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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.

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 [

Interference alignment is the most studied interference cancellation method [

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.

The system model proposed in this paper is shown in

C i = [ c 4 i − 3 + e j θ c 4 i − 1 − c 4 i − 2 * − e − j θ c 4 i * c 4 i − 2 + e j θ c 4 i c 4 i − 3 * + e − j θ c 4 i − 1 * ] S i = [ s 4 i − 3 + e j θ s 4 i − 1 − s 4 i − 2 * − e − j θ s 4 i * s 4 i − 2 + e j θ s 4 i s 4 i − 3 * + e − j θ s 4 i − 1 * ] (1)

where c k ( k = 1 , 2 , ⋯ , 8 ) and s k are the modulated signals. The elements of C i and S i are un-zero with the proper value of θ . ( ⋅ ) ∗ denotes the conjugate.

The two users combine C i and S i respectively to get C and S as follows

C = [ C 1 + C 2 C 1 − C 2 C 1 − C 2 C 1 + C 2 ] S = [ S 1 + S 2 S 1 − S 2 S 1 − S 2 S 1 + S 2 ] (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 [ C 1 + C 2 C 1 − C 2 C 1 − C 2 C 1 + C 2 ] + G 1 [ S 1 + S 2 S 1 − S 2 S 1 − S 2 S 1 + S 2 ] + N (3)

Z = H 2 [ C 1 + C 2 C 1 − C 2 C 1 − C 2 C 1 + C 2 ] + G 2 [ S 1 + S 2 S 1 − S 2 S 1 − S 2 S 1 + S 2 ] + W (4)

where, N and Z 8 are 4 × 4 Gaussian noise matrices.

The interference alignment method is presented, taking R 1 as example. Let

H 1 = [ H 11 H 21 ] , G 1 = [ G 11 G 21 ] , N = [ N 11 N 21 ] and Y = [ Y 11 Y 21 ] . The dimension of H i 1 , G i 1 , Y i 1 and N i 1 are all 4 × 2. From (2) (3) (5) and (6) can be obtained.

Y 11 = H 11 ( C 1 + C 2 ) + H 21 ( C 1 − C 2 ) + G 11 ( S 1 + S 2 ) + G 21 ( S 1 − S 2 ) + N 11 (5)

Y 21 = H 11 ( C 1 − C 2 ) + H 21 ( C 1 + C 2 ) + G 11 ( S 1 + S 2 ) + G 21 ( S 1 + S 2 ) + N 21 (6)

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

Y 11 + Y 21 ︸ Y 1 = 2 ( H 11 + H 21 ) ︸ H 1 C 1 + 2 ( G 11 + G 21 ) ︸ H 2 S 1 + N 11 + N 21 ︸ N 1 (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 = [ h 11 1 h 12 1 h 21 1 h 22 1 ] and H 2 = [ h 11 2 h 12 2 h 21 2 h 22 2 ] . Use y i j and n i j to denote the

elements of Y 1 and N 1 , respectively, i = 1 , 2 , 3 , 4 , j = 1 , 2 . From (7), we have

[ y 11 y 12 * ] ︸ y 1 = Z 1 [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + Z 2 [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] + [ n 11 n 12 * ] ︸ n 1 (8)

[ y 21 y 22 * ] ︸ y 2 = Z 3 [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + Z 4 [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] + [ n 21 n 22 * ] ︸ n 2 (9)

[ y 31 y 32 * ] ︸ y 3 = Z 5 [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + Z 6 [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] + [ n 31 n 32 * ] ︸ n 3 (10)

[ y 41 y 42 * ] ︸ y 4 = Z 7 [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + Z 8 [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] + [ n 41 n 42 * ] ︸ n 4 (11)

where, Z 2 i − 1 = [ h i 1 1 h i 2 1 h i 2 1 * − h i 1 1 * ] , Z 2 i = [ h i 1 2 h i 2 2 h i 2 2 * − h i 1 2 * ] , 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 Z i H Z i ‖ Z i ‖ 2 = I 2 , i = 1 , 2 , 3 , 4 ,

where I 2 denotes the 2 × 2 unit matrix. ( ⋅ ) H and ‖ ⋅ ‖ 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 = y 1 Z 2 H ‖ Z 2 ‖ 2 − y 2 Z 4 H ‖ Z 4 ‖ 2 = Q 1 [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + n 1 Z 2 H ‖ Z 2 ‖ 2 − n 2 Z 4 H ‖ Z 4 ‖ 2 ︸ P 1 (12)

