19

Multi-user MIMO for Power Line Communications

Yago Sánchez Quintas, Daniel M. Schneider and Andreas Schwager

CONTENTS

19.1  Introduction

19.2  MU-MIMO Scenarios

19.3  MU-MIMO Precoding

19.3.1  Block Diagonalisation

19.3.1.1  MU Scenario

19.3.1.2  SDD Scenario

19.3.2  Multi-user Orthogonal Space Division Multiplexing

19.3.2.1  MU Scenario

19.3.2.2  NuSVD: Iterative Computation of MOSDM

19.3.2.3  SDD Scenario

19.3.3  SU-Precoding: Eigenbeamforming

19.4  Simulation Results

19.4.1  System Parameters

19.4.2  MU Scenario

19.4.3  SDD Scenario

19.5  Conclusions

References

19.1  Introduction

There are different channel access methods for shared medium networks. Different network users might be separated in the time domain (time division duplex [TDD]), in the frequency domain (frequency division duplex [FDD]) or in the code domain (code division multiple access [CDMA]). However, if the transmitters and receivers have multiple transmit and receive ports (multiple-input multiple-output [MIMO]), different users might also be separated by the spatial dimension. Using MIMO in the multi-user (MU) context is called MU-MIMO. MU-MIMO has already proven to be a successful way of enhancing the MIMO performance of wireless transmissions [1,2,3,4]. The use of MIMO algorithms allows the transmission of several simultaneous spatial streams to different users on the same frequency band and time slot. Potentially, the total throughput is increased.

MIMO has been successfully applied to power line communications (PLCs); see Chapters 8 and 9 for more details on MIMO signal processing strategies and the resulting capacity gains. Current implementations of MIMO PLC systems are presented in Chapters 12 and 14. However, so far, MIMO has only been applied to the single-user (SU) scenario, that is, for links between one transmitter and one receiver. This chapter aims to study the feasibility of MU-MIMO techniques for PLC.

There are several scenarios where MU-MIMO algorithms can be applied in a PLC network. For example, two different high-definition video streams could be transmitted simultaneously from a router to two different TVs placed in different rooms. This scenario is similar to the downlink from a base station (BS) of a cellular network to several users. Other scenarios might comprise several user pairs communicating in the same frequency band at the same time, where the multiplexing is achieved in the spatial domain.

One issue that one encounters when trying to adapt MU-MIMO strategies to the PLC scenario is the reduced number of transmit ports. As explained in Chapter 1, the number of transmit ports is limited to two for inhome PLC. This limited number of transmit ports reduces the possibilities of MU-MIMO coding strategies. On the other hand, having a relatively large number of up to four receive ports (see Chapter 1) offers several possibilities to cancel the multi-user interference (MUI).

Different solutions can be applied to deal with the spatial interference generated by several users transmitting on the same frequency band and time slot. On the one hand, MUI could be precancelled by the transmitter, who necessarily needs channel state information (CSI) to exploit the advantages of MU-MIMO strategies. In this way, no – or significantly reduced – interference is seen at the receiver side. The resulting MU-MIMO system will be decomposed into parallel uncoupled channels or streams, and users’ data can be transmitted in disjoint spaces. On the other hand, due to the relatively large number of receive ports, the MUI could be cancelled at the receiver. This is similar to the uplink from several users to one BS in a cellular network, where usually the BS has more antennas than the mobile users. Both techniques will have a limited number of simultaneous spatial streams and therefore a limited number of simultaneous users due to the limited number of transmit and receive ports.

Information theory discussed in [5,6,7] shows that it is necessary to use Costa’s ‘dirty-paper’ coding (DPC) or Tomlinson–Harashima precoding to reach the sum capacity of an MU-MIMO downlink system where the sum capacity is the sum of the capacities of all independent links. However, these techniques require the use of a complex sphere decoder or an approximate closest-point solution, which makes them hard to implement in practice [8]. Thus, the MU-MIMO algorithms investigated in this chapter are limited to linear algorithms.

Figure 19.1 gives a top-level overview of the different channel access methods and shows where MIMO and MU-MIMO fit in. For TDD and FDD, the MIMO algorithms are applied to point-to-point or SU connections. As shown in Chapter 8, different MIMO schemes might be applied, for example, space–time–frequency codes (STFC) or spatial multiplexing (SMX) without and with precoding, that is, beamforming (BF). TDD with BF serves as reference for the performance evaluation of the MU-MIMO algorithms in this chapter. The MU-MIMO algorithms might be separated into algorithms which cancel MUI at the receiver and algorithms that apply a precoding at the transmitter. The precoding might be either nonlinear (e.g. DPC) or linear. One form of linear precoding is based on block diagonalisation (BD), and this chapter focuses in particular on multi-user orthogonal space division multiplexing (MOSDM) and an iterative computation of this algorithm which is NuSVD.

