Rayleigh fading

 ( Broadcast Engineering ) 

$\backslash mathrm\{AFD\}\; =\; \backslash frac\{e^\{\backslash rho^2\}-1\}\{\backslash rho\; f\_d\; \backslash sqrt\{2\backslash pi\}\}.$

The level crossing rate and average fade duration taken together give a useful means of characterizing the severity of the fading over time.

For a particular normalized threshold value $\backslash rho$, the product of the average fade duration and the level crossing rate is a constant and is given by

$\backslash mathrm\{AFD\}\; \backslash times\; \backslash mathrm\{LCR\}\; =\; 1\; -\; e^\{-\backslash rho^2\}.$

$S(\backslash nu)\; =\; \backslash frac\{1\}\{\backslash pi\; f\_d\; \backslash sqrt\{1\; -\; \backslash left(\backslash frac\; \backslash nu\; \{f\_d\}\backslash right)^2\}\},$

where $\backslash ,\backslash !\backslash nu$ is the frequency shift relative to the carrier frequency. This equation is valid only for values of $\backslash ,\backslash !\backslash nu$ between $\backslash pm\; f\_d$; the spectrum is zero outside this range. This spectrum is shown in the figure for a maximum Doppler shift of 10 Hz. The 'bowl shape' or 'bathtub shape' is the classic form of this Doppler spectrum.

$\backslash ,\backslash !f\_n\; =\; f\_d\backslash cos\backslash alpha\_n$

Here, $\backslash ,\backslash !\backslash alpha$ and the $\backslash ,\backslash !\backslash beta\_n$ and $\backslash ,\backslash !\backslash theta\_\{n,k\}$ are model parameters with $\backslash ,\backslash !\backslash alpha$ usually set to zero, $\backslash ,\backslash !\backslash beta\_n$ chosen so that there is no cross-correlation between the real and imaginary parts of $R(t)$:

$\backslash ,\backslash !\backslash beta\_n\; =\; \backslash frac\{\backslash pi\; n\}\{M+1\}$

$\backslash theta\_\{n,k\}\; =\; \backslash beta\_n\; +\; \backslash frac\{2\backslash pi(k-1)\}\{M+1\}.$

In fact, it has been shown that the waveforms are correlated among themselves — they have non-zero cross-correlation — except in special circumstances. The model is also deterministic (it has no random element to it once the parameters are chosen). A modified Jakes's model chooses slightly different spacings for the scatterers and scales their waveforms using Walsh matrix to ensure zero cross-correlation. Setting

$\backslash alpha\_n\; =\; \backslash frac\{\backslash pi(n-0.5)\}\{2M\}\; \backslash text\{\; and\; \}\backslash beta\_n\; =\; \backslash frac\{\backslash pi\; n\}\; M,$

results in the following model, usually termed the Dent model or the modified Jakes model:

$R(t,k)\; =\; \backslash sqrt\{\backslash frac\; 2\; M\}\; \backslash sum\_\{n=1\}^M\; A\_k(n)\backslash left(\; \backslash cos\backslash beta\_n\; +\; j\backslash sin\backslash beta\_n\; \backslash right)\backslash cos\backslash left(2\backslash pi\; f\_d\; t\; \backslash cos\backslash alpha\_n\; +\; \backslash theta\_n\backslash right).$

The weighting functions $A\_k(n)$ are the $k$^th Walsh–Hadamard sequence in $n$. Since these have zero cross-correlation by design, this model results in uncorrelated waveforms. The phases $\backslash ,\backslash !\backslash theta\_n$ can be initialised randomly and have no effect on the correlation properties. The fast Walsh transform can be used to efficiently generate samples using this model.

The Jakes's model also popularised the Doppler spectrum associated with Rayleigh fading, and, as a result, this Doppler spectrum is often termed Jakes's spectrum.

$S\_\{Doppler\}(f)\; =\; |H\_\{Butterworth\}|^2\; =\; \backslash frac\{B\}\{\backslash sqrt\{1\; -\; (f/f\_0)^\{2k\}\}\}$

where f is a frequency, $H\_\{Butterworth\}$ is the Butterworth filter response, B is the normalization constant, k is the filter order and $f\_0$ is the Cutoff frequency which should be selected with respect to maximum Doppler shift.

• Fading
• Rayleigh scattering
• Rician fading
• Non-line-of-sight propagation
• Line-of-sight propagation
• Wireless
• Rayleigh distribution
• Two-wave with diffuse power (TWDP) fading
• Lord Rayleigh

• Rayleigh fading channel signal generator using the Dent model (Matlab)