Joint pre-processing co-channel interference cancellation for single user MIMO

Abstract

In general, multiplexing and diversity gains of single user MIMO systems are restricted by min(M,N) where M, N denote the number of antenna elements at a transmitter and receiver, respectively. In order to increase the multiplexing/diversity gains and improve the performance of single user MIMO systems, a joint pre-processing co-channel interference cancellation (JPCIC) method is proposed. The JPCIC is analyzed in both the perfect and the imperfect channel state information. The dependence of channel capacity on the number of antenna elements in every subset, the number of subsets, transmit powers and channel estimation errors is discussed. As theoretical calculation result, the channel capacity increases when the multiplexing/diversity gains and/or the transmit power increase in a certain channel model whether the channel estimation error is absent or present. Compared to the conventional zero-forcing method, the channel capacity of JPCIC is considerably higher because of higher multiplexing/diversity gains, however, it is less robust and decreased more rapidly due to incomplete cancellation of interference terms when the channel estimation error increases. There is a trade-off between the channel capacity and the complexity of system, however, according to quick development in circuit techniques and miniaturization of devices, the JPCIC is expected to be an attractive technology for MIMO system.

Appendices

Appendix 1: Creation method of orthogonal matrices

In Sect. 2.2, the ${\varvec{\Phi }}_{\mathrm{12,2}}{\varvec{\Phi }}_{\mathrm{12,1}}$ is indicated to be orthogonal to $\mathbf{H}_{\mathrm{12}}\mathbf{P}_{\mathrm{12}}$. the design of ${\varvec{\Phi }}_{\mathrm{12,1}}$ and ${\varvec{\Phi }}_{\mathrm{12,2}}$ are represented as follows. Let

$$\begin{aligned} \mathbf{Z}& \equiv \mathbf{H}_{\mathrm{12}}\mathbf{P}_{\mathrm{12}}, \\&= \begin{bmatrix} {\mathrm{z}}_\mathrm{11}&{\mathrm{z}}_\mathrm{12} \\ {\mathrm{z}}_{\mathrm{21}}&{\mathrm{z}}_{\mathrm{22}}\end{bmatrix}. \end{aligned}$$

(27)

There are many methods to design ${\varvec{\Phi }}_{\mathrm{12,1}}$ and ${\varvec{\Phi }}_{\mathrm{12,2}}$. One of them is to create ${\varvec{\Phi }}_{\mathrm{12,1}}$ subject to

$$\begin{aligned} {\varvec{\Phi }}_{\mathrm{12,1}}(\mathrm 1)\left[ \begin{array}{c}{\mathrm{z}}_{\mathrm{11}} \\ {\mathrm{z}}_{\mathrm{21}} \end{array} \right] =0, \\ {\varvec{\Phi }}_{\mathrm{12,1}}(\mathrm 2)\left[ \begin{array}{c}{\mathrm{z}}_{\mathrm{11}} \\ {\mathrm{z}}_{\mathrm{21}} \end{array} \right] =0. \end{aligned}$$

(28)

Therefore,

$$\begin{aligned} {\varvec{\Phi }}_{\mathrm{12,1}} \mathbf{H}_{\mathrm{12}}\mathbf{P}_{\mathrm{12}} =\begin{bmatrix} 0&{\hat{\mathrm{z}}}_\mathrm{12} \\ 0&{\hat{\mathrm{z}}}_{\mathrm{22}}\end{bmatrix}. \end{aligned}$$

(29)

And then, create ${\varvec{\Phi }}_{\mathrm{12,2}}$ subject to

$$\begin{aligned} {\varvec{\Phi }}_{\mathrm{12,2}}(\mathrm{1})\left[ \begin{array}{c} {\hat{\mathrm{z}}}_{\mathrm{12}} \\ \hat{{\mathrm{z}}}_{\mathrm{22}} \end{array} \right] =0, \\ {\varvec{\Phi }}_{\mathrm{12,2}}(\mathrm{\mathrm 2})\left[ \begin{array}{c}{\hat{\mathrm{z}}}_{\mathrm{12}} \\ {\hat{\mathrm{z}}}_{\mathrm{22}} \end{array} \right] =0. \end{aligned}$$

(30)

