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Combining precoding and equalization for interference cancellation in MU-MIMO systems with high density users

Abstract

In multiple users (MU) multiple input multiple output (MIMO) systems, the non-orthogonal multiple access (NOMA) method can provide multiple access. However, a spectrum efficiency of NOMA method is restricted because of remaining an interference signal of another user. A beamforming method can also be applied into MU-MIMO systems for implementation of multiple access without the interference with the other users; however, it is unavailable for a high density user environment in which multiple users locate close to each other. In order to further improve the spectrum efficiency, we propose a novel signal processing method which joints a precoding and an equalization. First, a precoding matrix is prepared for every user at base station, and we design an equalization which is orthogonal to the precoding matrix. Second, a set of linear weights is proposed to calculate at users for cancelling any interference signal. Therefore, signals for other users are eliminated according to the orthogonal feature and linear signal processing, and then, every user can receive its own signal without the interference signal. The proposed method is compared with the beamforming and NOMA methods in several scenarios, and the calculation result shows that our method outperforms other methods. Especially, our novel method works properly in the case of high density users.

Introduction

Related works

In recent years, many advanced technologies for multiple input multiple output (MIMO) communication systems have been proposed. A byword is the beamforming technology, which has attracted increasing research interest recently. In order to exploit the full potential of MIMO system and its expansion, the massive MIMO system, many beamforming algorithms were developed. Several well-known beamforming methods which generate multiple beams to serve multiple users (MU) in MU-MIMO systems were proposed, e.g. Zero-Forcing (ZF) beamforming [1,2,3], minimum mean square error (MMSE) beamforming [4,5,6], dirty paper coding (DPC) [7, 8], singular value decomposition (SVD) [9,10,11]. However, most algorithms have assumed that perfect channel state information (CSI) is available at both transceivers and responds insensitively to change of transmission environment and relative position between transceivers. To solve these issues, an intelligent beamforming scheme was suggested. The intelligent beamforming is achievable by deployment of artificial intelligence (AI) techniques to generate beams for every user. The machine learning-based beamforming design was first introduced for two-user MIMO system with interference channels by Hyung et al. [12]. The proposed machine learning structure takes transmit power and channel vectors as input and then recommends two users to choice between MRT and ZF as output. An intelligent recommendation method also was proposed to support users in big data environments [13, 14]. Recently, in [15], the deep learning-assisted detection method has been proposed for MU-MIMO systems. As addressed by the authors, the proposed detection method outperforms the maximum likelihood detection (MLD) in the case of channel impairments with low complexity. Compared to conventional beamforming methods, the deep learning and machine learning-based beamforming methods can improve the performance of system, especially in rapid alteration of transmission environment. However, the improvement of spectrum efficiency of these method has not yet met the demand of growth of wireless devices.

In order to further improve the performance of massive MIMO systems, a full-space spectrum-sharing strategy [16], a unified transmission strategy for TDD/FDD [17], and the hybrid beamforming as a combination of digital baseband and analog RF phase shifters [18, 19] were proposed. The hybrid beamforming method increases an efficiency of beamforming, and hence, the performance of MU systems is improved. However, beamforming algorithms depend on the number of antennas at a base station (BS) as well as the number of users, and they are inefficient when the number of users is much more than the number of antennas at the BS. In millimeter-wave wireless communication systems, BSs are densely distributed in order to cover a wide service area [6, 20, 21]. Microcell and macrocell cover a wide area, and their BSs are equipped with many antennas. However, the number of antennas at the BS of picocell and femtocell is insufficient to generate an individual beam for every user because of limited setting space and many users. Especially, when many users locate close to each other, i.e., high density user systems, multiple users are affected by a beam of the BS which is generated for another user. Therefore, because of overlap of beams, it is unavailable to serve every user without the interference with the other users.

The non-orthogonal multiple access (NOMA) is a widely used approach to deal with the problem of multiple nearby users by providing the multiple access based on power domain [22, 23]. The NOMA method allocates more transmit power for users of low channel gain and less transmit power for users of high channel gain. At the user of low channel gain, it detects its own signal while considering signals of other users as interference. Whereas, at the user of high channel gain, it applies the successive interference cancellation (SIC) following the principle of NOMA method to eliminate signals of other users, and then detect its own signal. As mentioned in [24, 25], the combination of NOMA and beamforming methods has been proved as an effective method to increase the spectral efficiency of system. However, the disadvantage of NOMA method is to maintain the interference at the user of low channel gain, it lets a signal-to-interference-plus-noise ratio (SINR) be low, and hence, the performance of NOMA systems is deteriorated, even for the combination of NOMA and beamforming methods.

Scope and contributions

As mentioned above, many technologies have developed to combine with the beamforming method in order to improve the spectral efficiency of wireless systems. However, the interference signal is still remained, especially in high density user systems. One of main reasons can be explained as follows. The beamforming method is applied only to the transmitter site or the receiver site. Therefore, the interference signal cannot be completely cancelled because of lack of rigor. We suggest that a combination of precoding at the transmitter site and equalization at the receiver site with utilization of orthogonal feature can further improve the spectral efficiency of MU-MIMO systems, even for high density user systems.

Consequently, in this paper, we propose a novel signal processing method to improve the performance of system in case the number of antennas at the BS is lack for generating individual beam for every users without overlapping. Our proposal method takes advantage of orthogonality between precoding and equalization to completely cancel interference signals of other users and maintain high quality of service.

The contributions of this paper can be summarized as follows:

  • A systematic approach is presented for designing an equalization which is orthogonal to precoding matrix for cancelling the interference signal.

  • A novel linear signal processing approach is proposed in order to calculate weights for detecting the designed signal.

  • The correlation coefficient of channel matrices of nearby users is derived and taken into consideration. The derived correlation coefficient can be used as a means to evaluate the performance of proposed method in several scenarios, especially for the case of multiple nearby users.

  • Various calculation and simulation results have been carried out to compare the efficiency of the proposed method with other methods, such as the NOMA and beamforming methods.

  • we discuss on ability of extension and application of proposed method to actual systems.

The rest of paper is organized as follows. We briefly overview the model of MU-MIMO systems in high density user environment in Sect. 2. Also in this section, the conventional beamforming and NOMA methods are explained. The novel signal processing method for interference cancellation in downlink MU-MIMO systems is proposed in Sect. 3. Section 4 presents simulation results and analysis. Finally, the conclusion is drawn in Sect. 5.

