Editing Rayleigh fading (section)

==Generating Rayleigh fading==
As described [[#The model|above]], a Rayleigh fading channel itself can be modelled by generating the real and imaginary parts of a complex number according to independent normal Gaussian variables. However, it is sometimes the case that it is simply the amplitude fluctuations that are of interest (such as in the figure shown above). There are two main approaches to this. In both cases, the aim is to produce a signal that has the Doppler power spectrum given above and the equivalent autocorrelation properties.

===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.

===Filtered white noise===
Another way to generate a signal with the required Doppler power spectrum is to pass a [[white noise|white]] Gaussian [[signal noise|noise]] signal through a Gaussian filter with a frequency response equal to the square-root of the Doppler spectrum required. Although simpler than the models above, and non-deterministic, it presents some implementation questions related to needing high-order filters to approximate the irrational square-root function in the response and sampling the Gaussian waveform at an appropriate rate.

===Butterworth filter as Doppler power spectral density===

[[File:Rayleigh fading.png|thumb|320px|Filtered by Butterworth filter Rayleigh time series (Sampling frequency is 120 Hz.)]]

According to <ref>Fernando Pérez-Fontán, Iria Sanchez-Lago, Roberto Prieto Cerdeira, andAna Bolea-Alama nac. Consolidation of a multi-state narrowband land mo-bile satellite channel model. In The Second European Conference on Anten-nas and Propagation, EuCAP 2007., pages 1 –6, nov. 2007</ref><ref>Fontæn, F.P. and Espiæeira, P.M., 2008. [https://d1wqtxts1xzle7.cloudfront.net/60491689/Modelling.the.Wireless.Propagation.Channel20190904-14283-1j7pfux-with-cover-page-v2.pdf?Expires=1634677755&Signature=R~6XK97YX7dUMo8uXdvNePiL4XPYpUVAPmzv5UoMotLHtuFPsAS776PgSP80s-aNs1CKZXN95YCwgz5CNf5GMLTJTRqp1YAYKnEX0urYDqO~w62jiFR-ruKNjUJoYUvB~I2uUXPty3Swa7P19axRmxvf2ZWpMZe83ewMCQ3PmC6UbtTpdSUyGNFGl-RHgAPDCFosfgLcyVqZz8RJTrmClZpj8huspz3YBmlnTmgYcD7wOqFu2trDG4qAy5mEU-uXGbiOpD98qYF7suR3h43Ld3Ft3toGon7ENOaHrz0NLTJ9c-9SQHqbBV1yuI5qiWXqQktFdj-GS0Swpo7trBBIXg__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA Modelling the wireless propagation channel: a simulation approach with Matlab] (Vol. 5). John Wiley & Sons. -  p. 123 - 129</ref><ref>[https://www.db-thueringen.de/rsc/viewer/dbt_derivate_00032343/ilm1-2015000358.pdf?page=40 Arndt, D., 2015. On channel modelling for land mobile satellite reception] (Doctoral dissertation). - p. 28</ref> Doppler [[Spectral density|PSD]] can also be modeled via [[Butterworth filter]] as:

:<math>S_{Doppler}(f) = |H_{Butterworth}|^2 = \frac{B}{\sqrt{1 - (f/f_0)^{2k}}}</math>

where ''f'' is a frequency, <math>H_{Butterworth}</math> is the Butterworth filter response, ''B'' is the normalization constant, ''k'' is the filter order and <math>f_0</math> is the [[Cutoff frequency]] which should be selected with respect to maximum Doppler shift.