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In spatial division multiple access systems, a BS can perform transmit beamforming so that multiuser interference can be reduced.

## Transmit BeamformingEdit

More than one antennas must be employed at the transmitter, to perform beamforming at the transmitter. Then, the received signal at a mobile can be modeled as

$\mathbf{x} = \sum_{i=1}^K \mathbf{w}_i s_i = \mathbf{W} \mathbf{s}$

where $s_i$ in $\mathbf{s} = [s_1, s_2, \ldots, s_K]$ is the message signal (chosen from a complex Gaussian alphabet with power $P/M$) intended for the $i$th receiver and $\mathbf{w}_i \in \mathbb{C}^{M \times 1}$ in $\mathbf{W} = [\mathbf{w}_1, \mathbf{w}_2, \ldots, \mathbf{w}_M]$ is the corresponding beamforming vector.

## Zero-Forcing BeamformingEdit

In zero-forcing (ZF) beamforming, each beamforming vector is set to be orthogonal to other user's channel vector and normalized to have an unit value. ZF beamforming yields the received signal at MS 1 as follows:

$y_{b,m} = \sqrt{P_{b,m}} \mathbf{h}_{b,m}^H \mathbf{W} \mathbf{s} + n_{b,m} = \sqrt{P_{b,m}} \mathbf{h}_{b,m}^H \mathbf{w}_1 s_1 + n_{b,m}$

where all interference terms are removed since we assume that full channel state information is available at the transmitter (CSIT). The average throughput performance of a ZF beamforming system with full CSIT is given by

$R_\mathrm{ZF} = E[ \log_2(1+ P_{b,m} |\mathbf{h}_{b,m}^H \mathbf{w}_1|^2)]$

## Digital Channel Feedback ModelEdit

Since BS determines beamforming vectors based on the downlink channel state information, MS should feed back the channel state information to BS by the feedback signaling from. In a digital transmission system, the feedback signaling uses a common codebook sharing between BS and MS. We assume that the beamforming codebook consists of $2^{B_\mathrm{FB}}$ $M$-dimensional unit norm vectors, $\mathcal{C}_\mathrm{FB}(B_\mathrm{FB}) \triangleq \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{2^{B_\mathrm{FB}}}\}$, which is used for channel quantization. When all receivers are are equipped with single antenna, the $i$th receiver quantizes its normalized channel $\tilde{\mathbf{h}}_d = \mathbf{h}_d / ||\mathbf{h}_d||$ to $\hat{\mathbf{h}}_d$ in accordance with:

$\hat{\mathbf{h}}_{b,m} = \arg \min_{\mathbf{v} \in \mathcal{C}_\mathrm{FB}(B_\mathrm{FB}) } \sin^2 ( \angle( \mathbf{h}_d, \mathbf{v})).$

MS sends the selected quantization index of the above equation to BS. Note that the magnitude of the channel vector is not quantized since the number of users are smaller than the number of transmit antennas so that user scheduling with channel magnitude values are not considered.

When a receiver has more than one antennas, we can use the quantization based combining (QBC) method where the $i$th receiver quantizes its channel $\mathbf{H}_d$ to $\hat{\mathbf{h}}_d$ in accordance with:

$\hat{\mathbf{h}}_d = \arg \min_{\mathbf{v}=\mathbf{v}_1, \ldots, \mathbf{v}_{2^{B_\mathrm{FB}}}} \sin^2 ( \angle( \mathrm{span}(\mathbf{H}_d), \mathbf{v})) = \arg \max_{\mathbf{v} = \mathbf{v}_1, \ldots, \mathbf{v}_{2^{B_\mathrm{FB}}}} || \mathbf{Q}_1^H \mathbf{v}||^2.$

## Feedback vs. PerformanceEdit

Quantify the amount of the feedback resource required to maintain at least 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 a fact that the required feedback bits as a simplied metric of the required resource 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 MS 1's downlink data channel.

To feedback B bits though uplink channel, the throughput performance of the uplink channel should be larger than or equal to B

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

where $T_{FB}$ is the feedback temporal resource and $\rho_{FB}$ is SNR of MS 1's uplink feedback channel. Then, the required feedback resource to satisfy $\Delta R \leq \log_2 g$ is

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

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 proportion 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 feeback resource ($b_{FB}$) is not necessary to scale acording 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 reductioned situation.

### Feedback time Edit

The required feedback time to transmit the number of given feedback bits is written as

$T_{FB} = \frac{B}{ \min(W,W_c) C_{FB} T_c}.$

For example, if

$B = 10bits, W = 100Hz, W_c > W, Tc = 1/f_{D}, and f_{D} = 1 to 5Hz$

$T_{FB}=f_D/10 =$ 1/10 to 5/10.

## ReferencesEdit

1. N. Jindal (Nov. 2006). "MIMO Broadcast Channels with Finite Rate Feedback". IEEE Trans. Information Theory. 52: 5045–5059. doi:10.1109/TIT.2006.883550.