Notations: Regular and bold styles denote a scalar and a vector/matrix, respectively. \({{\mathbf{X }}}^{\mathrm{{T}}}\), \({{\mathbf{X }}}^{\mathcal {H}}\) and \(\mathbf{X }^{\mathrm{{-1}}},\) respectively, depict the transpose, Hermitian and pseudo-inverse operations of \(\mathbf{X }\). Because a component-wise form is utilized for analyzing the proposed system performance, \(\mathbf{X} ({\mathrm{l}})\) or \(\mathbf{X }\mathbf{Y }({\mathrm{l}})\) represents the \({\mathrm{l}}^{th}\) row of matrix \(\mathbf{X }\) or the \({\mathrm{l}}^{th}\) row of matrix \(\mathbf{Z }= XY\). \(\mathbb {C}^{{{\mathrm{m}}\times {\mathrm{n}}}}\) denotes a \({\mathrm{m}}\) rows, \({\mathrm{n}}\) columns independent and identically distributed (i.i.d.) complex Gaussian entries matrix which is normalized with zero mean unit variance. \(\mathbb {E}\{\cdot \}\) denotes the average operator.

System model of MU-MIMO systems

MU-MIMO system model with high density users

Figure 1 shows a MU-MIMO communication system configuration which is equipped with a base station (BS) and K users. The number of antennas at the BS is denoted by M, whereas the number of antennas at every user is assumed to be two (2) due to limitation space of personal devices. In order to communicate with all users, the zero forcing beamforming (ZFBF) algorithm is applied at the BS due to its simplicity. Thanks to the beamforming method, the BS can communicate to each user via individual beams. However, despite the aforementioned advantage, this method suffers from overlap between beams when K users are in the same direction. This means that the MU-MIMO system still faces the problem of interference, which possibly degrades the system performance.

Fig. 1
figure 1

System model of MU-MIMO systems consisting of K users in the same beam

Under the assumptions that the channel response matrix between the BS and User i, \(\mathbf{H }_i\), is block fading, and the entries of \(\mathbf{H }_i\) are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. Let \(\mathbf{H }_i =\{h^i_{l,m}\}\in \Omega _{i}\mathbb {C}^{{{\mathrm{2}}\times M}}\) with \(i \in {1,\ldots ,K}\). \(h^i_{l,m}\) represents the channel coefficient between the \(m^{\mathrm{th}}\) antenna of BS and the \(l^{\mathrm{th}}\) antenna of User i, and \(\Omega _{i}\) denotes the variance of channel gain between the BS and User i. Without loss of generality, we assume that \(\Omega _{K}>\cdots>\Omega _{2}>\Omega _{1}\). In other words, the Users \(K,\ldots ,2,1\) are arranged farther away, the User 1 is the farthest user, whereas the User K is the nearest user.

Furthermore, the channel state information at transmitter (CSIT) is assumed to be outdated due to the delay of feedback transmission from users to the BS. The true and the error channel matrices of User i are, respectively, denoted by \(\mathbf{H }^t_i\) and \(\mathbf{H }^e_i\). Therefore, the estimated channel matrix \(\mathbf{H }_i\) is calculated via a correlation coefficient of the estimated and true channel matrices, \(\rho\), as follows [26, 27]:

$$\begin{aligned} \mathbf{H }_i=\sqrt{1-\rho ^2}\mathbf{H }^e_i+\rho \mathbf{H }^t_i. \end{aligned}$$
(1)

According to the ZFBF algorithm, the weight \({\mathbf{w }}_i\) for the User i is given by [23, 28]

$$\begin{aligned} \mathbf{w }_i=\frac{\mathbf{W }_i \mathbf{H }_i}{\Vert \mathbf{W }_i \mathbf{H }_i\Vert }, \end{aligned}$$
(2)

where

$$\begin{aligned}&\mathbf{W }_i=\mathbf{I }-\mathbb {H}_{i}(\mathbb {H}_{i}^H \mathbb {H}_{i})^{-1}\mathbb {H}_{i}^H,\\&\mathbb {H}_{i}=[\mathbf{H }_1,\mathbf{H }_2,\ldots ,\mathbf{H }_{i-1},\mathbf{H }_{i+1},\ldots ,\mathbf{H }_K],\\&\text {and }\mathbf{I }\text { is an identity matrix.} \end{aligned}$$

The weight \({\mathbf{w }}_i\) ensures that the BS can communicate with the User i via its own beam, and the interference with other users can be avoided if the beam is sharp enough or the user is far from each other. However, the number of antennas at the BS is not enough to generate a sharp beam and many users are close to each other in high density user environment. Therefore, the interference signal is still remained and depends on an overlap factor of beams.

Conventional beamforming and NOMA methods

Beamforming method

In the system utilizing beamforming method, the BS applies the ZFBF method, and the weight is generated by (2). Firstly, information signals of all users are multiplied by their own weight and transmitted via the transmit antenna of BS [23, 28].

$$\begin{aligned} \mathbf{X }^\mathrm{BEAM}=\sum ^K_{k=1}\mathbf{w }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k, \end{aligned}$$
(3)

where \({\mathrm{p}}\) denotes the transmit power of BS, \(a_{k}\) is the power allocation coefficient of the User k, and \({\mathbf{S }}_k=[s_{\mathrm{k1}},~s_{\mathrm{k2}}]^{\mathrm{T}}\) represents the information signal of User k. In order to provide the same quality of service (QoS) for every user, we assume that \(a_{k}\) is inversely proportional to the channel gain and subject to \(\sum _{k=1}^{K} a_{k} =1\). The power allocation coefficient is proposed to calculate by

$$\begin{aligned} a_k=\frac{\sum \nolimits _{i,j=1}\limits ^{K}\prod \limits ^{i,j\ne k}\Omega _i\Omega _j}{\sum \nolimits _{i,j=1}\limits ^{K}\prod \Omega _i\Omega _j}. \end{aligned}$$
(4)

In spite of application of ZFBF method, the received signal at the User i still includes the signal of others because of overlap of beams.

