Maybe I could approximate the following integral related to the modified Bessel function of the first kind ?
$$\mathbb{E}_{Z_{1},Z_{2},\hat{h}} \log \frac{I_{n_{\mathrm{d}}-1}\left(2 \left|\hat{h}\right| ||y|| \sqrt{n_{\mathrm{d}} \rho_{\mathrm{d}}}\right)}{(\left|\hat{h}\right| ||y|| \sqrt{n_{\mathrm{d}} \rho_{\mathrm{d}}})^{n_{\mathrm{d}}-1}} , \quad (1)$$
where $I_{n_{d}−1}(\cdot)$ the modified Bessel function of the first kind with $n_{d}−1$ degrees of freedom, $Z_{1}$ obeys the $\Gamma(1,1)$ distribution and $Z_{2}$ obeys the $\Gamma(n_{d}-1,1)$ distribution, $||y||^{2}$ equals to $(1+n_{d}\rho_{d})Z_{1}+Z_{2}$ in the distribution sense, $n_{d}$ , $\rho_{d}$ are all known, $h$ obeys the complex Gaussian distribution $\mathbf{CN}(0,1)$ and $\hat{h}|h$ obeys the complex Gaussian distribution $\mathbf{CN}(h,1/(n_{p}\rho_{p}))$ , $|\hat{h}|$ is the norm of $h$.
Let $a=\sqrt{n_{\mathrm{d}}\rho_{\mathrm{d}}}$, $\nu=n_{\mathrm{d}}-1$ and $b=1+n_{\mathrm{d}}\rho_{\mathrm{d}}$, then I could get the following equation
$$\mathbb{E}_{\hat{h}}\int_{0}^{\infty}\int_{0}^{\infty} z_{2}^{\nu-1}e^{-z_{1}-z_{2}}\log\frac{I_{\nu}\left(2 a\left|\hat{h}\right| ||y|| \right)}{(a\left|\hat{h}\right| ||y|| )^{\nu}}dz_{1}dz_{2}. \quad (2)$$
Maybe I can use the following inequality related to Bessel function ?
$$I_{\nu}(x)\leq \frac{(\frac{x}{2})^{\nu}e^{x}}{\Gamma(1+\nu)}, \quad (3)$$ where $I_{\nu}(\cdot)$ denotes the modified Bessel function of the first kind and $\Gamma(\cdot)$ reprents the gamma function.
Nevertheless, there is no idea about how to deal with the above integral $(2)$.
How can I calculate the upper bound of the integral $(2)$ or if it has the closed form ？
I would be grateful if someone could help solve this problem .
Thanks, Liu .