Non-Frequency Selective Fading With Direct Component

Channel model and Rice PDF

The  Rayleigh distribution  describes the mobile communication channel under the assumption that there is no direct path and thus the multiplicative factor  $z(t) = x(t) + {\rm j} \cdot y(t)$  is solely composed of diffusely scattered components.

If a direct component  $($Line of Sight,  $\rm LoS)$  is present, it is necessary to add direct components   $x_0$  and/or  $y_0$  to the zero mean Gaussian processes   $x(t)$  and  $y(t)$:

$x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$
$z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm} z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$

The graph shows this  Rice fading channel model.  As a special case, the Rayleigh model results when   $x_0 = y_0= 0$.

• The real part  $x(t)$  is gaussian distributed with mean value  $x_0$  and variance  $\sigma ^2$.
• The imaginary part  $y(t)$  is also gaussian distributed  $($mean  $y_0$,  equal variance  $\sigma ^2)$  and independent of  $x(t)$.
• For  $z_0 \ne 0$  the value  $|z(t)|$  has a Rice PDF, from which the term  "Rice fading"  is derived.
• To simplify the notation we set  $|z(t)| = a(t)$.   For  $a < 0$  it's PDF is  $f_a(a) \equiv 0$,  for  $a \ge 0$ the following equation applies, where  $\rm I_0(\cdot)$  denotes the  "modified Bessel–function" of zero order:
$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm exp} \big [ -\frac{a^2 + |z_0|^2}{2\sigma^2}\big ] \cdot {\rm I}_0 \left [ \frac{a \cdot |z_0|}{\sigma^2} \right ]\hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) = \sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)} \hspace{0.05cm}.$
• The greater the direct path power  $(|z_0|^2)$  compared to the power of the stray components  $(2\sigma^2)$  the better suited for digital signal transmission is the mobile communication channel.
• If   $|z_0| \gg \sigma$  $($factor  $3$  or more$)$, the Rice PDF can be approximated accurately by a Gaussian distribution with mean  $|z_0|$  and variance  $\sigma^2$.
• In contrast to  Rayleigh fading   ⇒   $z_0 \equiv 0$, the phase at  Rice fading  is not equally distributed, but there is a preferred direction  $\phi_0 = \arctan(y_0/x_0)$.  Often one sets  $y_0 = 0$   ⇒   $\phi_0 = 0$.

Example of signal behaviour with Rice fading

The diagram shows typical signal characteristics and density functions of two mobile communication channels:

• Rayleigh fading  (blue curves)  with
$${\rm E}\big [|z(t))|^2\big ] = 2 \cdot \sigma^2 = 1,$$
• Rice fading  (red curves)  with same  $\sigma$  and
$$x_0 = 0.707,\ \ y_0 = -0.707.$$

For the generation of the signals according to the above model, the  maximum Doppler frequency  $f_\text{D, max} = 100 \ \rm Hz$  was used as reference.

The auto-correlation function  $\rm (ACF)$  and power-spectral density  $\rm (PSD)$  of Rayleigh and Rice differ only slightly, other than adjusted parameter values.  The following applies:

$\varphi_z ({\rm \Delta}t)\Bigg |_{\hspace{0.1cm}{\rm Rice}} \hspace{-0.5cm} = \varphi_z ({\rm \Delta}t)\Bigg |_{\hspace{0.1cm}{\rm Rayleigh}} \hspace{-0.8cm} + |z_0|^2 \hspace{0.05cm},$
${\it \Phi}_z(f_{\rm D})\Bigg |_{\hspace{0.1cm}{\rm Rice}} \hspace{-0.5cm} = {\it \Phi}_z(f_{\rm D})\Bigg |_{\hspace{0.1cm}{\rm Rayleigh}} \hspace{-0.8cm} + |z_0|^2 \cdot \delta (f_{\rm D}) \hspace{0.05cm}.$

It is taken into account that the spectral representation of a DC component leads to a Dirac function.

• The real parts  $x(t)$  of Rayleigh (blue) and Rice (red) only differ by the constant  $x_0 = 0.707$.   The statistical properties are otherwise the same:   Gaussian PDF $f_x(x)$  with standard deviation  $\sigma = 0.707$, either zero-mean (Rayleigh) or with mean  $x_0$  (Rice).
• In the imaginary part  $y(t)$  of the Rice distribution one can additionally recognize the direct component  $y_0 = -0.707$.  The (here not shown) PDF $f_y(y)$  is thus a Gaussian curve with the standard deviation  $\sigma = 0. 707$  around the mean value  $y_0 = -0.707$, thus axisymmetrical to the shown PDF $f_x(x)$.
• The (logarithmic) representation of   ⇒   $a(t) =|z(t)|$ shows that the red curve is usually above the blue one.  This can also be read from the PDF $f_a(a)$ .
• The PDF $f_\phi(\phi)$  shows the preferred angle  $\phi \approx -45^\circ$  of the given Rice channel   The complex factor  $z(t)$  is located mainly in the fourth quadrant because of  $x_0 > 0$  and  $y_0 < 0$ , whereas in the Rayleigh channel all quadrants are equally probable.