  Joint communications and sensing (JCAS) is envisioned as a key feature in
future wireless communications networks. In massive MIMO-JCAS systems, hybrid
beamforming (HBF) is typically employed to achieve satisfactory beamforming
gains with reasonable hardware cost and power consumption. Due to the coupling
of the analog and digital precoders in HBF and the dual objective in JCAS,
JCAS-HBF design problems are very challenging and usually require highly
complex algorithms. In this paper, we propose a fast HBF design for JCAS based
on deep unfolding to optimize a tradeoff between the communications rate and
sensing accuracy. We first derive closed-form expressions for the gradients of
the communications and sensing objectives with respect to the precoders and
demonstrate that the magnitudes of the gradients pertaining to the analog
precoder are typically smaller than those associated with the digital precoder.
Based on this observation, we propose a modified projected gradient ascent
(PGA) method with significantly improved convergence. We then develop a deep
unfolded PGA scheme that efficiently optimizes the communications-sensing
performance tradeoff with fast convergence thanks to the well-trained
hyperparameters. In doing so, we preserve the interpretability and flexibility
of the optimizer while leveraging data to improve performance. Finally, our
simulations demonstrate the potential of the proposed deep unfolded method,
which achieves up to 33.5% higher communications sum rate and 2.5 dB lower
beampattern error compared with the conventional design based on successive
convex approximation and Riemannian manifold optimization. Furthermore, it
attains up to a 65% reduction in run time and computational complexity with
respect to the PGA procedure without unfolding.
