- #1
JamesGoh
- 140
- 0
Im aware that the generalised form of the Marcum Q function, which is
Q_{M}(\alpha,\beta)=1/(\alpha)^{M-1}\int_{\beta}x^{M}.exp(-x^{2} +\alpha^{2})/2.I_{M-1}(\alphax)dx
and the pdf of the amplitude in rice distribution is
f_{r}(r)=r/\sigma^{2}exp( (-r^{2}-A^{2})/2\sigma^{2} )I_{0}(rA/\sigma^{2})
where I_{0}(x) is a modified bessel function of first kind, zero order
and the cdf of the rice distribution is
F_{r}(r) =1-Q_{M}(A/\sigma,r_{min}/\sigma)
Using the formula for Qm and the rice pdf, I have tried to get the rice cdf, however I have not had much success. I have tried the following
Let x=r/\sigma, \alpha=A/\sigma and \beta=0
Q_{1}(\alpha,\beta)=\int_{0}^{r_{min}}=(r/\sigma)exp( (-r^{2}-A^{2})/2\sigma^{2} )I_{0}(r/\sigmaA/\sigma)d(r/\sigma)
Im aware that the cdf is the integral of the pdf and \sigma is a constant (which means it cannot change), so is my approach correct ?
Q_{M}(\alpha,\beta)=1/(\alpha)^{M-1}\int_{\beta}x^{M}.exp(-x^{2} +\alpha^{2})/2.I_{M-1}(\alphax)dx
and the pdf of the amplitude in rice distribution is
f_{r}(r)=r/\sigma^{2}exp( (-r^{2}-A^{2})/2\sigma^{2} )I_{0}(rA/\sigma^{2})
where I_{0}(x) is a modified bessel function of first kind, zero order
and the cdf of the rice distribution is
F_{r}(r) =1-Q_{M}(A/\sigma,r_{min}/\sigma)
Using the formula for Qm and the rice pdf, I have tried to get the rice cdf, however I have not had much success. I have tried the following
Let x=r/\sigma, \alpha=A/\sigma and \beta=0
Q_{1}(\alpha,\beta)=\int_{0}^{r_{min}}=(r/\sigma)exp( (-r^{2}-A^{2})/2\sigma^{2} )I_{0}(r/\sigmaA/\sigma)d(r/\sigma)
Im aware that the cdf is the integral of the pdf and \sigma is a constant (which means it cannot change), so is my approach correct ?