In this paper, we investigate how to cancel multi-user interference without feedback amount over multiple-input multiple-output (MIMO) interference channel using space-time code. Three transmitter-receiver pairs are considered and each node is equipped with four antennas. Space-time codeword with dimension 4 × 4 is designed for three transmitters, respectively. The number of interfering codeword is reduced through the de sign of codeword. Then, multi-user interference is cancelled by a linear operation on the received signals. Compared with the existing scheme for the same scene, the transmission efficiency is improved, and the unidirectional cooperative link is not required. Simulation results demonstrate the validity of theoretical analysis.
Reliability and effectiveness are important factors for wireless communications. Multiple-input multiple-output (MIMO) systems have the ability to improve both and hence, are immensely popular. In theory, MIMO systems can achieve extremely high spectral efficiency by transmitting independent data stream simultaneously. However, multiple users, transmitting different codewords to multiple receivers simultaneously using the same band, are not able to collaborate with each other [
Several kinds of interference cancellation methods have been conducted, such as pre-coding at the transmitters, multi-user detection of the receivers, and interference alignment [
A popular way to approximating the sum-capacity for interference alignment is using the concept of degrees of freedom (DoF), in terms of transmission efficiency. Many works have been conducted to evaluate DoF [
Under this motivation, interference cancellation method without feedback is proposed for MIMO interference channel with three transmitter-receiver pairs where each node is equipped with four antennas. Space-time codeword with dimension 4×4 is designed for three transmitters, respectively. Each codeword includes eight independent modulated symbols and zero matrices are introduced into each codeword, through which the number of interfering codeword is reduced. Then, multi-user interference is cancelled by a linear operation on the received signals. The transmission efficiency of the proposed scheme is higher than that of [
Notation: The operators ( ⋅ ) ∗ and ( ⋅ ) H stand for complex conjugation and conjugation transpose, respectively. ‖ ⋅ ‖ denotes Frobenius norm of the enclosed term. E ( ⋅ ) is the expectation operations.
The system model is shown in
a k , b k and c k , which take values from a set such that E [ | a k | 2 ] = E [ | b k | 2 ] = E [ | c k | 2 ] = 1 , k = 1 , 2 , ⋯ , 8 , represent the modulated signals at transmitter 1, transmitter 2 and transmitter 3, respectively. a k , b k and c k are encoded into 2 × 2 space-time codeword, denoted by A i , B i and C i , respectively, i = 1 , 2 . A i , B i and C i are expressed as
A i = [ a 4 i − 3 + e j θ a 4 i − 1 − a 4 i − 2 ∗ − e − j θ a 4 i ∗ a 4 i − 2 + e j θ a 4 i a 4 i − 3 ∗ + e − j θ a 4 i − 1 ∗ ] B i = [ b 4 i − 3 + e j θ b 4 i − 1 − b 4 i − 2 ∗ − e − j θ b 4 i ∗ b 4 i − 2 + e j θ b 4 i b 4 i − 3 ∗ + e − j θ b 4 i − 1 ∗ ] 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 ∗ ] (1)
where, j = − 1 , 0 < θ < 2 π . The elements of A i , the elements of B i and the elements of C i are nonzero by the proper value of θ . Each element of A i , each element of B i and each element of C i includes two independent symbols, with the purpose of improving transmission efficiency. If we see the elements of A i , the elements of B i and the elements of C i as the modulated signals, i = 1 , 2 , A i , B i and C i have an Alamouti structure. The motive for such design is that the effective channel matrix of Alamouti code has orthogonal columns, which can be used to reduce decoding complexity.
At transmitter 1, A 1 and A 2 are combined according to a certain rule to get codeword A . Similarly, B 1 and B 2 are combined to obtain codeword B while C 1 and C 2 are combined to obtain codeword C . A , B and C are given by
A = [ A 1 0 2 A 2 0 2 ] , B = [ B 1 0 2 0 2 B 2 ] , C = [ 0 2 C 1 0 2 C 2 ] (2)
where, 0 2 represents 2 × 2 zero matrix. 0 2 is introduced because it can be used to mitigate partial multi-user interference, as shown later. A , B and C are the desired codewords for the three receivers, respectively.
As shown in
Y 1 = H 11 A + H 12 B + H 13 C + N 1 (3)
Y 2 = H 21 A + H 22 B + H 23 C + N 2 (4)
Y 3 = H 31 A + H 32 B + H 33 C + N 3 (5)
where, i = 1 , 2 , 3 , H i k is the channel from transmitter k to receiver i and N i is the complex additive white Gaussian noise at receiver i. The dimension of H i k and N i are both 4 × 4 . To avoid degenerate channel conditions, we assume that the channel coefficient values are drawn i.i.d. from a continuous distribution and the absolute value of all the channel coefficients is bounded between a nonzero minimum value and a finite maximum value.
