Short note: Rayleigh and Ricean fading distribution

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:

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:

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.

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