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EQSM-based multiuser MIMO downlink transmission for correlated fading channels

Abstract

This paper presents three multiuser multiple input-multiple output (MU-MIMO) downlink transmission strategies based on the extended quadrature spatial modulation (EQSM) system for mobile communication. The three MU-MIMO precoding strategies utilised are block diagonalisation (BD), dirty paper coding (DPC), and a combined BD-DPC strategy. We analyse and compare the performance of these three MU-MIMO-EQSM schemes with the conventional MU-MIMO spatial multiplexing (MU-MIMO-SMux) system in terms of bit error rate (BER) and detection complexity considering correlated and uncorrelated fading channels. Results show that the BD-MIMO-EQSM and DPC-MIMO-EQSM systems outperform by 2−3 dB in BER performance their conventional counterparts with the additional advantage of a detection complexity reduction of up to 62% for the analysed cases. For the uncorrelated fading channel, the BD technique has better BER performance for a spectral efficiency (SE) of 12 bits per channel use (bpcu), while the DPC technique has better BER performance for an SE of 8 bpcu. Considering the correlated channel, DPC suffers a deep BER degradation of up to 30 dB compared with BD. However, still in this scenario, both proposed MU-MIMO-EQSM systems outperform their conventional counterparts. In terms of detection complexity, DPC has a complexity reduction of 77% as compared with the BD technique. The BD-DPC-MIMO-EQSM hybrid system heritages the low detection complexity of DPC with the advantage of all users in the system having the same BER performance.

Introduction

Multiple input-multiple output (MIMO) transmission technique has been recognised as a key technology for the implementation of future wireless communication systems [1]. Spatial modulation (SM) is a MIMO transmission technique where the array of transmit (Tx) antennas is considered as a spatial constellation, then, each Tx antenna represents a point in that constellation [2]. Ideally, from each Tx antenna to each receive (Rx) antenna exists a different channel that identifies this path. Therefore, by assuming that wireless channel changes slowly and the channel state information (CSI) is known, the receiver can determine which Tx antenna has been activated at any given time [3]. It has been shown that in comparison with the conventional spatial multiplexing (SMux) techniques, such as vertical-Bell Laboratories Layered Space-Time (V-BLAST), SM provides a better bit error rate (BER) performance and a reduced detection complexity.

A further extension of SM, known as quadrature spatial modulation (QSM), is based on the expansion of the spatial constellation by using the real and imaginary parts of the quadrature amplitude modulation (QAM) symbols as independent information-bearing carriers on the spatial constellation [4] [5]. As a result, QSM has the advantage of doubling the number of bits that can be transmitted in the spatial constellation. Proposals for enhancing the performance of the SM/QSM scheme include for example; precoding [6], trellis code modulation [7], differential spatial modulation [8], antenna selection [9], combined QSM-spatial multiplexing (SMux) [10], and low complexity detection algorithms [11] among others.

These previous research have motivated novel investigations to study the performance of SM/QSM on multiuser MIMO (MU-MIMO) schemes. Recently in [12], a generalised spatial modulation (GSM) MU-MIMO scheme was proposed for the up-link channel, and it is shown that the MU-MIMO-GSM scheme outperforms in BER and/or spectral efficiency (SE) the MU-MIMO-SM and the conventional MU-MIMO systems. In [13], a precoding matrix is used to transmit an MU-SM signal for the downlink transmission. In this case, the total number of Tx antennas is divided into Nu blocks, which are independently modulated for each user by using only one Rx antenna per user. It is shown in this case that the utilised precoding matrix is an effective way to avoid MU interference. An MU-MIMO transmission system that uses SM to transmit additional broadcast information over the conventional MU-MIMO scheme was proposed in [14]. Results in [14] show that some broadcast transmission bits can be added to the conventional MU-MIMO-SMux scheme transmission without performance impairments at the cost of a slightly increased detection complexity. In [15] and [16], the concept of SM at the receiver side was effectively implemented in MU-MIMO systems. The potential applications of the SM/QSM-based schemes for MU-MIMO systems include the design of future networks that demand energy and spectral efficiencies, for example, Internet of Thing (IoT) applications with rate and energy efficiency improvements [17] or beyond 5G systems where the spatial dimension could be used as a second layer to transfer additional information [18] [19].

More recently, an extended QSM (EQSM) transmission system which combines K QSM constellations to boost the SE by a factor of K was proposed. The EQSM system employs the average powers of the QSM constellations as an additional modulation dimension for the transmission of information bits [20]. As a result, EQSM has improved SE and also shows BER performance improvements. However, the EQSM scheme was evaluated only for a Rayleigh channel model and the single-user case.

Motivated by the emerging concept of massive MU-MIMO systems, where spatial correlation is a common scenario, in this paper, two questions related to EQSM are addressed. First, what is the performance of the EQSM scheme in multiuser scenarios? And second, what is the performance of the EQSM transmission scheme for more realistic channels? Considering these questions, in this work, three different downlink MU-MIMO strategies are evaluated and compared for the EQSM transmission scheme. The three MU-MIMO downlink strategies evaluated are block diagonalisation (BD) [14], dirty paper coding (DPC) [21] and a hybrid BD-DPC strategy. The proposed systems are analysed and compared in terms of BER performance and detection complexity versus their conventional counterparts considering two different scenarios and the optimal maximum likelihood (ML) detection criterion.

