Content deleted Content added
Line 62:
The suboptimal MRT approach removes the channel inversion and only selects
:<math>\mathbf{w}^{\mathrm{MRT}}
while the suboptimal [[zero-forcing precoding|ZF precoding]] makes sure that <math>\mathbf{h}_i^H\mathbf{w}^{\mathrm{ZF}}_k=0</math> for all i ≠ k and thus the interference can be removed in the SINR expression:
:<math>\textrm{SINR}^{\mathrm{ZF}}_k = | \mathbf{h}_k^H \mathbf{w}^{\mathrm{ZF}}_k |^2.</math>
Line 73:
:<math>\textrm{SINR}^{\mathrm{uplink}}_k = \frac{q_k|\mathbf{h}_k^H\mathbf{v}_k|^2}{1+\sum_{i \neq k} q_i |\mathbf{h}_i^H\mathbf{v}_k|^2}</math>
where <math>\mathbf{v}_k</math> is the unit-norm receive filter for this user. Observe that the only difference is that the indices are switched in the interference term. Remarkably, the optimal receive filters are the same as the MMSE beamforming vectors, up to a scaling factor:
:<math>\mathbf{v}
Observe that the coefficients <math>q_1,\ldots,q_K</math> that was used in the MMSE beamforming are exactly the optimal power coefficients in the uplink (that maximize the weighted sum rate). This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality<ref>M. Schubert and H. Boche, [http://dx.doi.org/10.1109/TVT.2003.819629 Solution of the multiuser downlink beamforming problem with individual SINR constraints], IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp. 18-28, 2004.</ref><ref>A. Wiesel, Y.C. Eldar, S. Shamai, [http://dx.doi.org/10.1109/TSP.2005.861073 Linear precoding via conic optimization for fixed MIMO receivers], IEEE Transactions on Signal Processing, vol. 54, no. 1, pp. 161-176, 2006.</ref>. As the downlink precoding problem usually is more difficult to solve, it often useful to first solve the corresponding uplink problem.
| |||