We can indicate an example as

$$\begin{aligned} {\varvec{\Phi }}_{\mathrm{12,1}}\equiv & {} \begin{bmatrix} {\mathrm{-z}}_{\mathrm{21}}&{\mathrm{z}}_{\mathrm{11}} \\ {\mathrm{-z}}_{\mathrm{21}}&{{\mathrm{z}}}_{\mathrm{11}}\end{bmatrix}. \\ {\varvec{\Phi }}_{\mathrm{12,2}}\equiv & {} \begin{bmatrix} -1&1 \\ -1&1 \end{bmatrix}. \end{aligned}$$

(31)

As explained above, a vector with length of p can be designed to be orthogonal to $\mathrm{p}-1$ vectors with the same length. Consequently, every row of matrix ${\varvec{\Phi }}_{\mathrm{p}_{\mathrm{sub}},1}$ is created to be orthogonal to the first $\mathrm{p}-1$ columns of $\mathbf{Z}$. Therefore,

$$\begin{aligned} {\varvec{\Phi }}_{\mathrm{p}_{\mathrm{sub}},1} \mathbf{H}_{{\mathrm{p}}_{\mathrm{sub}}}\mathbf{P}_{{\mathrm{p}}_{\mathrm{sub}}}= \begin{bmatrix} 0&0&\cdots&{\hat{\mathrm{z}}}_\mathrm{1p} \\&\cdots&\\ 0&0&\cdots&{\hat{\mathrm{z}}}_{\mathrm{p}\mathrm{p}}\end{bmatrix}, \end{aligned}$$

(32)

and ${\varvec{\Phi }}_{\mathrm{p}_{\mathrm{sub}},2}$ is designed to be orthogonal to the last column of ${\varvec{\Phi }}_{\mathrm{p}_{\mathrm{sub}},1} \mathbf{H}_{{\mathrm{p}}_{\mathrm{sub}}}\mathbf{P}_{{\mathrm{p}}_{\mathrm{sub}}}$, $\left[ \begin{array}{c}{\hat{\mathrm{z}}}_\mathrm{1p}\, {\hat{\mathrm{z}}}_\mathrm{2p}\,\cdots \, {\hat{\mathrm{z}}}_{\mathrm{p}\mathrm{p}} \end{array} \right] ^{\mathrm{T}}$, as similar to (30).

Appendix 2: Calculation method of q₁₁, q₁₂

The (10) can be represented as

$$\begin{aligned}&\begin{bmatrix} \hat{\mathrm{p}}_\mathrm{11}&\hat{\mathrm{p}}_\mathrm{12} \\ \hat{\mathrm{p}}_{\mathrm{21}}&\hat{\mathrm{p}}_\mathrm{22}\end{bmatrix} \left[ \begin{array}{c}\mathrm{s}_{\mathrm{3}} \\ \mathrm{s}_{\mathrm{4}} \end{array} \right] (1) \\&\quad +{ q}_{11} \begin{bmatrix} \mathrm{k}_\mathrm{11}&\mathrm{k}_\mathrm{12} \\ \mathrm{k}_\mathrm{21}&\mathrm{k}_\mathrm{22}\end{bmatrix} \left[ \begin{array}{c}\mathrm{s}_{\mathrm{3}} \\ \mathrm{s}_{\mathrm{4}} \end{array} \right] (1) \\&\quad +\mathrm{q}_{12} \begin{bmatrix} \mathrm{k}_\mathrm{11}&\mathrm{k}_\mathrm{12} \\ \mathrm{k}_\mathrm{21}&\mathrm{k}_\mathrm{22}\end{bmatrix} \left[ \begin{array}{c}\mathrm{s}_{\mathrm{3}} \\ \mathrm{s}_{\mathrm{4}} \end{array} \right] (2)=0, \end{aligned}$$

(33)

here $\hat{{\mathbf{P}}} \equiv \mathbf{P}_{\mathrm{12}}^\mathrm{-1} \mathbf{P}_{\mathrm{23}}$. Therefore, it is changed as follows.

$$\begin{aligned} \left( \hat{\mathrm{p}}_{11} + \mathrm{q}_{11}\mathrm{k}_{11} + \mathrm{q}_{12}\mathrm{k}_{21}\right) \mathrm{s}_{3}+ \left( \hat{\mathrm{p}}_{12} + \mathrm{q}_{11}\mathrm{k}_{12} + \mathrm{q}_{12}\mathrm{k}_{22}\right) \mathrm{s}_{4} = 0. \end{aligned}$$