Results show that for all analysed cases, the proposed BD-MIMO-EQSM and DPC-MIMO-EQSM systems outperform in BER performance and detection complexity of their conventional counterparts. In general, the BD technique has better BER performance while the DPC technique has the lowest detection complexity. For the correlated fading channel, DPC suffers a deep degradation in BER performance compared with BD. However, for the specific case of an SE of 8 bits per channel use (bpcu) and the uncorrelated fading channel, the proposed DPC-MIMO-EQSM system is a better BER performance for three out of four users in the system. In terms of detection complexity, the DPC technique has the lowest detection complexity. The combined BD-DPC-MIMO-EQSM system heritages the low detection complexity of DPC with the advantage of all users in the system having the same BER performance. The main contributions of this paper are as follows:

1. The EQSM transmission scheme is proposed for three different multiuser MIMO systems.

2. The three proposed MU-MIMO-EQSM systems are evaluated with respect to their detection complexity and BER performance for correlated fading channels.

The remainder of this work is organised as follows. In Section 2, the general system model of the MU-MIMO-EQSM scheme is introduced. In Section 3, MU interference cancellation for MU-MIMO-EQSM systems is presented. Section 4 describes the utilised channel model. In Section 5, optimal ML detection is described. Results of detection complexity are offered in Section 6. In Section 7, results of BER performance are analysed and discussed. Finally, in Section 8, we conclude the work.

Notation: Uppercase boldface letters denote matrices whereas lowercase boldface letters denote vectors. The transpose, Hermitian transpose, complex conjugate, and Frobenius norm of A are denoted by AT, AH, A, and AF, respectively. The statistical expectation is represented by E[ ·], and 0 denotes an all-zero matrix (or vector). Finally, \(\mathcal {CN}(0,\sigma ^{2})\) is used to represent the circularly symmetric complex Gaussian distribution with variance σ2.

Methods

The aim of this work is to apply and compare multiuser MIMO strategies for the recently proposed EQSM transmission scheme. First, BD and DPC multiuser strategies are implemented for the EQSM scheme, then a BD-DPC-MIMO-EQSM hybrid scheme is proposed. These three schemes are compared in BER performance using Monte Carlo simulations. The detection complexity of the used algorithms is analysed using floating point operations. We proved the proposed systems utilising correlated and uncorrelated fading channels based on a geometrical model where multipath propagation is caused by the interaction of the transmitted signal with local scatterers randomly located around the users. In order to show the advantage of the proposed schemes, we compared the performance of the proposed systems with the conventional systems based on SMux under the same conditions.

System model

The system model of the proposed MU-MIMO-EQSM downlink transmission scheme is presented in Fig. 1. We consider a base station (BS) with Nt transmit antennas and K independent mobile stations (MS) or users, each one with Nr-receiving antennas. Thus, the end-to-end configuration can be considered as a (K·NrNt downlink MU-MIMO transmission system.

Fig. 1
figure1

General MU-MIMO-EQSM system model. The system is composed of individual EQSM blocks for each user and the precoding stage in the transmitter, a MIMO fading channel and a receiver for each user

The sequence of input bits ak intended for the kth user is first modulated by one EQSM block. Then, the output of all EQSM blocks is fed into to the precoding stage where all signals are combined using DPC or BD techniques in order to cancel inter-user interference. Figure 2 shows the EQSM block. The EQSM block combines two QSM signals to generate an EQSM signal intended for a particular user. In this manner, the spectral efficiency of the EQSM scheme is twice of the QSM scheme.

Fig. 2
figure2

EQSM transmission block. This subsystem combines the output of two QSM basic modulators

The system can transmit mEQSM=2(log2(M)+2 log2(L)) bits in each time slot for each user, where M is the size of the M-ary quadrature amplitude modulation (QAM) constellation \(\mathcal {S}=\{s_{1}, s_{2}, \cdots, s_{M}\}\) and L is the length of the EQSM block.

The mathematical model of the MU-MIMO-EQSM scheme is defined as

$$ \left[\begin{array}{cccc} \mathbf{y}_{1} \\ \mathbf{y}_{2} \\ \vdots \\ \mathbf{y}_{K} \\ \end{array}\right]= \left[\begin{array}{cccc} \mathbf{H}_{1} & \mathbf{H}_{1} &\cdots & \mathbf{H}_{1}\\ \mathbf{H}_{2} & \mathbf{H}_{2} &\cdots & \mathbf{H}_{2}\\ \vdots && \ddots & \vdots \\ \mathbf{H}_{K} & \mathbf{H}_{K} & \cdots& \mathbf{H}_{K} \\ \end{array}\right] \left[\begin{array}{cccc} \mathbf{W}_{1}\tilde{\mathbf{x}}_{1} \\ \mathbf{W}_{2}\tilde{\mathbf{x}}_{2} \\ \vdots \\ \mathbf{W}_{K}\tilde{\mathbf{x}}_{K} \\ \end{array}\right] + \left[\begin{array}{cccc} \mathbf{n}_{1} \\ \mathbf{n}_{2} \\ \vdots \\ \mathbf{n}_{K} \\ \end{array}\right]. $$
(1)

The received signal \(\mathbf {y}_{k} \in \mathbb {C}^{N_{r}\times 1}\) by the kth user is

$$\ \mathbf{y}_{k} = \mathbf{H}_{k} \sum_{i=1}^{K}\mathbf{W}_{i}\tilde{\mathbf{x}}_{i} + \mathbf{n}_{k}, $$
(2)

where \(\mathbf {H}_{k} \in \mathbb {C}^{N_{r}\times N_{t}}\) is the channel matrix between the BS and the kth user, \(\mathbf {n}_{k} \in \mathbb {C}^{N_{r}\times 1}\) is the noise vector at the kth user. In this work, we use Nt=KNr which is a typical restriction for MU-MIMO systems based on BD precoding [14]. The noise samples are assumed to be independent and identically distributed (i.i.d.) with \(\mathcal {CN}(0, 1)\).

