Improved Interference Cancellation Scheme for X Channels with Four Antennas

Xinji Tian, Dong Yang, Haotian Zhang, Wenjie Jia

School of Physics & Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, China.

DOI: 10.4236/jcc.2017.56004                

Abstract

Interference cancellation scheme without feedback is proposed for X channels with four antennas at each user. Space-time codeword with Alamouti structure is designed for each user. Codewords are combined according a certain rule. The unwanted codewords are cancelled by linear operation on the received signals. Then, multi-user interference is mitigated by the orthogonal property of the Alamouti code. Comparing with the existing scheme for the same scene, feedback information is not required in the proposed scheme. So the transmission efficiency is improved.

Keywords

X Channel, Space-Time Codeword, Multi-User Interference, Transmission Efficiency, Feedback Information

1. Introduction

Multi-input multi-output (MIMO) systems have the ability to improve the reliability and the effectiveness by transmitting independent data stream simultaneously. There are single user MIMO and multi-user MIMO [1] [2] . For multi-user MIMO system, each user transmits signals to one or more receivers simultaneously using the same band. Since these users can’t cooperate with each other, there is serious interference at each receiver, which seriously affects the system performance [3] [4] .

Interference alignment is the most studied interference cancellation method [5] [6] . Interference alignment aligns the interference signals along the same subspace while the wanted signals occupy linearly independent signal dimensions, such that the wanted signals can be separated [7] [8] . Later, interference alignment and space-time code are combined to obtain diversity gain [9] [10] . For example, interference alignment and space-time code are introduced into two users MIMO X channels, in which multi-user interference is eliminated by interference alignment and linear processing at the receivers [11] . In [11] , the transmission efficiency and the diversity gain are 8/3 symbol/channel and 2, respectively. The idea in [11] is extended into [12] , in which 4 × 4 space-time code is used at each user and two columns of the zero vector are introduced into each codeword. [13] proposes a method of interference cancellation using space-time and pre-coding for MIMO X channels, where each user has four antennas. However, in [12] and [13] , channel station information (CSI) is required at the two users. Since some time slots are required to send CSI, the transmission efficiency needs to be improved.

Interference cancellation method without feedback is proposed for X channels with 4 antennas at each user. Codewords with Alamouti structure are designed, which contain 4 independent modulated symbols, and then the codewords are combined with a certain rule. The unwanted codewords are eliminated by linear operation on the received signals, and then the interference between wanted codewords is cancelled using the orthogonal property of the Alamouti code. So the multi-user interference is mitigated. Compared with the same scheme for the same scene, our proposed scheme greatly reduces feedback amount, while keeping the same diversity gain. Simulation results demonstrate the validity of the proposed scheme.

2. System Model

The system model proposed in this paper is shown in Figure 1. Consider a two-user and two-receiver X channel, where each node has four antennas. Use $R_i\left(i=1,2\right)$ to denote the two receivers. Both users want to send different codewords to $R_1$ and $R_2$ on the same frequency band at the same time. As shown in Figure 1, $C_1$ and $S_1$ are the wanted codewords for $R_1$ , hence they become interference for $R_2$ . $C_2$ and $S_2$ are the wanted codewords for $R_2$ , hence they become interference for $R_1$ . $C_i\left(i=1,2\right)$ and $S_i$ can be written as

$\begin{array}{l}{C}_i=\left[\begin{array}{cc}{c}_{4i-3}+e^{j\theta }{c}_{4i-1}& -c_{4i-2}^{*}-e^{-j\theta }{c}_{4i}^{*}\\ c_{4i-2}+e^{j\theta }{c}_{4i}& c_{4i-3}^{*}+e^{-j\theta }{c}_{4i-1}^{*}\end{array}\right]\\ S_i=\left[\begin{array}{cc}{s}_{4i-3}+e^{j\theta }{s}_{4i-1}& -s_{4i-2}^{*}-e^{-j\theta }{s}_{4i}^{*}\\ s_{4i-2}+e^{j\theta }{s}_{4i}& s_{4i-3}^{*}+e^{-j\theta }{s}_{4i-1}^{*}\end{array}\right]\end{array}$ (1)

where $c_k\left(k=1,2,\cdots ,8\right)$ and $s_k$ are the modulated signals. The elements of $C_i$ and $S_i$ are un-zero with the proper value of $\theta$ . ${(\cdot )}^{\ast }$ denotes the conjugate.

