*2.1.2. LOS channels: Ricean distribution*

An important assumption in the Rayleigh fading model is that all the arriving reflections have a mean of zero. This will not be the case if there is a dominant path—for example, a LOS path — between the transmitter and the receiver. For a LOS signal, the received envelope distribu‐ tion is more accurately modeled by a Ricean distribution, which is given by

$$I\_{\|\cdot\|}(\mathbf{x}) = \frac{\mathbf{x}}{\sigma^2} e^{-(\mathbf{x}^2 + \mu^2)/2\sigma^2} I\_0(\frac{\mathbf{x}\mu}{\sigma^2}) \qquad , \quad \mathbf{x} \ge \mathbf{0}, \tag{3}$$

where *μ* <sup>2</sup> is the power of the LOS component, *σ* <sup>2</sup> is the variance and *I*<sup>0</sup> is the 0th-order, modified Bessel function of the first kind. Although more complicated than a Rayleigh distribution, this expression is a generalization of the Rayleigh distribution. This can be confirmed by observing that

$$
\mu = 0 \Rightarrow I\_0(\frac{\varkappa \mu}{\sigma^2}) = 1 \quad \text{or} \quad
$$

Therefore the Ricean distribution reduces to the Rayleigh distribution in the absence of a LOS component.

Since the Ricean distribution depends on the LOS component's power *μ* <sup>2</sup> , a common way to characterize the channel is by the relative strengths of the LOS and scattered paths. This factor, *K*, is quantified as

$$\mathcal{K} = \frac{\mu^2}{2\sigma^2}$$

amplitude |*r* | = *rI*

38 Selected Topics in WiMAX

and

tially distributed. Formally [10],

<sup>2</sup> <sup>+</sup>*rQ* 2

than 0.1 nanoseconds in phase, which corresponds to about 3 cm.

*r*

2

*<sup>r</sup> <sup>r</sup>*

is Rayleigh and that the received power |*<sup>r</sup>* <sup>|</sup> <sup>2</sup> <sup>=</sup>*rI*

**Figure 2.** The difference between (a) constructive interference and (b) destructive interference at *fc* = 2.5GHz is less



<sup>2</sup> <sup>2</sup> / ( ) , 0, *<sup>r</sup> x p*

<sup>2</sup> / ( ) , 0, *<sup>r</sup> x p*

*<sup>x</sup> fx e x p*

*r <sup>x</sup> fx e x p*

<sup>2</sup> <sup>+</sup> *rQ* 2

is exponen‐

and is a natural description of how strong the LOS component is relative to the NLOS components.

For *K* =0, the Ricean distribution again reduces to Rayleigh, and as *K* →*∞*, the physical meaning is that there is only a single LOS path and no other scattering. Mathematically, as *K* grows large, the Ricean distribution is quite Gaussian about its mean *μ* with decreasing variance, physically meaning that the received power becomes increasingly deterministic.

The average received power under Ricean fading is the combination of the scattering power and the LOS power: *Pr* =2*<sup>σ</sup>* <sup>2</sup> <sup>+</sup> *<sup>μ</sup>* <sup>2</sup> . Although it is not straightforward to directly find the Ricean power distribution *f* <sup>|</sup>*r*|2(*x*), the Ricean envelope distribution in terms of K can be found by substituting *<sup>μ</sup>* <sup>2</sup> <sup>=</sup> *<sup>K</sup> Pr* / (*<sup>K</sup>* <sup>+</sup> 1) and 2*<sup>σ</sup>* <sup>2</sup> <sup>=</sup>*Pr* / (*<sup>K</sup>* <sup>+</sup> 1) into Equation (3).

Although its simplicity makes the Rayleigh distribution more amenable to analysis than the Ricean distribution, the Ricean distribution is usually a more accurate depiction of wireless broadband systems, which typically have one or more dominant components. This is espe‐ cially true of fixed wireless systems, which do not experience fast fading and often are deployed to maximize LOS propagation.
