Ricianfördelns

Ricianfördelns, or the Rician distribution, is a continuous probability distribution that describes the magnitude of a randomly varying vector in a Euclidean space when the vector is composed of a deterministic component and a zero-mean random component. It arises in various fields, particularly in signal processing and wireless communications, to model the received signal strength when there is a line-of-sight component along with scattered components.

The Rician distribution is characterized by two parameters: a shape parameter, often denoted by k, which represents the ratio of the power of the deterministic component to the power of the random component, and a scale parameter, often denoted by Ω (Omega), which is related to the variance of the random component. When k = 0, the Rician distribution reduces to the Rayleigh distribution, which models the magnitude of a random vector with only scattered components. As k approaches infinity, the Rician distribution approaches a Gaussian distribution.

The probability density function (PDF) of a Rician distribution for a random variable X is given by:

f(x; m, σ) = (x/σ²) * exp(-(x² + m²)/(2σ²)) * I₀(xm/σ²)

where m is the amplitude of the deterministic component, σ is the standard deviation of the random component, and I₀ is the modified Bessel function of the first kind with order zero. The parameter k is related to m and σ by k = m²/ (2σ²), and Ω is related to m and σ by Ω = m² + 2σ².

The Rician distribution is used in modeling fading phenomena in wireless channels, such as Ricean fading, which occurs when there is a strong direct path between the transmitter and receiver, alongside multipath propagation. It is also applied in fields like acoustics, image processing, and statistics.