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Nakagami distribution

What it describes

Nakagami models signal amplitude under fast fading. m controls fading severity and omega mean power; the family adapts to different propagation environments.

Amplitude is non-negative. m=1 is Rayleigh and m=1/2 is Half Normal under a common convention; larger values concentrate the signal around mean power.

History and names

Minoru Nakagami proposed the model in 1960 while studying rapid radio fading. It spread into wireless-channel and performance analysis.

How it connects to other distributions

It is a Gamma transform applied to squared amplitude. Rayleigh and approximate Rician behaviours are related cases; Weibull is another empirical fading alternative.

Where it appears

  • wireless channels and multipath fading
  • radar, sonar, and signal-intensity models

Fitting cautions

Enforce m ≥ 1/2 and omega > 0; compare omega with mean power only when package conventions agree.

The amplitude whose power becomes Gamma

Minoru Nakagami proposed the m distribution for rapid fading of radio waves. Its key construction is simple: squared amplitude is Gamma. Thus omega represents mean power while m governs fading severity.

In the classical form, m=1 recovers Rayleigh and m=1/2 is related to Half Normal. Larger values concentrate amplitude and can approximate some Rice channels, although Nakagami does not explicitly model a line-of-sight component. This approximation ability explains much of its computational success. It also warns against reading m as a physical count without evidence from the channel.

Decision guide

A good candidate when: nonnegative signal amplitude is modelled and its power can be described by a Gamma law.

Compare it with: Rayleigh with no line of sight and Rice with an explicit coherent component. Fit and diagnose power as well as amplitude.

References

Nakagami Distribution: equations and calculator

Distribution defintion

X∼Nakagami(m,Ω)X\sim\mathrm{Nakagami}\left(m,\Omega\right)

Distribution domain

x∈(0,∞)x\in\left(0,\infty\right)

Parameters domain and parameters constraints

m∈R⩾12+,Ω∈R+m\in\mathbb{R}^{+}_{\geqslant \frac{1}{2}},\Omega\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=γ(m,mΩx2)Γ(m)=P(m,mΩx2)F_{X}\left(x\right)=\frac{\gamma\left(m,\frac{m}{\Omega} x^2\right)}{\Gamma(m)}=\text{P}\left(m,\frac{m}{\Omega} x^2\right)

Probability density function

fX(x)=2mmΓ(m)Ωmx2m−1exp⁡(−mΩx2)f_{X}\left(x\right)=\frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1} \exp\left(-\frac{m}{\Omega}x^2\right)

Percent point function/Sample

FX−1(u)=ΩmP−1(m,u)F^{-1}_{X}\left(u\right)=\sqrt{\frac{\Omega}{m}\text{P}^{-1}\left(m,u\right)}

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=Γ(m+12)Γ(m)(Ωm)1/2\mathrm{Mean}(X)=\mu'_{1}=\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2}

Parametric variance

Variance(X)=μ2′−μ1′2=Ω(1−1m(Γ(m+12)Γ(m))2)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\Omega\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=Γ(m+12)Γ(m)m(1−4m(1−1m(Γ(m+12)Γ(m))2))2m(1−1m(Γ(m+12)Γ(m))2)3/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)\sqrt{m}}\left(1-4m\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)\right)}{2m\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)^{3/2}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+−6(Γ(m+12)Γ(m)m)4m+(8m−2)(Γ(m+12)Γ(m)m)2−2m+1m(1−1m(Γ(m+12)Γ(m))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}}=3+\frac{-6\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)\sqrt{m}}\right)^{4}m+\left(8m-2\right)\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)\sqrt{m}}\right)^{2}-2m+1}{m\left(1-\frac{1}{m}\left(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\right)^2\right)^{2}}

Parametric median

Median(X)=ΩmP−1(m,12)\mathrm{Median}(X)=\sqrt{\frac{\Omega}{m}\text{P}^{-1}\left(m,\frac{1}{2}\right)}

Parametric mode

Mode(X)=22((2m−1)Ωm)1/2\mathrm{Mode}(X)=\frac{\sqrt{2}}{2}\left(\frac{(2m-1)\Omega}{m}\right)^{1/2}

Additional information and definitions

u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}