Wireless Notes

Rayleigh Fading Model in Wireless Communication

Learn Rayleigh fading with statistical model, PDF CDF formulas, deep fades, BER degradation over Rayleigh channel, level crossing rate, diversity mitigation, and when Rayleigh applies for engineering students.

Rayleigh fading is a statistical model that describes how the signal envelope varies when there is no dominant Line-of-Sight (LOS) component – only multiple scattered/reflected paths exist.

🎯 What is Rayleigh Fading?

When many reflected/scattered signal copies (no strong LOS) arrive at the receiver, the combined signal's envelope (amplitude) follows a Rayleigh distribution.

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📍 When Does Rayleigh Fading Occur?

Condition	Rayleigh?	Example
Dense urban, mobile	✅ Yes	City streets, NLoS
Indoor NLoS	✅ Yes	Different rooms
Urban, ground level	✅ Yes	Phone in city
Rural with LOS	❌ No (Rician)	Open fields
Satellite (clear sky)	❌ No (Rician/AWGN)	Direct path exists
Indoor same room LOS	❌ No (Rician)	Device sees router

Rule of Thumb:

No LOS → Rayleigh fading
Strong LOS + scattering → Rician fading
Pure LOS (no scattering) → AWGN channel (no fading)

📐 Mathematical Model

In the Rayleigh fading model, the received signal is the sum of many reflected paths:

Signal Model:

In-phase: X = Σ aᵢ×cos(φᵢ)	Gaussian(0, σ²)
Quadrature: Y = Σ aᵢ×sin(φᵢ)	Gaussian(0, σ²)
Envelope: r = √(X² + Y²)	Rayleigh distributed
Phase: θ = arctan(Y/X)	Uniform [0, 2π]

📊 PDF and CDF

Rayleigh PDF (Probability Density Function):

│ Mean	E[r] = σ√(π/2) ≈ 1.253σ │
│ Variance	Var[r] = (2 - π/2)σ² ≈ 0.429σ² │
│ RMS	r_rms = σ√2 │
│ Median	r_median = σ√(2×ln2) ≈ 1.177σ │

PDF Shape:

f(r) ↑

│ ╱╲

│ ╱ ╲

│ ╱ ╲╲

│╱ ╲╲╲

┼──────────────╲╲╲──── r

0 σ σ√2 3σ

↑peak

CDF (Probability of r ≤ R):

F(R) = P(r ≤ R) = 1 - exp(-R²/2σ²)

Probability of being below threshold R:

P(r ≤ R) = 1 - exp(-R²/2σ²)

🔑 Key Properties

Property	Value	Hindi
Mean	σ√(π/2) ≈ 1.25σ	Average envelope
Mode (peak)	σ	Most probable value
Median	1.177σ	50% time above this
RMS	σ√2	Root mean square
Mean power	2σ²	Average power (Ω)
Variance	(2-π/2)σ²	Spread around mean

Power Distribution (in dB relative to mean):

Level relative to RMS	% Time above	% Time below
+10 dB	0.5%	99.5%
+5 dB	8%	92%
0 dB (mean)	37%	63%
-5 dB	72%	28%
-10 dB	90%	10%
-20 dB	99%	1%
-30 dB	99.9%	0.1%

The signal is 10 dB below the mean for 10% of the time, and 20 dB below the mean for 1% of the time. These deep fades seriously affect the system!

📉 Deep Fades

In Rayleigh fading, the signal sometimes becomes very weak (deep fade). This occurs when multipath copies combine destructively.