Let us consider the overall transmission vector \(\mathbf {x} \in \mathbb {C}^{N_{t}\times 1}\) defined as

$$\ \mathbf{x} = \sum_{i=1}^{K}\mathbf{W}_{i}\tilde{\mathbf{x}}_{i}. $$
(3)

Then, the received signal by the kth user can be represented as

$$\ \mathbf{y}_{k} = \mathbf{H}_{k} \mathbf{x} + \mathbf{n}_{k}. $$
(4)

EQSM transmission block

Each EQSM transmission block generates the signal intended for one particular user (mobile station) in the system. A simple two branches EQSM scheme is mainly used because of its good BER performance and SE. Each QSM branch independently modulates a bit sequence in order to have the double of the SE in the transmission system.

For the kth user, the sequence of bits \(a_{k}=\lbrace b_{n} \rbrace _{n=1}^{m}\), with bn{0,1}, is fed into each EQSM block, whose output vector \(\boldsymbol {\tilde {x}}_{k} \in \mathbb {C}^{L\times 1}\) is represented by

$$ \boldsymbol{\tilde{x}}_{k}= \left[\tilde{x}_{k}^{(1)}, \tilde{x}_{k}^{(2)}, \cdots, \tilde{x}_{k}^{(L)}\right]^{T}. $$
(5)

The QSM output vectors x1 and x2 are weighted by the factors B1=1 and B2=0.5. Then, this signals are combined to generate the EQSM output (Fig. 2). Table 1 shows the EQSM mapping rule for the first 16 values of the input bit sequence. The EQSM output signals are generated as s‚=B1x1+B2x2.

Table 1 Example of EQSM signals generation with L=2

QSM technique

In order to generate the QSM signals xi, the input sequence of bits is divided into three flows. One flow is utilised to modulate a M-QAM signal and the other two flows (spatial bits) are utilised to modulate the position in the output vector as shown in Fig. 3. For an input bit sequence of length mQSM=mEQSM/2, the first log2(M) bits modulate an M-QAM symbol. The remaining 2 log2(L)=mQSM− log2(M) bits are divided into two flows of log2(L) spatial bits. This spatial bits modulate the position in the output vector \(\mathbf {x}\textsc {\char 13}_{i} \in \mathbb {C}^{L\times 1}\) as follows: the real part of the QAM symbol is assigned to one specific position in the output vector. The remaining L−1 positions are set to zero. The imaginary part of the QAM symbol can be assigned to another one or even the same position in the output vector. Finally, these two SM signals are combined to obtain the QSM output vector [4].

Fig. 3
figure3

QSM signal generation. The QSM subsystem combines two SM signals based on the real and the imaginary parts of the transmitted QAM symbol

MU interference cancellation of EQSM signals

In order to avoid the multiuser interference of the EQSM signals, two different strategies of precoding are presented in this subsection. Additionally, these two basic strategies are combined to work together on the MU-MIMO interference cancellation. The two precoding strategies used are based on the CSI knowledge on the transmission side. The first strategy utilised is BD, which is based on SVD matrix decomposition. The other interference cancellation strategy used, known as DPC, is based on LQ matrix decomposition. The main difference is that in the BD technique, all users have the same BER performance while in DPC, all users in the system have different BER performances.

Block diagonalisation

The BD technique is based on the construction of the precoding matrix \(\mathbf {W}_{k} \in \mathbb {C}^{N_{t}\times N_{r}}\). This precoding matrix uses CSI in order to cancel all the interfering signals in the system. As shown in Fig. 4, first, the output vector \(\tilde {\mathbf {x}}_{k}\) intended for the kth user is precoded using the matrix Wk. Then, all these precoded signals are linearly combined to generate the output vector x.

Fig. 4
figure4

BD precoder. The signal intended for each user is precoded by an individual matrix. All precoded signals are then combined in order to generate the transmission vector x

Rearranging terms in (2), the received signal for the kth user can be rewritten as

$$ \mathbf{y}_{k}= \mathbf{H}_{k} \mathbf{W}_{k} \tilde{\mathbf{x}}_{k} + \mathbf{H}_{k}\sum_{i=1,i \neq k}^{K}\mathbf{W}_{i}\tilde{\mathbf{x}}_{i} + \mathbf{n}_{k}. $$
(6)

The first term in (6) is the signal sent to the kth user while the second term is the interference produced by the other users in the system and the third term is the noise. The interference term can be cancelled by the channel if the precoding matrix Wk is designed to satisfy

$$ \mathbf{\overline{H}}_{k} \mathbf{W}_{k}=\mathbf{0},~~~~ k=1,2,\cdots, K, $$
(7)

where the matrix \(\mathbf {\overline {H}}_{k}\) contains all system users’ matrices except that of the kth user. Thus,

$$ \mathbf{\overline{H}}_{k} =\left[ \mathbf{H}_{1}^{H},\cdots, \mathbf{H}_{k-1}^{H},\mathbf{H}_{k+1}^{H},\cdots,\mathbf{H}_{K}^{H} \right]^{H}. $$
(8)

In this manner, the received signal in (6) is reduced to

$$ \mathbf{y}_{k}=\mathbf{H}_{k} \mathbf{W}_{k} \tilde{\mathbf{x}}_{k} + \mathbf{n}_{k}, $$
(9)

which is an interference-free signal.

