*2.1.1. Rayleigh fading*

Suppose that the number of scatters is large and that the angles of arrival between them are uncorrelated. From the Central Limit Theorem, it can be shown that the in-phase (cosine) and quadrature (sine) components of *r*(t), denoted as *rI* (*t*) and *rQ*(*t*), follow two independent time correlated Gaussian random processes.

Consider a snapshot value of at time *t* =0, and note that *r*(0)= *rI* (0) +*rQ*(0). Since the values *rI* (0) and *rQ*(0) are Gaussian random variables, it can be shown that the distribution of the envelope

**Figure 2.** The difference between (a) constructive interference and (b) destructive interference at *fc* = 2.5GHz is less than 0.1 nanoseconds in phase, which corresponds to about 3 cm.

amplitude |*r* | = *rI* <sup>2</sup> <sup>+</sup>*rQ* 2 is Rayleigh and that the received power |*<sup>r</sup>* <sup>|</sup> <sup>2</sup> <sup>=</sup>*rI* <sup>2</sup> <sup>+</sup> *rQ* 2 is exponen‐ tially distributed. Formally [10],

$$f\_{\left[\mathbf{r}\right]}(\mathbf{x}) = \frac{2\boldsymbol{\omega}}{p\_r} e^{-\boldsymbol{\chi}^2/p\_r} \qquad , \quad \boldsymbol{\chi} \ge 0,\tag{1}$$

where *pr* is the average received power.

*r*

s

is the power of the LOS component, *σ* <sup>2</sup>

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‐

m


is the variance and *I*<sup>0</sup> is the 0th-order, modified

PAPR Reduction in WiMAX System http://dx.doi.org/10.5772/55380 39

, a common way to

 s

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

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

characterize the channel is by the relative strengths of the LOS and scattered paths. This factor,

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

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

power distribution *f* <sup>|</sup>*r*|2(*x*), the Ricean envelope distribution in terms of K can be found by

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‐

. Although it is not straightforward to directly find the Ricean

tion is more accurately modeled by a Ricean distribution, which is given by

Since the Ricean distribution depends on the LOS component's power *μ* <sup>2</sup>

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).

22 2 ( )/2 2 2 <sup>0</sup> ( ) ( ) , 0, *<sup>x</sup>*

*x x fx e I x* m s

*2.1.2. LOS channels: Ricean distribution*

where *μ* <sup>2</sup>

*μ* =0⇒*I*0(

component.

*<sup>K</sup>* <sup>=</sup> *<sup>μ</sup>* <sup>2</sup> 2*σ* <sup>2</sup>

components.

and the LOS power: *Pr* =2*<sup>σ</sup>* <sup>2</sup> <sup>+</sup> *<sup>μ</sup>* <sup>2</sup>

*K*, is quantified as

*xμ <sup>σ</sup>* <sup>2</sup> )=1 ,

that

and

$$\|f\_{\|\mathbf{r}\|^2}(\mathbf{x}) = \frac{2\chi}{p\_r} e^{-\chi/p\_r} \qquad , \quad \chi \ge 0,\tag{2}$$

where *pr* is the average received power.
