Channel论文阅读笔记7-4之multipath interference by Jakes

信号频率分布
衰落信号（信道）在同相和正交相分量上的时变意味着信号的频率是随机变量，即信号经历随机频率调制，这种随机调频可以用概率密度函数和功率谱来描述。

由联合概率密度函数 $$p(r,\dot{r},\theta ,\dot{\theta })=\frac{r^2}{4{\pi }^2{b}_0{b}_2}\exp \left[-\frac{1}{2}\left(\frac{r^2}{b_0}+\frac{{\dot{r}}^2}{b_2}+\frac{r^2{\dot{\theta }}^2}{b_2}\right)\right]$$ 得到 $\dot{\theta }$ 的概率密度函数
$$p(\dot{\theta })={\int }_0^{\infty }dr{\int }_{-\infty }^{\infty }{\int }_0^{2\pi }d\theta p(r,\dot{r},\theta ,\dot{\theta })d\dot{\theta }=\frac{1}{2}\sqrt{\frac{b_0}{b_2}}{\left(1+\frac{b_0{\dot{\theta }}^2}{b_2}\right)}^{-3/2}\text{\hspace{1em}(1.4-1)}$$
随机调频的均方值为无穷大
$$\begin{array}{rl}& {\int }_{-\infty }^{\infty }{\dot{\theta }}^{2}p(\dot{\theta })=\frac{1}{2}\sqrt{\frac{b_0}{b_2}}{\int }_{-\infty }^{\infty }-\frac{b_2}{b_0\dot{\theta }}\cdot {\dot{\theta }}^{2}d{\left(1+\frac{b_0{\dot{\theta }}^2}{b_2}\right)}^{-1/2}\\ & =\frac{1}{2}\sqrt{\frac{b_2}{b_0}}\left\{-{\dot{\theta }{\left(1+\frac{b_0{\dot{\theta }}^2}{b_2}\right)}^{-1/2}|}_{-\infty }^{\infty }+{\int }_{-\infty }^{\infty }{\left(1+\frac{b_0{\dot{\theta }}^2}{b_2}\right)}^{-1/2}d\dot{\theta }\right\}\\ & =\frac{b_2}{b_0}\left[-1+\underset{\dot{\theta }\to \infty }{\lim }\ln \left(2\sqrt{\frac{b_0}{b_2}}\dot{\theta }\right)\right]=\frac{b_2}{b_0}[-1+\infty ]\end{array}\text{\hspace{1em}(1.4-2)}$$
(系数与原文中不同，有待核查）
最后等式的第一项是因为
$$\underset{\dot{\theta }\to \infty }{\lim }\dot{\theta }{\left(1+\frac{b_0{\dot{\theta }}^2}{b_2}\right)}^{-1/2}=\underset{\dot{\theta }\to \infty }{\lim }\dot{\theta }{\left(\frac{b_0{\dot{\theta }}^2}{b_2}\right)}^{-1/2}=\sqrt{\frac{b_2}{b_0}}$$
最后等式的第二项是因为
$$\int \frac{1}{\sqrt{a^2+x^2}}=\ln (x+\sqrt{a^2+x^2})+C$$ $$\underset{\dot{\theta }\to \infty }{\lim }\ln \left(\sqrt{\frac{b_0}{b_2}}\dot{\theta }+\sqrt{1+\frac{b_0{\dot{\theta }}^2}{b_2}}\right)=\underset{\dot{\theta }\to \infty }{\lim }\ln \left(2\sqrt{\frac{b_0}{b_2}}\dot{\theta }\right)=\infty$$