R_\mathsf{ts}{\mathopen{}\mathclose\bgroup\originalleft}({\mathbb{H}(X)}  - \sum\limits_{i=1}^{m}{\mathbb{H}(X^{\mathsf{B}}_i|Y^{\mathsf{B}}_i)} {\aftergroup\egroup\originalright}) \nonumber\\ 
&\quad + {\mathopen{}\mathclose\bgroup\originalleft}(1-R_\mathsf{ts}{\aftergroup\egroup\originalright}){\mathopen{}\mathclose\bgroup\originalleft}(m_\mathsf{par} - \sum\limits_{i=1}^{m_\mathsf{par}}{\mathbb{H}(X^{\mathsf{B}}_{\mathsf{par},i}|Y^{\mathsf{B}}_{\mathsf{par},i})}{\aftergroup\egroup\originalright}).\label{eqn:air}\end{aligned}$$

The achievable rate for has been derived in [@bocherer2017_achievable_rate_ps]. For a system employing time-sharing, the resulting achievable rate is the average of the achievable rate of the two transmission schemes.

![Achievable rates for different code rates with $\Lambda_\mathsf{par}$ according to [Algorithm]{} for a cutoff parameter $\delta=0.005$.[]{data-label="fig:shaping:achievable_rate_air"}](air_time_share)

In [Fig.]{}, we plot the capacity and the achievable rate [()]{} for different code rates $R_\mathsf{c}=\{$$1/4$, $1/3$, $2/5$, $1/2$, $3/5$, $2/3$, $3/4$, $4/5$, $5/6$, $8/9$, $9/10\}$ and modulation orders for a cutoff parameter $\delta=0.005$. $\Lambda_\mathsf{par}$ and hence $M_\mathsf{par}$ are chosen according to the results of [Algorithm]{}. For each modulation order, we notice that the curves cross at a certain . For below this point, the lowest code rate (corresponding to the highest curve) gives the best performance whereas for above this point, the highest code rate (corresponding to the highest curve) gives the best performance. We note the influence of time-sharing, which results in a gap between the achievable rate and capacity. The gap increases for lower code rates $R_\mathsf{c}$ as the channel input distribution deviates more from the optimal one.

Numerical Evaluation {#sec:performance}
====================

In this section, we evaluate the performance of the 