that [()]{} is of exponential shape with an additional point mass at zero. Furthermore, we note that the shape of the distribution is mostly caused by the variable pulse duration. The noise then determines the optimal location and optimal number of constellation points.

For a transmission system, the is an upper bound on the achievable rate. In [Fig.]{} we evaluate the time-scaled for various input distributions for a cutoff parameter $\delta=0.005$. The capacity is depicted with a black solid line. To reduce the complexity of implementation, we constrain the constellation $\Lambda$ to $M$ linearly spaced points from $\lambda_1=0$ to $\lambda_M$, i.e., $$\begin{aligned}
    \lambda_i &= (i-1)\frac{\lambda_M}{M-1} \text{ for } i=1,\ldots,M,\end{aligned}$$ and plot the corresponding time-scaled in colored solid lines with markers. We note that the time-scaled is very close to the capacity curve until it saturates. Increasing the modulation order $M$ shows significant increase in the time-scaled . For comparison purposes, we also plot the time-scaled for a system with fixed symbol duration and conventional uniform distribution on a linearly spaced constellation as in [@shevchenko2018tcom]. We observe that the rate saturates at very low values and that increasing the modulation order $M$ shows only slight improvement.

![Block diagram of the scheme.[]{data-label="fig:shaping:system_block_diagram_pes"}](block_diagram_pes)

