  This paper considers a cell-free integrated sensing and communication (ISAC)
MIMO system, where distributed MIMO access points (APs) jointly serve the
communication users and sense the target. For this setup, we derive a sensing
SNR for multi-static sensing where both joint communication and sensing signals
transmitted by different APs are utilized. With this sensing objective, we
develop two baseline approaches that separately design the sensing and
communication beamforming vectors, namely communication-prioritized sensing
beamforming and sensing-prioritized communication beamforming. Then, we
consider the joint sensing and communication (JSC) beamforming design and
derive the optimal structure of these beamforming vectors based on a max-min
fairness formulation. In addition, considering any pre-determined JSC beam
design, we devise a power allocation approach. The results show that the
developed JSC beamforming is capable of achieving nearly the same communication
signal-to-interference-plus-noise ratio (SINR) of the communication-prioritized
sensing beamforming solution with almost the same sensing SNR of the
sensing-prioritized communication beamforming approach. The proposed JSC
beamforming optimization also provides a noticeable gain over the power
allocation with regularized zero-forcing beamforming, yielding a promising
strategy for cell-free ISAC MIMO systems.