The outline of this chapter is as follows. Section 19.2 introduces two different PLC scenarios where MU-MIMO algorithms could be applied to PLC. An MU-MIMO algorithm based on BD, which is MOSDM, is introduced in Section 19.3 where the algorithm is applied to the two scenarios introduced in Section 19.2. The performance of the introduced algorithms is investigated and discussed in Section 19.4. Depending on the underlying MU-MIMO scenario, a performance gain of MU-MIMO compared to TDD with BF of up to 20% might be expected. The basis of these simulations is the MIMO PLC channels obtained in the European Telecommunications Standards Institute (ETSI) STF410 measurement campaign (refer to Chapter 5 [9,10,11]).

FIGURE 19.1
Channel access methods and MIMO and MU-MIMO algorithms.

19.2  MU-MIMO Scenarios

In recent years, research and development of MU-MIMO algorithms focused mostly on wireless applications. In order to highlight the similarities and differences of MU-MIMO in the context of PLC compared to the wireless scenario, first, the typical wireless setup is recalled. Here, a BS typically offers service to a high number of users or mobile stations (MS). The BS might be the cell tower in a cellular network (e.g. a 4G or LTE network) or an access point on a wireless local area network (LAN). MU-MIMO algorithms at the BS allow for the transmission of spatial streams to the different users using the same frequency and time slot. Usually, the BS has a high number of antennas while the (mobile) users have a smaller and limited number of antennas. When applying MU-MIMO algorithms, the downlink (from the BS to the users) is considered to be the more challenging part, since the BS has to ensure that the spatial streams to one user do not cause interference to the other users [2,4]. For the uplink, the BS can use the larger number of antennas to cancel the interference from the different users.

In a PLC network, examples of the scenario described previously might be as follows: A home server transmits several, different video streams to different users in the home network or a router equipped with a PLC modem communicates to several users on the network. As explained in Chapters 1 and 5, up to 2 × 4 MIMO might be used for inhome PLC. The scenario of one transmitter and several receivers is illustrated in Figure 19.2a for two receivers where the transmitter comprises two transmit ports and the receivers have four receive ports. This scenario is referred to as MU in the following. As described in Chapter 8, the maximum number of spatial streams in SU-MIMO is limited by the minimum number of transmit and receive ports. In this example, two spatial streams are available where each spatial stream is assigned to one of the two users. The channels to users 1 and 2 are called H₁ and H₂ in Figure 19.2a. The transmission to user 1 causes interference to user 2 and vice versa. This interference needs to be handled either at the transmitter (T_x) or the receivers (R_x1 and R_x2). In the uplink (from R_x1 and R_x2 to T_x in Figure 19.2a), basically more spatial streams are available since up to four receive ports may be used in receive mode.

FIGURE 19.2
MU-MIMO scenarios. (a) MU system and (b) SDD system.

In a meshed PLC network, the scenario described previously can be extended as shown in Figure 19.2b. Here, two independent communication links are illustrated where transmitter 1 (T_x1) communicates to receiver 1 (R_x1) and transmitter 2 (T_x2) sends data simultaneously to receiver 2 (R_x2). The transmission of T_x1 causes interference to R_x2 and the transmission of T_x2 causes interference to R_x1. The idea in this scenario is to apply MU-MIMO algorithms to spatially multiplex the two links while transmitting at the same time and on the same frequency band. This scenario is called spatial division duplex (SDD) in the following.

The term MU-MIMO will be used throughout this chapter to describe MU-MIMO communication systems in a generic way, that is, simultaneous users receiving data in the same time slot and frequency by means of SMX. The term MU is used to describe the particular scenario where only one transmitter communicates with several receivers. The second scenario will always be referred to as SDD.

The MU and SDD scenarios yield differences in the implementation of the precoding algorithms. Also, the differences between the scenarios influence the obtained results with the same precoding techniques. Therefore, during the study of coding techniques in the next sections, a distinction will always be made between both cases. This difference will also be pointed out in the results section (Section 19.4).

Combinations between the described scenarios are also possible. However, due to the already large number of possibilities these two scenarios offer, combinations among them have not been included in the scope of this chapter. It has to be noticed that any other type of multiplexing, like TDD, could be applied together with MU-MIMO algorithms in case it becomes necessary to increase the flexibility of the system.

19.3  MU-MIMO Precoding

The adaptation of MU precoding techniques to the PLC environment was studied in [12]. Some investigated algorithms were discarded without the need of being implemented and simulated, mainly due to dimensionality constraints which did not fit the PLC channel. For instance, the algorithms suggested in [13,14,15,16], which have shown to yield successful results for the wireless environment are not feasible for PLC due to the limited number of ports at the transmitter. Among the investigated and simulated precoding algorithms in [12], two groups might be differentiated: interference cancellation algorithms at the transmitter, like BD, and interference cancellation at the receiver.