The matrix Wk is obtained decomposing \(\mathbf {\overline {H}}_{k}\) into its singular values as

$$ \mathbf{\overline{H}}_{k}= \mathbf{U}_{k} \left[ \mathbf{\Sigma}_{k},\mathbf{0} \right] \left[\! \mathbf{V}_{k}^{(1)} \mathbf{V}_{k}^{(0)} \right]^{H}, $$
(10)

where Uk is a unitary matrix, Σk is a diagonal matrix containing the non-negative singular values of \(\mathbf {\overline {H}}_{k}\) with dimension equals to the rank of \(\mathbf {\overline {H}}_{k}\) and 0 is an all-zero matrix. \(\mathbf {V}_{k}^{(1)}\) contains vectors corresponding to the nonzero singular values and \(\mathbf {V}_{k}^{(0)}\) contains vectors corresponding to the zero singular values. The matrix \(\mathbf {V}_{k}^{(0)}\) contains the last Nr columns of Vk, which form an orthogonal basis that is in the null space of \(\mathbf {\overline {H}}_{k}\) and can be used as the precoding matrix Wk.

The BD-MIMO-EQSM system can be mathematically modelled as

$$ {}\left[\begin{array}{cccc} \mathbf{y}_{1} \\ \mathbf{y}_{2} \\ \vdots \\ \mathbf{y}_{K} \\ \end{array}\right]= \left[\begin{array}{cccc} \mathbf{H}_{1}\mathbf{W}_{1} & \mathbf{0} &\cdots & \mathbf{0}\\ \mathbf{0} & \mathbf{H}_{2}\mathbf{W}_{2} &\cdots & \mathbf{0}\\ \vdots &\vdots& \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots& \mathbf{H}_{K} \mathbf{W}_{K} \\ \end{array}\right] \left[\begin{array}{cccc} \tilde{\mathbf{x}}_{1} \\ \tilde{\mathbf{x}}_{2} \\ \vdots \\ \tilde{\mathbf{x}}_{K} \\ \end{array}\right] + \left[\begin{array}{cccc} \mathbf{n}_{1} \\ \mathbf{n}_{2} \\ \vdots \\ \mathbf{n}_{K} \\ \end{array}\right] $$
(11)

Dirty paper coding

The DPC technique is based on the subtraction of the potential interference for each user before transmission. More specifically, the interference due to the first up to (k−1)th user signals is cancelled in the course of precoding the kth user signal [21]. As in the BD technique, DPC can be implemented when channel gains are completely known on the transmitter. In DPC, full channel cancellation can be carried out; however, SM techniques require to have some information in the reception related to the channel, which is used for carrying extra information. Then, in the procedure proposed here, the eigenvalues of matrix H are transmitted along with the QAM symbols.

First, the complete matrix channel \(\mathbf {H} \in \mathbb {C}^{KN_{r}\times N_{t}}\) is LQ decomposed as H=LQ. In matrix form

$$ \left[\begin{array}{cccc} \mathbf{H}_{1} \\ \mathbf{H}_{2} \\ \vdots\\ \mathbf{H}_{K} \\ \end{array}\right]= \left[\begin{array}{cccc} \mathit{l}_{1,1} & 0 & \cdots & 0\\ \mathit{l}_{2,1}& \mathit{l}_{2,2} & \cdots & 0\\ \vdots & & \ddots & 0 \\ \mathit{l}_{KN_{r},1} & \mathit{l}_{KN_{r},2} &\cdots& \mathit{l}_{KN_{r},N_{t}} \\ \end{array}\right] \left[\begin{array}{cccc} \mathit{\mathbf{q}}_{1} \\ \mathit{\mathbf{q}}_{2} \\ \vdots \\ \mathit{\mathbf{q}}_{N_{t}} \\ \end{array}\right], $$
(12)

where L is a lower triangular matrix and Q is an unitary matrix composed by orthogonal row vectors \(\mathbf {q}_{k} \in \mathbb {C}^{1\times N_{r}}\) such as QHQ=QQH=I.