$$\begin{aligned} \mathbf{Y }_i^{\mathrm{BEAM}}&=\mathbf{H }_i\mathbf{X }^{\mathrm{BEAM}}+\mathbf{n }_i \nonumber \\&=\mathbf{H }_i\sum ^K_{k=1}\mathbf{w }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k+\mathbf{n }_i \nonumber \\&=\underbrace{\mathbf{H }_i\mathbf{w }_i\sqrt{a_{i} {\mathrm{p}}} \mathbf{S }_i}_{\text{ Desired } \text{ signal } \text{ of } \text{ User } {i} }+\underbrace{\mathbf{H }_i\sum _{k=1,\ne i}^K\mathbf{w }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k}_{\text {Interference signals}}+\mathbf{n }_i, \end{aligned}$$
(5)

where \(\mathbf{n }_i \in \sigma _i^{2}\mathbb {C}^{{{\mathrm{2}} \times 1}}\) denotes the thermal noise vector at the User i, and \(\sigma _i^{2}\) represents the variance of noise vector. We assumed that the variance of noise vector for all users is the same, \(\sigma _1^2=\sigma _2^2 =\cdots =\sigma _K^2 =\sigma ^2\).

As defined in (2), the weight depends on the channel response matrix and \(\mathbf{H }_i\mathbf{w }_k=\mathbf{0 }~(~i\ne k), ~\mathbf{H }_i\mathbf{w }_i=\mathbf{I }\) in case the users are far from each other. However, in this work, we assumed that the users are close to each other. Therefore, \(\gamma _{ik},~0\le \gamma _{ik}\le 1\), is defined as the overlap factor of beams of Users i and k. \(\gamma _{ik}=1\) means that the User i locates on the center line of the beam for the User k. The \(\gamma _{ik}\) decreases when they move forward to the edge of the other’s beam, and \(\gamma _{ik}=0\) when they are located out site of the other’s beam. Moreover, the CSIT is outdated, and then, the first term in (5) is represented by

$$\begin{aligned} \mathbb {E}\{\mathbf{H }_i\mathbf{w }_i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i\}=\rho ^2\Omega _i a_{i} {\mathrm{p}}+(1-\rho ^2)\Omega _i a_{i} {\mathrm{p}}. \end{aligned}$$
(6)

In the right side of (6), the first term is the useful one which is used to detect the designed signal, whereas the second term is the interference one which is established due to the estimation error of channel matrix. The interference term in (5) is depicted as

$$\begin{aligned} \mathbb {E}\{\mathbf{H }_i\sum _{k=1,\ne i}^K\mathbf{w }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k\}=\sum _{k=1,\ne i}^K\left( (1-\rho ^2)+\gamma _{ik}\rho ^2\right) \Omega _i a_{k} {\mathrm{p}}. \end{aligned}$$
(7)

The term of \((1-\rho ^2)\) is due to channel estimation error, and the term of \(\gamma _{ik}\rho ^2\) is due to the overlap of beams of Users i and k.

As a result, the signal-to-interference-plus-noise ratio (SINR) of User i is depicted as

$$\begin{aligned} {\mathrm{SINR}}_{i}^{\mathrm{BEAM}}=&\frac{\rho ^2\Omega _i a_{i} {\mathrm{p}}}{(1-\rho ^2)\Omega _i a_{i} {\mathrm{p}}+\sum \nolimits _{k=1,\ne i}^K\left( 1-\rho ^2+\gamma _{ik}\rho ^2\right) \Omega _i a_{k} {\mathrm{p}}+\sigma ^2} \nonumber \\ =&\frac{\rho ^2\Omega _i a_{i} \delta }{(1-\rho ^2)\Omega _i a_{i} \delta +\sum \nolimits _{k=1,\ne i}^K\left( 1-\rho ^2+\gamma _{ik}\rho ^2\right) \Omega _i a_{k} \delta +1}, \end{aligned}$$
(8)

where \(\delta =\frac{{\mathrm{p}}}{\sigma ^2}\).

NOMA method

In the system utilizing NOMA methods, the BS provides multiple access base on power domain. The higher transmit power is allocated for the user of low channel gain, and the lower transmit power is allocated for the user of high channel gain. We assume that the power correlation coefficient in (4) is applied into the NOMA system. The BS simultaneously broadcasts information signals to K users with different power levers [23, 24].

$$\begin{aligned} \mathbf{X }^{\mathrm{NOMA}}=\sum ^K_{k=1}\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k. \end{aligned}$$
(9)

The user i receives the information signal of all users [23, 24].

$$\begin{aligned} \mathbf{Y }_i^{\mathrm{NOMA}}&=\mathbf{H }_i\sum ^K_{k=1}\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k+\mathbf{n }_i \nonumber \\&=\underbrace{\mathbf{H }_i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i}_{\text{ Desired } \text{ signal } \text{ of } \text{ User } {i} }+\underbrace{\mathbf{H }_i\sum ^K_{k=1,\ne i}\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k}_{\text{ Interference } \text{ signals } }+\mathbf{n }_i . \end{aligned}$$
(10)

According to the principle of NOMA, the power allocation coefficient is inversely proportional to the channel gain, i.e., \(a_1> a_2>\cdots >a_K\), and hence, the User i firstly utilizes the SIC operation to cancel the signal of users 1, 2, \(\cdots\), \(i-1\) and then detects its own signal while considering the signal of users \(i+1\), \(\cdots\), K as interference. However, because of the outdated CSIT, the interference from users 1, 2, \(\cdots\), \(i-1\) is incompletely cancelled. Therefore, the SINR of User i is represented as

$$\begin{aligned} {\mathrm{SINR}}_{i}^{\mathrm{NOMA}}=\frac{\rho ^2a_i\Omega _i\delta }{\sum \nolimits ^{i-1}_{k=1}(1-\rho ^2)a_k\Omega _i\delta +\sum \limits ^K_{k=i+1}a_k\Omega _i\delta +1} \end{aligned}$$
(11)

Proposal of novel signal processing method

In order to cancel the interference signal, we propose the novel signal processing method which combines the precoding and equalization, furthermore, utilizes a linear signal processing at the user site. Our method can extract the desired signal of every user without the interference signal. However, the CSIT is assumed to be outdated, and the residual interference at the user in the proposed system should be investigated. The detail of proposed method for the desired User i while considering the signal for other users as the interference signal is explained below.

