modulation orders and . For $M=4$, we note that for low gives the best result. Increasing the results in a third level being added. The same behavior is observed for $M=8$. Compared to $M=4$, the third level is introduced at a slightly lower . This results from the fact that for $M=8$, different constellation points are available. For $M=16$, we note that again a third level appears when increasing the . When further increasing it, this third level moves to an eigenvalue with larger imaginary part and consequently a fourth level at an eigenvalue with lower imaginary part appears. This behavior can be observed repeatedly. To map the binary parity bits to the constellation points, we require $M_\mathsf{par}$ to be a power of two. As this is not always the case (see [Fig.]{}), we pick the largest power of two that is smaller or equal than the number of constellation points given by [Algorithm]{}.

Achievable Rate of Probabilistic Eigenvalue Shaping {#sec:achievable_rate}
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To characterize the performance of , we derive the achievable rate of , denoted by $\mathsf{R}_\mathsf{ps}$. We assume that the channel is memoryless and that the decoder performs bit-metric decoding.

The achievable rate of is $$\begin{aligned}
\mathsf{R}_\mathsf{ps} &= 