# Nakagami distribution

< List of probability distributions < Nakagami distribution.

The Nakagami distribution, named after Japanese engineer Minoru Nakagami , is a relatively recent addition to the world of probability distributions — it was introduced in 1960. This versatile distribution is especially useful for modeling small scale fading in settings with dense signal scatters, and is heavily used to model right-skewed data sets.

Its flexibility and wide-ranging applicability make the Nakagami distribution an important tool for statistical modeling in a range of fields. It has a broad range of real-world applications – such as characterizing breast tumors using ultrasound imaging, modeling wireless signal and radio wave propagation, and meteorological forecast modeling.

## Nakagami distribution PDF

The probability density function (PDF) for the Nakagami distribution is

Where:

The two parameters combine to determine the height, slope and concavity of the probability density curve. The PDF is valid on the interval (0,∞). In other words, if x ≤ 0, then p = 0.

The Nakagami distribution is closely related to the gamma distribution : If x has shape parameter μ and scale parameter ω, then x2 has a gamma distribution with shape parameter μ and scale parameter ω/μ.

## Nakagami distribution variants

If μ is restricted to the interval q=μ; 0 > q > 1, the distribution is called Nakagami-q, also known as the Hoyt distribution or Nakagami-Hoyt distribution. Close to the mean, it is a good approximation to the Rician distribution. However, towards the tails the two distributions differ significantly.

The Nakagami-m distribution has gained popularity in modeling of physical fading radio channels because of its good fit to empirical fading data. It introduces a parameter m and includes the one-sided Gaussian distribution (𝑚 = 0.5) and the Rayleigh distribution (𝑚 = 1) as special cases. In a Nakagami-𝑚 fading environment, 𝑅𝑖 and 𝑟𝑖 (𝑖 = 1, 2) follow the distributions 

The Nakagami-μ variant is the probability distribution of the absolute value of X — a 2µ-dimensional normal random variable with mean of zero mean µ and variance Ω .

Suppose that RNaka is the amplitude magnitude, defined as the norm of X :

Then RNaka is a Nakagami-µ random variable with distribution

## References

 Nakagami, M. (1960) “The m-Distribution, a general formula of intensity of rapid fading”. In William C. Hoffman, editor , Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18–20, 1958, pp 3-36. Pergamon Press.

 Pavlovovic, D. et al. Statistics for Ratios of Rayleigh, Rician, Nakagami-m, and Weibull Distributed Random Variables. Mathematical Problems in Engineering, 2013.

 Sanchez, J. (2011). Analysis of SC-FDMA and OFDMA Performance over Fading Channels. Doctoral Thesis.

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