The structure of proposed method

The proposed signal processing method is illustrated in Fig. 2, and information signals of all users firstly are multiplied by the prepared precoding matrix, \({\mathbf{P }}_k\), and then broadcast. Hence, the transmit signal at the transmit antenna of BS is represented by

$$\begin{aligned} \mathbf{X }=\sum ^K_{k=1}\mathbf{X }_k=\sum ^K_{k=1}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k, \end{aligned}$$
(12)

and, the received signal of User i is depicted as

$$\begin{aligned} \mathbf{Y }_i&=\mathbf{H }_i\sum ^K_{k=1}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k+\mathbf{n }_i,\nonumber \\&=\underbrace{\mathbf{H }_i\mathbf{P }_i\sqrt{a_{i} {\mathrm{p}}} \mathbf{S }_i}_{\text {Desired signal of User } {i} }+\underbrace{\mathbf{H }_i\sum ^K_{k=1,\ne i}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k}_{\text {Interference signals} }+\mathbf{n }_i, \end{aligned}$$
(13)
Fig. 2
figure 2

The structure of proposed method of combined precoding and equalization

Without loss of generality, we are going to explain the proposed method at the User i while considering the signal of other users as the interference signal.

Designing equalization orthogonal to precoding

As shown in Fig. 2, after being multiplied by \(\mathbf{H }_i^{-1}\) to cancel the effect of propagation channel, the received signal of User i in (13) is processed parallelly to cancel the signal of the others. In the first stream, the received signal is multiplied by \(\mathbf{P }_{\mathrm{i}}^{-1}\) to eliminate the precoding process, and we have

$$\begin{aligned} \mathbf{R }_1&\equiv \left[ \begin{array}{c}{\mathrm{r}}_{\mathrm{11}} \\ {{ \mathrm r}}_{\mathrm{12}} \end{array} \right] = (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{Y }_i, \nonumber \\&=\Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i + \Omega _i\mathbf{P }^{-1} _i\sum ^K_{k=1,\ne i}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k + (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i. \end{aligned}$$
(14)

The term of \(\sum ^K_{k=1,\ne i}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k\) represents by an \(2\times 1\) vector, and hence, we can rewrite it in another form which includes an amplification factor \(\Gamma\), and a vector \({\varvec{\tilde{{\mathrm{S}}}}}= \left[ {\tilde{s}}_{\mathrm{1}} ~~ {\tilde{s}}_{\mathrm{2}} \right] ^T \in \mathbb {C}^{\mathrm{2\times 1}}\).

$$\begin{aligned} \Gamma {\varvec{\tilde{{\mathrm{S}}}}}\equiv \sum ^K_{k=1,\ne i}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k. \end{aligned}$$
(15)

To notice that \(\mathbb {E}\{\sum \limits ^K_{k=1,\ne i}\mathbf{P }_k\sqrt{a_{k}{\mathrm{p}}}{\mathbf{S }}_k\}=\sum \limits ^K_{k=1,\ne i}{a_{k} {\mathrm{p}}}\). Consequently, Eq. (14) is rewritten as

$$\begin{aligned} \mathbf{R }_1 =\Omega _i\sqrt{a_{i} {{\mathrm{p}}}} \mathbf{S }_i + \Omega _i\mathbf{P }^{-1} _i \Gamma {\varvec{\tilde{{\mathrm{S}}}}}+ (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i. \end{aligned}$$
(16)

As shown in (13), the second term is the interference signal that should be eliminated. In order to cancel the interference signal, in the second stream, the received signal is multiplied by two equalization matrices, \(\mathbf{E }_{\mathrm{1}}\in \mathbb {C}^{{{\mathrm{2}}\times 2}}\) and \(\mathbf{E }_{\mathrm{2}}\in \mathbb {C}^{\mathrm{2\times 2}}\), which are designed to be orthogonal to the precoding matrix of User i, i.e. \({\mathbf{E }}_{\mathrm{2}} \mathbf{E }_{\mathrm{1}}\mathbf{P }_i=0\).

The detail of designing these two equalization matrices is explained as follows. Let

$$\begin{aligned} \mathbf{P }_i = \begin{bmatrix} {\mathrm{p}}_{\mathrm{11}} &{} {\mathrm{p}}_{\mathrm{12}} \\ {\mathrm{p}}_{\mathrm{21}} &{} {\mathrm{p}}_{\mathrm{22}}\end{bmatrix}. \end{aligned}$$
(17)

To create the matrix \(\mathbf{E }_{\mathrm{1}}\) subject to

$$\begin{aligned} \mathbf{E }_{\mathrm{1}}(\mathrm{1})\left[ \begin{array}{c}{\mathrm{p}}_{\mathrm{11}} \\ {{ \mathrm p}}_{\mathrm{21}} \end{array} \right] =0,\nonumber \\ \mathbf{E }_{\mathrm{1}}(\mathrm{2})\left[ \begin{array}{c}{\mathrm{p}}_{\mathrm{11}} \\ {\mathrm{p}}_{\mathrm{21}} \end{array} \right] =0. \end{aligned}$$
(18)

Consequently,

$$\begin{aligned} \mathbf{E }_{\mathrm{1}} \mathbf{P }_i= \begin{bmatrix} 0 &{} {{\hat{p}}}_{\mathrm{12}} \\ 0 &{} {{\hat{p}}}_{\mathrm{22}}\end{bmatrix}. \end{aligned}$$
(19)

And then, to create \(\mathbf{E }_{\mathrm{2}}\) subject to

$$\begin{aligned} \mathbf{E }_{\mathrm{2}}(\mathrm{1})\left[ \begin{array}{c} {{\hat{p}}}_{\mathrm{12}} \\ {{\hat{p}}}_{\mathrm{22}} \end{array} \right] =0,\\ \nonumber \mathbf{E }_{\mathrm{2}}(\mathrm{2})\left[ \begin{array}{c} {{\hat{p}}}_{\mathrm{12}} \\ {{\hat{p}}}_{\mathrm{22}} \end{array} \right] =0. \end{aligned}$$
(20)