Figure 1. System model of the proposed X channel.

The two users combine $C_i$ and $S_i$ respectively to get $C$ and $S$ as follows

$\begin{array}{l}C=\left[\begin{array}{cc}{C}_1+C_2& C_1-C_2\\ C_1-C_2& C_1+C_2\end{array}\right]\\ S=\left[\begin{array}{cc}{S}_1+S_2& S_1-S_2\\ S_1-S_2& S_1+S_2\end{array}\right]\end{array}$ (2)

Let $H_i$ and $G_i$ to denote the 4 × 4 channel matrices from user 1 to $R_i$ and from user 2 to $R_i$ , respectively. These two users transmit $C$ and $S$ respectively at the same time. The received signals at $R_1$ and $R_2$ , denoted by $Y$ and $Z$ with dimension 4 × 4 respectively, which are written as follows

$Y=H_1\left[\begin{array}{cc}{C}_1+C_2& C_1-C_2\\ C_1-C_2& C_1+C_2\end{array}\right]+G_1\left[\begin{array}{cc}{S}_1+S_2& S_1-S_2\\ S_1-S_2& S_1+S_2\end{array}\right]+N$ (3)

$Z=H_2\left[\begin{array}{cc}{C}_1+C_2& C_1-C_2\\ C_1-C_2& C_1+C_2\end{array}\right]+G_2\left[\begin{array}{cc}{S}_1+S_2& S_1-S_2\\ S_1-S_2& S_1+S_2\end{array}\right]+W$ (4)

where, $N$ and $Z_8$ are 4 × 4 Gaussian noise matrices.

3. The Method of Interference Cancellation

The interference alignment method is presented, taking $R_1$ as example. Let

$H_1=\left[\begin{array}{cc}{H}_{11}& H_{21}\end{array}\right]$ , $G_1=\left[\begin{array}{cc}{G}_{11}& G_{21}\end{array}\right]$ , $N=\left[\begin{array}{cc}{N}_{11}& N_{21}\end{array}\right]$ and $Y=\left[\begin{array}{cc}{Y}_{11}& Y_{21}\end{array}\right]$ . The dimension of $H_{i1}$ , $G_{i1}$ , $Y_{i1}$ and $N_{i1}$ are all 4 × 2. From (2) (3) (5) and (6) can be obtained.

$Y_{11}=H_{11}\left(C_1+C_2\right)+H_{21}\left(C_1-C_2\right)+G_{11}\left(S_1+S_2\right)+G_{21}\left(S_1-S_2\right)+N_{11}$ (5)

$Y_{21}=H_{11}\left(C_1-C_2\right)+H_{21}\left(C_1+C_2\right)+G_{11}\left(S_1+S_2\right)+G_{21}\left(S_1+S_2\right)+N_{21}$ (6)

From (5) (6), (7) can be obtained.

$\underset{Y_1}{\underset{︸}{Y_{11}+Y_{21}}}=\underset{H_1}{\underset{︸}{2\left(H_{11}+H_{21}\right)}}{C}_1+\underset{H_2}{\underset{︸}{2\left(G_{11}+G_{21}\right)}}{S}_1+\underset{N_1}{\underset{︸}{N_{11}+N_{21}}}$ (7)

The wanted codewords of $R_1$ are included in (7), while the unwanted codewords are not included. Thus, the unwanted codewords are mitigated through linear operation on the received signals, and the number of interfering codewords is reduced. However, interference between $C_1$ and $S_1$ still exists, as shown in (7). In what follows, the method to separate $C_1$ and $S_1$ is presented.

Let $H_1=\left[\begin{array}{cc}{h}_{11}^1& h_{12}^1\\ h_{21}^1& h_{22}^1\end{array}\right]$ and $H_2=\left[\begin{array}{cc}{h}_{11}^2& h_{12}^2\\ h_{21}^2& h_{22}^2\end{array}\right]$ . Use $y_{ij}$ and $n_{ij}$ to denote the

elements of $Y_1$ and $N_1$ , respectively, $i=1,2,3,4$ , $j=1,2$ . From (7), we have

$\underset{y_1}{\underset{︸}{\left[\begin{array}{c}{y}_{11}\\ y_{12}^{*}\end{array}\right]}}=Z_1\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+Z_2\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]+\underset{n_1}{\underset{︸}{\left[\begin{array}{c}{n}_{11}\\ n_{12}^{*}\end{array}\right]}}$ (8)

