We describe multi-user MIMO systems with limited feedback precoding. To achieve the capacity of a multi-user MIMO channel, the accurate channel state information must be available at the transmitter. Because the channel state information is fed back from each UE, higher accuracy in feedback information requires larger uplink resource. A receiver in practical system is unable to feed back the perfect channel state information to the transmitter because of uplink resource limitation. Instead, a receiver feeds back the partial channel state information, which improve the efficiency of the uplink resource use. We define such system as the Multi-user MIMO system with limited feedback precoding.

The received signal in MIMO BC with limited feedback precoding at UE $ 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 $

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 reduce this interference, we should use the higher accuracy channel information feedback which results in decreasing the uplink resource.

Performance of a Digital Feedback System Edit

We quantify the amount of the feedback resource required to maintain performance higher than a given throughput performance gap between zeroforcing with perfect feedback and with limited feedback, i.e.,,

$ \Delta R = R_{ZF} - R_{FB} \leq log_2 b $.

Jindal showed that the required feedback bits should be scaled acording to SNR of the downlink channel, which is given by[1]:

$ B = (M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (b-1) $

where M, is the number of transmit antennas and $ \rho_{b,m} $ is SNR of the downlink channel.

In order that the mobile feeds back B bits to the basestation though uplink channel, the throughput performance of the uplink channel throughput should be larger than or equal to B

$ b_{FB} \log_2(1+\rho_{FB}) \geq B $

where $ b = \Omega_{FB} T_{FB} $ is the feedback resource which is equal tothe feedback frequency resource multiplied by the frequency temporal resource subsequently and $ \rho_{FB} $ is SNR of the feedback channel. Then, the required feedback resource to satisfy $ \Delta R \leq \log_2 g $ is

$ b_{FB} \geq \frac{B}{\log_2(1+\rho_{FB})} = \frac{(M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1)}{\log_2(1+\rho_{FB})} $.

Note that differently from the feedback bits case, the required feedback resource is function of both downlink and uplink chanel conditions. It is resonable to incldue the uplink channel status in the calcuration of the feedback resource since the uplink channel status determines the capacity, i.e., bits/second per unit frequency band (Hz), of the feedback link. Considedr a case when SNR of the downlink and uplink are proportion such that $ \rho_{b,m} / \rho_{FB}) = C_{up,dn} $ is constant and both SNRs are sufficiently high. Then, the feedback resource will be only proportional to the number of transmit antennas

$ b_{FB,min}^* = \lim_{\rho_{FB} \to \infty } \frac{(M-1) \log_2 \rho_{b,m} - (M-1) \log_2 (g-1)}{\log_2(1+\rho_{FB})} = M - 1 $.

It follows the above equation that the feedback resource ($ b_{FB} $) is not necessary to scale according to SNR of the downlink channel, which is almost contradict to the case of the feedback bits. We, hence, see that the whole systematic analysis can reverse the facts resulted from each reduced situation.

If the AP can utilize also $ M $ receive antennas, the frequency-time feedback resource can be reduced to $ (M - 1)/M $ as the SNR goes to infinity by applying uplink SDMA[2]. Based on this result, we can say that the amount of feedback resource can be reduced as the number of AP antennas goes to high assuming the transmit antennas are compatibly used as the receiver antennas at the AP.


  1. N. Jindal, MIMO Broadcast Channels with Finite Rate Feedback, IEEE Trans. Information Theory, Vol. 52, No. 11, pp. 5045-5059, Nov. 2006.
  2. G. Caire, N. Jindal, M. Kobayashi, and N. Ravindran, Multiuser MIMO Achievable Rates with Downlink Training and Channel State Feedback, Submitted to IEEE Trans. Information Theory, Nov. 2007. (Revised May 2009)

See Also