First, the transmission vectors \(\tilde {\mathbf {x}}_{k}\) intended for each user are concatenated to form the vector \(\tilde {\mathbf {x}} \in \mathbb {C}^{1\times N_{r}}\). Then, as shown in Fig. 5, the output vector x is obtained as \(\mathbf {x}=\mathbf {W} \tilde {\mathbf {x}} \). The precoding matrix W for the DPC technique is defined as

$$ \mathbf{W}=\mathbf{Q}^{H} \mathbf{P}, $$
(13)
Fig. 5
figure5

DPC precoder. The signals intended for each user are first concatenated. Then, these signals are multiplied by a general precoding matrix in order to generate the transmission vector x

where the matrix P is a lower triangular matrix defined as

$$ \mathbf{P}= L^{-1} \left[\begin{array}{ccc} \mathit{l}_{1,1} & 0& 0\\ 0& \ddots & 0 \\ 0 & 0& \mathit{l}_{KN_{r},N_{t}} \\ \end{array}\right]. $$
(14)

The matrix QH cancels the orthogonal component of the channel, while the matrix P is intended to deal with the inter-user interference.

Considering the transmission vector \(\mathbf {x}=\mathbf {W}\tilde {\mathbf {x}} = \mathbf {Q}^{H} \mathbf {P}\tilde {\mathbf {x}}\). The complete array of received signals can be mathematically modelled as

$$ \left[\begin{array}{cccc} \mathbf{y}_{1} \\ \mathbf{y}_{2} \\ \vdots \\ \mathbf{y}_{K} \\ \end{array}\right]= \left[\begin{array}{cccc} \mathbf{H}_{1} \\ \mathbf{H}_{2} \\ \vdots\\ \mathbf{H}_{K} \\ \end{array}\right] (\mathbf{Q}^{H} \mathbf{P} \tilde{\mathbf{x}})+ \left[\begin{array}{cccc} \mathbf{n}_{1} \\ \mathbf{n}_{2} \\ \vdots \\ \mathbf{n}_{K} \\ \end{array}\right]. $$
(15)

The received signal for the kth user is

$$ \mathbf{y}_{k}=\mathbf{H}_{k} \mathbf{Q}^{H} \mathbf{P} \tilde{\mathbf{x}} + \mathbf{n}_{k} = \mathbf{D}_{k} \tilde{\mathbf{x}}_{k} + \mathbf{n}_{k}, $$
(16)

where Dk=HkQHP is a diagonal matrix. In matrix form, the received signal for the kth user is

$$ \mathbf{y}_{k}= \left[\begin{array}{cccc} \mathit{l}_{1,1} & 0& \cdots& 0\\ 0 &\mathit{l}_{2,2} & \cdots & 0 \\ \vdots &\vdots & \ddots & \vdots \\ 0& 0& \cdots& \mathit{l}_{N_{r},N_{t}} \\ \end{array}\right] \left[\begin{array}{cccc} \mathit{\mathit{x}}_{1} \\ \mathit{\mathit{x}}_{2} \\ \vdots \\ \mathit{\mathit{x}}_{N_{r}} \\ \end{array}\right] + \left[\begin{array}{cccc} \mathit{\mathit{n}}_{1} \\ \mathit{\mathit{n}}_{2} \\ \vdots \\ \mathit{\mathit{n}}_{N_{r}} \\ \end{array}\right], $$
(17)

which is free from multiuser interference.

BD-DPC-MIMO-EQSM hybrid scheme

In this subsection, an MU-MIMO transmission system which combines BD with DPC for EQSM signals transmission is presented. It has been shown in previous works that the BD technique has good MU interference cancellation properties which result in excellent BER performance [14]. However, the main drawback of the BD technique is its high detection complexity which can be critical mainly in massive MIMO systems. On the other hand, the DPC technique has low detection complexity with relative good to medium BER performance. However, the main drawback of DPC is that all users in the system have different BER performance. Then, in order to reduce the detection complexity and at the same time obtain the same BER performance of all users in the system, a combined BD-DPC-MIMO-EQSM scheme can be implemented. In what follows, the proposed BD-DPC-MIMO-EQSM system is described.

Let us consider the signal received by the kth user using BD precoding,

$$ \mathbf{y}_{k}=\mathbf{Z}_{k} \tilde{\mathbf{x}}_{k} + \mathbf{n}_{k}. $$
(18)

where Zk=HkWk is the utilised precoding.

The received signal in BD technique removes the multiuser interference. However, the inter-antenna interference produced by the MIMO channel is still present on the received signal. In order to reduce the inter-antenna interference, the transmitted signal \(\tilde {\mathbf {x}}_{k}\) can be additionally precoded utilising a simplified version of the DPC strategy.

Utilising QR decomposition, of the matrix \(\mathbf {Z}_{k}^{H}\), the precoded noiseless received signal by the kth user in (18) can be represented as

$$ \mathbf{y}_{k}=\mathbf{Z}_{k} \mathbf{Q(Z)}\mathbf{P(Z)}\tilde{\mathbf{x}}_{k}=\text{diag}(\mathbf{R^{T}})\tilde{\mathbf{x}}_{k}, $$
(19)

where P(Z)=(RT)−1diag(RT). The obtained signal is more simple and therefore has a reduced detection complexity. In Sections 6 and 7, respectively, the detection complexity and BER performance of the three MU-MIMO-EQSM systems is analysed and compared considering two different SE configurations for correlated and uncorrelated fading channels.

