Nakagami distribution

< List of probability distributions < Nakagami distribution.

The Nakagami distribution PDF.
The Nakagami distribution PDF.

The Nakagami distribution, named after Japanese engineer Minoru Nakagami [1], 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

nakagami distribution


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 [2]

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 [3].

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

Then RNaka is a Nakagami-µ random variable with distribution


[1] 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.

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

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

Scroll to Top