Simple and Accurate Approximations for the Sum of Nakagami-m and Related Random Variables

Abstract

Simple and accurate closed-form approximations to the probability density function of the sum of independent identically distributed random variables are derived. We propose a unified approach capable of providing the pdf of the sum when the summands involve the generalized Nakagami-m, generalized Gamma, Rayleigh and Weibull random variables. Unlike others work, the approximations do not use any infinite series or require any complicated special functions. Numerical results show that the new approximations have satisfactory accuracies. As an application, the outage probability and the M-QAM average SER performance of pre-detection EGC diversity are presented.

Appendices

Appendix A–Recurrence Relations

Let $f(z)=\sum _{k=0}^{\infty } a_k\,z^k$ and $g(z)=\sum _{k=0}^{\infty } c_k\,z^k$ be two absolutely convergent power series. Let $g=f^N$, with N a nonnegative integer, by differentiating with respect to z we find $fg'=Nf'g$. Then

$$\sum_{i=0}^{\infty} a_iz^i \sum_{j=0}^{\infty} c_j j z^{j-1} = N \sum_{i=0}^{\infty} a_i iz^{i-1} \sum_{j=0}^{\infty} c_jz^j.$$

(A1)

Applying the Cauchy product to both sides we have that

$$\begin{aligned}{} & {} \sum _{j=0}^{k} a_j c_{k-j} (k-j) = N \sum _{j=0}^{k} a_j c_{k-j} j, \; \; \text {or} \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} a_0 c_k k+\sum _{j=1}^{k} a_j c_{k-j}k = (N+1) \sum _{j=1}^{k} a_j c_{k-j} j. \end{aligned}$$

(A3)

After some rearrangements, the coefficients $c_k$ are given by

$$\begin{aligned}{} & {} c_k =\frac{1}{a_0} \sum _{j=1}^{k} \left[ \frac{(N+1)j}{k} -1 \right] a_j c_{k-j}, \; k\ge 1 , \end{aligned}$$

(A4)

where $c_0=(a_0)^N$.

Now, if we set $a_k = E[X^k]/k!$, $c_k = E[Y^k]/k!$ and $z\in \mathbb {C}$, we can see f and g as moment generating functions. Then, follows immediately that the kth-moments of the Y RV can be obtained recursively by

$$\begin{aligned}{} & {} E[Y^k] = \sum _{j=1}^{k} \left( {\begin{array}{c}k\\ j\end{array}}\right) \left[ \frac{(N+1)j}{k} -1 \right] E[X^j] E[Y^{k-j}], \; k\ge 1 . \end{aligned}$$

(A5)

Appendix B–Multivariate Nonlinear Regression

Multivariate nonlinear regression, i.e., a nonlinear regression with two or more variables, is a very difficult task. In order to find equations for $\Omega _a$ and $m_a$, as a function of both N and m, we appeal to a divide-and-conquer approach. In a first step, we obtained approximants for $\Omega _a$ as a function of N and $n_c$ regression coefficients $p_i(m)$, $i=\{1,2, \dots , n_c\}$. Note that there are regression coefficients for each value of m. In the following, we obtained approximants for $p_i(m)$, reaching the final regression model. We considered $N=\{2,4,\dots ,20\}$ and $m = \{0.5,1,2,\dots , 5\}$. Exploiting identical approach we obtained the approximants for $m_a$. The maximum absolute error is less than 1.5% and 2.0%, for $\Omega _a$ and $m_a$, respectively.

An illustrative example follows. Let us consider that $\Omega _a(N,.)$ is well approximated by a quadratic polynomial model and $p_i(m)$ by a power model, respectively, then

$$\begin{aligned} \begin{matrix} \Omega _a(N,m) = p_1(m) + p_2(m).N + p_3(m).N^2, \; \; \text {with} \\ \\ p_i(m) = q_{1,i} + q_{2,i}.m^{q_{3,i}}, \; \; i=\{1,2,3\}. \end{matrix} \end{aligned}$$

(B6)