Adaptive Subcarrier Pulse Shaping for GFDM-Based 6G ISAC: Riemannian Manifold PAPR Reduction and Power Allocation

Generalized frequency division multiplexing (GFDM) shapes every subcarrier with a prototype filter, which makes the pulse shape a natural design variable for sixth-generation (6G) integrated sensing and communication (ISAC). This paper designs one filter per subcarrier, together with the subcarrier powers, to reduce the peak-to-average power ratio (PAPR) while safeguarding sensing and communication. Using the fourth-order cumulant of the constellation, the PAPR of the oversampled block and the ergodic sensing mutual information (MI) of a monostatic receiver that knows its own transmitted block are expressed in closed form over the random data. A link-MI guard keeps the modulator well conditioned, and a rate constraint governs power allocation. The filters are optimised by a Riemannian augmented-Lagrangian conjugate-gradient method on a band-limited oblique manifold whose support parameter trades per-subcarrier spectral confinement against PAPR. The powers are updated by an exact successive-convex-approximation step with a closed-form KKT solution, and all gradients are validated against finite differences. In 3GPP TDL-C simulations at 28 GHz with 100 MHz bandwidth, the designed filter banks reduce the PAPR at a CCDF of 10−2 relative to raised-cosine GFDM by 0.1 to 2.1 dB, depending on the spectral support (1.1 dB for a moderately relaxed support), without side information or in-band distortion. Sensing MI stays within 0.5% and link MI within about 1% of raised-cosine GFDM, and the method outperforms GA, PSO and NSGA-II on the same objective at lower cost. At equal mean power the detection performance is unchanged, as the analysis predicts. Under a peak-power-limited amplifier, however, the smaller back-off needed for 8% EVM raises the OS-CFAR detection probability at a false-alarm probability of 10−4, from 0.38 to 0.40 for the moderate support and to 0.52 for the widest support.