We investigate the downlink of a cell-free massive multiple-in multiple-out system in which all access points (APs) are connected in a linear-topolpgy fronthaul with constrained capacity and send a common message to a single receiver. By modeling the system as an extension of the multiple-access channel with partially cooperating encoders, we derive the channel capacity of the two-AP setting and then extend the results to arbitrary N-AP scenarios. By developing a cooperating mode concept, we investigate the optimal cooperation among the encoders (APs) when we limit the total fronthaul capacity, and the total transmit power is constrained as well. It is demonstrated that achieving capacity requires a water-pouring distribution of the total available fronthaul capacity over the fronthaul links. Our study reveals that a linear growth of total fronthaul capacity results in a logarithmic growth of the beamforming capacity. Moreover, even if the number of APs would be unlimited, only a finite number of them need to be activated. We found an expression for this number.
Keywords: cell-free MIMO; channel capacity; constrained fronthaul; distributed beamforming.