Limited feedback is a signaling approach using non-perfect information so as to meet the uplink resource available for feedback signaling.

To achieve the full capacity of a multi-user MIMO channel, the perfect channel state information must be available at the transmitter. Because the channel state information is fed back from each UE through the uplink channel, the higher accuracy in feedback information requires the larger uplink resource including time and spatial resources. A receiver in practical systems is unable to feed back the perfect channel state information to the transmitter because of uplink resource limitation. Rather, a receiver feeds back the partial channel state information to the base station, resulting in improving the efficiency of the uplink resource use subject to downlink performance improvement. We define above case as the limited feedback precoding system.

The received signal at UE in MIMO BC with limited feedback precoding k is mathematically described as

y_k = \mathbf{h}_k^T \sum_{i=1}^K s_i P_i \hat{\mathbf{w}}_i +n_k, \quad k=1,2, \ldots, K

We assume that the feedback link is suffered no error for analytical simplicity. Since the transmit vector for limited feedback precoding is \hat{\mathbf{w}}_i = \mathbf{w}_i + \mathbf{e}_i where \mathbf{e}_i is the error vector caused by the limited feedback such as quantization, the received signal can be rewritten as

y_k = \mathbf{h}_k^T \sum_{i=1}^K s_i P_i \mathbf{w}_i + \mathbf{h}_k^T \sum_{i=1}^K s_i P_i \mathbf{e}_i + n_k, \quad k=1,2, \ldots, K

where \mathbf{h}_k^T \sum_{i=1}^K s_i P_i \mathbf{e}_i is the residual interference according to the limited feedback precoding. To minimize this quantization interference, the mobile station should use more accurate channel state information at the expense of decreasing the uplink resource.

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