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:<math>\mathbf{x} = \sum_{i=1}^K \mathbf{w}_i s_i,</math>
where <math>s_i</math> is the (normalized) data symbol and <math>\mathbf{w}_i</math> is the <math>N \times 1</math> linear precoding vector. The [[Signal-to-noise plus interference|signal-to-interference-and-noise]] ratio (SINR) at user <math>k</math> becomes
:<math>\textrm{SINR}_k = \frac{|\mathbf{h}_k^H\mathbf{w}_k|^2}{
where <math>\sigma_k^2</math> is the noise variance for channel to user <math>k</math> and the corresponding achievable information rate is <math>\log_2(1+\textrm{SINR}_k)</math> bits per channel use. The transmission is limited by power constraints. This can, for example, be a total power constraint <math>\sum_{i=1}^K \|\mathbf{w}_i\|^2 \leq P</math> where <math>P</math> is the power limit.
A common performance metric in multi-user systems is the weighted sum rate
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:<math>\mathbf{w}^{\mathrm{MRT}}_k = \sqrt{p_k} \frac{\mathbf{h}_k}{\|\mathbf{h}_k\|}, </math>
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 = \frac{|
====Uplink-downlink duality====
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:<math>\mathbf{y} = \sum_{k=1}^{K} \mathbf{h}_k \sqrt{q_k} s_k + \mathbf{n}</math>
where <math>s_k</math> is the transmitted symbol for user <math>k</math>, <math>q_k</math> is the transmit power for this symbol, <math>\mathbf{y}</math> and <math>\mathbf{n}</math> are the <math>N \times 1</math> vector of received signals and noise respectively, <math>\mathbf{h}_k</math> is the <math>N \times 1</math> vector of channel coefficients. If the base station uses linear receive filters to combine the received signals on the <math>N</math> antennas, the SINR for the data stream from user <math>k</math> becomes
:<math>\textrm{SINR}^{\mathrm{uplink}}_k = \frac{q_k|\mathbf{h}_k^H\mathbf{v}_k|^2}{
where <math>\mathbf{v}_k</math> is the unit-norm receive filter for this user. Compared with the downlink case, the only difference in the SINR expressions is that the indices are switched in the interference term. Remarkably, the optimal receive filters are the same as the weighted MMSE precoding vectors, up to a scaling factor:
:<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>
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