Editing Rayleigh fading (section)

===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.