  Uplink-downlink duality refers to the fact that under a sum-power constraint,
the capacity regions of a Gaussian multiple-access channel and a Gaussian
broadcast channel with Hermitian transposed channel matrices are identical.
This paper generalizes this result to a cooperative cellular network, in which
remote access-points are deployed as relays in serving the users under the
coordination of a central processor (CP). In this model, the users and the
relays are connected over noisy wireless links, while the relays and the CP are
connected over noiseless but rate-limited fronthaul links. Based on a
Lagrangian technique, this paper establishes a duality relationship between
such a multiple-access relay channel and broadcast relay channel, under the
assumption that the relays use compression-based strategies. Specifically, we
show that under the same total transmit power constraint and individual
fronthaul rate constraints, the achievable rate regions of the Gaussian
multiple-access and broadcast relay channels are identical, when either
independent compression or Wyner-Ziv and multivariate compression strategies
are used. The key observations are that if the beamforming vectors at the
relays are fixed, the sum-power minimization problems under the achievable rate
and fronthaul constraints in both the uplink and the downlink can be
transformed into either a linear programming or a semidefinite programming
problem depending on the compression technique, and that the uplink and
downlink problems are Lagrangian duals of each other. Moreover, the dual
variables corresponding to the downlink rate constraints become the uplink
powers; the dual variables corresponding to the downlink fronthaul constraints
become the uplink quantization noises. This duality relationship enables an
efficient algorithm for optimizing the downlink transmission and relaying
strategies based on the uplink.