Channel model

Following the modelling approach presented in [22, 23], the channel matrix Hk is characterised considering multipath propagation caused by the interaction of the transmitted signal with local scatterers randomly located around the kth MS. In addition, it is assumed that the BS and the MSs are equipped with uniform linear antenna (ULA) arrays. Under these conditions, Hk can be written as

$$ \mathbf{H}_{k} = \sum_{\ell=1}^{L} g_{\ell,k} e^{-j(\theta_{\ell,k} + \psi_{\ell,k})} \mathbf{a}_{r}^{T}(\phi_{\ell}^{r}) \mathbf{a}_{t}(\phi_{\ell}^{t}), $$
(20)

for k{1,2,,K}. In (20), L denotes the number of local scatterers surrounding the kth MS, g,k and θ,k stand for the amplitude attenuation and random phase shift, respectively, resulting from the interaction of the transmitted signal with the th local scatterer. The attenuation factors g,k are deterministic quantities equal to \(g_{\ell,k} = \sqrt {1/L}\), ,k, whereas the phases ψ,k are independent and identically distributed (i.i.d.) random variables having a uniform distribution in the interval [−π,π) [23]. In turn, the phase ψk, is given as

$$ \psi_{\ell,k} = \frac{2\pi}{\lambda} \left(d_{\ell,k}^{t}+d_{\ell,k}^{r}\right), $$
(21)

where \(d_{\ell,k}^{t}\) and \(d_{\ell,k}^{r}\) denote the distances from the BS to the th local scatterer, and from there to the kth MS, respectively, and λ stands for the carrier signal’s wavelength. The distances \(d_{\ell,k}^{t}\) and \(d_{\ell,k}^{r}\) can be determined based on the propagation scenario’s geometrical configuration. In this paper, the distance from the kth MS to its th local scatterer is modelled by a constant quantity equal to \(d_{\ell }^{r} = r_{k}\). On the other hand, the distance from the BS to the th interfering object is given as

$$ d_{\ell}^{t} = \sqrt{D_{k}^{2}+r_{k}^{2}-2rD\cos(\phi_{\ell}^{r})}, $$
(22)

where Dk is the distance from the BS to the kth MS, and \(\phi _{r}^{k}\) is the angle of arrival of the th multipath component of the received signal [23]. Concerning the ULA array vectors \(\mathbf {a}_{r}^{T}(\phi _{\ell }^{t})\) and \(\mathbf {a}_{t}(\phi _{\ell }^{r})\), they are equal to

$$ \mathbf{a}_{t}(\phi_{\ell}^{t}) = \left[\begin{array}{ccc} e^{-j A_{1}^{t} \sin(\phi_{\ell}^{t})}, & \cdots &,e^{-j A_{N_{t}}^{t} \sin(\phi_{\ell}^{t})} \end{array}\right]^{T}, $$
(23)
$$ \mathbf{a}_{r}(\phi_{\ell}^{r}) = \left[\begin{array}{ccc} e^{-j A_{1}^{r} \cos(\phi_{\ell}^{r})}, &\cdots&,e^{-j A_{N_{r}}^{r} \cos(\phi_{\ell}^{r})} \end{array}\right]^{T}, $$
(24)

where \(\phi _{\ell }^{t}\) denotes the angle of departure (AOD) of the th multipath component of the received signal, and

$$ A_{t} = \frac{\pi\Delta_{t} (1-N_{t})}{\lambda}, \quad A_{r} = \frac{\pi\Delta_{r} (1-N_{r})}{\lambda}. $$
(25)

In the previous equations, Δt denotes the spacing between the BS antenna elements. Likewise, Δr stands for the distance between the MS antenna elements. It is assumed that the angle of arrivals (AOA) \(\phi _{\ell }^{r}\) are i.i.d. random variables characterised by a uniform distribution on the circle. The AODs, on the other hand, are computed from the AOAs according to

$$ \phi_{\ell}^{t}= \arctan\left(\frac{r_{k}\sin(\phi_{\ell}^{r})}{D_{k}+r_{k}\cos(\phi_{\ell}^{r})}\right). $$
(26)

In the limit when L, the elements of the channel matrix defined by (20) can be modelled by complex-valued Gaussian random processes, each having a mean value equal to zero and an average power equal to one. The envelope of the resulting MIMO sub-channels is characterised by a Rayleigh distribution [24] and the spatial cross-correlation between any pair of elements of Hk follows the correlation model proposed in [22].

Reception

In the reception, the optimal ML detection criterion is utilised in order to recover the transmitted signals. The ML criterion compares the received signal yk with all possible Tx signals in the reception in order to find the most likely one. The ML criterion for the BD-MIMO-QSM scheme is defined as

$$\textit{} \widehat{\tilde{\mathbf{x}}}_{k} = \arg\min_{j} \left\| \mathbf{y}_{k} - \mathbf{H}_{k} \mathbf{W}_{k} \tilde{\mathbf{x}}_{k}^{(j)} \right\|^{2}. $$
(27)

This formulation is simplified for the DPC-MIMO-QSM and the BD-DPC-MIMO-QSM schemes as

$$\textit{} \widehat{\tilde{\mathbf{x}}}_{k} = \arg\min_{j} \left\| \mathbf{y}_{k} - \mathbf{D}_{k} \tilde{\mathbf{x}}_{k}^{(j)} \right\|^{2}. $$
(28)

In Sections 6 and 7, these formulas are used in order to evaluate the detection complexity and BER performance of all analysed schemes.

