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New transmission scheme based on space-time code is proposed for massive MIMO system, which consists of K users and a base station. Each user is equipped with four antennas and two of them are selected to send space-time codeword with Alamouti coded by using the method of generalized spatial modulation. The base station estimates the index of transmit antennas used by each user at first, and then uses an iterative method to separate each space-time codeword and decode modulated signals of each code-word. Compared with the existing scheme for the same scene, the proposed scheme reduces the c omplexity of decoding, while keeping the same transmission rate. Simulation results demonstrate that the proposed scheme outperforms the existing scheme in the same scene when the base station is equipped with large amount of antennas.

Transmission rate and spectrum efficiency are particularly important for multi- cell communication. Massive MIMO technology have the ability to improve spectrum efficiency, cut down transmission power, reduce intra cell interference and inter-cell interference of the system without increasing spectrum resources [

In massive MIMO system, the base station is equipped with hundreds of antennas. K users transmit modulated signals simultaneously, or transmit the signals after precoding at the same time in a single cell uplink system [

In order to reduce the decoding complexity, the transmission based on space- time coding is proposed in single-cell massive MIMO uplink system. All of the users are equipped with 4 antennas and generalized spatial modulation is used at each user. According to the information sequence, two of the antennas are selected to transmit the space-time code which has Alamouti structure. Firstly, the serial number of the transmit antenna used at each user are estimated at the base station. Then, each space-time codeword is separated using the orthogonality of Alamouti code. Finally, modulation symbols in each codeword are detected. The ML decoding complexity of the proposed scheme is proportional to the square of the modulation order. The simulation results show that the reliability of the proposed scheme is better than [

The system model of the proposed scheme is shown in

Rate-2 space-time code proposed in [

X ′ k = [ x 1 x 2 ] = [ x ′ k , 1 − x ′ k , 2 ∗ x ′ k , 2 x ′ k , 1 ∗ ] = [ x k , 1 + e j θ x k , 2 − x k , 3 ∗ − e − j θ x k , 4 x k , 3 k + e j θ x k , 4 x k , 1 ∗ + e − j θ x k , 2 ∗ ] (1)

where, x k , λ represents the modulation symbols, λ = 1 , 2 , 3 , 4 , j = − 1 , ( ⋅ ) * represents conjugate. The value of θ makes the elements of X ′ k nonzero.

Assume that

The received signals at the base station, denoted by Y , can be expressed as

where, dimension of Y and N are both N × 2 . N represents the complex Gauss noise matrix with zero mean and variance of σ 2 . H k , with dimension N × 4 , represents the channel fading coefficient from user k to the base station. The dimension of x k , 1 and x k , 2 are both 4 × 1 . The μ k -th and ν k -th elements of x k , 1 are x ′ k , 1 and

The detection at the base station can be divided into two steps. Firstly, the serial number of the transmit antennas at each user is estimated at the base station according to the received signals and the channel state information (CSI). Then, each codeword is separated and each element of the codeword is estimated using iterative method at the base station.

Use y ′ 1 to represent the first column of Y . Let g = p i n v ( H ) y ′ 1 , which p i n v ( ⋅ ) represents pseudo inverse. g is a vector with a dimension of 4 K × 1 . Use g ¯ k to represent the vector with a dimension of 4 × 1 , which consists of the 4 ( k − 1 ) + 1 -th element to 4 k -th emelent of g , k = 1 , 2 , ⋯ , K . According to the estimation method of transmitting antennas’ number in (2) and [

where, μ k 0 ≠ ν k 0 ∈ { 1 , 2 , 3 , 4 } , g μ ′ k and g ν ′ k represents the μ ′ -th and ν ′ -th element of g ¯ k . ‖ ⋅ ‖ represents norm.

