The non-central Nakagami distribution

Søren Hauberg
Section for Cognitive Systems
Technical University of Denmark
sohau@dtu.dkAbstract
The Nakagami distribution describe the square root of a random variable drawn
from a Gamma distribution. Equivalently, the Gamma distribution can be seen as a
one-dimensional Wishart distribution. In this note, we consider the distribution of
the square root of a random variable drawn from a non-central one-dimensional
Wishart distribution. We present the probability density function of this distribution
along with closed-form expressions for its moments.

This document is just a short technical note. Comments can be directed at sohau@dtu.dk1

The Nakagami distribution

Let zd ∼ N (0, σ 2 ), d = 1, . . . , D be iid samples from a zero-mean normal distribution. Then
x=DX

zd2 ∈ R+(1.1)

d=1follows a Gamma distribution. Equivalently, and more suitable for our purposes, x also follows a
one-dimensional Wishart distribution [1],x ∼ W1 (D, σ 2 )(1.2)
rp(x) = W1 (x | D, σ 2 ) =xD−2−1x Γ (D/2) exp − 2 .
(1.3)
2D σ 2D
2σThis distribution can be seen as describing the squared√ norm of a zero-mean normally distributedvector. The norm of this vector is then given by y = x, which follows a Nakagami distribution [2],y ∼ Nakagami D/2, Dσ 2(1.4)

2

2y
√
p(y) = Nakagami y | D/2, Dσ 2 =y D−1 exp − 2 .
(1.5)
2σΓ (D/2) ( 2σ)DThis expression is easily derived by the change of value theorem; see next section.
The expectation and variance of a Nakagami distributed variable is√Γ D+12
E [y] =2σD
Γ 2
2 !
Γ ((D + 1)/2)
2var [y] = σ D − 2.
Γ (D/2)
We will derive a more general version of these results in the next section.
Preprint. Work in progress.

(1.6)
(1.7)

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The non-central Nakagami distribution

Now, let zd ∼ N (µd , σ 2 ), d = 1, . . . , D be independent samples from normal distributions withvariance σ 2 . ThenDX
x=zd2 ∈ R+(2.1)
d=1follows a one-dimensional non-central Wishart distribution [1],x ∼ W1 (D, σ 2 , Ω)
(2.2)
PD
2µd(2.3)Ω = d=12σp(x) = W1 (x | D, σ 2 , Ω)r x (2.4)
xD−2
Ω
−1−2D
D1=
Γ ( /2) 0 F1 /2, /4Ωσ x exp − 2 exp −.
2D σ 2D2σ2Here 0 F1 is a generalized hypergeometric function; see Sec. 7.3 of Muirhead’s book [1]. This
distribution can be seen as describing the squared norm of √a normal distributed vector with isotropicvariance. The norm of this vector is then given by y = x, which we say follows a non-central
Nakagami distribution. This distribution can be derived by change-of-variables to bep (y) = W1 (y 2 | D, σ 2 , Ω) · 2y(2.5)




D−122yyDyΩ= (D −2)/2 D D 0 F1, Ω 2 exp − 2 exp −.
(2.6)
24σ
2σ22σ Γ ( /2)√To compute the moments of x we recall the following result from Muirhead [1, Theorem 10.3.7].
q×q
Let X ∈ Rand X ∼ Wq (D, Σ, Ω) thenΓq (D/2 + k)
D
1E (det(X))k = (det(Σ))k 2qk(2.7)
1 F1 (−k, /2, − /2Ω),
Γq (D/2)√
where 1 F1 is a generalized hypergeometric function. Since y = x is a positive scalar, its determinantis merely y, and we geth√ iΓ(D/2 + k)
2kD
1E x= σ 2k 2k(2.8)
1 F1 (−k, /2, − /2Ω).
Γ(D/2)√From this we see that the mean and the variance of y = x is√ 
Γ((D+1)/2)
11D
1E x = σ2 /2(2.9)
1 F1 (− /2, /2, − /2Ω)
Γ(D/2)
Γ((D+2)/2)D
1E [x] = σ 2 2(2.10)
1 F1 (−1, /2, − /2Ω)
Γ(D/2)√  √ 2var x = E x − E x(2.11)

2σ 2
D+2
D
1= D
Γ
1 F1 (−1, /2, − /2Ω)
Γ( /2)
2!
(2.12)
Γ((D+1)/2)221D
1−.
1 F1 (− /2, /2, − /2Ω)
Γ(D/2)
Acknowledgments. The author was supported by a research grant (15334) from VILLUM
FONDEN. This project has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (grant agreement no 757360).

References[1] R. J. Muirhead. Aspects of Multivariate Statistical Theory. John Wiley & Sons, 2005.
[2] M. Nakagami. The m-distribution, a general formula of intensity of rapid fading, iwc hoffman, ed, 1960.

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