Rician distribution (Rice distribution)

< List of probability distributions

A Rician distribution, also called a Rice distribution or Nakagami-n distribution, is often used to model scattered signals that reach a receiver via multiple paths. Frequently found in the analysis of time variant systems — where multiple moving signals must be analyzed, they give more control over the extent of fading than the Rayleigh distribution.

Rician distribution definition and properties

The probability density function (PDF) is:


  • ν and σ are parameters,
  • I0(z) is a modified Bessel function of the first kind with order zero.

The support of the Rician distribution is real positive numbers; the interval is {a, ∞) where a is any real positive number (i.e., positive numbers that can be found on the number line).

Uses of the Rice distribution

A Rician distribution models the paths that scattered signals take to a receiver. This distribution models line-of-sight scatter, such as microwaves or FM radio waves, that occurs when two stations in view of each other have an unobstructed path between them.

The Rician distribution can also model Rician fading which show how signal cancellations can affect radio propagation. Rician fading occurs when a radio signal is transmitted through a multipath environment. When the signal travels through multiple paths of varying lengths and attenuations, it is received as the sum of multiple signals, each with its own amplitude and phase. The amplitude of the received signal follows a Rician distribution, which is more peaked than the Rayleigh distribution seen in single-path environments. As a result of this peaked distribution, the probability of receiving a large amplitude signal is higher while the probability of receiving a small amplitude signal is lower.

Rician distributions can only model non-dense, line-of-sight signals. Distributions that model dense scattered signals include the Nakagami distribution as well as the Rayleigh distribution.

Similar Distributions

The Rice Distribution has close ties to several other distributions, including:

  • The PDF of the Rician distribution [ν, σ] is identical to that of the Beckmann Distribution of parameters [ν/√2, ν/√2, σ, σ].
  • The PDF of the Rician distribution [0, σ] is identical to that of the Rayleigh distribution [σ].

Rician distributions only serve as viable models for non-dense, line-of-sight signals. To model dense, scattered signals use the Nakagami distribution or Rayleigh distribution instead.


[1] PAR~commonswiki  licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Scroll to Top