  The capacity limits of continuous-aperture array (CAPA)-based wireless
communications are characterized. To this end, an analytically tractable
transmission framework is established for both uplink and downlink CAPA
systems. Based on this framework, closed-form expressions for the single-user
channel capacity are derived. The results are further extended to a multiuser
case by characterizing the capacity limits of a two-user channel and proposing
the associated capacity-achieving decoding and encoding schemes. 1) For the
uplink case, the sum-rate capacity and capacity region, as well as the
capacity-achieving detectors, are derived. 2) For the downlink case, the
uplink-downlink duality is established by deriving the uplink-to-downlink and
downlink-to-uplink transformations under the same power constraint, based on
which the optimal power allocation policy and the achieved sum-rate capacity
and capacity region are characterized. To gain further insights, several case
studies are presented by specializing the derived results into various array
structures, including the planar CAPA, linear CAPA, and planar spatially
discrete array (SPDA). Numerical results are provided to reveal that: i) the
channel capacity achieved by CAPAs converges towards a finite upper bound as
the aperture size increases; and ii) CAPAs offer significant capacity gains
over the conventional SPDAs.
