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:<math>\mathbf{v}^{\textrm{MMSE}}_k = \frac{( \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k}{\|( \mathbf{I} + \sum_{i \neq k} q_i \mathbf{h}_i \mathbf{h}_i^H )^{-1} \mathbf{h}_k\|} </math>
Observe that the coefficients <math>q_1,\ldots,q_K</math> that was used in the weighted MMSE precoding are not exactly the optimal power coefficients in the uplink (that maximize the weighted sum rate) except under certain conditions. This important relationship between downlink precoding and uplink receive filtering is known as the uplink-downlink duality.<ref>M. Schubert and H. Boche, [https://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, [https://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.
==== Limited feedback precoding ====
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