BD will be explained in detail in the course of this chapter (Sections 19.3.1 and 19.3.2), since the algorithms yielding the best simulation results for the PLC environment belong to this group. The interference decoding algorithms at the receiver are briefly discussed in Section 19.3.3.

As mentioned in the introduction, the sum capacity of the MIMO broadcast (BC) channel can be only achieved with DPC [6]. However, a practical scheme that approaches DPC is still unavailable, and worse, the encoding process to achieve the sum capacity is data dependent. This means that the cancellation needs to be done independently for every symbol. Several algorithms that approach the sum capacity exposed by DPC have been proposed in [17,18]; however, they are considered to be too complicated for cost-effective implementation. An alternative linear precoding technique to DPC (nonlinear), widely applied in the wireless environment, is BD. The main concept of BD consists in precoding each user’s data with a linear matrix before transmission. This particular matrix lies in the null space of all other simultaneous user channel matrices. Hence, assuming the channel matrices of all simultaneous users are known at the transmitter, with perfect CSI, zero interuser interference is achievable at every receiver. This enables the use of simple receiver structures. This group of algorithms has been described as a suboptimal solution in terms of total achievable throughput but also as a feasible solution in terms of complexity.

19.3.1  Block Diagonalisation

First, the BD system model is described for the MU scenario, that is, only one transmitter in the system. The differences to the SDD scenario will be shown in a second step. The system model presented here will be applied to the precoding algorithm described in this chapter. Several papers like [19,20] use this model for BD systems. Note that the matrix operations shown in the following are described for a single carrier system. However, it can be easily extended to an orthogonal frequency division multiplexing (OFDM) system where all the matrix operations have to be applied for each subcarrier separately.

19.3.1.1  MU Scenario

Consider a downlink MU-MIMO system with M users, where N_T indicates the number of transmit ports, N_R indicates the total number of receive ports among all users and N_R,j denotes the number of receive ports at the jth user. The transmitted symbol vector of user j is denoted as a k_j-dimensional vector s_j. Note that k_j indicates the number of spatial modes directed to the user j. s_j is precoded by a N_T × k_j precoding matrix T_j for each particular user. At the receiver j, a detection matrix R_j of size N_R,j × k_j is applied to the received signal in order to obtain the desired symbol. Thus, the post-detection symbol vector y_j for user j can be written as

$\begin{array}{l}{y}_j=R_j^H\left(H_j{T}_j{s}_j+{\displaystyle \sum _{m=1,m\ne j}^M{H}_j{T}_m{s}_m+n_j}\right),\\ =R_j^H{H}_j{T}_j{s}_j+R_j^H{\displaystyle \sum _{m=1,m\ne j}^M{H}_j{T}_m{s}_m+R_j^H{n}_j},\end{array}$

(19.1)

where

n_j denotes the noise vector for user j

(⋅)^H indicates the Hermitian operator

The matrix $H_j\in {ℂ}^{N_{R,j}\times N_T}$ indicates the channel matrix to the jth user. T_j and R_j are constructed to be unitary matrices (as explained in Section 19.3.2).

The overall number of spatial streams is limited by the number of transmit ports:

${\displaystyle \sum _{j=1}^M{k}_j\le N_T},$

(19.2)

Details of the constraint in Equation 19.2 are discussed in Section 19.3.2.

Figure 19.3 shows the application to the MU scenario with M = 2 users. The transmitter has N_T = 2 transmit ports. According to Equation 19.2, the number of spatial streams for each user is k₁ = k₂ = 1, that is, one spatial mode is activated for each of the two users. The symbols s₁ and s₂ to the two users are weighted by the 2 × 1 precoding vectors T₁ and T₂, respectively, and the 2 × 1 symbol vector x = T₁s₁ + T₂s₂ is transmitted to the channel. At the receivers, estimates of the transmitted symbols are obtained according to Equation 19.1.

Note that in Figure 19.3, ${\widehat{y}}_j$ represents the received vector before being decoded by R_j.

The goal of BD is to find precoding matrices T_j for each user j such that no interference is generated to the other users. For the example shown in Figure 19.3 and according to Equation 19.1, an estimate of the symbol for user 1 is given by

$y_1=R_1^H{H}_1{T}_1{s}_1+R_1^H{H}_1{T}_2{s}_2+R_1^H{n}_1$

(19.3)

and for user 2 by

$y_2=R_2^H{H}_2{T}_2{s}_2+R_2^H{H}_2{T}_1{s}_1+R_2^H{n}_2,$

(19.4)

In order to fully cancel the interference, the conditions $R_1^H{H}_1{T}_2=0$ and $R_2^H{H}_2{T}_1=0$ in Equations 19.3 and 19.4, respectively, have to be fulfilled. Note that the receivers in this example could comprise only one receive port since only one spatial mode is used per user. Of course, more receive ports would increase the performance due to the increased receive diversity.

