lmomco (version 1.7.3)

pdfemu: Probability Density Function of the Eta-Mu Distribution

Description

This function computes the probability density of the Eta-Mu ($\eta:\mu$) distribution given parameters ($\eta$ and $\mu$) of the distribution computed by paremu. The probability density function of the distribution is $$f(x) = \frac{4\sqrt{\pi}\mu^{\mu - 1/2}h^\mu}{\gamma(\mu)H^{\mu - 1/2}}x^{2\mu}\mathrm{exp}(-2\mu h x^2)\mathrm{I}_{\mu-1/2}(2\mu H x^2)$$ where $f(x)$ is the nonexceedance probability for quantile $x$ and the modified Bessel function of the first kind is $I_k(x)$. The $h$ and $H$ are $$h = \frac{1}{1-\eta^2}$$ and $$H = \frac{\eta}{1-\eta^2}$$ for Format 2 of the distribution as described by Yacoub (2007). This format is exclusively used in the algorithms of the lmomco package.

If $\mu=1$, then the Rice distribution results, although pdfrice is not used. If $\kappa \rightarrow 0$, then the exact Nakagami-m density function results with a close relation to the Rayleigh distribution.

Define $m$ as $$m = 2\mu\biggl[1 + {\biggr(\frac{H}{h}\biggl)}^2 \biggr]$$ and for a given $m$, the parameter $\mu$ must lie in the range $$m/2 \le \mu \le m$$

Usage

pdfemu(x, para, paracheck=TRUE)

Arguments

x

A real value.

para

The parameters from paremu or similar.

paracheck

A logical controlling whether the parameters and checked for validity.

Value

• Probability density ($f$) for $x$.

References

Yacoub, M.D., 2007, The kappa-mu distribution and the eta-mu distribution: IEEE Antennas and Propagation Magazine, v. 49, no. 1, pp. 68--81

See Also

cdfemu, quaemu, paremu

Examples

x <- seq(0,4, by=.1)
para <- vec2par(c(.5, 1.4), type="emu")
F <- cdfemu(x, para)
X <- quaemu(F, para)
plot(F, X, type="l", lwd=8)
lines(F, x, col=2)


delx <- 0.005
x <- seq(0,3, by=delx)

plot(c(0,3), c(0,1), xaxs="i", yaxs="i",
     xlab="RHO", ylab="pdfemu(RHO)", type="n")
mu <- 0.6
# Note that in order to produce the figure correctly using the etas
# shown in the figure that it must be recognized that these are the etas
# for format1, but all of the algorithms in lmomco are built around
# format2
etas.format1 <- c(0, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5, 1)
etas.format2 <- (1 - etas.format1)/(1+etas.format1)
H <- etas.format2 / (1 - etas.format2^2)
h <-            1 / (1 - etas.format2^2)
for(eta in etas.format2) {
   lines(x, pdfemu(x, vec2par(c(eta, mu), type="emu")),
         col=rgb(eta^2,0,0))
}
mtext("Yacoub (2007, figure 5)")


plot(c(0,3), c(0,2), xaxs="i", yaxs="i",
     xlab="RHO", ylab="pdfemu(RHO)", type="n")
eta.format1 <- 0.5
eta.format2 <- (1 - eta.format1)/(1 + eta.format1)
mus <- c(0.25, 0.3, 0.5, 0.75, 1, 1.5, 2, 3)
for(mu in mus) {
   lines(x, pdfemu(x, vec2par(c(eta, mu), type="emu")))
}
mtext("Yacoub (2007, figure 6)")


plot(c(0,3), c(0,1), xaxs="i", yaxs="i",
     xlab="RHO", ylab="pdfemu(RHO)", type="n")
m <- 0.75
mus <- c(0.7425, 0.75, 0.7125, 0.675, 0.45, 0.5, 0.6)
for(mu in mus) {
   eta <- sqrt((m / (2*mu))^-1 - 1)
   print(eta)
   lines(x, pdfemu(x, vec2par(c(eta, mu), type="emu")))
}
mtext("Yacoub (2007, figure 7)")