(34)

Since this equation is for all $\mathrm{s}_{3}$ and $\mathrm{s}_{4}$, we have

$$\begin{aligned} \hat{\mathrm{p}}_{11}+\mathrm{q}_{11}\mathrm{k}_{11}+\mathrm{q}_{12}\mathrm{k}_{21}&= 0, \\ \hat{\mathrm{p}}_{12}+\mathrm{q}_{11}\mathrm{k}_{12}+\mathrm{q}_{12}\mathrm{k}_{22}&= 0. \end{aligned}$$

(35)

As a result,

$$\begin{aligned} \mathrm{q}_{11}&= \frac{\hat{\mathrm{p}}_{12}\mathrm{k}_{21}-\hat{\mathrm{p}}_{11}\mathrm{k}_{22}}{\mathrm{k}_{11}\mathrm{k}_{22}-\mathrm{k}_{12}\mathrm{k}_{21}}, \\ \mathrm{q}_{12}&= \frac{\hat{\mathrm{p}}_{12}\mathrm{k}_{11}-\hat{\mathrm{p}}_{11}\mathrm{k}_{12}}{\mathrm{k}_{21}\mathrm{k}_{12}-\mathrm{k}_{22}\mathrm{k}_{11}}, \end{aligned}$$

(36)

and matrices $\mathbf{P}_{\mathrm{12}}$ and $\mathbf{P}_{\mathrm{23}}$ should be designed subject to $\hat{\mathrm{p}}_{12}\mathrm{k}_{21}-\hat{\mathrm{p}}_{11}\mathrm{k}_{22}\ne 0$, $\hat{\mathrm{p}}_{12}\mathrm{k}_{11}-\hat{\mathrm{p}}_{11}\mathrm{k}_{12}\ne 0$ and ${k}_{21}{ k}_{12}-{k}_{22}{k}_{11}\ne 0$. Notice that $\mathbf{K}_{\mathrm{23}}={\varvec{\Phi }}_{\mathrm{12,2}} {\varvec{\Phi }}_{\mathrm{12,1}} \mathbf{H}_{\mathrm{12}}\mathbf{P}_{\mathrm{23}}$.

Appendix 3: Taylor expansion of pseudo inverse matrix

We are going to prove that

$$\begin{aligned} \left( \mathrm{\rho } {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}\! + \!{\xi } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}\right) ^{\mathrm{-1}}\!=\!{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\left( \mathrm{\rho } \mathbf{I}_{\mathrm{p}} \! +\! {\xi } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\right) . \end{aligned}$$

(37)

We have

$$\begin{aligned}&\left( \mathrm{\rho } {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}} + {\xi } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}\right) ^{\mathrm{-1}} \\&\quad =\left( \mathrm{\rho } {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}} {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}} {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}\! + \!{\xi } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}} {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}} {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}\right) ^{\mathrm{-1}}, \\&\quad =\left( \left( {\rho } \mathbf{I}_{\mathrm{p}} + \!{\xi } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\right) {\hat{H}}_{{\mathrm{p}}_{\mathrm{sub}}}\right) ^{\mathrm{-1}}, \\&\quad ={\rho } {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\left( \mathbf{I}_{\mathrm{p}} + \frac{\xi }{\mathrm{\rho } } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\right) ^{\mathrm{-1}}, \end{aligned}$$

(38)

and the Taylor expansion is applied to.

$$\begin{aligned}&\left( \mathbf{I}_{\mathrm{p}} + \frac{\xi }{{\rho }} {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\right) ^{\mathrm{-1}} \\&\quad =\mathbf{I}_{\mathrm{p}} - \frac{\xi }{{\rho }} {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}+\left( \frac{\xi }{{\rho }} {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}} {\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}\right) ^{\mathrm{2}} -\cdots , \\&\quad \approx \mathbf{I}_{\mathrm{p}} - \frac{\xi }{{\rho } } {\bar{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}{\hat{\mathbf{H}}}_{{\mathrm{p}}_{\mathrm{sub}}}^{\mathrm{-1}}. \end{aligned}$$

(39)

From (38) and (39), the (37) is obtained.