Frequency selective fading and Rayleigh distribution

Frequency selective fading

Frequency selective fading occurs when the channel has a constant gain and linear phase over a bandwidth smaller than the signal bandwidth, i.e. when

Bs > Bc   and   Ts < στ

Different frequency components of the signal fade by different amounts, so the received signal is distorted. In time, multiple delayed copies of each symbol overlap the next symbols, causing inter-symbol interference (ISI). It needs equalizers, RAKE receivers or OFDM. A common rule: the channel is frequency selective if Ts < 10στ.

Rayleigh distribution

In a mobile channel with many scattered paths and no dominant (LOS) path, the in-phase and quadrature components of the received signal are independent Gaussian random variables with zero mean and variance σ². The envelope r = √(I² + Q²) then follows the Rayleigh distribution:

p(r) = (r/σ²)·exp(−r²/(2σ²)),  0 ≤ r ≤ ∞
p(r) = 0,                       r < 0

• σ = rms value of received voltage before envelope detection; σ² = time-average power.
• CDF: P(R ≤ r) = 1 − exp(−r²/(2σ²))

p(r)
 |   .-.
 |  /   '.
 | /      '.
 |/         '--.____
 +-----σ--------------- r
   (peak at r = σ)

The phase is uniformly distributed over 0 to 2π. Rayleigh fading is used to model flat fading in dense urban areas. When a strong LOS path exists, the envelope follows the Ricean distribution instead.