The term H 11 A in Equation (3) is the desired signals for receiver 1 and the term H 12 B + H 13 C in Equation (3) is the interference signals for receiver 1. The term H 22 B in Equation (4) is the desired signals for receiver 2 and the term H 21 A + H 23 C in Equation (4) is the interference signals for receiver 2. The term H 33 C in Equation (5) is the desired signals for receiver 3 and the term H 31 A + H 32 B in Equation (5) is the interference signals for receiver 3.
We describe the interference cancellation method at R 1 . It can be similarly used to mitigate interference at R 2 and to mitigate interference R 3 .
Let H 11 = [ H 1 H 2 ] , H 12 = [ H 3 H 4 ] , H 13 = [ H 5 H 6 ] , Y 1 = [ Y 11 Y 21 ] and N 1 = [ N 11 N 21 ] . The dimension of H l , N i 1 and Y i 1 are both 4 × 2 , l = 1 , 2 , ⋯ , 6 ， i = 1 , 2 . Then, (3) can be expressed as
[ Y 11 Y 21 ] = [ H 1 A 1 + H 2 A 2 + H 3 B 1 H 4 B 2 + H 5 C 1 + H 6 C 2 ] + [ N 11 N 21 ] (6)
(6) is equivalent to the following two equations,
Y 11 = H 1 A 1 + H 2 A 2 + H 3 B 1 + N 11 (7)
Y 21 = H 4 B 2 + H 5 C 1 + H 6 C 2 + N 21 (8)
The codeword of transmitter 1 and the partial codeword of transmitter 2 are encompassed by Y 11 . The codeword of transmitter 3 and the partial codeword of transmitter 2 are encompassed by Y 21 . So, the number of interfering codeword is reduced by the introduction of zero matrices. However, both the desired codeword for receiver 1 and the undesired codeword for receiver 1 in terms of B 1 are included by Y 11 . In what follows, we represent how to cancel the undesired codeword B 1 .
Let H 1 = [ h 11 1 h 12 1 h 21 1 h 22 1 h 31 1 h 32 1 h 41 1 h 42 1 ] , H 2 = [ h 11 2 h 12 2 h 21 2 h 22 2 h 31 2 h 32 2 h 41 2 h 42 2 ] and H 3 = [ h 11 3 h 12 3 h 21 3 h 22 3 h 31 3 h 32 3 h 41 3 h 42 3 ] . Use y i j and
n i j to denote the element of Y 11 and N 11 , i = 1 , 2 , 3 , 4 , j = 1 , 2 . The elements of Y 11 can be effectively written as
y 1 = G 11 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + G 12 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + G 13 [ b 1 + e j θ b 3 b 2 + e j θ b 4 ] + n 1 (9)
y 2 = G 21 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + G 22 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + G 23 [ b 1 + e j θ b 3 b 2 + e j θ b 4 ] + n 2 (10)
y 3 = G 31 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + G 32 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + G 33 [ b 1 + e j θ b 3 b 2 + e j θ b 4 ] + n 3 (11)
y 4 = G 41 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + G 42 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + G 43 [ b 1 + e j θ b 3 b 2 + e j θ b 4 ] + n 4 (12)
where, G i j = [ h i 1 j h i 2 j h i 2 j ∗ − h i 1 j ∗ ] , y i = [ y i 1 y i 2 ∗ ] , n i = [ n i 1 n i 2 ∗ ] , i = 1 , 2 , 3 , 4 , j = 1 , 2 , 3 . y i
and n i are the effective received signals and the effective noise at the i-th an-
tenna, respectively. [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] , [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] and [ b 1 + e j θ b 3 b 2 + e j θ b 4 ] are the effec-
tive transmitted signals of the three transmitters. G i j is the effective channel matrix corresponding to the effective transmitted signals, i = 1 , 2 , 3 , 4 ,
j = 1 , 2 , 3 . G i j , having orthogonal columns, satisfies G i j H G i j ‖ G i j ‖ 2 = I 2 , where I 2 is
unit matrix with dimension 2 × 2 . The third terms at the right side of Equation (9-12) are the interference signals, which can be mitigated using interference cancellation method as follows
z 1 = G 13 H y 1 ‖ G 13 ‖ 2 − G 23 H y 2 ‖ G 23 ‖ 2 = P 1 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + P 2 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + w 1 (13)
z 2 = G 33 H y 3 ‖ G 33 ‖ 2 − G 43 H y 4 ‖ G 43 ‖ 2 = P 3 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + P 4 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + w 2 (14)
where, P i = G 13 H G 1 i ‖ G 13 ‖ 2 − G 23 H G 2 i ‖ G 23 ‖ 2 , P j = G 33 H G 3 ( j − 2 ) ‖ G 33 ‖ 2 − G 43 H G 4 ( j − 2 ) ‖ G 43 ‖ 2 ,
w 1 = G 13 H n 1 ‖ G 13 ‖ 2 − G 23 H n 2 ‖ G 23 ‖ 2 , w 2 = G 33 H n 3 ‖ G 33 ‖ 2 − G 43 H n 4 ‖ G 43 ‖ 2 , i = 1 , 2 , j = 1 , 2 . z i and w i are
the signals and the noise after linear operation, respectively, i = 1 , 2 . P i is effective channel matrix after linear operation, i = 1 , 2 , 3 , 4 . The dimension of z i and the dimension w i are both 2 × 1 , i = 1 , 2 . The dimension of P i is 2 × 2 , i = 1 , 2 , 3 , 4 . Symbols transmitted from transmitter 2 and symbols transmitted from transmitter 3 are not included in (13-14). So multi-user interference is mitigated through linear operation on the received signals.
