Editing Rayleigh fading (section)

===Jakes's model===
In his book,<ref>{{cite book|title=Microwave Mobile Communications|editor=William C. Jakes|publisher=John Wiley & Sons Inc|location=New York|date=February 1, 1975|isbn=978-0-471-43720-8}}</ref>  Jakes popularised a model for Rayleigh fading based on summing [[Sine wave|sinusoid]]s. Let the scatterers be uniformly distributed around a circle at angles <math>\alpha_n</math> with <math>k</math> rays emerging from each scatterer. The Doppler shift on ray <math>n</math> is
:<math>\,\!f_n = f_d\cos\alpha_n </math>
and, with <math>M</math> such scatterers, the Rayleigh fading of the <math>k^\text{th}</math> waveform over time <math>t</math> can be modelled as:

: <math>
\begin{align}
R(t,k) = 2\sqrt{2}\left[\sum_{n=1}^M \right. & \left(\cos\beta_n + j\sin\beta_n\right)\cos\left(2 \pi f_n t + \theta_{n,k}\right) \\[4pt]
& \left. {} + \frac 1 {\sqrt{2}} \left(\cos\alpha + j\sin\alpha\right)\cos(2 \pi f_d t)\right].
\end{align}
</math>

Here, <math>\,\!\alpha</math> and the <math>\,\!\beta_n</math> and <math>\,\!\theta_{n,k}</math> are model parameters with <math>\,\!\alpha</math> usually set to zero, <math>\,\!\beta_n</math> chosen so that there is no cross-correlation between the real and imaginary parts of <math>R(t)</math>:
:<math>\,\!\beta_n = \frac{\pi n}{M+1}</math>
and <math>\,\!\theta_{n,k}</math> used to generate multiple waveforms. If a single-path channel is being modelled, so that there is only one waveform then <math>\,\!\theta_n</math> can be zero. If a multipath, frequency-selective channel is being modelled so that multiple waveforms are needed, Jakes suggests that uncorrelated waveforms are given by

:<math>\theta_{n,k} = \beta_n + \frac{2\pi(k-1)}{M+1}.</math>

In fact, it has been shown that the waveforms are correlated among themselves — they have non-zero cross-correlation — except in special circumstances.<ref>{{cite book|title=Kanalmodeller för radiotransmission (Channel models for radio transmission)|type=Master's thesis|author1=Von Eckardstein, S.  |author2=Isaksson, K. |name-list-style=amp |publisher=Royal Institute of Technology|location=Stockholm, Sweden|date=December 1991|language=sv}}</ref> The model is also [[deterministic]] (it has no random element to it once the parameters are chosen). A modified Jakes's model<ref>{{cite journal|title=Jakes Fading Model Revisited|author=P. Dent, G. E. Bottomley and T. Croft|journal=Electronics Letters|volume=29|issue=13|pages=1162–1163|date=24 June 1993 | doi = 10.1049/el:19930777|bibcode=1993ElL....29.1162D}}</ref> chooses slightly different spacings for the scatterers and scales their waveforms using [[Walsh matrix|Walsh–Hadamard sequences]] to ensure zero cross-correlation. Setting

:<math>\alpha_n = \frac{\pi(n-0.5)}{2M} \text{ and }\beta_n = \frac{\pi n} M,</math>

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

: <math>R(t,k) = \sqrt{\frac 2 M} \sum_{n=1}^M A_k(n)\left( \cos\beta_n + j\sin\beta_n \right)\cos\left(2\pi f_d t \cos\alpha_n + \theta_n\right).</math>

The weighting functions <math>A_k(n)</math> are the <math>k</math><sup>th</sup> Walsh–Hadamard sequence in <math>n</math>. Since these have zero cross-correlation by design, this model results in uncorrelated waveforms. The phases <math>\,\!\theta_n</math> 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.