FIGURE 19.3
BD system model for PLC MU scenario.

Generally, the precoding matrices T_j of dimensions N_T × k_j and the decoding matrices R_j of dimensions N_R,j × k_j have to fulfil

$R_j^H{H}_j{T}_m=0\text{\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}j\ne m\text{\hspace{0.17em}and\hspace{0.17em}1}\le j,m\le M$

(19.5)

Then, the post-detection symbol vector for user j is reduced to

$\begin{array}{l}{y}_j=R_j^H{H}_j{T}_j{s}_j+R_j^H{\displaystyle \sum _{\underset{=0}{\underbrace{m=1,m\ne j}}}^M{H}_j{T}_m{s}_m+R_j^H{n}_j},\\ =R_j^H{H}_j{T}_j{s}_j+R_j^H{n}_j,\end{array}$

(19.6)

In the noise-free case and for an appropriate design of T_j and R_j (see later), the entries of y_j are scaled versions of the corresponding entries of s_j, that is, y_j needs to be equalised by a diagonal matrix to obtain estimates of s_j.

As it can be observed in Equations 19.5 and 19.6, with the proper precoding matrices T_j and detection matrices R_j, the inter-user interference can be cancelled. It is useful to define the total MU transmit weight matrix as

$T=\left[T_1\cdots T_M\right]$

(19.7)

and the MU transmitted vector as

$s=\left[\begin{array}{c}{s}_1\\ ⋮\\ s_M\end{array}\right].$

(19.8)

For the example shown in Figure 19.3, the dimensions of T are 2 × 2 and the dimensions of s are 2 × 1.

If BD is applied successfully, an equivalent block diagonal model is obtained for the MU system. Figure 19.4 shows the result for the example introduced in Figure 19.3. λ_j indicates the equivalent channel gain for user j, and ${\tilde{n}}_j$ represents the equivalent noise sample after filtering.

FIGURE 19.4
Equivalent block diagonal model for MU scenario.

19.3.1.2  SDD Scenario

Several differences should be considered between the adaptation of BD to the MU scenario and the adaptation to the SDD scenario. In the SDD scenario, several modems transmit on the same frequency band and time slot. In order to simplify the description of the model, a system with only two transmitters and two receivers is explained here. However, this model can easily be extended to the case where three or more transmitter–receiver pairs coexist together.

Figure 19.5 shows an example of M = 2 links, where transmitter 1 communicates to receiver 1 and transmitter 2 to receiver 2, respectively. In this example, each transmitter is equipped with two transmit ports and each receiver has four receive ports.

In the following, it is always assumed that transmitter 1 communicates to receiver 1, and transmitter 2 to receiver 2. In the SDD setup, the number of ports at the transmitter and the receiver remains the same as for the MU scenario, that is, up to N_T,j = 2 and N_R,j = 4 where j is the index of the link (j = 1, …, 2). The channel between transmitter 1 and receiver 1 is represented by the channel matrix H₁₁, and the channel between transmitter 2 and receiver 2 is represented by the channel matrix H₂₂, respectively. Here, two interferences are possible. Transmitter 1, which is attempting to communicate with receiver 1, will generate interference to receiver 2, and transmitter 2 will generate interference to receiver 1. The interfering channels are denoted by H₂₁ and H₁₂ where the first index denotes the index of the receiver and the second index denotes the transmitter’s index. As assumed for the MU scenario, each transmitter applies a precoding matrix T_j (j = 1, …, M). According to this description, the precoding of transmitter 1 should be able to cancel any interference produced to receiver 2, and transmitter 2 should be able to cancel any interference generated to receiver 1.

In the SDD scenario, the transmitters need to be synchronised in order for the receivers to estimate the channels from the interfering modems. Also, the signals used for channel estimation have to be designed in a way that each receiver can estimate the channel from each transmitting modem. Similar to the channel estimation in MIMO systems, the training symbols need to be orthogonal to separate the different channels.

As a difference to the MU scenario, it should be noted that in this setup, the transmission of k = 2 spatial streams per user is physically possible. For example, two symbols could be sent simultaneously to user 1, at the same time and same frequency used for transmission of two other symbols sent to user 2. However, as it will be discussed in the following sections, the number of spatial streams per link is limited due to constraints imposed by the interference cancellation.

FIGURE 19.5
BD system model for PLC SDD scenario.

