Precoding: Difference between revisions

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where <math>\mathbf{x}</math> is the <math>N \times 1</math> transmitted vector signal, <math>y_k</math> is the received signal, and <math>n_k</math> is the zero-mean unit-variance noise.

Under linear precoding, the transmitted vector signal is
:<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-noiseinterference-and-interferencenoise]] ratio (SINR) at user <math>k</math> becomes

:<math>\textrm{SINR}_k = \frac{|\mathbf{h}_k^H\mathbf{w}_k|^2}{1+\sum_{i \neq k} |\mathbf{h}_k^H\mathbf{w}_i|^2}</math>

and the corresponding communicationachievable information rate is <math>\log_2(1+\textrm{SINR}_k)</math> bits per channel use. The transmission is limited by a power constraintconstraints. 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 measuremetric in multi-user systems is the weighted sum rate

:<math>\underset{\{\mathbf{w}_k\}:\sum_i \|\mathbf{w}_i\|^2 \leq P}{\mathrm{maximize}} \sum_{k=1}^K a_k \log_2(1+\textrm{SINR}_k)</math>

for some positive weights <math>a_k</math> that represent the user priority. The weighted sum rate is maximized by weighted MMSE precoding that selects

:<math>\mathbf{w}^{\textrm{W-MMSE}}_k = \sqrt{p_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>

for some positive coefficients <math>q_1,\ldots,q_K</math> (related to the user weights) that satisfy <math>\sum_{i=1}^K q_i = P</math> and <math>p_i</math> is the optimal power allocation.<ref name=bjornson />

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====Uplink-downlink duality====

For comparison purposes, it is instructive to compare the downlink results with the corresponding uplink MIMO channel where the same single-antenna users transmit to the same base station, having <math>N</math> receive antennas. The input-output relationship can be described as

:<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}{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. ObserveCompared thatwith 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 beamformingprecoding 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>

Observe that the coefficients <math>q_1,\ldots,q_K</math> that was used in the weighted MMSE beamformingprecoding 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.

==== Limited feedback precoding ====

The precoding strategies described above was based on havehaving perfect [[channel state information]] at the transmitter. However, in real systems, receivers can only feed back quantized information that is described by a limited number of bits. If the same precoding strategies are applied, but now based on inaccurate channel information, additional interference appears. This is an example on limited feedback precoding.

The received signal in multi-user MIMO with limited feedback precoding is mathematically described as

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:<math>y_k = \mathbf{h}_k^H \sum_{i=1}^K \mathbf{w}_i s_i + \mathbf{h}_k^H \sum_{i=1}^K \mathbf{e}_i s_i+ n_k, \quad k=1,2, \ldots, K</math>

where <math>\mathbf{h}_k^H \sum_{i=1 \neq k}^K \mathbf{e}_i s_i</math> is the residualadditional interference at user <math>k</math> according to the limited feedback precoding. To reduce this interference, higher accuracy in the [[channel state information|channel information]] feedback is required, which in turn reduces the throughput in the uplink.

==See also==