Effective channel control random beamforming for single user MIMO systems

Abstract

In general, it has been demonstrated that the performance of conventional random beamforming (RBF) approaches that of ideal eigen beamforming when the number of users is large in a cell by exploiting multi-user diversity. However, if the number of users decreases, such as femto or pico cell in the 3GPP Long Term Evolution-Advanced standard, the performance degradation occurs in the RBF scheme due to the lack of multi-user diversity. In this paper, we present a novel precoder based on singular value decomposition (SVD) using feedback of weight values in an RBF environment. For achieving performance improvement in the femto cell environment, we generate a precoding matrix appropriate to user channel by controlling the feedback values with an equivalent feedback quantity. In the simulation results, we verify the outstanding performance of the proposed scheme for a small number of users.

Appendix: Derivation of the closed form for w k,1 and w k,2 for two transmit antennas

For deriving a closed form solution from Eq. (13), the intermediate equations are the following:

$$ \widetilde{v}_{k,11} = v_{b,11} $$

(17)

$$ \left(1+\left| {\frac{{C_{k}}^{2}}{(D_{k}-\widetilde{\lambda}_{k,1})^{2}} } \right|\right)^{-\frac{1}{2}} = v_{b,11} $$

(18)

$$ \frac{ \left| { {C_{k}}^{2}} \right|}{v_{b,11}^{-2} - 1} = (D_{k}-\widetilde{\lambda}_{k,1})^{2}. $$

(19)

Assume $a_{k} = h_{k,11}^{\ast}h_{k,11}+h_{k,21}^{\ast}h_{k,21},$ $b_{k} = h_{k,11}^{\ast}h_{k,12}+h_{k,21}^{\ast}h_{k,22}, \,c_{k} = h_{k,11}h_{k,12}^{\ast}+h_{k,21}h_{k,22}^{\ast}$ and $d_{k} = h_{k,12}^{\ast}h_{k,12}+h_{k,22}^{\ast}h_{k,22}.$ Then, we can rewrite Eq. (19) as

$$ Qw_{k,1}w_{k,2} = A_{k} - D_{k} - \sqrt{(A_{k} + D_{k})^{2} - 4(A_{k}D_{k}-B_{k}C_{k})} $$

(20)

$$ Qw_{k,1}w_{k,2} = a_{k}w_{k,1}^{2} - d_{k}w_{k,2}^{2} - \sqrt{a_{k}^{2}w_{k,1}^{4} + d_{k}^{2}w_{k,2}^{4} - 2(a_{k}d_{k} - 2b_{k}c_{k})w_{k,1}^{2}w_{k,2}^{2}} $$

(21)

where $Q = \left(\frac{4\left| { {c_{k}}^{2} } \right|}{v_{b,11}^{-2} - 1}\right)^{\frac{1}{2}}.$ Here, Q is a positive real value because w _k,1,  w _k,2 and $\left| { {c_{k}}^{2} } \right|$ are positive real values and v ⁻² _b,11  > 1. Then, we can rewrite Eq. (21) as

$$ 2Qa_{k}w_{k,1}^{2} - 2Qd_{k}w_{k,2}^{2} + (4b_{k}c_{k} - Q^{2})w_{k,1}w_{k,2} = 0. $$

(22)

For obtaining a closed form of w _k,1, use $w_{k,2} = \sqrt{P - w_{k,1}^{2}}$ from Eq. (12). Then,

$$ w_{k,1}^{4}\{(Q^{2} - 4b_{k}c_{k})^{2} + (2Qa_{k} + 2Qd_{k})^{2}\}\\ + w_{k,1}^{2}\{P(Q^{2} - 4b_{k}c_{k})^{2} + 4PQd_{k}(2Qa_{k} + 2Qd_{k})\} - 4P^{2}Q^{2}d_{k}^{2} = 0 $$

(23)

$$ \hat{a_{k}}w_{k,1}^{4} + \hat{b_{k}}w_{k,1}^{2} + \hat{c_{k}} = 0 $$

(24)

where $\hat{a_{k}} = (Q^{2} - 4b_{k}c_{k})^{2} + (2Qa_{k} + 2Qd_{k})^{2}, \,\hat{b_{k}} = P(Q^{2} - 4b_{k}c_{k})^{2} + 4PQd_{k}(2Qa_{k} + 2Qd_{k})$ and $\hat{c_{k}} = -4P^{2}Q^{2}d_{k}^{2}.$ Then, we can solve Eq. (24) by using the square root of quadratic formula or quartic formula as

$$ w_{k,1} = \sqrt{\frac{-\hat{b_{k}}+\sqrt{\hat{b_{k}}^{2} - 4\hat{a_{k}}\hat{c_{k}}}}{2\hat{a_{k}}}} \quad \hbox{and} \; w_{k,2} = \sqrt{P - w_{k,1}} $$

(25)

where w _k,1 and w _k,2 are positive real values.