I learnt in $\textit{Elements of Information Theory -Cover & Thomas}$ that $\underline{\text{memory increases the capacity of a channel}}$.
A transmitter sends $X_i$ and the receiver receives $h_iX_i+w_i$, for all $i \in \{1,\ldots,n\}$. $h_i$'s and $w_i$'s are unknown to the receiver as well as transmitter except for its statistics.
Consider a binary channel with output $\boxed{ Y_i= 1, \text{if } |h_iX_i+w_i|^{2} > 1, \text{ else } Y_i=0}$
Here $X_i, Y_i \in\{0,1\}$. $w_i$'s are iid with $ w_i \sim \mathcal{CN}(0,1)$ and is independent of $X_i'$s. Also, $h_{i} \equiv h \sim \mathcal{C} \mathcal{N}(0,1)$ and is independent of $w_i$'s and $X_i$'s. Here $h_i$'s are equal (random but fixed across the channel uses). This means there is memory.
I am interested in finding the capacity of this channel with memory given by
\begin{aligned} &\max I\left(X_{1}, X_{2}, \ldots, X_{n} ; Y_{1}, Y_{2}, \ldots, Y_{n}\right) \\ &\quad=H\left(X_{1}, X_{2}, \ldots, X_{n}\right)-H\left(X_{1}, X_{2}, \ldots, X_{n} \mid Y_{1}, Y_{2}, \ldots, Y_{n}\right). \end{aligned}
My goal is to show that this capacity will be more than the capacity of the same channel model with i.i.d $h_i$'s.
I'm interested in the non-coherent case where $h_i$'s are not known.
Can someone help me solve this?