On the Capacity of Cluster-Based Cooperative MIMO Cellular System with Universal or Fractional Frequency Reuse

Abstract

The information-theoretic capacity for the cluster-based multicell cooperative processing (MCP) network with inter-cluster interference is investigated in this paper. An upper bound for the ergodic capacity is derived by the QR decomposition of the channel matrix, which is concisely expressed and close to the results from numerical simulations. Capacity results for the universal frequency reuse (UFR) network show that the cluster-based MCP system has great advantages over the non-cooperated system, and the capacity gain depends mainly on the size of the cooperative cluster, the radius of the cell and the path loss exponent (PLE). However, the cluster-based UFR system is still interference limited, whose capacity declines sharply due to the inter-cluster interference. Therefore, a cluster-based fractional frequency reuse (FFR) network is proposed to improve the system capacity, simulation results show that the cluster-based FFR system can outperform the UFR system, especially for the cases of small radius of cell or small PLE.

Appendix A

1.1 Derivation for (11)

Rewrite $\mathbf{H}_X$ as $\mathbf{H}_X =\left[ {A_1 ,A_2 ,\ldots ,A_N } \right]$, we can construct $N$ mutually orthogonal of norm 1 vectors $Q_{1},{\ldots },Q_{N}$ from $A_{1},{\ldots },A_{N}$ by applying the Gram–Schmidt algorithm.

Let the normalized result of $A_{1}$ equal to $Q_{1}$, then

$$\begin{aligned}&\displaystyle r_{11} =\left\| {A_1 } \right\| =\left[ {\left| {m_{11} p_{11} } \right| ^{2}+\left| {m_{21} p_{21} } \right| ^{2}+\ldots +\left| {m_{N1} p_{N1} } \right| ^{2}} \right] ^{1/2}\end{aligned}$$

(29)

$$\begin{aligned}&\displaystyle Q_1 =A_1 /r_{11} \end{aligned}$$

(30)

Subtract the component parallel to $A_{1}$ from $A_{2}$, and then normalize it,

$$\begin{aligned}&\displaystyle r_{12} =Q_1 ^\mathrm{H}A_2\end{aligned}$$

(31)

$$\begin{aligned}&\displaystyle r_{22}=\left\| {A_2 -Q_1 r_{12} } \right\| =\left\| {A_2 -Q_1 Q_1 ^\mathrm{H}A_2 } \right\| =\left\| {A_2 (\mathbf{E}-Q_1 Q_1 ^\mathrm{H})} \right\| \le \left\| {A_2 } \right\| C_2\end{aligned}$$

(32)

$$\begin{aligned}&\displaystyle Q_2=(A_2 -Q_1 r_{12} )/r_{22} \end{aligned}$$

(33)

Similarly,

$$\begin{aligned} r_{33}&= \left\| {A_3 -Q_1 r_{13} -Q_2 r_{23} } \right\| =\left\| {A_3 -Q_1 Q_1 ^\mathrm{H}A_3 -Q_2 Q_2 ^\mathrm{H}A_3 } \right\| \nonumber \\&= \left\| {A_3 (\mathbf{E}-Q_1 Q_1 ^\mathrm{H}-Q_2 Q_2 ^\mathrm{H})} \right\| \le \left\| {A_3 } \right\| C_3 \end{aligned}$$

(34)

By analogy, for $r_{ii}, 1\le i\le N$

$$\begin{aligned} r_{ii} =\left\| {A_i -\sum _{j=1}^{i-1} {Q_j r_{ji} } } \right\|&= \left\| {A_i -\sum _{j=1}^{i-1} {Q_j Q_j ^{\mathrm{H}}A_i } } \right\| =\left\| {A_i (\mathbf{E}-\sum _{j=1}^{i-1} {Q_j Q_j ^{\mathrm{H}})} } \right\| \nonumber \\&\qquad \le \left\| {A_i } \right\| C_i\end{aligned}$$

(35)

$$\begin{aligned} \hbox {In which}\, C_i&= \left\| {\mathbf{E}-\sum _{j=1}^{i-1} {Q_j Q_j ^{\mathrm{H}}} } \right\| ,\;2\le i\le N \end{aligned}$$

(36)

When $N=7,\;\left[ C_1 ,C_2 ,\ldots , C_7 \right] = \left[ 1.0, 2.4495, 2.2361, 2.0, 1.7321, 1.4142, 1.0 \right]$.