Nakagami-:math:`m`
==================

.. author   jgaeddert Joseph D. Gaeddert <joseph@liquidsdr.org>
.. date     January 1, 1975
.. location Boston, MA
.. header   /doc/random/banner.png
.. keywords random number generator, RNG, randn, randnf, randf


The Nakagami-:math:`m` distribution is a versatile stochastic model for
modeling radio links :cite:`Braun:1991` and has often been regarded as the
best distribution to model land mobile propagation due to its ability to
describe fading situations worse than Rayleigh, including one-sided
Gaussian :cite:`Simon:1998`.
Empirical evidence regarding the efficacy the Nakagami-:math:`m` distribution
has on fading profiles been presented in :cite:`Turin:1980,Suzuki:1977`.
Thus statistical inference of the Nakagami-:math:`m` fading parameters are of
interest in the design of adaptive radios such as optimized transmit
diversity modes :cite:`Cavers:1999,Ko:2003` and adaptive modulation schemes
:cite:`Catreux:2002`.
The Nakagami-:math:`m` probability density function is given by
:cite:`Papoulis:2002`

.. math::
    :name: eqn-random-nakagamim-pdf

    f_X(x;m,\Omega) =
    \begin{cases}
        \frac{2}{\Gamma(m)}
        \left( \frac{m}{\Omega} \right)^m
        x^{2m-1}
        e^{ -(m/\Omega)x^2}             & x \ge 0 \\
        0                               & \text{else}.
    \end{cases}

where
:math:`m \ge 1/2` is the shape parameter and
:math:`\Omega \gt 0` is the spread parameter.
Nakagami-:math:`m` random numbers are generated from the gamma distribution.
Specifically if :math:`R` follows a gamma distribution defined by
:eq:`eqn-random-gamma-pdf` with parameters :math:`\alpha` and :math:`\beta`,
then :math:`X=\sqrt{R}` has a Nakagami-:math:`m` distribution with
:math:`m=\alpha` and :math:`\Omega=\beta/\alpha`.

.. qplot:: random random/random.c
    :args: -d nak -m 0.75 -o 4
    :width: 80%
    :name: fig-nakagami-m
    :caption: Nakagami-:math:`m` distribution with :math:`m=0.75` and :math:`\Omega=4`

Interface
~~~~~~~~~

Nakagami random number generator interface:

.. code-block:: c

    // generate a Nakagami-m random variable
    float randnakmf(float _m,
                    float _omega);

    // compute a Nakagami-m probability density
    float randnakmf_pdf(float _x,
                        float _m,
                        float _omega);

    // compute a Nakagami-m cumulative distribution
    float randnakmf_cdf(float _x,
                        float _m,
                        float _omega);