Deriving cdf of ricean distribution + help

  • Thread starter JamesGoh
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Im aware that the generalised form of the Marcum Q function, which is

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

and the pdf of the amplitude in rice distribution is

[tex]f_{r}(r)=[/tex][tex]r/\sigma^{2}[/tex][tex]exp( (-r^{2}-A^{2})/2\sigma^{2} )[/tex][tex]I_{0}[/tex][tex](rA/\sigma^{2})[/tex]

where [tex]I_{0}(x)[/tex] is a modified bessel function of first kind, zero order

and the cdf of the rice distribution is

[tex]F_{r}(r) = [/tex][tex]1-Q_{M}(A/\sigma,r_{min}/\sigma)[/tex]

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 [tex]x=r/\sigma[/tex], [tex]\alpha=A/\sigma[/tex] and [tex]\beta=0[/tex]

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

Im aware that the cdf is the integral of the pdf and [tex]\sigma[/tex] is a constant (which means it cannot change), so is my approach correct ?
 

Answers and Replies

  • #2
143
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Im aware that the generalised form of the Marcum Q function, which is

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

and the pdf of the amplitude in rice distribution is

[tex]f_{r}(r)=[/tex][tex]r/\sigma^{2}[/tex][tex]exp( (-r^{2}-A^{2})/2\sigma^{2} )[/tex][tex]I_{0}[/tex][tex](rA/\sigma^{2})[/tex]

where [tex]I_{0}(x)[/tex] is a modified bessel function of first kind, zero order

and the cdf of the rice distribution is

[tex]F_{r}(r) = [/tex][tex]1-Q_{M}(A/\sigma,r_{min}/\sigma)[/tex]

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 [tex]x=r/\sigma[/tex], [tex]\alpha=A/\sigma[/tex] and [tex]\beta=0[/tex]

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

Im aware that the cdf is the integral of the pdf and [tex]\sigma[/tex] is a constant (which means it cannot change), so is my approach correct ?
 

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