瑞利衰落信道

百科解釋

瑞利衰落能有效描述存在能夠大量散射無線電信號的障礙物的無線傳播環(huán)境。若傳播環(huán)境中存在足夠多的散射，則沖激信號到達接收機后表現(xiàn)為大量統(tǒng)計獨立的隨機變量的疊加，根據(jù)中心極限定理，則這一無線信道的沖激響應將是一個高斯過程。如果這一散射信道中不存在主要的信號分量，通常這一條件是指不存在直射信號（LoS），則這一過程的均值為0，且相位服從0 到2π 的均勻分布。即，信道響應的能量或包絡服從瑞利分布。設隨機變量R，于是其概率密度函數(shù)為：
其中σ = E(R2
若信道中存在一主要分量，例如直射信號（LoS），則信道響應的包絡服從萊斯分布，對應的信道模型為萊斯衰落信道。
通常將信道增益以等效基帶信號表示，即用一復數(shù)表示信道的幅度和相位特性。由此瑞利衰落即可由這一復數(shù)表示，它的實部和虛部服從于零均值的獨立同分布高斯過程。

模型的適用

建筑密集的曼哈頓地區(qū)的無線信道符合瑞利衰落信道模型。

最大多普勒頻移為10Hz的瑞利衰落信道。

最大多普勒頻移為100Hz的瑞利衰落信道。

瑞利衰落模型適用于描述建筑物密集的城鎮(zhèn)中心地帶的無線信道。密集的建筑和其他物體使得無線設備的發(fā)射機和接收機之間沒有直射路徑，而且使得無線信號被衰減、反射、折射、衍射。在曼哈頓的實驗證明，當?shù)氐臒o線信道環(huán)境確實接近于瑞利衰落。[3] 通過電離層和對流層反射的無線電信道也可以用瑞利衰落來描述，因為大氣中存在的各種粒子能夠?qū)o線信號大量散射。
瑞利衰落屬于小尺度的衰落效應，它總是疊加于如陰影、衰減等大尺度衰落效應上。
信道衰落的快慢與發(fā)射端和接收端的相對運動速度的大小有關。相對運對導致接收信號的多普勒頻移。圖中所示即為一固定信號通過單徑的瑞利衰落信道后，在1秒內(nèi)的能量波動，這一瑞利衰落信道的多普勒頻移最大分別為10Hz和100Hz，在GSM1800MHz的載波頻率上，其相應的移動速度分別為約6千米每小時和60千米每小時。特別需要注意的是信號的“深衰落”現(xiàn)象，此時信號能量的衰減達到數(shù)千倍，即30～40分貝。

性質(zhì)

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.

相關性

瑞利衰落信道的自相關函數(shù)，其多普勒頻移為10Hz。

無線終端的發(fā)射端和接收端之間若以恒定的相對速度移動，則這一瑞利衰落信道的歸一化自相關函數(shù)為零階貝塞爾函數(shù)：

其中延時為，最大多普勒頻偏為f<sub>d</sub>。如圖所示，為最大多普勒頻移為10Hz的瑞利衰落信道的自相關函數(shù)，它關于延時是周期的，而且其包絡在第一個零點之后緩慢衰減。

幅度穿越率
幅度穿越率（LCR，level crossing rate）是對衰落快慢的一種度量。LCR 給出衰落信號的幅度以怎樣的頻率穿越某一門限，通常按照正向穿越方向進行計算。瑞利衰落的LCR為：[5]

其中是最大多普勒頻偏，為對信號的均方根進行歸一化的信號門限值：

平均衰落時間
平均衰落時間（AFD，average fade duration）這一參數(shù)是指信號在門限以下持續(xù)的時間。瑞利衰落的平均衰落時間為：
幅度穿越速率和平均衰落時間這兩個參數(shù)給出了衰落在時間上嚴重程度的描述。對于一定的門限值ρ，平均衰落時間和幅度穿越速率的積為常數(shù)，并且可以表示為：

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
where is the frequency shift relative to the carrier frequency. Clearly, this equation is only valid for values of between ; the spectrum is zero outside this range. This spectrum is shown in the figure for a maximum Doppler shift of 10Hz. The &＃39;&＃39;&＃39;&＃39;bowl shape&＃39;&＃39;&＃39;&＃39; or &＃39;&＃39;&＃39;&＃39;bathtub shape&＃39;&＃39;&＃39;&＃39; is the classic form of this doppler spectrum.

瑞利衰落信道的仿真

根據(jù)上文所述，瑞利衰落信道可以通過發(fā)生實部和虛部都服從獨立的高斯分布變量來仿真生成。不過，在有些情況下，研究者只對幅度的波動感興趣。針對這種情況，有兩種方法可以仿真產(chǎn)生瑞利衰落信道。這兩種方法的目的是產(chǎn)生一個信號，有著上文所示的多普勒功率譜或者等效的自相關函數(shù)。這個信號就是瑞利衰落信道的沖激響應。

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 α<sub>n</sub> with k rays emerging from each scatterer. The Doppler shift on ray n is
<dl>
<dd></dd>
</dl>
and, with M such scatterers, the Rayleigh fading of the kth waveform over time t can be modelled as:
<dl>
<dd>.</dd>
</dl>
Here, and the and are model parameters with usually set to zero, chosen so that there is no cross-correlation between the real and imaginary parts of R(t):
<dl>
<dd></dd>
</dl>
and used to generate multiple waveforms. If a single-path channel is being modelled, so that there is only one waveform then 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:
<dl>
<dd>.</dd>
</dl>
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&＃39;&＃39;&＃39;&＃39; model[8] chooses slightly different spacings for the scatterers and scales their waveforms using Walsh-Hadamard sequences to ensure zero cross-correlation. Setting
<dl>
<dd> and ,</dd>
</dl>
results in the following model, usually termed the Dent model or the modified Jakes model:
<dl>
<dd>.</dd>
</dl>
The weighting functions A<sub>n</sub>(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 can be initialised randomly and have no effect on the correlation properties.
The Jakes&＃39;&＃39;&＃39;&＃39; model also popularised the Doppler spectrum associated with Rayleigh fading, and, as a result, this Doppler spectrum is often termed Jakes&＃39;&＃39;&＃39;&＃39; 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.

通信詞典解釋

<xmp id="0cbi2">