Detection complexity

The complexity γ of the detection algorithm is measured using the total number of floating-point operations (flops) [25]. For real additions and multiplications, one flop is carried out. For complex additions and multiplications, two and six flops are carried out, respectively, while for subtractions and divisions, it will take the same value in flops as in addition and multiplication, respectively. Multiplication of m×n and n×p complex matrices uses 8 mnp flops [25]. Table 2 summarises all operations used for the complexity calculation of the MU-BD-ML scheme.

Table 2 Comparison of detection complexity (γ)

For the conventional BD-MIMO-SMux scheme, the lattice for the ML detector is composed as

$$ \mathbf{G}_{k} = \mathbf{H}_{k}\mathbf{W}_{k}\mathbf{B}, $$
(29)

where \(\mathbf {H}_{k} \in \mathbb {C}^{N_{r} \times N_{t} }\) and \(\mathbf {W}_{k} \in \mathbb {C}^{N_{t} \times N_{r} }\). Multiplication of HkWk requires \(8 N_{t} N_{r}^{2} \) flops and generates a square matrix of dimension Nr×Nr. This matrix multiplies the matrix \(\mathbf {B} \in \mathbb {C}^{N_{r} \times 2^{m} }\) and requires \(8 N_{r}^{2} (2^{m})\) flops. Each point in matrix B is an M-QAM signal. Then, generating matrix Gk requires \(8 N_{r}^{2}(N_{t} + 2^{m})\) flops. Each row in this matrix is used for a different Rx antenna.

Differences use 2(2m)Nr flops. Obtaining the magnitude requires 3(2m)Nr flops. Combining all results in a maximum ratio combiner (MRC) requires 2(Nr−1)2m flops and finding the minimum requires 2(2m) flops. Adding all this results, the complexity of BD-MIMO-SMux scheme can be approximated as

$$ \eta_{\mathrm{BD, ML}} \approx 8 N_{r}^{2}(N_{t} +2^{m} \beta) + 7N_{r}(2^{m}\beta)\: \text{flops}. $$
(30)

In (30), we have introduced the factor β to consider the cases where the size of the constellation can be reduced by the inserted zeros, which is the case of EQSM. For the BD-MIMO-SMux system β=1.

The detection complexity of the conventional DPC-MIMO-SMux system can be easily evaluated utilising the detection complexity of the BD-MIMO-SMux scheme. The main difference is that DPC avoids the multiplication of Hk and Wk matrices, which is substituted by only a real value derived from the diagonal matrix diag(L). The detection lattice is conformed by the multiplication of matrix B and a real number, which uses 2Nr(2m) flops. As in BD-SMux, the MRC uses 7Nr(2m) flops. Then, the complexity of the DPC-MIMO-SMux system can be approximated as

$$ \eta_{\mathrm{DPC, ML}} \approx 9N_{r}(2^{m}\beta)\: \text{flops}. $$
(31)

Again, β=1 for the conventional scheme.

The detection complexity of the BD-MIMO-EQSM and DPC-MIMO-EQSM schemes can be easily obtained using (30) and (31) and considering the number of zeros inserted in the detection lattice B. The reduction factor β for EQSM can be computed as the number zeros used by the EQSM constellation divided by the total quantity of complex signals in the conventional constellation.

In the EQSM-based systems, two M-QAM symbols are transmitted at a time. However, the same Tx antenna can be used for both symbol transmissions which increases the number of zeros in B. It means that the complexity of EQSM is at most the double of the QSM/SM systems complexity.

Since EQSM-based systems transmit only the real or imaginary part of a M-QAM symbol, they operate with a complexity reduced 50%. Considering the above assumptions, the reduction factor can be approximated as

$$ \beta \approx \frac{3}{2L}. $$
(32)

Due to the precoding technique used, the complexity of the BD-DCP-MIMO-EQSM becomes the same as the DPC-MIMO-EQSM scheme. Table 2 compares the detection complexity for all schemes. In Table 2, we consider the conventional BD-MIMO-SMux system as a reference with a 100% of detection complexity.

Results and discussion

In this Section, (4·2)×8, (8 bpcu) and (4·4)×16, (12 bpcu) configurations are utilised over correlated and uncorrelated fading channels in order to compare the BER performance of the five systems. All systems are compared considering the same SE and a normalised transmission power of E[ xHx]=K. For simulation, we target a BER of 10−4.

For the uncorrelated fading channel and an SE of 8 bpcu (Fig. 6), the BD-MIMO-EQSM system has 2 dB gain compared to the conventional BD-MIMO-SMux system used as a reference. However, the best performance is obtained by three out of four users of the DPC-MIMO-EQSM system which has gains of 6 dB, 5 dB, and 2 dB compared with the BD-MIMO-EQSM system. On the other hand, the last user in the DPC-MIMO-EQSM scheme has noticeable BER losses. Finally, the BD-DPC-MIMO-EQSM scheme has loses of 7 dB compared with the conventional BD-MIMO-SMux scheme.

Fig. 6
figure6

BER performance comparison for an (4·2)×8 configuration and the uncorrelated fading channel. This figure compares the three proposed MU-MIMO-EQSM schemes with the conventional ones considering an spectral efficiency of 8 bpcu for all systems

For the uncorrelated fading channel and an SE of 12 bpcu (Fig. 7), the BD-MIMO-EQSM scheme has the best BER performance with 3 dB gain when compared with the conventional BD-MIMO-SMux and the best user (first user) of the DPC-MIMO-EQSM scheme. The DPC-MIMO-EQSM scheme outperforms its conventional counterpart for 2 DB approximately. Also, the BD-DPC-EQSM scheme has losses of 14 dB compared with the conventional BD-MIMO-SMux scheme.

