Rayleigh and Ricean Distributions

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5.6 Rayleigh and Ricean Distributions

      5.6.1 Rayleigh Fading Distribution

In mobile radio channels, the Rayleigh distribution is commonly used to describe the statistical time

varying nature of the received envelope of a flat fading signal, or the envelope of an individual multipath

component. It is well known that the envelope of the sum of two quadrature Gaussian noise

signals obeys a Rayleigh distribution. Figure 5.15 shows a Rayleigh distributed signal envelope as a

function of time. The Rayleigh distribution has a probability density function (pdf) given by

where σ is the rms value of the received voltage signal before envelope detection, and σ² is the

time-average power of the received signal before envelope detection. The probability that the

[+] Enlarge Image

Figure 5.15 A typical Rayleigh fading envelope at 900 MHz [from [Fun93] © IEEE].

envelope of the received signal does not exceed a specified value R is given by the corresponding

cumulative distribution function (CDF)

The mean value r_mean of the Rayleigh distribution is given by

and the variance of the Rayleigh distribution is given by σ²_r , which represents the ac power in

the signal envelope

The rms value of the envelope is the square root of the mean square, or 

where σ is the standard deviation of the original complex Gaussian signal prior to

envelope detection.

The median value of r is found by solving

and is

Thus the mean and the median differ by only 0.55 dB in a Rayleigh fading signal. Note

that the median is often used in practice, since fading data are usually measured in the field and

a particular distribution cannot be assumed. By using median values instead of mean values, it is

easy to compare different fading distributions which may have widely varying means.

Figure 5.16 illustrates the Rayleigh pdf. The corresponding Rayleigh cumulative distribution

function (CDF) is shown in Figure 5.17.

5.6.2 Ricean Fading Distribution

When there is a dominant stationary (nonfading) signal component present, such as a line-ofsight

propagation path, the small-scale fading envelope distribution is Ricean. In such a situation,

random multipath components arriving at different angles are superimposed on a stationary

dominant signal. At the output of an envelope detector, this has the effect of adding a dc component

to the random multipath.

Just as for the case of detection of a sine wave in thermal noise [Ric48], the effect of a

dominant signal arriving with many weaker multipath signals gives rise to the Ricean distribution.

As the dominant signal becomes weaker, the composite signal resembles a noise signal

which has an envelope that is Rayleigh. Thus, the Ricean distribution degenerates to a Rayleigh

distribution when the dominant component fades away.

Figure 5.16 Rayleigh probability density function (pdf).

Figure 5.17 Cumulative distribution for three small-scale fading measurements and their fit to

Rayleigh, Ricean, and log-normal distributions [from [Rap89] © IEEE].

The Ricean distribution is given by

The parameter A denotes the peak amplitude of the dominant signal and I₀(•) is the

modified Bessel function of the first kind and zero-order. The Ricean distribution is often

described in terms of a parameter K which is defined as the ratio between the deterministic signal

power and the variance of the multipath. It is given by K= A²/ (2σ²⁾ or, in terms of dB

Figure 5.18 Probability density function of Ricean distributions:K = – ∞ dB (Rayleigh) and

K=6 dB. For K >> 1, the Ricean pdf is approximately Gaussian about the mean.

The parameter K is known as the Ricean factor and completely specifies the Ricean distribution.

As A → 0, K → –∞ dB, and as the dominant path decreases in amplitude, the Ricean

distribution degenerates to a Rayleigh distribution. Figure 5.18 shows the Ricean pdf. The

Ricean CDF is compared with the Rayleigh CDF in Figure 5.17.

Relevant NI products

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