# 瑞利衰落信道相关知识[zz from wikipedia.com]

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## 模型

$p_R(r) = \frac{r}{\sigma}e^{-r^2/2\sigma},\ r\geq{}0$

## 性质

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

Note that the parameters discussed here are for a non-static channel. If a channel is not changing with time, clearly 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.

### 相关性

$\,\! R(\tau) = J_0(2\pi f_d \tau)$

### Level crossing rate

The level crossing rate is a measure of the rapidity of the fading. It quantifies how often he fading crosses some threshold, usually in the positive-going direction. For Rayleigh fading, the level crossing rate is:[5]

$LCR = \sqrt{2\pi}f_d\rho e^{-\rho^2}$

where fd is the maximum Doppler shift and $\,\!\rho$ is the threshold level normalised to the root mean square (RMS) signal level:

$\rho = \frac{R_{thresh}}{R_{rms}}$.

The average fade duration quantifies how long the signal spends below the threshold $\,\!\rho$. For Rayleigh fading, the average fade duration is:[5]

$AFD = \frac{e^{\rho^2} - 1}{\rho f_d \sqrt{2\pi}}$.

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

For a particular normalised threshold value ρ, the product of the average fade duration and the level crossing rate is a constant and is given by

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

### Doppler power spectral density

The normalized Doppler power spectrum of Rayleigh fading with a maximum Doppler shift of 10Hz.

The Doppler 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 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:[4]

$S(\nu) = \frac{1}{\pi f_d \sqrt{1 - \left(\frac{\nu}{f_d}\right)^2}}$,

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

## 瑞利衰落信道的仿真

### Jakes模型

In his book,[6] Jakes popularised a model for Rayleigh fading based on summing sinusoids. Let the scatterers be uniformly distributed around a circle at angles αn with k rays emerging from each scatterer. The Doppler shift on ray n is

$\,\!f_n = f_d\cos{\alpha_n}$

and, with M such scatterers, the Rayleigh fading of the kth waveform over time t can be modelled as:

$R(t,k) = 2\sqrt{2}\left[\sum_{n=1}^{M}\left(\cos{\beta_n} + j\sin{\beta_n}\right)\cos{\left(2 \pi f_n t + \theta_{n,k}\right)} + \frac{1}{\sqrt{2}}\left(\cos{\alpha} + j\sin{\alpha}\right)\cos{2 \pi f_d t}\right]$.

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

$\,\!\beta_n = \frac{\pi n}{M+1}$

and $\,\!\theta_{n,k}$ used to generate multiple waveforms. If a single-path channel is being modelled, so that there is only one waveform then $\,\!\theta_{n}$ 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:

$\theta_{n,k} = \beta_n + \frac{2\pi(k-1)}{M+1}$.

In fact, it has been shown that the waveforms are correlated among themselves — they have non-zero cross-correlation — except in special circumstances.[7] The model is also deterministic (it has no random element to it once the parameters are chosen). A modified Jakes’ model[8] chooses slightly different spacings for the scatterers and scales their waveforms using Walsh-Hadamard sequences to ensure zero cross-correlation. Setting

$\alpha_n = \frac{\pi(n-0.5)}{M+1}$ and $\beta_n = \frac{\pi n}{M}$,

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

$R(t,k) = \sqrt{\frac{2}{M}} \sum_{n=1}^{M} A_k(n)\left( \cos{\beta_n} + j\sin{\beta_n} \right)\cos{\left(2\pi f_n t + \theta_{n,k}\right)}$.

The weighting functions An(k) are the kth Walsh-Hadamard sequence in n. Since these have zero cross-correlation by design, this model results in uncorrelated wavforms. The phases $\,\!\theta_{n,k}$ can be initialised randomly and have no effect on the correlation properties.

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

### Filtered white noise

Another way to generate a signal with the required Doppler power spectrum is to pass a white Gaussian noise signal through a 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 Guassian waveform at an appropriate rate.

## References

1. John G. Proakis (1995). Digital Communications, 3rd edition, 767–768, Singapore: McGraw-Hill Book Co. ISBN 0-07-113814-5.
2. Bernard Sklar (July 1997). “Rayleigh Fading Channels in Mobile Digital Communication Systems Part I: Characterization”. IEEE Communications Magazine 35 (7): 90–100. DOI:10.1109/35.601747 ISSN 0163-6804.
3. Dmitry Chizhik, Jonathan Ling, Peter W. Wolniansky, Reinaldo A. Valenzuela, Nelson Costa, and Kris Huber (April 2003). “Multiple-Input–Multiple-Output Measurements and Modeling in Manhattan”. IEEE Journal on Selected Areas in Communications 21 (3): 321–331. DOI:10.1109/JSAC.2003.809457.
4. ^ 4.0 4.1 R. H. Clarke (July–August 1968). “A Statistical Theory of Mobile Radio Reception”. Bell Systems Technical Journal 47 (6): 957–1000.
5. ^ 5.0 5.1 T. S. Rappaport (December 31 2001). Wireless Communications: Principles and Practice, 2nd edition, Prentice Hall PTR. ISBN 0130422320.
6. William C. Jakes, Editor (February 1 1975). Microwave Mobile Communications, New York: John Wiley & Sons Inc. ISBN 0471437204.
7. Von Eckardstein, S. and Isaksson, K. (December 1991). Kanalmodeller för radiotransmission (Channel models for radio transmission) (Master’s thesis) (in Swedish), Stockholm, Sweden: Royal Institute of Technology.
8. P. Dent, G. E. Bottomley and T. Croft (24 June 1993). “Jakes Fading Model Revisited”. Electronics Letters 29 (13): 1162–1163.

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