Fig. 7
figure7

BER performance comparison for an (4·4)×16 configuration and the uncorrelated fading channel. This figure compares the three proposed MU-MIMO-EQSM schemes with the conventional ones considering a spectral efficiency of 12 bpcu for all systems

Considering a spatial correlated fading channel with Tx antennas separated 1.5 λ and SE of 8 bpcu (Fig. 8), BD-basedschemes are affected in BER around 15 dB. Meanwhile, schemes using DPC are affected by 20 dB approximately.It is worth noting that the two best users of the DPC-MIMO-EQSM scheme, the BD-MIMO-EQSM, and the conventional BD-MIMO-SMux system have approximately the same BER performance.

Fig. 8
figure8

BER performance comparison for an (4·2)×8 configuration and a correlated fading channel. This figure compares the three proposed MU-MIMO-EQSM schemes with the conventional ones considering a spectral efficiency of 8 bpcu for all systems and a spatial correlated channel with Tx antennas separated by 1.5 λ

Considering the correlated fading channel with Tx antennas separated 1.5 λ and SE of 12 bpcu (Fig. 9), the BD technique clearly outperforms the DPC technique. The correlated channel adds losses of 10 dB and 30 dB for the BD and the DPC techniques, respectively. In this scenario, the best performance is achieved by the BD-MIMO-EQSM scheme, which has 4 dB gain compared with the conventional scheme and at least 31 dB gain compared with the DPC-based multiuser schemes. Again, the BD-DPC-MIMO-EQSM hybrid system has a BER performance between the third and fourth user of the DPC-MIMO-EQSM scheme.

Fig. 9
figure9

BER performance comparison for an (4·4)×16 configuration and a correlated fading channel. This figure compares the three proposed MU-MIMO-EQSM schemes with the conventional ones considering a spectral efficiency of 12 bpcu for all systems and a spatial correlated channel with Tx antennas separated by 1.5 λ

It is worth noting that the DPC technique is more affected by the correlated channel than the BD technique, and this is mainly because the correlation affects the singular values of the channel which are used by the DPC-based systems as the unique way to identify a particular path in the SM-based transmission. Also, the advantage of using more Tx antennas in the (4·4)×16 configuration is severely affected when a correlated channel is used, resulting in a degradation of the BER performance.

Conclusion

In this paper, three MU-MIMO strategies have been evaluated and compared for the recently proposed EQSM transmission scheme. The three MU-MIMO-EQSM schemes have been compared in terms of BER performance and detection complexity with their conventional counterparts for correlated and uncorrelated fading channels and optimal ML detection algorithms.

Results show that for all configurations and types of channels, the proposed BD-MIMO-EQSM and DPC-MIMO-EQSM systems outperform in BER performance and detection complexity of their conventional counterparts. More specifically, for the (4·2)×8 configuration and considering uncorrelated fading channels, the DPC strategy outperforms the BD strategy. However, for the rest of the analysed cases, the BD-MIMO-EQSM system has the best BER performance. In particular, for the correlated fading channel, DPC schemes suffer deep degradation in BER performance, mainly for the (4·4)×16 configuration. The hybrid BD-DPC-MIMO-EQSM scheme heritages the low detection complexity of the DPC scheme, with the advantage of all users having the same BER performance, however, suffers a deep BER performance degradation for correlated fading channels.

In conclusion, for future massive MIMO systems where a high SE is utilised and spatial correlation is typically present in the system, the proposed BD-MIMO-EQSM scheme can be the best option. In this case, low complexity detection algorithms could be implemented in order to reduce even more the detection complexity of the system.

Availability of data and materials

Not applicable.

Abbreviations

AOA:

Angle of arrivals

AOD:

Angle of departure

BD:

Block diagonalisation

BER:

Bit error rate

BS:

Base station

bpcu:

Bits per channel use

CSI:

Channel state information

DPC:

Dirty paper coding

EQSM:

Extended quadrature spatial modulation

flops:

Floating point operations

GSM:

Generalised spatial modulation

MIMO:

Multiple input-multiple output

ML:

Maximum likelihood

MU:

Multiuser

MS:

Mobile station QAM: Quadrature amplitude modulation

Rx:

Receive

SE:

Spectral efficiency

SMux:

Spatial multiplexing

Tx:

Transmit

ULA:

Uniform linear antenna

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Acknowledgements

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Funding

Francisco R. Castillo Soria thanks the financial support obtained by the project FAI-UASLP-2019.

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FRCS and AAC conceived the main idea and analysed the results. CAG proposed the channel and AGB performed the experiments. FRCS and JS wrote the paper. All authors have read and approved the final manuscript.

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Correspondence to Francisco R. Castillo-Soria.

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Castillo-Soria, F.R., Gutierrez, C.A., García-Barrientos, A. et al. EQSM-based multiuser MIMO downlink transmission for correlated fading channels. J Wireless Com Network 2020, 30 (2020). https://doi.org/10.1186/s13638-020-1648-1

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Keywords

  • Extended QSM
  • Spatial modulation
  • Multiuser MIMO
  • Correlated channels