Editing Rayleigh fading (section)

==Properties==
Since it is based on a well-studied distribution with special properties, the Rayleigh distribution lends itself to analysis, and the key features that affect the performance of a wireless network have [[analytic expression]]s.

Note that the parameters discussed here are for a non-static channel. If a channel is not changing with time, it does not fade and instead remains at some particular level. Separate instances of the channel in this case will be uncorrelated with one another, owing to the assumption that each of the scattered components fades independently. Once relative motion is introduced between any of the transmitter, receiver, and scatterers, the fading becomes correlated and varying in time.

===Level crossing rate===
The level crossing rate is a measure of the rapidity of the fading. It quantifies how often the fading crosses some threshold, usually in the positive-going direction. For Rayleigh fading, the level crossing rate is:<ref name="rappaport">{{cite book|title=Wireless Communications: Principles and Practice|edition=2nd|author=T. S. Rappaport|date=December 31, 2001|publisher=Prentice Hall PTR|isbn=978-0-13-042232-3}}</ref>
:<math>\mathrm{LCR} = \sqrt{2\pi}f_d\rho e^{-\rho^2}</math>
where <math>f_d</math> is the maximum Doppler shift and <math>\,\!\rho</math> is the threshold level normalised to the [[root mean square]] (RMS) signal level:
:<math>\rho = \frac{R_\mathrm{threshold}}{R_\mathrm{rms}}.</math>

===Average fade duration===
The average fade duration quantifies how long the signal spends below the threshold <math>\,\!\rho</math>. For Rayleigh fading, the average fade duration is:<ref name="rappaport" />
:<math>\mathrm{AFD} = \frac{e^{\rho^2}-1}{\rho f_d \sqrt{2\pi}}.</math>

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 <math>\rho</math>, the product of the average fade duration and the level crossing rate is a constant and is given by

:<math>\mathrm{AFD} \times \mathrm{LCR} = 1 - e^{-\rho^2}. </math>

===Doppler power spectral density===
[[File:Rayleigh Doppler PSD 10Hz.svg|thumb|250px|right|The normalized Doppler power spectrum of Rayleigh fading with a maximum Doppler shift of 10 Hz]]
The Doppler [[spectral density|power spectral density]] of a fading channel describes how much spectral broadening it causes. This shows how a pure frequency, e.g., a pure sinusoid, which is an [[impulse function|impulse]] in the frequency domain, is spread out across frequency when it passes through the channel. It is the Fourier transform of the time-autocorrelation function. For Rayleigh fading with a vertical receive antenna with equal sensitivity in all directions, this has been shown to be:<ref name="clarke">{{cite journal | title=A Statistical Theory of Mobile Radio Reception | author=R. H. Clarke | journal=Bell System Technical Journal | volume=47 | issue=6 | date=July–August 1968 | pages=957–1000 | doi=10.1002/j.1538-7305.1968.tb00069.x}}</ref>

:<math>S(\nu) = \frac{1}{\pi f_d \sqrt{1 - \left(\frac \nu {f_d}\right)^2}},</math>

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