According to the matrix operations illustrated in Figure 19.5 and similar to Equation 19.3, the equalised symbol vector of receivers 1 and 2 is given as

$\begin{array}{l}{y}_1=R_1^H\left(H_{11}{T}_1{s}_1+H_{12}{T}_2{s}_2+n_1\right),\\ =R_1^H{H}_{11}{T}_1{s}_1+R_1^H{H}_{12}{T}_2{s}_2+R_1^H{n}_1,\end{array}$

(19.9)

and

$\begin{array}{l}{y}_2=R_2^H\left(H_{21}{T}_1{s}_1+H_{22}{T}_2{s}_2+n_2\right),\\ =R_2^H{H}_{21}{T}_1{s}_1+R_2^H{H}_{22}{T}_2{s}_2+R_2^H{n}_2.\end{array}$

(19.10)

Generally, the equalised symbol vector of receiver n is given by

$y_n=R_n^H\left({\displaystyle \sum _{m=1}^M\left(H_{nm}{T}_m{s}_m\right)+n_n}\right),$

(19.11)

$=R_n^H{H}_{nn}{T}_n{s}_n+R_n^H{\displaystyle \sum _{m=1,m\ne n}^M\left(H_{nm}{T}_m{s}_m\right)+R_n^H{\text{n}}_{n,}}$

(19.12)

where

T_m is the N_T,m × k_m precoding matrix

s_m is the k_m × 1 transmit symbol vector of the mth transmitter which activates k_m spatial streams

R_n is the N_R,n × k_m receive matrix

H_nm is the channel matrix from transmitter m to receiver n

The aim of the precoding is to cancel the interference to other receivers. In the example shown in Figure 19.5, transmitter 1 should not cause interference to receiver 2 and transmitter 2 should not cause any interference to receiver 1. In terms of Equations 19.9 and 19.10, this requires $R_1^H{H}_{12}{T}_2=0$ and $R_2^H{H}_{21}{T}_1=0$, respectively.

Generally, the following equation has to be fulfilled for interference cancellation:

$R_n^H{H}_{nm}{T}_m=0\text{\hspace{0.17em}for\hspace{0.17em}}n,m=1,\dots ,M,n\ne m.$

(19.13)

If Equation 19.13 is fulfilled, Equation 19.11 reduces to

$y_n=R_n^H\left(H_{nn}{T}_n{s}_n+n_n\right).$

(19.14)

and the equalised symbol vector depends only on the desired transmit symbol vector.

FIGURE 19.6
Equivalent block diagonal model of the jth user.

The equivalent block diagonal model for the SDD scenario is shown in Figure 19.6, where ${\text{λ}}_k^{\left(j\right)}$ indicates the equivalent gain for spatial stream k and link j. ${\tilde{n}}_{jk}$ is the equivalent noise sample after filtering of spatial stream k and link j.

The following section explains how the precoding and decoding matrices have to be designed in order to cancel the interference. The algorithm MOSDM based on BD will be presented.

19.3.2  Multi-user Orthogonal Space Division Multiplexing

As defined in [21], by jointly optimising the transmitter and receivers in the MU system, the MU signals can be projected onto orthogonal subspaces, so that the MUI experienced by each independent user is eliminated. The weight matrices in the transmitter and receivers are iteratively jointly optimised. This method assumes CSI at the transmitter side for all MU channels. This is necessary in order to have a mathematical representation of the subspaces where the different channel matrices lie and hence be able to project them onto equivalent orthogonal subspaces. The objective is to obtain an MU channel diagonalisation by finding the joint weight matrices T, R₁, R₂, …, R_M. In the following, the algorithm is described for the MU scenario (i.e. with a single transmitter) according to [21] and is extended later to the SDD scenario.

19.3.2.1  MU Scenario

Recall Equation 19.5 which defined the necessary condition for interference cancellation. Equation 19.5 can be written in matrix notation as follows:

$R_j^H{H}_{j}T=\left[\begin{array}{ccc}\begin{array}{ccc}{0}_1& \cdots & 0_{j-1}\end{array}& \underset{j\text{th\hspace{0.17em}subblock\hspace{0.17em}matrix}}{\underbrace{{\text{Λ}}_j}}& \begin{array}{ccc}{0}_{j+1}& \cdots & 0_M\end{array}\end{array}\right]$

(19.15)

for the users j = 1, …, M, where

${\text{Λ}}_j=\text{diag}\left(\begin{array}{cccc}{\text{λ}}_1^{\left(j\right)}& {\text{λ}}_2^{\left(j\right)}& \cdots & {\text{λ}}_{k_j}^{\left(j\right)}\end{array}\right)$

(19.16)

is of dimension k_j × k_j and ${\text{λ}}_m^{\left(j\right)}$ represents the channel gain of the mth spatial stream of the jth user. Equations 19.15 and 19.16 not only show the interference cancellation but also show that the unitary precoding T_j and unitary decoding $R_j^H$ decompose the link of user j into k_j parallel independent spatial streams. This is similar to eigenbeamforming (see e.g. Chapter 8).