Even if multi-user interference is cancelled, the desired codeword at receiver 1 includes 8 independent modulated symbols, namely a 1 , a 2 , ⋯ , a 8 , as shown in (13-14). If we decode a 1 , a 2 , ⋯ , a 8 based on (13-14), the complexity of maximum likelihood (ML) algorithm is proportional to the 8th power of modulation order, which is extremely high. So, it is necessary to divide these symbols into several groups. Take operation on z 1 and z 2 to get,
x 1 = P 2 H z 1 ‖ P 2 ‖ 2 − P 4 H z 2 ‖ P 4 ‖ 2 = Q 1 [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + w 3 (15)
x 2 = P 1 H z 1 ‖ P 1 ‖ 2 − P 3 H z 2 ‖ P 3 ‖ 2 = Q 2 [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + w 4 (16)
where, Q 1 = P 2 H P 1 ‖ P 2 ‖ 2 − P 4 H P 3 ‖ P 4 ‖ 2 , Q 2 = P 1 H P 2 ‖ P 1 ‖ 2 − P 3 H P 4 ‖ P 3 ‖ 2 , w 3 = P 2 H w 1 ‖ P 2 ‖ 2 − P 4 H w 2 ‖ P 4 ‖ 2 ,
w 4 = P 1 H w 1 ‖ P 1 ‖ 2 − P 3 H w 2 ‖ P 3 ‖ 2 . a 1 , a 2 , a 3 and a 4 are encompassed by A 1 whereas
a 5 , a 6 , a 7 and a 8 are encompassed by A 2 . So, the elements of A 1 and the elements of A 2 are separated, as shown in (15-16). The decoding complexity can be reduced.
From the above analysis, not only multi-user interference is cancelled, but also the two desired codewords A 1 and A 2 are separated. Similar operation can be performed at the other two receivers. Note that neither feedback information nor the unidirectional cooperative link is required.
In this section, we will show how to decode, taking receiver 1 as example.
G i j has an Alamouti structure, i = 1 , 2 , 3 , 4 , j = 1 , 2 , 3 . Due to the completeness of matrix addition, matrix multiplication, and the scalar multiplication of the Alamouti matrix, Q 1 and Q 2 still have the Alamouti structure. Let
Q 1 = [ q 1 − q 2 ∗ q 2 q 1 ∗ ] and Q 2 = [ q 3 − q 4 ∗ q 4 q 3 ∗ ] . Multiply both sides of (15) with Q 1 H to
get
Q 1 H x 1 = [ x 1 x 2 ] = [ | q 1 | 2 + | q 2 | 2 0 0 | q 1 | 2 + | q 2 | 2 ] [ a 1 + e j θ a 3 a 2 + e j θ a 4 ] + Q 1 H w 3 (17)
Multiply both sides of (16) with Q 2 H to get
Q 2 H x 2 = [ x 3 x 4 ] = [ | q 3 | 2 + | q 4 | 2 0 0 | q 3 | 2 + | q 4 | 2 ] [ a 5 + e j θ a 7 a 6 + e j θ a 8 ] + Q 2 H w 4 (18)
In (17), the channel corresponding to a 1 + e j θ a 3 is orthogonal to the channel corresponding to a 2 + e j θ a 4 . So we can decode a 1 + e j θ a 3 and a 2 + e j θ a 4 separately. Similarly, we can decode a 5 + e j θ a 7 and a 6 + e j θ a 8 separately as well. According to the analysis mentioned above, the decoding method is as follows.
