  Integrated sensing and communications is regarded as a key enabling
technology in the sixth generation networks, where a unified waveform, such as
orthogonal frequency division multiplexing (OFDM) signal, is adopted to
facilitate both sensing and communications (S&C). However, the random
communication data embedded in the OFDM signal results in severe variability in
the sidelobes of its ambiguity function (AF), which leads to missed detection
of weak targets and false detection of ghost targets, thereby impairing the
sensing performance. Therefore, balancing between preserving communication
capability (i.e., the randomness) while improving sensing performance remains a
challenging task. To cope with this issue, we characterize the random AF of
OFDM communication signals, and demonstrate that the AF variance is determined
by the fourth-moment of the constellation amplitudes. Subsequently, we propose
an optimal probabilistic constellation shaping (PCS) approach by maximizing the
achievable information rate (AIR) under the fourth-moment, power and
probability constraints, where the optimal input distribution may be
numerically specified through a modified Blahut-Arimoto algorithm. To reduce
the computational overheads, we further propose a heuristic PCS approach by
actively controlling the value of the fourth-moment, without involving the
communication metric in the optimization model, despite that the AIR is
passively scaled with the variation of the input distribution. Numerical
results show that both approaches strike a scalable performance tradeoff
between S&C, where the superiority of the PCS-enabled constellations over
conventional uniform constellations is also verified. Notably, the heuristic
approach achieves very close performance to the optimal counterpart, at a much
lower computational complexity.
