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Rice distribution

What it represents

Rice models signal amplitude with a deterministic component plus two-dimensional Gaussian noise. v represents dominant-signal magnitude and sigma diffuse noise spread.

Historical clue

Stephen O. Rice studied the statistical properties of a sinusoid plus noise at Bell Labs. The distribution became standard for line-of-sight fading.

Relationships that clarify its use

Rayleigh is the no-deterministic-component case and Nakagami is a flexible fading alternative. The modified Bessel function appears after integrating the direction of a shifted Gaussian vector.

Data examples

  • wireless channels with partial line of sight
  • radar, sonar, medical imaging, and signal detection

Modelling warning

Separating specular signal from diffuse noise needs a measurement model; a good Rice fit alone does not prove a dominant path.

A fixed signal emerges between two noise components

Rice measures the length of a two-dimensional Gaussian vector whose centre is displaced from the origin. The distance v represents a deterministic component and sigma the noise spread. In communications, this is the story of a dominant path accompanied by many reflections.

At v=0, the family reduces to Rayleigh. When signal greatly dominates noise, amplitude becomes approximately Normal around v. Its square is related to Noncentral Chi Square with two degrees of freedom. These connections offer diagnostics and computational routes. Estimating v near zero is difficult because sign and direction have already been lost when only magnitude is observed.

Decision guide

A good candidate when: the magnitude of two Gaussian components is observed and a nonzero signal or line-of-sight component is present.

Compare it with: Rayleigh when that component vanishes and Nakagami as a flexible approximation. Interpret noncentrality through signal physics, not only fit.

References

  • SciPy reference: scipy.stats.rice — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Rice, S. O. (1945). Mathematical analysis of random noise. Bell System Technical Journal, 24, 46–156.
  • Papoulis, A. (1965). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.

Rice Distribution: equations and calculator

Distribution defintion

X∼Rice(v,σ)X\sim\mathrm{Rice}\left(v,\sigma\right)

Distribution domain

x∈[0,∞)x\in [0,\infty)

Parameters domain and parameters constraints

v∈R+,σ∈R+v\in\mathbb{R}^{+},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−Q1(vσ,xσ)F_{X}\left(x\right)=1-Q_1\left(\frac{v}{\sigma},\frac{x}{\sigma }\right)

Probability density function

fX(x)=xσ2exp⁡(−(x2+v2)2σ2)I0(xvσ2)f_{X}\left(x\right)=\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)}{2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)

Percent point function/Sample

SampleX=Φ−1(u1,v,σ)2+Φ−1(u2,0,σ)2\text{Sample}_{X}=\sqrt{\Phi^{-1}(u_{1},v,\sigma)^{2}+\Phi^{-1}(u_{2},0,\sigma)^{2}}

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx=σk2k/2 Γ(1+k/2) Lk/2(−v2/2σ2)\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx=\sigma^k2^{k/2}\,\Gamma(1+k/2)\,L_{k/2}(-v^2/2\sigma^2)

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=FX−1(12)\mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right)

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
Φ−1(u,mean,variance):Inverse of cumulative function from normal distribution\Phi^{-1}\left(u,mean,variance\right):\text{Inverse of cumulative function from normal distribution}
Lr(x):Laguerre polynomials of order r∈RL_{r}\left(x\right): \text{Laguerre polynomials of order }r\in\mathbb{R}
L12(x)=ex/2(x)I1(x2)−ex/2(x−1)I0(x2)L_{\frac{1}{2}}\left(x\right)=e^{x/2} (x) I_{1}\left(\frac{x}{2}\right)-e^{x/2} (x-1) I_{0}\left(\frac{x}{2}\right)
L32(x)=13ex/2(2x2−6x+3)I0(x/2)−23ex/2(x−2)xI1(x/2)L_{\frac{3}{2}}\left(x\right)=\frac{1}{3} e^{x/2} (2 x^2-6 x+3) I_0(x/2)-\frac{2}{3} e^{x/2} (x-2) x I_1(x/2)
Iα(x):Modified Bessel function of the first kind of order α∈NI_{\alpha}\left(x\right): \text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N}
Qk(a,b):Marcum Q-function of order k ∈NQ_{k}(a,b): \text{Marcum Q-function of order k }\in\mathbb{N}
u1:Uniform[0,1] random varibleu_{1}:\text{Uniform[0,1] random varible}
u2:Uniform[0,1] random varibleu_{2}:\text{Uniform[0,1] random varible}