Uplink-Downlink SINR Duality via Lagrange Duality

The uplink-downlink SINR duality theorem is a key tool which simplify substantially the problem of joint design of the linear transmit and receive beamformers in multiple-input multiple-output (MIMO) downlink channels. The theorem has been proved previously under the assumption that the cross- coupling matrix between the users is primitive or, alternatively, by postulating the nonnegativity of the resolvent. By using the Lagrange duality theory, we first give an alternative proof which holds for arbitrary cross-coupling matrices. The proof does not only extend the result to a larger set of practical applications, but it also reveal more insight on the sum power minimization problem under a set of minimum SINR requirements for data streams. As a practical application, we apply the uplink-downlink SINR duality to derive a general method for MIMO downlink linear transceiver optimization according to different system performance criteria, including weighted sum rate maximization, weighted sum mean square error minimization, and minimum SINR maximization. The proposed method can handle multiple antennas at the BS and at the mobile user with single and/or multiple data streams per scheduled user. The numerical simulations show that the sum rate achieved by the sum rate maximization algorithm is within 0.5-1.5 bits/second/Hz close to the sum capacity. When compared to the traditional zero forcing based solutions, the proposed method provides more than 4 dB SNR gain and up to 3.5 bits/sec/Hz better spectral efficiency.