MT Rayleigh Selective Fading profile (Jakes)

Rayleigh Distribution Equation

The Rayleigh distribution describes a flat-fading channel characterized by a single-tap impulse response with a time-varying envelope that is Rayleigh-distributed. This model describes the statistical time-varying nature of the received envelope of a flat fading channel or the envelope of an individual multipath component. The Rayleigh distribution has a probability density function (PDF) given by the following equation:

$p\left(r\right)=\frac{r}{{\sigma }^2}\exp (-\frac{r^2}{{2\sigma }^2})u\left(r\right)$

where r is the specified fading variance.

Jakes Model Equation

The Jakes model is a deterministic method that simulates time-correlated Rayleigh fading waveforms. The model assumes that N equal-strength rays arrive at a moving receiver with uniformly distributed arrival angles, such that ray n experiences a Doppler shift defined by the following equation:

${\omega }_n={\omega }_m\cos \left({\alpha }_n\right)$

where

${\omega }_n={2\pi f}_m$

and

${\alpha }_m=2\pi \frac{(n-0.5)}{N}$ represents the arrival angle of the ray n.