Theorem

Given a fading channel where $Y$ and $A$ are known at the receiver and at the transmitter at each time instant. Then we define the channel capacity with FSI, $C_{FSI}(P)$ as $$C_{FSI}(P)=E_{A}\left[\frac{1}{2}\log_{2}\left(1+ \frac{A^{2}p^{*}(A)}{\sigma^{2}}\right) \right]$$where we define the optimal power allocation, $p^{*}$, as $$p^{*}(a)=\max\left(0,\frac{1}{\lambda}- \frac{\sigma^{2}}{a^{2}}\right)$$and $\lambda$ satisfies $E_{A}[p^{*}(A)]=P$.

Note

Fading decreases capacity for the AWGN Channel $$C_{FSI}(P)\le C_{G}(P)$$