
│
Symmetric bell curve centered at 0 dB
σ = standard deviation (4-12 dB typically)
📐 Shadowing Parameters
Standard Deviation (σ):
| Environment | σ (dB) | Hindi |
|---|---|---|
| Open/rural | 4-6 dB | Khula maidan |
| Suburban | 6-8 dB | Outside city |
| Urban | 8-10 dB | Sheher mein |
| Dense urban | 10-12 dB | Very dense buildings |
| Indoor | 3-6 dB | Inside building |
| Indoor (multi-floor) | 6-10 dB | Between floors |
Probability Table (Gaussian):
| Deviation | Probability | Meaning |
|---|---|---|
| Within ±1σ | 68.3% | Most common variation |
| Within ±2σ | 95.4% | Almost always within this |
| Within ±3σ | 99.7% | Practically always |
| Beyond +2σ | 2.3% | Signal much stronger than expected |
| Beyond -2σ | 2.3% | Signal much weaker than expected |
If σ = 8 dB, then 95% of the time the signal will be within ±16 dB of the predicted value.
📈 Path Loss with Shadowing
In the complete path loss model, shadowing is added as a random term:
┌──────────────────────────────────────────────────────┐
│ │
│ PL(d) = PL̄(d) + Xσ │
│ │
│ PL(d) = PL(d₀) + 10n×log₁₀(d/d₀) + Xσ │
│ │
│ Where: │
│ PL̄(d) = Mean path loss at distance d │
│ PL(d₀) = Path loss at reference distance │
│ n = Path loss exponent │
│ Xσ ~ N(0, σ²) = Shadowing random variable │
│ │
│ Received Power: │
│ Pr(d) = Pt + Gt + Gr - PL(d₀) - 10n×log(d/d₀) - Xσ│
│ │
└──────────────────────────────────────────────────────┘
📶 Coverage Probability
An important question in network planning: at what percentage of locations will the signal be sufficient?
Coverage Probability at distance d:
P(Pr ≥ Sensitivity) = P(Xσ ≤ Margin)
= Q((Sensitivity - Pr_mean) / σ)
= 1 - Q(Margin / σ)
Where Q(x) = complementary Gaussian CDF
Coverage vs Margin Table (σ = 8 dB):
| Shadow Margin | Coverage Probability | Q-value |
|---|---|---|
| 0 dB | 50% | Q(0) = 0.5 |
| 4 dB (0.5σ) | 69% | |
| 8 dB (1σ) | 84% | Q(-1) |
| 12 dB (1.5σ) | 93% | |
| 16 dB (2σ) | 97.7% | Q(-2) |
| 24 dB (3σ) | 99.9% | Q(-3) |
To achieve 90% coverage, typically 8-10 dB shadow margin is needed (depending on σ).
🛡️ Shadow Margin
Shadow margin is the extra signal strength kept in the design so that even after shadowing, the signal remains sufficient.
| │ Link Budget with Shadow Margin | │ |
| │ For 90% coverage | Margin ≈ 1.28σ │ |
| │ For 95% coverage | Margin ≈ 1.65σ │ |
| │ For 99% coverage | Margin ≈ 2.33σ │ |
Shadow Margin Examples:
| Target Coverage | σ = 6 dB | σ = 8 dB | σ = 10 dB |
|---|---|---|---|
| 75% | 4.0 dB | 5.4 dB | 6.7 dB |
| 90% | 7.7 dB | 10.2 dB | 12.8 dB |
| 95% | 9.9 dB | 13.2 dB | 16.5 dB |
| 99% | 14.0 dB | 18.6 dB | 23.3 dB |
📏 Correlation Distance
Shadowing is spatially correlated – nearby locations experience similar shadowing. Correlation distance indicates how far apart locations need to be for shadowing to become independent.
Autocorrelation
R(Δd) = σ² × exp(-Δd / d_corr)
Or: R(Δd) = σ² × 2^(-Δd/X_c)
Where
Δd = Separation distance
d_corr = Correlation distance (decorrelation length)
X_c = Decorrelation distance for 0.5 correlation
| Environment | Correlation Distance |
|---|---|
| Urban | 20-50 m |
| Suburban | 50-100 m |
| Rural | 100-200 m |
| Indoor | 5-10 m |
In urban areas, shadowing changes after moving 20-50 meters. This means consecutive measurements are correlated – some distance must be traveled for independent samples.
🏗️ Design Implications
For Cell Planning:
| Aspect | Impact of Shadowing |
|---|---|
| Cell radius | Must reduce to meet coverage target |
| Power budget | Need shadow margin (8-15 dB extra) |
| Handover | Shadowing causes premature/late handovers |
| Interference | Random – hard to predict exactly |
| Coverage holes | Inevitable in deep-shadow areas |
| Base station density | Need more for higher coverage % |
Strategies to Combat Shadowing:
- Increase Shadow Margin – More TX power or better antennas
- Macro Diversity – Multiple base stations, soft handover
- Small Cells – Fill coverage holes with small cells
- Relay Nodes – Extend coverage around obstacles
- Adaptive Power Control – Increase power when shadowed
- Site-Specific Design – Ray tracing for critical areas
📝 Summary
| Concept | Key Point |
|---|---|
| Definition | Large-scale random signal variations due to big obstacles |
| Distribution | Log-Normal (Gaussian in dB) |
| Standard deviation | 4-12 dB (environment dependent) |
| Scale | Changes over 10s-100s of meters |
| Path loss model | PL(d) = PL_mean(d) + Xσ |
| Coverage at 90% | Need ~1.28σ margin |
| Correlation | 20-100m (spatially correlated) |
| Mitigation | Shadow margin, macro diversity, small cells |
❓ FAQ
Q: What is the difference between shadowing and multipath fading? A: Shadowing is slow (large scale, 10s-100s meters), caused by buildings/hills. Multipath fading is fast (cm level), caused by constructive/destructive interference of reflected signal copies.
Q: How is shadowing handled in network planning? A: Shadow margin is added to the link budget. For 90% outdoor coverage, typically 8-10 dB margin is used. Plus macro diversity (multiple BSs) is employed.
Q: Indoor shadowing different hai outdoor se? A: Yes – indoors, σ is lower (3-6 dB) and correlation distance is shorter (5-10m). But there is more loss between floors (floor penetration loss).
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Shadowing Log-Normal Fading in Wireless Communication.
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