An example of the matrices is presented by

$$\begin{aligned} \mathbf{E }_{\mathrm{1}}\equiv & {} \frac{1}{\sqrt{p_{\mathrm{21}}^2 +{\mathrm{p}}_{\mathrm{11}}^2}}\begin{bmatrix} {\mathrm{-p}}_{\mathrm{21}} &{}{\mathrm{p}}_{\mathrm{11}} \\ {\mathrm{-p}}_{\mathrm{21}} &{} {\mathrm{p}}_{\mathrm{11}}\end{bmatrix}.\nonumber \\ \mathbf{E }_{\mathrm{2}}\equiv & {} \frac{1}{\sqrt{2}}\begin{bmatrix} -1 &{} 1 \\ -1 &{} 1 \end{bmatrix}. \end{aligned}$$
(21)

Since the matrix \({\mathbf{E }}={\mathbf{E }}_{\mathrm{2}} \mathbf{E }_{\mathrm{1}}\) is orthogonal to \(\mathbf{P }_i\), we have

$$\begin{aligned} {\mathbf{R }}_{2}\equiv & {} \mathbf{E }_{\mathrm{2}} \mathbf{E }_{\mathrm{1}} \mathbf{H }_i^{-1} \mathbf{Y }_i, \nonumber \\= & {} \mathbf{E }_{\mathrm{2}} \mathbf{E }_{\mathrm{1}} \mathbf{H }_i^{-1}\left( \mathbf{H }_i\mathbf{P }_i\sqrt{a_{i} {\mathrm{p}}} \mathbf{S }_i+\mathbf{H }_i\sum ^K_{k=1,\ne i}\mathbf{P }_k\sqrt{a_{k} {\mathrm{p}}} {\mathbf{S }}_k+\mathbf{n }_i\right) ,\nonumber \\= & {} \mathbf{E }_{\mathrm{2}} \mathbf{E }_{\mathrm{1}} \left( \mathbf{P }_i\Omega _i\sqrt{a_{i} \mathrm{p}} {\mathbf{S }}_i+\Omega _i\Gamma {\varvec{\tilde{{\mathrm{S}}}}}+\mathbf{H }_i^{-1}\mathbf{n }_i\right) ,\nonumber \\= & {} \mathbf{E }_{\mathrm{2}} \mathbf{E }_{\mathrm{1}}\Omega _i\Gamma {\varvec{\tilde{{\mathrm{S}}}}}+\mathbf{E }_{\mathrm{2}} E_{\mathrm{1}}\mathbf{H }_i^{-1}\mathbf{n }_i,\nonumber \\= & {} {\mathbf{E }}\Omega _i\Gamma {\varvec{\tilde{{\mathrm{S}}}}}+{\varvec{\tilde{{\mathrm{n}}}}}, \end{aligned}$$
(22)

where \({\varvec{\tilde{{\mathrm{n}}}}}=\mathbf{E }_{{\mathrm{2}}} E_\mathrm{1}\mathbf{H }_i^{-1}\mathbf{n }_i\).

Linear signal processing for interference cancellation

After that, the \(\mathbf{R }_{2}\) signal is controlled linearly by four weights, \(q_{ij},~i,j\in \{1,2\}\), as shown in Fig. 2.

$$\begin{aligned} k_{11}&=q_{11}\mathbf{R }_{2}(\mathrm{1}), ~~ k_{12}=q_{12}\mathbf{R }_{2}(\mathrm{2}),\\ k_{21}&=q_{21}\mathbf{R }_{2}(\mathrm{1}), ~~ k_{22}=q_{22}\mathbf{R }_{2}(\mathrm{2}). \end{aligned}$$

Therefore, we have

$$\begin{aligned} {{\hat{y}}}_{11}\equiv & {} r_{11}+k_{11}+k_{12},\nonumber \\= & {} \mathbf{R }_1(1)+q_{11}\mathbf{R }_{2}(\mathrm{1})+q_{12}\mathbf{R }_{2}(\mathrm{2}),\nonumber \\= & {} \{\Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i + \Omega _i\mathbf{P }^{-1} _i \Gamma {\varvec{\tilde{\mathrm{S}}}}+ (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i\}({ \mathrm 1}) \nonumber \\&+ q_{11}\{{\mathbf{E }}\Omega _i\Gamma {\varvec{\tilde{\mathrm{S}}}}+{\varvec{\tilde{\mathrm{n}}}}\}(\mathrm{1})+q_{12}\{{\mathbf{E }}\Omega _i \Gamma {\varvec{\tilde{\mathrm{S}}}}+{\varvec{\tilde{{\mathrm{n}}}}}\}(\mathrm{2}),\nonumber \\= & {} \Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i (\mathrm{1}) + \Omega _i\Gamma \left( \mathbf{P }^{-1}_i {\varvec{\tilde{{\mathrm{S}}}}}(1) + q_{11}{\mathbf{E }} {\varvec{\tilde{{\mathrm{S}}}}}(1)+q_{12}{\mathbf{E }} {\varvec{\tilde{{\mathrm{S}}}}}(2)\right) \nonumber \\&+ (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i(\mathrm{1})+q_{11}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{1})+q_{12}{\varvec{\tilde{{\mathrm{n}}}}}(\mathrm{2}) \end{aligned}$$
(23)

The term of \(\Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i (\mathrm{1})\) is the designed signal which should be detected, the term of \((\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i(\mathrm{1})+q_{11}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{1})+q_{12}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{2})\) is the noise, and the remained term in the right site of (23) is the interference signal. It is completely removed when the linear weights \(q_{11}\) and \(q_{12}\) satisfy with following condition for any \({\varvec{\tilde{{{\mathrm{S}}}}}}\).

$$\begin{aligned} \Omega _i\Gamma \left( \mathbf{P }^{-1}_i {\varvec{\tilde{\mathrm{S}}}}(1) + q_{11}{\mathbf{E }} {\varvec{\tilde{\mathrm{S}}}}(1)+q_{12}{\mathbf{E }} {\varvec{\tilde{\mathrm{S}}}}(2)\right) =0. \end{aligned}$$
(24)