$\underset{y_2}{\underset{︸}{\left[\begin{array}{c}{y}_{21}\\ y_{22}^{*}\end{array}\right]}}=Z_3\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+Z_4\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]+\underset{n_2}{\underset{︸}{\left[\begin{array}{c}{n}_{21}\\ n_{22}^{*}\end{array}\right]}}$ (9)

$\underset{y_3}{\underset{︸}{\left[\begin{array}{c}{y}_{31}\\ y_{32}^{*}\end{array}\right]}}=Z_5\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+Z_6\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]+\underset{n_3}{\underset{︸}{\left[\begin{array}{c}{n}_{31}\\ n_{32}^{*}\end{array}\right]}}$ (10)

$\underset{y_4}{\underset{︸}{\left[\begin{array}{c}{y}_{41}\\ y_{42}^{*}\end{array}\right]}}=Z_7\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+Z_8\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]+\underset{n_4}{\underset{︸}{\left[\begin{array}{c}{n}_{41}\\ n_{42}^{*}\end{array}\right]}}$ (11)

where, $Z_{2i-1}=\left[\begin{array}{cc}{h}_{i1}^1& h_{i2}^1\\ h_{i2}^{1*}& -h_{i1}^{1*}\end{array}\right]$ , $Z_{2i}=\left[\begin{array}{cc}{h}_{i1}^2& h_{i2}^2\\ h_{i2}^{2*}& -h_{i1}^{2*}\end{array}\right]$ , $i=1,2,3,4$ , $y_1$ , $y_2$ , $y_3$ and

$y_4$ are the effective received signals. $n_1$ , $n_2$ , $n_3$ and $n_4$ are the effective

noise. $Z_i$ , having orthogonal characteristic, satisfies $\frac{Z_i^H{Z}_i}{{\Vert Z_i\Vert }^2}=I_2,i=1,2,3,4$ ,

where $I_2$ denotes the 2 × 2 unit matrix. ${(\cdot )}^H$ and $\Vert \cdot \Vert$ denote the conjugate- transpose and the norm, respectively. Taking operation on $y_1$ , $y_2$ , $y_3$ and $y_4$ according to (12)-(15), it is easy to derive $z_1$ , $z_2$ , $z_3$ and $z_4$ .

$z_1=\frac{y_1{Z}_2^H}{{\Vert Z_2\Vert }^2}-\frac{y_2{Z}_4^H}{{\Vert Z_4\Vert }^2}=Q_1\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+\underset{P_1}{\underset{︸}{\frac{n_1{Z}_2^H}{{\Vert Z_2\Vert }^2}-\frac{n_2{Z}_4^H}{{\Vert Z_4\Vert }^2}}}$ (12)

$z_2=\frac{y_1{Z}_1^H}{{\Vert Z_1\Vert }^2}-\frac{y_2{Z}_3^H}{{\Vert Z_3\Vert }^2}=Q_2\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]+\underset{P_2}{\underset{︸}{\frac{n_1{Z}_1^H}{{\Vert Z_1\Vert }^2}-\frac{n_2{Z}_3^H}{{\Vert Z_3\Vert }^2}}}$ (13)

$z_3=\frac{y_3{Z}_6^H}{{\Vert Z_6\Vert }^2}-\frac{y_4{Z}_8^H}{{\Vert Z_8\Vert }^2}=Q_3\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+\underset{P_3}{\underset{︸}{\frac{n_3{Z}_6^H}{{\Vert Z_6\Vert }^2}-\frac{n_4{Z}_8^H}{{\Vert Z_8\Vert }^2}}}$ (14)

$z_4=\frac{y_3{Z}_5^H}{{\Vert Z_5\Vert }^2}-\frac{y_4{Z}_7^H}{{\Vert Z_7\Vert }^2}=Q_4\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]+\underset{P_4}{\underset{︸}{\frac{n_3{Z}_5^H}{{\Vert Z_5\Vert }^2}-\frac{n_4{Z}_7^H}{{\Vert Z_7\Vert }^2}}}$ (15)

where, $Q_i=\frac{Z_{2i-1}{Z}_{2i}^H}{{\Vert Z_{2i}\Vert }^2}-\frac{Z_{2i+1}{Z}_{2i+2}^H}{{\Vert Z_{2i+2}\Vert }^2},i=1,3$ ,