z 2 = y 1 Z 1 H ‖ Z 1 ‖ 2 − y 2 Z 3 H ‖ Z 3 ‖ 2 = Q 2 [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] + n 1 Z 1 H ‖ Z 1 ‖ 2 − n 2 Z 3 H ‖ Z 3 ‖ 2 ︸ P 2 (13)

z 3 = y 3 Z 6 H ‖ Z 6 ‖ 2 − y 4 Z 8 H ‖ Z 8 ‖ 2 = Q 3 [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + n 3 Z 6 H ‖ Z 6 ‖ 2 − n 4 Z 8 H ‖ Z 8 ‖ 2 ︸ P 3 (14)

z 4 = y 3 Z 5 H ‖ Z 5 ‖ 2 − y 4 Z 7 H ‖ Z 7 ‖ 2 = Q 4 [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] + n 3 Z 5 H ‖ Z 5 ‖ 2 − n 4 Z 7 H ‖ Z 7 ‖ 2 ︸ P 4 (15)

where, Q i = Z 2 i − 1 Z 2 i H ‖ Z 2 i ‖ 2 − Z 2 i + 1 Z 2 i + 2 H ‖ Z 2 i + 2 ‖ 2 , i = 1 , 3 ,

Q k = Z 2 k − 1 H Z 2 i ‖ Z 2 k − 1 ‖ 2 − Z 2 K + 1 H Z 2 k + 2 ‖ Z 2 k + 1 ‖ 2 , k = 2 , 4 , Q i and Q k are the effective channel matrices of [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] or [ s 1 + e j θ s 3 s 2 + e j θ s 4 ] , 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.

In this section, the decoding method is presented, taking R 1 as example. By

calculating, we can get Z 1 Z 2 H = [ h 1 1 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 ] . 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 = [ q 1 − q 2 ∗ q 2 q 1 ∗ ] and c 4 , we process

z 1 and z 3 according to (16)

Q 1 H z 1 + Q 3 H z 3 = [ q 0 0 0 q 0 ] [ c 1 + e j θ c 3 c 2 + e j θ c 4 ] + Q 1 H P 1 + Q 3 H P 3 (16)

where q 0 = | q 1 | 2 + | q 2 | 2 + | q 3 | 2 + | q 4 | 2 . Let Q 1 H z 1 + Q 3 H z 3 = [ z 1 z 2 ] and

P 1 H z 1 + P 3 H z 3 = [ p 1 p 2 ] , (16) can be rewritten as

z 1 = q 0 ( c 1 + e j θ c 3 ) + p 1 (17)

z 2 = q 0 ( c 2 + e j θ c 4 ) + p 2 (18)

It can be seen from (17) (18) that c 1 + e j θ c 3 and c 2 + e j θ c 4 are separated. So we can decode ( c 1 , c 3 ) and ( c 2 , c 4 ) , 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 ( H 11 + H 21 ) and H 2 = 2 ( G 11 + G 21 ) 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 , ⋯ , 8 , and let

Q 1 = [ q 1 − q 2 * q 2 q 1 * ] and Q 3 = [ q 3 − q 4 * q 4 q 3 * ] ;

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 = [ z 1 z 2 ] ;

Step 6, with the aid of the effective transmit signal c 1 + e j θ c 3 , the effective channel matrix | q 1 | 2 + | q 2 | 2 + | q 3 | 2 + | q 4 | 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 θ c 3 and c 2 + e j θ c 4 reaches R 1 by experiencing 4 independent paths. So the diversity gain is 4.

There are 32 modulated signals to be transmitted in 6 time slots in [

There are a comparison of the transmission efficiency, diversity gain, feedback and decoding complexity of the three schemes, as shown in

In

Scheme | Proposed scheme | Ref. [ | Ref. [ |
---|---|---|---|

Transmission efficiency | 4 symbol/channel | 16/9 symbol/channel | 8/3 symbol/channel |

Diversity gain | 4 | 4 | 8 |

Feedback amount | No | Global CSI | Global CSI |

Decoding complexity | M^{2} | M | M^{2} |

Gauss white noise. We can see that the reliability of the proposed scheme is not better than that of Ref. [

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.

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.

Tian, X.J., Yang, D., Zhang, H.T. and Jia, W.J. (2017) Improved Interference Cancellation Scheme for X Channels with Four Antennas. Journal of Computer and Communications, 5, 57-64. https://doi.org/10.4236/jcc.2017.56004