┌─────────────────────────────────────────────────────────────────────┐ │ RAYLEIGH FADING OVER TIME │ │ │ │ Signal ┌─ Mean signal level │ │ Power ↑ │ │ │ (dB) │ │ ╱╲ ╱╲ ╱╲ ╱╲ │ │ +5 │───┤─╱──╲─╱──╲╱──╲─────╱──╲────── │ │ │ │╱ ╲╱ ╲ ╱ ╲ │ │ 0 │───┤ ╲ ╱ ╲ ╱╲ │ │ │ │ ╳ ╲ ╱ ╲ │ │ -5 │───┤ ╱ ╲ ╳ ╲ │ │ │ │ ╱ ╲ ╱ ╲ ╲ │ │ -10 │───┤ ╱ ╲ ╱ ╲ ╲ │ │ │ │ ╲ ╱ ╲╱ │ │ -20 │───┤ ╲╱ ← DEEP FADE! │ │ │ │ │ │ ──────┼───┼────────────────────────────────── Time │ │ │ │ │ Deep fades occur randomly but inevitably! │ │ Duration depends on speed: faster movement = shorter fades │ └─────────────────────────────────────────────────────────────────────┘

Fade Statistics:

Metric	Formula	Meaning
Level Crossing Rate (LCR)	NR = √(2π)×fm×ρ×e^(-ρ²)	How often signal crosses threshold
Average Fade Duration (AFD)	τ̄ = (e^(ρ²)-1)/(ρ×fm×√(2π))	How long each fade lasts

Where: ρ = threshold/RMS, fm = max Doppler frequency

💀 Impact on Communication

BER Degradation:

Rayleigh fading dramatically worsens BER compared to an AWGN channel:

Average SNR	BER (AWGN)	BER (Rayleigh)	Degradation
10 dB	4×10⁻⁶	5×10⁻²	~10,000× worse!
20 dB	~0	5×10⁻³	Much worse
30 dB	~0	5×10⁻⁴	Still significant
40 dB	~0	5×10⁻⁵	Getting acceptable

BER Formula (BPSK over Rayleigh):

At high SNR	BER ≈ 1/(4×γ̄) (decreases as 1/SNR!)
Compare AWGN	BER decreases exponentially with SNR
Rayleigh: BER decreases only linearly (1/SNR)	MUCH slower!

In AWGN, increase SNR by 3 dB → BER drops rapidly. In Rayleigh, increase by 3 dB → BER only halves. Therefore, diversity is ESSENTIAL in fading channels.

🛡️ Mitigation

Technique	Diversity Gain	Used In
Spatial diversity (MRC)	L-branch: ~10log₁₀(L) dB	MIMO systems
Frequency diversity	Multiple carriers	OFDM
Time diversity	Coding + interleaving	All systems
Antenna selection	Pick strongest	Simple MIMO
Adaptive MCS	Adjust rate to channel	4G/5G AMC
Power control	Increase power in fades	CDMA, LTE

Diversity Effect on BER:

No diversity	BER ∝ 1/SNR (very slow improvement)
2-branch MRC	BER ∝ 1/SNR² (much better!)
4-branch MRC	BER ∝ 1/SNR⁴ (excellent!)
L-branch MRC	BER ∝ 1/SNR^L (exponential improvement)

📝 Summary

Concept	Key Point
When	No LOS, many scattered paths
Distribution	Envelope: Rayleigh, Phase: Uniform
PDF	f(r) = (r/σ²)×exp(-r²/2σ²)
Deep fades	10% time: -10 dB, 1% time: -20 dB
BER impact	1/SNR (much worse than AWGN)
Key mitigation	Diversity (spatial, freq, time)
Modern solution	MIMO + OFDM + Adaptive modulation
Environment	Urban NLoS, indoor NLoS

❓ FAQ

Q: When does Rayleigh fading not apply? A: When there is a strong LOS path (use Rician model), when the channel is static (no movement), or when signal bandwidth is very narrow relative to coherence bandwidth.

Q: Why does MIMO work well in Rayleigh fading? A: MIMO uses multiple antennas. The probability that ALL antennas are simultaneously in a deep fade is very low. Diversity gain dramatically improves BER.

Q: 5G mein Rayleigh fading relevant hai? A: Haan! 5G sub-6 GHz bands mein urban NLoS = Rayleigh fading. mmWave mein scenario thoda different hai (sparser multipath, clustered scattering), but fading still exists.

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