$Q_k=\frac{Z_{2k-1}^H{Z}_{2i}}{{\Vert Z_{2k-1}\Vert }^2}-\frac{Z_{2K+1}^H{Z}_{2k+2}}{{\Vert Z_{2k+1}\Vert }^2},k=2,4$ , $Q_i$ and $Q_k$ are the effective channel matrices of $\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]$ or $\left[\begin{array}{c}{s}_1+e^{j\theta }{s}_3\\ s_2+e^{j\theta }{s}_4\end{array}\right]$ , $i=1,2,3,4$ . $c_k,k=1,2,3,4$ , the elements

of $C_1$ , are included in $z_1$ and $z_3$ , while the elements of other codewords are not included in them. $s_k,k=1,2,3,4$ , the elements of $S_1$ , are included in $z_2$ and $z_4$ , while the elements of other codewords are not included in them. Therefore, $C_1$ and $S_1$ are separated. The interference between the wanted codewords is mitigated. So is the multi-user interference. Similar operations can be performed on $R_2$ to mitigate multi-user interference. No feedback information is required.

4. Decoding and Diversity Gain

In this section, the decoding method is presented, taking $R_1$ as example. By

calculating, we can get $Z_1{Z}_2^H=\left[\begin{array}{cc}{h}_{11}^1{h}_{11}^{2*}+h_{12}^1{h}_{12}^{2*}& h_{11}^1{h}_{12}^2-h_{12}^1{h}_{11}^2\\ h_{12}^{1*}{h}_{11}^{2*}-h_{11}^{1*}{h}_{12}^{2*}& h_{12}^{1*}{h}_{12}^2+h_{11}^{1*}{h}_{11}^2\end{array}\right]$ . If we consider

$h_{11}^1{h}_{11}^{2*}+h_{12}^1{h}_{12}^{2*}$ and $h_{12}^{1*}{h}_{11}^{2*}-h_{11}^{1*}{h}_{12}^{2*}$ as the modulated signals, $Z_1{Z}_2^H$ has the structure of the Alamouti code. Similarly, $Q_i$ has the structure of Alamouti code as well, $i=1,2,3,4$ . Let $Q_1=\left[\begin{array}{cc}{q}_1& -q_2{}^{\ast }\\ q_2& q_1{}^{\ast }\end{array}\right]$ and $c_4$ , we process

$z_1$ and $z_3$ according to (16)

$Q_1^H{z}_1+Q_3^H{z}_3=\left[\begin{array}{cc}{q}_0& 0\\ 0& q_0\end{array}\right]\left[\begin{array}{c}{c}_1+e^{j\theta }{c}_3\\ c_2+e^{j\theta }{c}_4\end{array}\right]+Q_1^H{P}_1+Q_3^H{P}_3$ (16)

where $q_0={\left|q_1\right|}^2+{\left|q_2\right|}^2+{\left|q_3\right|}^2+{\left|q_4\right|}^2$ . Let $Q_1^H{z}_1+Q_3^H{z}_3=\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]$ and

$P_1^H{z}_1+P_3^H{z}_3=\left[\begin{array}{c}{p}_1\\ p_2\end{array}\right]$ , (16) can be rewritten as

$z_1=q_0\left(c_1+e^{j\theta }{c}_3\right)+p_1$ (17)

$z_2=q_0\left(c_2+e^{j\theta }{c}_4\right)+p_2$ (18)

It can be seen from (17) (18) that $c_1+e^{j\theta }{c}_3$ and $c_2+e^{j\theta }{c}_4$ are separated. So we can decode $\left(c_1,c_3\right)$ and $\left(c_2,c_4\right)$ , respectively. The specific steps are given as follows.