Detailing (24), the metric can be expressed as

$$\begin{aligned} \begin{bmatrix} {\tilde{p}}_{\mathrm{11}} {\tilde{p}}_{\mathrm{12}} \\ {\tilde{p}}_{\mathrm{21}} {\tilde{p}}_{\mathrm{22}}\end{bmatrix} \left[ \begin{array}{c} {\tilde{s}}_{\mathrm{1}} \\ {\tilde{s}}_{\mathrm{2}} \end{array} \right] \!(1) + q_{11} \begin{bmatrix} E_{\mathrm{11}} E_{\mathrm{12}} \\ E_{\mathrm{21}} E_{\mathrm{22}}\end{bmatrix} \left[ \begin{array}{c} {\tilde{s}}_{\mathrm{1}} \\ {\tilde{s}}_{\mathrm{2}} \end{array} \right] \!(1) +{\mathrm{q}}_{12} \begin{bmatrix} E_{\mathrm{11}} E_{\mathrm{12}} \\ E_{\mathrm{21}} E_{\mathrm{22}}\end{bmatrix} \left[ \begin{array}{c} {\tilde{s}}_{\mathrm{1}} \\ {\tilde{s}}_{\mathrm{2}} \end{array} \right] \!(2)\!=\!0, \end{aligned}$$
(25)

where \(\mathbf{P }^{-1}_i\equiv \begin{bmatrix} {\tilde{p}}_{\mathrm{11}} &{} {\tilde{p}}_{\mathrm{12}} \\ {\tilde{p}}_{\mathrm{21}} &{} {\tilde{p}}_{\mathrm{22}}\end{bmatrix}\) and \(\mathbf{E }\equiv \begin{bmatrix} E_{\mathrm{11}} E_{\mathrm{12}} \\ E_{\mathrm{21}} E_{\mathrm{22}}\end{bmatrix}\). After some manipulations, we have

$$\begin{aligned} ( {\tilde{p}}_{11}\!+\!q_{11}E_{11}\!+\!q_{12}E_{21}){\tilde{s}}_\mathrm{1}+ ( {\tilde{p}}_{12}\!+\!q_{11}E_{12}\!+\!q_{12}E_{22}){\tilde{s}}_\mathrm{2}\!=\!0. \end{aligned}$$
(26)

Because the above equation should be satisfied for all \({\tilde{s}}_{\mathrm{1}}\) and \({\tilde{s}}_{\mathrm{2}}\), it is separated into two equations.

$$\begin{aligned} {\tilde{p}}_{11}+q_{11}E_{11}+q_{12}E_{21}= & {} 0, \nonumber \\ {\tilde{p}}_{12}+q_{11}E_{12}+q_{12}E_{22}= & {} 0. \end{aligned}$$
(27)

Consequently,

$$\begin{aligned} q_{11}= & {} \frac{{\tilde{p}}_{12}E_{21}-{\tilde{p}}_{11}E_{22}}{E_{11}E_{22}-E_{12}E_{21}}, \nonumber \\ q_{12}= & {} \frac{{\tilde{p}}_{12}E_{11}-{\tilde{p}}_{11}E_{12}}{E_{21}E_{12}-E_{22}E_{11}}. \end{aligned}$$
(28)

From (23)and (24), we have the first signal of User i without the interference signal.

$$\begin{aligned} {{\hat{y}}}_{11}=\Omega _i\sqrt{a_{i} {\mathrm{p}}} s_{11}+ (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i(\mathrm{1})+q_{11}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{1})+q_{12}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{2}). \end{aligned}$$
(29)

Similar to calculation of \({{\hat{y}}}_{11}\), the \({{\hat{y}}}_{12}\) is represented by

$$\begin{aligned} {{\hat{y}}}_{12}\equiv & {} r_{12}+k_{21}+k_{22},\nonumber \\= & {} \mathbf{R }_1(2)+q_{21}\mathbf{R }_{2}(\mathrm{1})+q_{22}\mathbf{R }_{2}(\mathrm{2}),\nonumber \\= & {} \{\Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i + \Omega _i\mathbf{P }^{-1} _i \Gamma {\varvec{\tilde{\mathrm{S}}}}+ (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i\}(\mathrm{2}) \nonumber \\&+ q_{21}\{{\mathbf{E }}\Omega _i\Gamma {\varvec{\tilde{\mathrm{S}}}}+{\varvec{\tilde{\mathrm{n}}}}\}(\mathrm{1})+q_{22}\{{\mathbf{E }}\Omega _i \Gamma {\varvec{\tilde{\mathrm{S}}}}+{\varvec{\tilde{\mathrm{n}}}}\}(\mathrm{2}),\nonumber \\= & {} \Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i (\mathrm{2}) + \Omega _i\Gamma \left( \mathbf{P }^{-1}_i{\varvec{\tilde{\mathrm{S}}}}(2) + q_{21}{\mathbf{E }} {\varvec{\tilde{\mathrm{S}}}}(1)+q_{22}{\mathbf{E }}{\varvec{\tilde{\mathrm{S}}}}(2)\right) \nonumber \\&+ (\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i(\mathrm{2})+q_{21}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{1})+q_{22}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{2}). \end{aligned}$$
(30)

Therefore, the \({{\hat{y}}}_{12}\) can be obtained without the interference signal under the following condition.

$$\begin{aligned} \mathbf{P }^{-1}_i{\varvec{\tilde{\mathrm{S}}}}(2) + q_{21}{\mathbf{E }} {\varvec{\tilde{\mathrm{S}}}}(1)+q_{22}{\mathbf{E }}{\varvec{\tilde{\mathrm{S}}}}(2)=0. \end{aligned}$$
(31)

The noise component is assumed to be uncorrelated with the other, by applying the average operation for the received signal in (29), and the SNR of User i is described by

$$\begin{aligned} {\mathrm{SNR}}_{i}=\frac{\mathbb {E}\{\Omega _i\sqrt{a_{i} {\mathrm{p}}} s_{11}\}}{\mathbb {E}\{(\mathbf{H }_i\mathbf{P }_i)^{-1}\mathbf{n }_i(\mathrm{1})+q_{11}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{1})+q_{12}{\varvec{\tilde{\mathrm{n}}}}(\mathrm{2})\}}=\frac{\Omega _i^2a_{i} {\mathrm{p}}}{\sigma ^2 (1+{\mathrm{q}}_{11}^2+{\mathrm{q}}_{12}^2)}. \end{aligned}$$
(32)

