Rayleighfading

Rayleigh fading is a statistical model used to describe rapid fluctuations in the amplitude of a received radio signal in a multipath environment that lacks a dominant line-of-sight component. It arises when a transmitted signal reaches the receiver through many scattered paths, each with a random phase, so the received complex baseband signal is the sum of many independent phasors.

If the in-phase and quadrature components are modeled as independent zero-mean Gaussian random variables with equal variance, the envelope r = |h| of the complex channel coefficient h follows a Rayleigh distribution. Its probability density function is p_R(r) = (r / sigma^2) exp(-r^2/(2 sigma^2)) for r ≥ 0, where sigma^2 is the average power per dimension. The instantaneous power, proportional to r^2, has an exponential distribution. The instantaneous SNR gamma thus follows p_gamma(g) = (1/gamma_bar) exp(-g/gamma_bar) with mean gamma_bar.

Temporal variation is produced by motion, causing Doppler spread and time-varying fading. The envelope and channel gain are typically modeled as stochastic processes; the Jakes model is a common reference, and the coherence time is related to the Doppler frequency (roughly 0.423 divided by the maximum Doppler frequency).

Rayleigh fading is a special case of broader fading models. If a nonzero line-of-sight component exists, the envelope is better described by the Rice (Rician) distribution; Rayleigh is also a special case of the Nakagami-m family with m = 1.

Applications include wireless system design and performance analysis, where Rayleigh fading informs outage probability, bit-error-rate calculations, and diversity gains in mobile and indoor environments.