After the serial number of transmit antenna is estimated, the zero elements in (2) can be removed. Then, (2) can be equivalently expressed as

where, X ′ k represents a 2 × 2 matrix after removing the zero elements in X k , which is expressed as (1),

Since X ′ k has Alamouti structure, (4) can be expressed as

where H i k = [ h i , 1 k h i , 2 k h i , 2 k ∗ − h i , 1 k ∗ ] ,,

According to the method of separating Alamouti codewords in [

Let

G l , j ( l ) = [ I 2 F l , j H ( l ) ] H = [ I 2 − H ¯ l , j K − l + 1 ( l − 1 ) H ¯ l , K − l + 1 K − l + 1 H ( l − 1 ) ‖ H ¯ l , K − l + 1 K − l + 1 ( l − 1 ) ‖ 2 ︸ F l , j H ( l ) ] , l ∈ { 1 , ⋯ , K − 1 } , j ∈ { 1 , ⋯ , K − l | l } (7)

where, ( ⋅ ) H represents conjugate transpose, I 2 represents a unit matrix with a dimension of 2 × 2 . From (4)-(8) in [

From (25) in [

G ^ l H y ¯ l = G ^ l H [ H ( l − 1 ) K + 1 1 ⋮ H l K 1 ] s 1 + G ^ l H n ¯ l = G ^ l H [ h ¯ l , 1 h ¯ l , 2 ] ︸ H ˜ l s 1 + G ^ l H n ¯ l (8)

where, G ^ l = [ A l , 1 A l , 2 A l , 3 ⋮ A l , K ] = [ I 2 F l , 1 ( K − 1 ) F l , 1 ( K − 2 ) + φ l ( 1 ) ( A l , 2 ) F l , 1 ( K − 1 ) ⋮ ∑ l = 1 K − 1 φ l ( K − l ) ( A l , l ) F l , 1 ( l ) ] , φ l ( n ) ( A l , k )

represents a function replace all of F l , ψ ( l ) in A l , k to F l , ψ + n ( l − n ) , n ∈ { 1 , ⋯ , K − 1 } , ψ ∈ { 1 , ⋯ , K − 1 } . φ l ( n ) ( I 2 ) = I 2 , A l , k is a matrix with dimension 2 × 2 , h ¯ l , 1 and h ¯ l , 2 represent the 1-th column and 2-th column of

[ ( H ( l − 1 ) K + 1 1 ) H ( H ( l − 1 ) K + 2 1 ) H ⋯ ( H l K 1 ) H ] H respectively.

Let H ˜ l = G ^ l H [ h ¯ l , 1 h ¯ l , 2 ] , and multiply both sides of (8) by H ˜ l H to get

H ˜ l H G ^ l H y ¯ l = H ˜ l H H ˜ l s 1 + H ˜ l H G ^ l H n ¯ l (9)

Since the matrix with Alamouti structure is closed for addition, multiplication, and conjugate transpose operations, H ˜ l has Alamouti structure, which result in H ˜ l H H ˜ l = ( h ¯ 1 , 1 H G ^ 1 G ^ 1 H h ¯ 1 , 1 + ⋯ + h ¯ Λ , 1 H G ^ Λ G ^ Λ H h ¯ Λ , 1 ) I 2 . Use r ˜ 1 to represent the first element of H ˜ l H G ^ l H y ¯ l , then (10) can be obtained by (6)-(9).

r ˜ 1 = ( h ¯ 1 , 1 H G ^ 1 G ^ 1 H h ¯ 1 , 1 + ⋯ + h ¯ Λ , 1 H G ^ Λ G ^ Λ H h ¯ Λ , 1 ) x ′ 1 , 1 + ( h ¯ 1 , 1 H G ^ 1 G ^ 1 H n ¯ 1 + ⋯ + h ¯ Λ , 1 H G ^ Λ G ^ Λ H n ¯ Λ ) (10)

The base station can get x ′ 1 , 1 from r ˜ 1 directly. The detection of the other modulation symbols is similar. Since X ′ k has Alamouti structure and each codeword consists of two modulation symbols, the complexity of ML decoding is proportional to the square of the modulation order.