However, in the case of outdate CSIT, the \(\mathbf{R }_1\) in (16) still has the inseparable term, \({(1-\rho ^2)}\Omega _i\sqrt{a_{i} {\mathrm{p}}}\mathbf{S }_i\), and it becomes an interference signal. Similarly, in \({\mathbf{R }}_{2}\) of (22), \(\mathbf{P }_i\Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i\) is not clearly cancelled by the condition of \({\mathbf{E }}_{\mathrm{2}} \mathbf{E }_\mathrm{1}\mathbf{P }_i=0\), and hence, the term of \((1-\rho ^2)\mathbf{P }_i\Omega _i\sqrt{a_{i} {\mathrm{p}}} {\mathbf{S }}_i\) is still remained. Moreover, \(\Omega _i\mathbf{P }^{-1} _i \Gamma {\varvec{\tilde{{\mathrm{S}}}}}\) of \(\mathbf{R }_1\) and \(\mathbf{E }\Omega _i\Gamma {\varvec{\tilde{{\mathrm{S}}}}}\) of \(\mathbf{R }_2\) are not removed completely, apart of them, i.e., \({(1-\rho ^2)}\Omega _i\mathbf{P }^{-1} _i \Gamma {\varvec{\tilde{{\mathrm{S}}}}}\) and \({(1-\rho ^2)}\mathbf{E }\Omega _i\Gamma {\varvec{\tilde{{\mathrm{S}}}}}\), is added into the received signal as interference. Consequently, the total of interference signal due to the outdated CSIT is represented as \((1-\rho ^2)\Omega _i\sqrt{a_{i} {{\mathrm{p}}}}(\mathbf{I }+\mathbf{P }_i)\mathbf{S }_i+ {(1-\rho ^2)}\Omega _i\Gamma (\mathbf{P }^{-1} _i +\mathbf{E }){\varvec{\tilde{{\mathrm{S}}}}}\).

As a result, the SINR of User i in the case of outdated CSIT is represented by

$$\begin{aligned} {\mathrm{SINR}}_{i}&=\frac{\rho ^4\Omega _i^2a_{i} {\mathrm{p}}}{{2(1-\rho ^2)^2}\Omega _i^2a_{i} {\mathrm{p}}+ 2{(1-\rho ^2)^2}\Omega _i^2\sum \limits ^K_{k=1,\ne i}{a_{k} {\mathrm{p}}} + (1+q_{11}^2+ q_{12}^2)\sigma ^2},\nonumber \\&=\frac{\rho ^4\Omega _i^2a_{i} {\mathrm{p}}}{2{(1-\rho ^2)^2}\Omega _i^2\sum \limits ^K_{k=1}{a_{k} \mathrm{p}} + (1+q_{11}^2+ q_{12}^2)\sigma ^2},\nonumber \\&=\frac{\rho ^4\Omega _i^2a_{i} {\mathrm{p}}}{2{(1-\rho ^2)^2}\Omega _i^2{\mathrm{p}} + (1+q_{11}^2+ q_{12}^2)\sigma ^2}. \end{aligned}$$
(33)

Moreover, from the (24), we have

$$\begin{aligned} \mathbb {E}\{q_{11}{\mathbf{E }}_1 {\mathbf{S }}_2(1)+q_{12}{\mathbf{E }}_1 \mathbf{S }_2(2)\}=\mathbb {E}\{\mathbf{P }^{-1}_1\mathbf{P }_2 {\mathbf{S }}_2(2)\}. \end{aligned}$$
(34)

To notice that all matrices are normalized, the component of information signal is uncorrelated with the other, and hence

$$\begin{aligned} q_{11}^2+q_{12}^2=1. \end{aligned}$$
(35)

Thus, the SINR in (33) is rewritten as

$$\begin{aligned} {\mathrm{SINR}}_{i}&=\frac{\rho ^4\Omega _i^2a_{i} {\mathrm{p}}}{2{(1-\rho ^2)^2}\Omega _i^2 {\mathrm{p}} + 2\sigma ^2},\nonumber \\&=\frac{\rho ^4\Omega _i^2a_{i} \delta }{2{(1-\rho ^2)^2}\Omega _i^2 \delta + 2}. \end{aligned}$$
(36)

The channel capacity of users is calculated by the Shannon theory.

$$\begin{aligned} C_i=\log _2(1+{\mathrm{SINR}}_{i})~~ [{\mathrm{bit/s/Hz}}]. \end{aligned}$$
(37)

Calculation results and discussion

In this section, we investigate the performance of proposed method via channel capacity based on several parameters and compare to the NOMA and beamforming methods. Their channel capacity is calculated by (37), whereas the SINR of beamforming, NOMA and proposed methods is, respectively, calculated by (8), (11) and (36). For simplicity, we assume that the factors \(\rho\) and \(\gamma _{ij}\) are the same for all users, \(\gamma _{ij}=\gamma\) for all i,  j. The parameters of system are changed in turn in order to investigate their impact on the performance of system while fixing the other parameters. The initial parameters are set as follows: \(K=4\), \(\Omega _K=0\) dB, \(\Omega _k=\Omega _{k+1}-1\) for \(k \in \{1,\ldots ,K-1\}\), \(\rho =0.8,~ \gamma =0.9\).

Comparison of channel capacity based on \(\delta\)

The channel capacity of all methods via \(\delta\) with above mentioned parameters is represented in Fig. 3. The channel capacity is increased when the \(\delta\) increases; however, it is saturated with high \(\delta\). The reason is that when the \(\delta\) increases, the designed signal power is increased; however, the interference signal power is also increased. Therefore, the SINR and the channel capacity become saturated. Compared with the NOMA and beamforming methods, the proposed method outperforms, especially in high \(\delta\) regime. It confirms the advantage of our novel method that can remove the interference signal by the orthogonal feature and the linear signal processing.

Fig. 3
figure 3

The channel capacity via the number of users in the case of \(\Omega _K= 0\,\hbox {dB}\), \(K=4\)

For the proposed method, in low \(\delta\) regime, the received signal of farther users is weak; hence, their channel capacity is low. However, in high \(\delta\) regime, the designed signal of farther users and the interference signal of nearer users are increased. Consequently, the SINR of farther users is improved considerably, whereas that of nearer users is improved slowly following the increase in \(\delta\). It is the reason why the channel capacity of User 1 increases rapidly and be the highest from 15 dB of the \(\delta\). The beamforming method is similar, the far user has the higher transmit power, and it interferes with the near user due to the overlap of beamforming. In contrast to both above methods, in the case of NOMA method, the User 1 detects its own signal while considering the signal of other users as interference. Whereas, the nearer user detects its own signal by applying SIC operation to remove the signal of other users. However, the SIC operation is imperfect because of the outdated CSIT. Therefore, the signal of other users is still remained as interference. To compare to the remaining interference caused by imperfect SIC operation, the power of nearer user signal is higher, and hence, the SINR of nearer user is higher.