Generally, if a solution to Equations 19.15 and 19.16 exists, there will be more than one solution. Between the possible solutions, the one that optimises the overall MIMO system performance is chosen [21]. This can be mathematically written as

${\left(T,R_1,R_M\right)}_{\text{opt}}=\arg \underset{T,R_1,\dots R_M}{\max }{{\displaystyle \sum _{j=1}^M\Vert {\text{Λ}}_j\Vert }}^2,$

(19.17)

where $\Vert \cdot \Vert$ denotes the Frobenius norm of a matrix.

Figure 19.6 illustrates the k_j parallel spatial streams of the jth user.

The calculation of the weighting matrices is approached as follows. First, the optimal transmit and receive weights are found for a particular user j, with the assumption that the receive weights for other users ${\left\{R_m\right\}}_{m\ne \text{j}}$ are given. The weights should be found in a way that the equivalent channel gains are maximised and the co-channel interference (CCI) caused to other users is eliminated. Next, the weight matrices of other users are updated in a step-by-step approach until convergence.

First, the equivalent MU channel matrix H_e of dimensions $\left({\displaystyle {\sum }_{m=1}^M{k}_m}\right)\times N_T$ is defined as

$H_e\triangleq \left[\begin{array}{c}{R}_1^H{H}_1\\ ⋮\\ R_M^H{H}_M\end{array}\right].$

(19.18)

Assuming that the receive matrices $R_1,\dots ,R_{j-1},\dots ,R_{j+1},\dots ,R_M\left(m\ne j\right)$ are known by the transmitter, it will try to find the best T_j and R_j that optimise the performance of the jth user link:

${\left(T_j,R_j\right)}_{\text{opt}}=\arg \underset{T_j,R_j}{\max }{\Vert {\text{Λ}}_j\Vert }^2,$

(19.19)

while no CCI is caused to the other users according to

$H_e{T}_j=\left[\begin{array}{c}{0}_1^T\\ ⋮\\ 0_{j-1}^T\\ {\text{Λ}}_j\\ 0_{j+1}^T\\ ⋮\\ 0_M\end{array}\right].$

(19.20)

Equation 19.20 is similar to Equation 19.15 and again shows the interference cancellation where each subblock represents the equivalent spatial streams associated with user j.

To satisfy Equations 19.19 and 19.20, the interference matrix ${\tilde{H}}_{\text{e}}^{\left(j\right)}$ of dimensions $\left({\displaystyle {\sum }_{m=1,m\ne j}^M{k}_m}\right)\times N_T$ is defined as

${\tilde{H}}_e^{\left(j\right)}\triangleq \left[\begin{array}{c}{R}_1^H{H}_1\\ ⋮\\ R_{j-1}^H{H}_{j-1}\\ R_{j+1}^H{H}_{j+1}\\ ⋮\\ R_M^H{H}_M\end{array}\right].$

(19.21)

Once these definitions have been set, the condition to satisfy Equation 19.20 and therefore to eliminate the inter-user interference is

$T_j\in \text{null}\left\{{\tilde{H}}_e^{\left(j\right)}\right\},$

(19.22)

where null{⋅} denotes the null space of a matrix.*

T_j according to Equation 19.22 only exists if the null space is not empty. This leads to the following constraint of the number of spatial streams. For a system with M users, each transmitting k_j spatial modes, the number of ports at the transmitter must be greater or equal to the total number of active spatial modes in the whole system [21]:

${\displaystyle \sum _{j=1}^M{k}_j\le N_T}.$

(19.23)

For each user taking part in the MU system, the number of ports on a particular receiver must be greater than or equal to the number of active spatial modes that this user is receiving [21]:

$k_j\le N_{R,j}\forall j.$

(19.24)

This is similar to the SMX case in SU-MIMO (see Chapter 8).

Consider the relevant case for inhome PLC of N_T = 2 transmit ports. According to Equation 19.23, two users are possible, each with one spatial stream. For one spatial stream, ${\tilde{H}}_{\text{e}}^{\left(j\right)}$ is of dimensions 1 × 2 and the precoding vector according to Equation 19.22 is of dimensions 2 × 1.

Now, in order to optimise the system, we can write the basis of the null space as $Q_j=\left[\begin{array}{ccc}{q}_1^{\left(j\right)}& q_2^{\left(j\right)}& \cdots \end{array}\right]$ where $q_m^{\left(j\right)}$ indicates the mth column vector of the matrix Q_j. Next, we decompose the precoding matrix T_j into two submatrices as follows T_j = Q_jB_j. In this decomposition, B_j indicates the coordinate transformation under the basis Q_j. The matrix Q_j will be in charge of cancelling the interference in the MU system, and once the interference is cancelled, the already independent channels can be optimised by applying the matrix B_j. The next step is then to choose a matrix B_j. such that

${\left(B_j,R_j\right)}_{\text{opt}}=\arg \underset{B_j,R_j}{\max }{\Vert {\text{Λ}}_j\Vert }^2,$

(19.25)

and

$R_j^H{H}_j{Q}_j{B}_j={\text{Λ}}_j.$

(19.26)

If the precoding matrix T_j. for user j has been properly computed, we have ensured that there is no inter-user interference in the system. Due to the properties of the null space, precoding with the matrix T_j. is actually projecting our channel matrix into a subspace which is orthogonal to all other users’ subspaces.

