Capacity loss of fast fading regarding AWGN
Standard Rayleigh fading is $C\mathcal N\left(0,1\right)$ then $E\{\lvert h \rvert^2\} = 1$. Using Jensen's inequality $$\begin{align}\Delta&amp;=E\left\{\log\left(\lvert h \rvert^2\right)\right\}\\ &amp;\ge \log(E\{\lvert h \rvert^2\}) \\&amp;= \log(1) \\&amp;= 0 \\&amp;&gt; -0.83\end{align}$$ and this $\Delta$ should not be $-0.83$? I think I made some mistakes but I don't know what it is. 


Rayleigh fading is when the amplitudes of the real and imaginary part of $h$ are normal distributed; so $|h|$ follows a Rayleigh distribution, which has a probability density function  of

$$
p(x=|h|) = \frac {x}{\sigma^2}e^{\frac{x^2}{2\sigma^2}}\text.
$$

Hence, $E\left\{\lvert h\rvert^2\right\}$ is not the variance of the Normal distribution with variance $\sigma^2$.

Instead, the value you're looking for is
$$\begin{align}
E\left\{\log_2\left(\lvert h\rvert^2\right)\right\} &amp;= \int_{-\infty}^\infty \log_2\left(\lvert x\rvert^2\right)\frac {x}{\sigma^2}e^{\frac{x^2}{2\sigma^2}}\,dx\\
&amp;= \int_0^\infty 2\log_2\left(\lvert x\rvert\right)\frac {x}{\sigma^2}e^{\frac{x^2}{2\sigma^2}}\,dx\\
&amp;= \frac2{\sigma^2}\int_0^\infty x\log_2\left( x\right) e^{\frac{x^2}{2\sigma^2}}\,dx
\end{align}$$