Comparison of channel capacity based on K

The investigation of channel capacity with \(K = 8\) is shown in Fig. 4 which includes the channel capacity of Users 2, 4, 6 and 8 of three methods. The system model is similar to the model of Fig. 3, and the result is also similar. However, the saturation value is different from the case of \(K = 4\). The reason is that the transmit power of every user is changed according to the number of users. Furthermore, the interference signal in the case of \(K = 8\) should include the signal of 7 other users. Therefore, the SINR and the channel capacity in this case are lower.

Fig. 4
figure 4

The channel capacity via the number of users in the case of \(\Omega _K= 0\,\hbox {dB}\), \(K=8\)

Channel capacity of all methods under different channel gain

Figures 5 and 6 illustrate the channel capacity of all methods in the case of \(\Omega _K= 5\,\hbox {dB}\) and − 5 dB, respectively. The increase in \(\Omega _K\) variance means that the channel gain is improved. Therefore, the channel capacity of high \(\Omega _K\) is higher; however, its saturation value is the same when the \(\delta\) is high enough. From the above figures, we can recognize the impact of \(\Omega _K\) on the performance of system. Our proposed method is sensitive to the channel gain, and it outperforms the NOMA and beamforming methods when the channel gain and/or the \(\delta\) are high. However, its channel capacity is rapidly decreased when the channel gain and the \(\delta\) become small.

Fig. 5
figure 5

The change of channel capacity according to \(\Omega _K= 5\,\hbox {dB}\)

Fig. 6
figure 6

The change of channel capacity according to \(\Omega _K= -5\, \hbox {dB}\)

The impact of overlap factor on the channel capacity

We investigate the impact of \(\gamma\) on the channel capacity at \(\delta = 15\,\hbox {dB}\), and the result is shown in Fig. 7. The change of \(\gamma\) affects the performance of only beamforming method. Therefore, the channel capacity of 2 other methods is constant. The reduction of \(\gamma\) means that the overlap of beams is mitigated. Consequently, the interference between users is extenuated, and the channel capacity is improved. As expected, the channel capacity of beamforming method is considerably improved when the \(\gamma\) reduces, and it becomes the highest with \(\gamma =0\). In this scenario, the overlap of beams is absent, and every user is communicated by its individual beam. However, the inseparable term is still remained because of outdated CSIT, and it is considered as interference and lets the SINR of conventional beamforming method down. As a result, the channel capacity of beamforming method is lower than that of proposed method even in the case of \(\gamma =0\). It confirms the advantage of our novel method.

Fig. 7
figure 7

The channel capacity via \(\gamma\) with \(\delta = 15\,\hbox {dB}\), \(\rho =0.8\), \(K=4\)

The effect of outdated CSIT

The effect of factor \(\rho\) on the channel capacity of considered methods is depicted in Figs. 8 and 9 with \(\gamma =0.2\) and 0.9, respectively. To notice that the factor \(\rho\) indicates the accuracy of estimated channel matrix comparing with the true one. From these figures we can say that, to estimate the channel matrix accurately is important, the channel capacity is considerably increased following the increase in factor \(\rho\). Besides, the reduction of \(\gamma\) lets the channel capacity of beamforming method be improved as explained above. However, the channel capacity of 3 methods is low and almost the same for \(\rho \le 0.8\). For \(\rho > 0.8\), the channel capacity of our novel method is increased more rapidly and becomes the highest one in both cases of \(\gamma\).

Fig. 8
figure 8

The channel capacity via \(\rho\) with \(\delta = 15\,\hbox {dB}\) and \(\gamma =0.2\)

Fig. 9
figure 9

The channel capacity via \(\rho\) with \(\delta = 15\,\hbox {dB}\) and \(\gamma =0.9\)

Conclusion and future works

In this work, the novel method is proposed for cancelling the interference of MU-MIMO systems with high density users, and the proposed system is investigated in the case of multiple users and outdated CSIT. The outdated CSIT lets the interference cancellation be incomplete, and it affects the system performance of proposed method. Our novel method is compared with the NOMA and beamforming methods based on several terms, such as the number of users, channel gain, and accuracy of channel matrix estimation. In the case of poor channel gain and low transmit power, the channel capacity of proposed method is slightly lower than that of 2 other methods. However, the proposed method outperforms in other considered scenarios. Our proposed method can provide a high quality of service for multiple users systems with few antennas. Therefore, it can be applied to advanced MU-MIMO systems to improve the channel capacity, especially for the system with compact base station, few antennas and serving many users.

In our future works, the proposed method will be discussed on other criteria, such as complexity, bit error rate and so on. Furthermore, the number of antennas at users was fixed to 2 for accommodating personal devices in MU-MIMO systems. The proposed method will be considered to apply to another system with different scenario, e.g., variable number of antennas at users.

Availability of data and materials

Data used to support the findings of this study are already available in the manuscript.

Abbreviations

BS:

Base station

MU:

Multiple users

MIMO:

Multiple input multiple output

NOMA:

Non-orthogonal multiple access

SIC:

Successive interference cancellation

SINR:

Signal-to-interference-plus-noise ratio

SNR:

signal-to-noise ratio

ZFBF:

Zero forcing beamforming

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Acknowledgements

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Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2017.311

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The contribution of Nguyen Thu Phuong contains methodology, investigation, and writing–review and editing. The contribution of Vu Van Son is investigation, formal analysis, and writing–original draft preparation. The contribution of Pham Thanh Hiep is conceptualization, formal analysis, and writing–original draft preparation.

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Correspondence to Pham Thanh Hiep.

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Phuong, N.T., Son, V.V. & Hiep, P.T. Combining precoding and equalization for interference cancellation in MU-MIMO systems with high density users. J Wireless Com Network 2022, 34 (2022). https://doi.org/10.1186/s13638-022-02118-2

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Keywords

  • Equalization design orthogonal to precoding
  • Multiple users interference cancellation
  • Multiple input multiple output
  • Channel capacity
  • Zero-forcing beamforming algorithm