Let P e 1 and P e 2 represents the error probability of estimating the number of transmit antennas and the symbol error rate (SER) of the second step. Then the SER of the system can be expressed as

P e = 1 − ( 1 − P e 1 ) ( 1 − P e 2 ) (11)

We deduce P e 2 firstly. Assume the mathematical expectation of the modulation signal is P 1 . The signal-noise ratio (SNR) during the process of decoding x ′ 1 , 1 , denoted by ρ , can be obtained from (10).

ρ = P 1 σ 2 ( h ¯ 1 , 1 H G ^ 1 G ^ 1 H h ¯ 1 , 1 ) 2 + ⋯ + ( h ¯ Λ , 1 H G ^ Λ G ^ Λ H h ¯ Λ , 1 ) 2 h ¯ 1 , 1 H G ^ 1 G ^ 1 H G ^ 1 G ^ 1 H h ¯ 1 , 1 + ⋯ + h ¯ Λ , 1 H G ^ Λ G ^ Λ H G ^ Λ G ^ Λ H h ¯ Λ , 1 (12)

According to the estimation method of SER in [

where, E ( ⋅ ) represents mathematical expectation. The value of | x ′ 1 , 1 − x ^ ′ 1 , 1 | depends on the modulation mode.

In what follows, we derive P e 1 . According to the consistent bound method proposed by (12) in [

P e 1 ≤ E [ ∑ ξ ^ N ( ξ , ξ ^ ) P ( x 0 → x ^ 0 ) ] ≤ ∑ ξ = 1 4 ∑ q = 1 M ∑ ξ ^ = 1 4 N ( ξ , ξ ^ ) P ( x 0 → x ^ 0 ) 4 M (14)

where, M represents the modulation order, ξ represents the serial number of transmit antennas, ξ ^ is the number of antennas estimated by the base station. x 0 is the first column of X 1 . x ^ 0 is the decoding results obtained by decoding the first column of X 1 . N ( ξ , ξ ^ ) represents the number of error bits between ξ and ξ ^ .

It is known from (18) in [

P ( x 0 → x ^ 0 ) = f ( α ) N ∑ β = 0 N − 1 ( N − 1 + β β ) ( 1 − f ( α ) ) β (15)

where, f ( α ) = 1 2 ( 1 − α 1 + α ) , α = 1 4 σ 2 ∑ ξ = 1 4 | x ξ − x ˜ ξ | 2 , x ξ represent the ξ -th element of x 0 , ξ ∈ { 1 , 2 , 3 , 4 } . Substitute (15) into (14) to get

P e 1 ≤ ∑ ξ = 1 4 ∑ q = 1 M ∑ ξ ^ = 1 4 N ( ξ , ξ ^ ) f ( α ) N ∑ β = 0 N − 1 ( N − 1 + β β ) ( 1 − f ( α ) ) β 4 M (16)

Take (13) and (16) into (11) to get the SER of the system, as shown in (17).

This chapter simulates the reliability of the proposed scheme and the reliability of [

The error probability P e 1 of estimating the transmit antenna number when the base station is equipped with 64 antennas is shown is

The SER curves of the proposed scheme and the scheme in [

Compared with the existing scheme, each user adopts rate-2 space-time code. The transmission efficiency is the same as the existing scheme. The proposed scheme uses the orthogonal characteristics of the equivalent channel matrix corresponding to the Alamouti codeword to gradually separate each codeword, and then decodes each symbol of the codewords. The decoding complexity is greatly reduced. However, the reliability of the proposed scheme is greatly dependent on the estimated of the transmission antenna. How to reduce the dependency of the reliability of the system on the estimated transmit antenna number requires further study.

Tian, X.J., Jia, W.J. and Zhang, H.T. (2017) New Transmission Scheme Based on Space-Time Code for Massive MIMO. Open Access Library Jour- nal, 4: e3712. https://doi.org/10.4236/oalib.1103712