# Distribution of Amplitude and Power for Rician fading

In Rician fading, a strong dominant component is present. Similar to the case of Rayleigh fading, the in-phase and quadrature phase component of the received signal are i.i.d. jointly Gaussian random variables. However, in Rician fading the mean value of (at least) one component is non-zero due to a deterministic strong wave.

## PDF of signal amplitude

The derivation is similar to the derivation for Rayleigh fading. In order to obtain the probability density of the signal amplitude we observe the random processes I(t) and Q(t) at one particular instant t0. If the number of scattered waves is sufficiently large, and are i.i.d., the central limit theorem says that I(t0) and Q(t0) are Gaussian, but, due to the deterministic dominant term, no longer zero mean. Transformation of variables shows that the amplitude and the phase have the joint pdf

Here, is the local-mean scattered power and is the power of the dominant component. The pdf of the amplitude is found from the integral

,

where is the modified Bessel function of the first kind and zero order, defined as

#### Hint for Numerical evaluation

The Bessel function can be approximated numerically using the following formulas:

``` function I0(z){
if (z > 4)
return Math.exp(z)/Math.sqrt(2*Math.PI*z)*(1+1/(8*z));
else
return 1 + z*z/4 + Math.pow(z,4)/64  + Math.pow(z,6)/2304 +   Math.pow(z,8)/147456;
}
```

## Rician K-factor

The Rician K-factor is defined as the ratio of signal power in dominant component over the (local-mean) scattered power. Thus

The total local-mean power is the sum of the power in the line-of-sight and the local-mean scattered power.

.

The local-mean scattered power equals s2 = /(K + 1). The amplitude of the line-of-sight is C = Ö( 2K/(K + 1) )

### Conversion

 Amplitude LOS: C Sigma s Calculate K factor (or dB) local-mean power

Expressed in terms of the local-mean power and the Rician K-factor, the pdf of the signal amplitude becomes

From the pdf of signal amplitude, one can derive the pdf of signal power using the standard mathematical methods. In particular, we need p = r2/2, so dp = r dr. This is used in fp(p) dp = fr(r) dr

```            (1+K)e-K         (1+K)                     p
fp (p) = --------  exp(- ---- p ) I0 (Ö (4K(1+K) ----) )

```

 Figure: Probability Density Function of received signal power for various Rician K-factors. The total local mean power (scattered plus line-of-sight) = 1. Source code: PlotRice.java Credits: The Java curves are computed using the Ptolemy Plot package, authored by Edward Lee et al. © University of California, Berkeley.

### Evaluate the Rician distribution

Use the Special Purpose Embedded Calculator (spreadsheet) to evaluate the outage probability and the BER in a Rician fading channel.

### Exercise

Show that for a large fade margin (h >> 1), the probability that the instantaneous power drops below the noise threshold pn tends to

Outage probability = (1 + K) exp(-K) h-1

where the fade margin h is defined as the local-mean power devided by the threshold pn.

Hint: Use only the first term of the approximation for the Bessel function given on this page. Then integrate the PDF for powers from zero to pn.

Compare this with the expression for the special case of Rayleigh fading (K = 0). What is different? The level, the slope or both level and slope of the curve of probability versus h?

### Exercise

Describe how you would measure the Rician K-factor. Answer