Step1, obtain G 11 , G 12 , ⋯ , G 43 from the channel matrices, and then compute P 1 , P 2 , P 3 and P 4 from G 11 , G 12 , ⋯ , G 43 ;
Step 2, compute Q 1 = P 2 H P 1 ‖ P 2 ‖ 2 − P 4 H P 3 ‖ P 4 ‖ 2 and Q 2 = P 1 H P 2 ‖ P 1 ‖ 2 − P 3 H P 4 ‖ P 3 ‖ 2 through di-
rect multiplications and subtraction ;
Step 3, obtain y 1 , y 2 , y 3 and y 4 according to the received signals, as in (9-12);
Step 4, take operation on y 1 , y 2 , y 3 and y 4 to get z 1 and z 2 ,
z 1 = G 13 H y 1 ‖ G 13 ‖ 2 − G 23 H y 2 ‖ G 23 ‖ 2 , z 2 = G 33 H y 3 ‖ G 33 ‖ 2 − G 43 H y 4 ‖ G 43 ‖ 2 ;
Step 5, take operation on z 1 and z 2 to get x 1 = P 2 H z 1 ‖ P 2 ‖ 2 − P 4 H z 2 ‖ P 4 ‖ 2 and
x 2 = P 1 H z 1 ‖ P 1 ‖ 2 − P 3 H z 2 ‖ P 3 ‖ 2 , furthermore, computer [ x 1 x 2 ] = Q 1 H x 1 and [ x 3 x 4 ] = Q 2 H x 2
through direct multiplications;
Step 6, estimate a 1 and a 3 by
( a ^ 1 , a ^ 3 ) = arg ︸ a 1 , a 3 min [ x 1 − ( | q 1 | 2 + | q 2 | 2 ) ( a 1 + e j θ a 3 ) ] , where a ^ 1 and a ^ 3 are the es-
timation of a 1 and a 3 , respectively;
Step 7, estimate a 2 and a 4 by
( a ^ 2 , a ^ 4 ) = arg ︸ a 2 , a 4 min [ x 2 − ( | q 1 | 2 + | q 2 | 2 ) ( a 2 + e j θ a 4 ) ] , where a ^ 2 and a ^ 4 are
the estimation of a 2 and a 4 , respectively;
Step 8, estimate a 5 and a 6 by
( a ^ 5 , a ^ 6 ) = arg ︸ a 5 , a 6 min [ x 3 − ( | q 1 | 2 + | q 2 | 2 ) ( a 5 + e j θ a 6 ) ] , where a ^ 5 and a ^ 6 are the
estimation of a 5 and a 6 , respectively;
Step 9, estimate a 7 and a 8 by
( a ^ 7 , a ^ 8 ) = arg ︸ a 7 , a 8 min [ x 4 − ( | q 1 | 2 + | q 2 | 2 ) ( a 7 + e j θ a 8 ) ] , where a ^ 7 and a ^ 8 are the
estimation of a 7 and a 8 , respectively;
The ML decoding complexity in step (6-9) is proportional to the square of modulation order.
It is easy to see from (17) that the channel corresponding to a 1 + e j θ a 3 is | q 1 | 2 + | q 2 | 2 , which includes two independent path with gain | q 1 | 2 and | q 2 | 2 . So a 1 + e j θ a 3 has a diversity gain of two, which is equal to that of [
6 independent modulated signals are sent over two time slots in [
Performance of the proposed scheme and performance of the scheme in [
In
| Scheme | Proposed | Ref. [ |
|---|---|---|
| Transmission efficiency | 1/2 symbol/antenna/time slot | 3/22 symbol/antenna/time slot |
| Diversity gain | 2 | 2 |
| Feedback | No | Local CSI |
| Unidirectional cooperative link | No | Transmit two complex number |
| Decoding complexity | Proportional to the square of modulation order | Proportional to the modulation order |
slope of BER curves represents diversity gain, which is 2 in these two schemes. It can be seen from this figure that the slope of BER curves of the proposed scheme is almost the same as that of [
In this paper, the multi-user interference is cancelled by the design of the codeword and linear operation on the received signals. Compared with the existing scheme for the same scene, the proposed scheme doesn’t require feedback information and doesn’t require unidirectional cooperative link, while keeping the same diversity gain. However, the scheme is limited to three transmitter-receiver pairs, where each node is equipped with four antennas. Future work on this scheme includes extending the application scene.
The authors would like to thank the reviewers for their detailed reviews and constructive comments, which have helped improve the quality of this paper. This work was supported in part by National Natural Science Foundation of China under Grant No. 51174263.
Tian, X.J., Zhang, H.T. and Jia, W.J. (2017) Interference Cancellation Method without Feedback Amount for Three Users Interference Channel. Open Access Library Journal, 4: e3574. https://doi.org/10.4236/oalib.1103574