Once the interference is cancelled (supposing the dimensionality constraints are satisfied), the rest of the problem is reduced to an optimisation task. Accordingly, this task is to find (B_j, R_j)_opt of an SU-MIMO system supporting multiple spatial streams.

As shown in Chapter 8, the best precoding for SU-MIMO supporting several spatial modes is eigenbeamforming, and the precoding matrix is obtained by means of a singular value decomposition (SVD):

$H_j{Q}_j=U_j{\text{Λ}}_j{V}_j^H.$

(19.27)

Then, R_j and B_j are given by

${\left(R_j\right)}_{\text{opt}}=U_j|{}_{1\leftrightarrow k_j}$

(19.28)

and

${\left(B_j\right)}_{\text{opt}}=V_j|{}_{1\leftrightarrow k_j},$

(19.29)

where the notation $|{}_{1\leftrightarrow k_j}$ is used to indicate that only the k_j first column vectors corresponding to the k_j largest singular values are included in the matrices R_j and B_j. Finally, the optimum precoding matrix is formulated as follows:

${\left(T_j\right)}_{\mathrm{opt}}=Q_j{\left(B_j\right)}_{\text{opt}}.$

(19.30)

The final precoding matrix T obtained with this algorithm is not unitary. However, it presents normalised columns, which ensures equal transmit powers in all transmit ports.

19.3.2.2  NuSVD: Iterative Computation of MOSDM

An iterative way of approaching MOSDM is proposed in [21] and is called iterative null space-directed SVD (iterative NuSVD). Using the definitions and equations introduced before, the algorithm works iteratively as follows [21]:

1.  The decoding matrices are initialised to the identity matrix, R_j = I.

2.  The matrix ${\tilde{H}}_e^{\left(j\right)}$ is formed for each user included in the MU system. The matrix Q_j is obtained as the null space basis of ${\tilde{H}}_e^{\left(j\right)}$. Subsequently each link is optimised computing the SVD of H_jQ_j, which yields the optimum precoding matrices B_j according to Equation 19.29 and R_j using Equation 19.28, respectively.

3.  The off-diagonal of the equivalent channel matrix H_e⋅T is computed. The off-diagonal norm indicates the level of interference in the system. As the iterations continue, the level of interference should decrease:

a.  Compute $\text{ε}=\text{off}\left(H_{\text{e}}\cdot T\right)$

b.  where $\text{off}\left(A\right)\triangleq {\displaystyle \sum _{k,l,k\ne l}\left|a_{k,l}\right|}$

in which $|\cdot |$ is the absolute value of the elements of the matrix A. If ∈ is below a certain threshold $\in \le {{\rm T}}_{\in }$, go to step 4, otherwise go to step 2. The threshold was selected to T_∈ = 10⁻¹² since this value showed good results in our simulations.

4.  For $\in \le {{\rm T}}_{\in }$, the convergence is said to be achieved. The columns of the precoding matrix T need to be then normalised in order to satisfy the power constraint.

It is important to note that this algorithm needs the CSI for all channels at the transmitter side in order to perform the described computations. This means that every receiver needs to feedback its corresponding channel matrix to the transmitter (if the channel is not reciprocal), before the whole process is started. Once convergence is reached and the decoding matrices are obtained, the transmitter must forward them to every receiver before the communication process starts.

19.3.2.3  SDD Scenario

The adaptation process of MOSDM and its iterative computation (NuSVD) to the SDD scenario share many similarities with the previous section for the MU scenario. The main difference is that two transmitters take part in SDD. Again, the derivations in the following are explained for the case of two transmitters and two receivers (refer to Figure 19.5).

Recall Equations 19.3 and 19.4 which show that the equalised symbol vector is disturbed by the interference from the second transmitter. Similar to Equation 19.18 in the previous subsection, equivalent channels can be defined as

$H_{e1}=\left[\begin{array}{cc}{R}_1^H{H}_{11}& R_1^H{H}_{12}\end{array}\right],$

(19.31)

$H_{e2}=\left[\begin{array}{cc}{R}_2^H{H}_{21}& R_2^H{H}_{22}\end{array}\right].$

(19.32)

In order to cancel the interference from the second transmitter, the precoding matrices are designed to fulfil the following conditions, analogue to Equation 19.20:

$\begin{array}{l}{H}_{e1}{T}_1=\left[\begin{array}{cc}{\text{Λ}}_1& 0\end{array}\right],\\ H_{e2}{T}_2=\left[\begin{array}{cc}0& {\text{Λ}}_2\end{array}\right].\end{array}$

