---
title: "Short note: Rayleigh and Ricean fading distribution"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:rayleigh-ricean-distribution-note"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/rayleigh-ricean-distribution-note
site: BCT Engineering
---

# Short note: Rayleigh and Ricean fading distribution

Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks

- Asked in: 2070 Magh, 4 marks
- Video: [Rayleigh and Ricean Distribution - Mobile Radio Propagation - Mobile Communication System](https://www.youtube.com/watch?v=kuNLZLTZNfw) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/rayleigh-ricean-distribution-note
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Write a short note on Rayleigh and Ricean fading distribution.

## Short answer

Rayleigh fading (no LOS) envelope: p(r) = (r/σ²)exp(-r²/2σ²) with deep fades. Ricean fading adds dominant LOS, K = A²/(2σ²); envelope includes I₀(Ar/σ²). Large K gives near-constant envelope; K → -∞ yields Rayleigh.

## Answer

#### Rayleigh fading distribution
When there is **no line-of-sight** path, the received signal is the sum of many scattered waves with random phases. By the central limit theorem its in-phase and quadrature parts are Gaussian, so the envelope r follows the Rayleigh pdf:
```text
p(r) = (r/σ²) exp(−r²/2σ²),   0 ≤ r ≤ ∞
     = 0,                     r < 0
```
- σ = rms value of the received signal before envelope detection; σ² = time-average power.
- Mean = 1.2533σ, median = 1.177σ, variance = 0.4292σ².
- Models the worst case: deep fades (20–40 dB) occur often. Typical of dense urban areas.

#### Ricean fading distribution
When a **dominant, steady component** (LOS path) exists along with random scattered waves, the envelope follows the Ricean pdf:
```text
p(r) = (r/σ²) exp(−(r² + A²)/2σ²) I₀(Ar/σ²),
        A ≥ 0, r ≥ 0
```
- A = peak amplitude of the dominant signal; I₀ = modified Bessel function of first kind, zero order.
- **Rician factor** K = A²/(2σ²), or K(dB) = 10 log(A²/2σ²).
- As A → 0 (K → −∞ dB) the dominant path fades away and the Ricean distribution becomes Rayleigh. Large K means a nearly constant (Gaussian-like) envelope.

```text
 p(r)   K = −∞ dB (Rayleigh)
  |   /\       K = 6 dB
  |  /  \      /\
  | /    \    /  \
  |/      \__/    \__
  +--------------------> r
```

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is the Rayleigh envelope obtained with no LOS?
   A. I and Q are Gaussian by the central limit theorem from many scattered waves
   B. There is one dominant wave
   C. Shadowing is log-normal
   D. The mobile is stationary
   Answer: A. Many random-phase scattered waves give Gaussian I and Q components, hence a Rayleigh envelope. (Key point: Why Rayleigh)

2. Rayleigh fading is considered:
   A. The mildest case
   B. A worst case with frequent deep fades
   C. Free of fades
   D. Only for LOS
   Answer: B. Deep fades of 20-40 dB occur often, typical of dense urban areas. (Key point: Rayleigh behaviour)

3. Large K in Ricean fading means:
   A. Deeper fades
   B. Nearly constant, Gaussian-like envelope
   C. No signal
   D. Pure Rayleigh
   Answer: B. A strong dominant component dominates the scattered part, giving a steady envelope. (Key point: Rician factor)

4. In the Ricean pdf, A represents:
   A. Peak amplitude of the dominant signal
   B. Average power of the scattering
   C. Doppler shift
   D. Delay spread
   Answer: A. A is the peak amplitude of the dominant (LOS) signal, with K = A^2/(2 sigma^2). (Key point: Rician parameters)

## Related in Wireless Communication

- [Frequency selective fading and Rayleigh distribution](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afrequency-selective-rayleigh) — Wireless Communication, chapter 3
- [Rayleigh versus Rician fading channel](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Arayleigh-vs-rician-fading) — Wireless Communication, chapter 3