  A bidirectional integrated sensing and communication (ISAC) system is
proposed, in which a pair of transceivers carry out two-way communication and
mutual sensing. Both full-duplex and half-duplex operations in narrowband and
wideband systems are conceived for the bidirectional ISAC. 1) For the
narrowband system, the conventional full-duplex and half-duplex operations are
redesigned to take into account sensing echo signals. Then, the transmit
beamforming design of both transceivers is proposed for addressing the sensing
and communication (S&C) tradeoff. A one-layer iterative algorithm relying on
successive convex approximation (SCA) is proposed to obtain Karush-Kuhn-Tucker
(KKT) optimal solutions. 2) For the wideband system, the new full-duplex and
half-duplex operations are proposed for the bidirectional ISAC. In particular,
the frequency-selective fading channel is tackled by delay pre-compensation and
path-based beamforming. By redesigning the proposed SCA-based algorithm, the
KKT optimal solutions for path-based beamforming for characterizing the S&C
tradeoff are obtained. Finally, the numerical results show that: i) For both
bandwidth scenarios, the existence of the interference introduced by sensing
results in full-duplex may not always outperform half-duplex, especially in the
sensing-prior regime or when the communication channel is
line-of-sight-dominated; and ii) For both duplex operations, it is sufficient
to reuse communication signals for sensing in the narrowband system, while an
additional dedicated sensing signal is required in the wideband system.