Step 1, obtain $y_1$ , $y_2$ , $y_3$ and $y_4$ from the received signals;

Step 2, obtain $H_1=2\left(H_{11}+H_{21}\right)$ and $H_2=2\left(G_{11}+G_{21}\right)$ from the channel matrices, and then obtain $Z_1$ , $Z_2$ , $Z_3$ , $Z_4$ , $Z_5$ , $Z_6$ , $Z_7$ and $Z_8$ ;

Step 3, calculate $Q_1$ and $Q_3$ from $Z_i$ , $k=1,2,\cdots ,8$ , and let

$Q_1=\left[\begin{array}{cc}{q}_1& -q_2^{*}\\ q_2& q_1^{*}\end{array}\right]$ and $Q_3=\left[\begin{array}{cc}{q}_3& -q_4^{*}\\ q_4& q_3^{*}\end{array}\right]$ ;

Step 4, obtain $z_1$ and $z_3$ by processing $y_1$ , $y_2$ , $y_3$ and $y_4$ using $Z_1$ , $Z_2$ , $Z_3$ , $Z_4$ , $Z_5$ , $Z_6$ , $Z_7$ and $Z_8$ ;

Step 5, obtain $Q_1^H{z}_1+Q_3^H{z}_3$ by combining $Q_1$ , $Q_3$ with $z_1$ , $z_3$ , and let

$Q_1^H{z}_1+Q_3^H{z}_3=\left[\begin{array}{c}{z}_1\\ z_2\end{array}\right]$ ;

Step 6, with the aid of the effective transmit signal $c_1+e^{j\theta }{c}_3$ , the effective channel matrix ${\left|q_1\right|}^2+{\left|q_2\right|}^2+{\left|q_3\right|}^2+{\left|q_4\right|}^2$ and the effective received signal $z_1$ , $c_1$ and $c_3$ can be estimated;

Step 7, similar operations can be performed to decode $c_2$ and $c_4$ .

From (17) (18), we can see that both $c_1+e^{j\theta }{c}_3$ and $c_2+e^{j\theta }{c}_4$ reaches $R_1$ by experiencing 4 independent paths. So the diversity gain is 4.

5. Performance Analysis and Simulation Results

There are 32 modulated signals to be transmitted in 6 time slots in [12] . Before transmitting the modulated symbols, two users need to know the feedback information which are 64 plurals. The feedback information is transmitted to the two users from receivers, which takes up some time slots. 12 time slots are required to send feedback information if these receivers adopt the same transmission and adopt the same code mode as the transmitters. So, 18 time slots are required to send 32 independent modulated signals in [12] , with a transmission efficiency of 16/9 symbol/channel. 16 independent modulated signals are sent over 4 time slots in [13] , in which the feedback information are 8 plurals. 2 time slots are required to send feedback information if these receivers adopt the same transmission and adopt the same code mode as the transmitters. So, 6 time slots are required to send 16 independent modulated signals in [13] , with a transmission efficiency of 8/3 symbol/channel. 16 independent modulated signals are sent over four time slots in the proposed scheme with a transmission efficiency of 4 symbol/channel. So the transmission efficiency of the proposed scheme is 2.25 times as much as that of Ref. [12] , and is 1.5 times as much as that of Ref. [13] .

There are a comparison of the transmission efficiency, diversity gain, feedback and decoding complexity of the three schemes, as shown in Table 1. M denotes the modulation order. As can be seen from the table, the advantage of the proposed scheme lies in improving the transmission efficiency without any feedback information. The disadvantages are that the decoding complexity is higher than that of Ref. [12] and the diversity gain is lower than that of Ref. [13] .

In Figure 2, we simulate the average BER curves of these three schemes with 4 QAM modulation and 16 QAM modulation. We consider uncoded systems, in which the channel is independent of the Rayleigh distribution and the noise is

Table 1. Performance comparison of the three schemes.

Figure 2. BER curves of the two schemes.

Gauss white noise. We can see that the reliability of the proposed scheme is not better than that of Ref. [13] with the same modulation. This is because the proposed scheme improves the transmission efficiency at the cost of decreasing of the diversity gain. The reliability of the proposed scheme is very close to that of Ref. [12] , because the two schemes both have a same diversity gain of 4, which demonstrate the validity of theoretical analysis.

6. Conclusions

For X channels, where each user has four antennas, the number of interfering time slots is reduced through the combination of codewords. Then, the multi-user interference is mitigated using the orthogonal property of the Alamouti code. Compared with the existing scheme, feedback information is not required, which greatly improves the transmission efficiency. Simulation results demonstrate that the reliability of the proposed scheme is not restricted to system full-rate full-diversity space-time block code. It can be extended to the other type of perfect space-time block code. However, the scheme is limited to the two users X channels. Future work on this scheme includes extending the application scene.

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 61202286; the National Natural Science Foundation of China under Grant No. 61104079.

Conflicts of Interest

The authors declare no conflicts of interest.

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