capacity, however, we neglect these effects. Hence, the results can be seen as an upper bound on the performance.

Capacity Achieving Distribution {#sec:optimum_input_distribution}
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From [Fig.]{}, it is intuitive that pulses with short duration should be transmitted more frequently than pulses with long duration. However, shorter pulses are more perturbed by noise than longer pulses. Hence, the optimal input distribution to the channel as described by the conditional [()]{} is not the conventional uniform distribution. The channel capacity is obtained by maximizing the , $$\begin{aligned}
    {\mathbb{I}(X;Y)} &\triangleq {\mathbb{E}_{X,Y}\!{\mathopen{}\mathclose\bgroup\originalleft}\{{\log_2{\mathopen{}\mathclose\bgroup\originalleft}({\frac{p_{Y|X}(Y|X)}{\sum_{\tilde{\lambda}\in\Lambda}p_{Y|X}(Y|\tilde{\lambda}) p_X(\tilde{\lambda})}}{\aftergroup\egroup\originalright})}{\aftergroup\egroup\originalright}\}
}\end{aligned}$$ over all possible input distributions $p_X(\lambda )$. Here, due to the variable transmission duration, we need to consider the under a variable cost constraint $\bar{T}(\cdot)$ [@verdu1990], $$\begin{aligned}
    {\mathsf{I}(X;Y)} &\triangleq \frac{{\mathbb{I}(X;Y)}}{\bar{T}(X)}.
    \label{eqn:mutual_information_time_scaled}\end{aligned}$$ To emphasize that the cost of a symbol is its corresponding pulse duration, we refer to the in the form of [()]{} as time-scaled . We can therefore define the capacity as $$\begin{aligned}
{\mathsf{C}}&\triangleq \sup_{p_X(\lambda)} {\mathsf{I}(X;Y)}
\label{eqn:def_cstar}\end{aligned}$$ where we set the supremum to zero if the set of distributions therein is empty. The capacity-achieving distribution, denoted by $p_X^*(\lambda )$, is in the set for which the supremum is non-zero.

As the ${\mathbb{I}(X;Y)}$ is concave in $p_X(\lambda )$ and $\bar{T}(X)$ 