# Deriving cdf of ricean distribution + help

1. Apr 15, 2009

### JamesGoh

Im aware that the generalised form of the Marcum Q function, which is

$$Q_{M}(\alpha,\beta)=$$$$1/(\alpha)^{M-1}$$$$\int_{\beta}$$$$x^{M}$$$$.exp(-x^{2} +\alpha^{2})/2$$$$.I_{M-1}$$$$(\alpha$$$$x)$$$$dx$$

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 $$\beta=0$$

$$Q_{1}(\alpha,\beta)$$$$=$$$$\int_{0}^{r_{min}}=$$$$(r/\sigma)$$$$exp( (-r^{2}-A^{2})/2\sigma^{2} )$$$$I_{0}$$$$(r/\sigma$$$$A/\sigma)$$$$d(r/\sigma)$$

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 ?

2. Apr 15, 2009

### JamesGoh

Im aware that the generalised form of the Marcum Q function, which is

$$Q_{M}(\alpha,\beta)=$$$$1/(\alpha)^{M-1}$$$$\int_{\beta}$$$$x^{M}$$$$.exp(-x^{2} +\alpha^{2})/2$$$$.I_{M-1}$$$$(\alpha$$$$x)$$$$dx$$

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 $$\beta=0$$

$$Q_{1}(\alpha,\beta)$$$$=$$$$\int_{0}^{r_{min}}=$$$$(r/\sigma)$$$$exp( (-r^{2}-A^{2})/2\sigma^{2} )$$$$I_{0}$$$$(r/\sigma$$$$A/\sigma)$$$$d(r/\sigma)$$

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 ?