(19.33)

Analogue to Equation 19.21, the interference matrices are defined as

$\begin{array}{l}{\tilde{H}}_{e1}=R_2^H{H}_{21},\\ {\tilde{H}}_{e2}=R_1^H{H}_{12},\end{array}$

(19.34)

Then, in order to cancel the interference at the transmitter, the precoding matrices have to lie in the null space of the corresponding equivalent inference matrix, analogue to Equation 19.22:

$\begin{array}{l}{T}_1\in \text{null}\left\{{\tilde{H}}_{e1}\right\},\\ T_2\in \text{null}\left\{{\tilde{H}}_{e2}\right\},\end{array}$

(19.35)

Equation 19.35 shows how many spatial streams can be transmitted. It is assumed again that each transmitter has two transmit ports. If two spatial streams are transmitted by the two transmitters, ${\tilde{H}}_{e1}$ and ${\tilde{H}}_{e2}$, which are 2 × 2 in dimension according to Equation 19.34, then the null space according to Equation 19.35 is empty. Thus, the transmission of two spatial streams is not possible. Only one spatial stream can be utilised by each transmitter in this scenario. Then, the dimensions of ${\tilde{H}}_{e1}$ and ${\tilde{H}}_{e2}$ according to Equation 19.34 are 1 × 2, and the precoding vectors according to Equation 19.35 are of dimension 2 × 1, and one spatial stream is used.

The precoding matrices T₁ and T₂ will block diagonalise the matrix subspaces H_e1 and H_e2, obtaining the equivalent channels and cancelling the interference that transmitter 1 would generate to receiver 2 (and transmitter 2 receiver 1) according to Equation 19.33. Once the precoding matrices for interference cancellation have been acquired, the same channel optimisation process applied to the MU scenario can be applied here, that is, the steps described by Equations 19.25 through 19.30 and iterative computation according to NuSVD.

19.3.3  SU-Precoding: Eigenbeamforming

Assume that the receivers have more receive ports than the transmitter has transmit ports, as in the 2 × 4 MIMO setup. The additional receive ports can then be used to cancel the interference. Each link might be optimised by SU precoding, that is, by eigenbeamforming as introduced in Chapter 8. First, assume the MU scenario of one transmitter and two receivers. If the transmitter has two transmit ports, two spatial streams might be used, one for each of the users. For each link, the optimum precoding vector is calculated, for example, each receiver feedbacks the optimum precoding vector based on the CSI. The transmitter composes the final precoding matrix by combining the two precoding vectors into a matrix. However, the columns of this matrix will not be orthogonal to each other. This will generate more interference among the users. Although each receiver is able to detect the spatial stream directed at it, the detection matrix might enhance the noise and interference. Sanchez [12] investigated this algorithm for zero-forcing (ZF) detection. No performance gain could be achieved compared to SU-MIMO. More sophisticated detection algorithms compared to the simple ZF detection (see Chapter 8) might improve the performance.

The same concepts can be applied to the SDD scenario. Each of the links uses eigenbeamforming to maximise the performance. Again, the receivers use the higher number of receive ports to detect the spatial stream intended for the specific receiver and simultaneously cancel the interference. So, for the 2 × 4 MIMO setup, the 4 receive ports can be used to decode 4 spatial streams, 2 from the intended transmitter and 2 to remove the interference from other transmitters. The detection matrix is different compared to the SU-MIMO since the detection also has to consider the interference from other users. For ZF, the detection matrix for the two receivers is calculated as

$\begin{array}{l}{W}_1=\text{pinv}\left(\left[H_{11}{T}_1,H_{12}{T}_2\right]\right),\\ W_2=\text{pinv}\left(\left[H_{22}{T}_2,H_{21}{T}_1\right]\right).\end{array}$

(19.36)

where pinv (A) is the pseudoinverse of the matrix A, that is, pinv (A) = (A^HA)⁻¹A^H.

Unfortunately, the signal to interference plus noise ratio (SINR) after detection is lower compared to the SINR for the SU-MIMO case, and the performance is not as good as for the BD algorithms [12].

19.4  Simulation Results

19.4.1  System Parameters

Simulations were performed in order to investigate the performance of the MU-MIMO algorithms introduced in Section 19.3. The MIMO-OFDM system introduced in Chapter 9 was used. The main system parameters are summarised in the following. The system deploys 1296 carriers in the frequency range between 4 and 30 MHz. The precoding and detection matrices are computed for each subcarrier. From there, the SINR after detection was calculated for each subcarrier according to the equivalent channels as introduced in Chapter 8. The adaptive modulation algorithm introduced in Chapter 9 was then used to derive the throughput rates for an uncoded bit error ratio (BER) of P_b = 10⁻³. No forward error correction (FEC) is applied. Perfect channel knowledge is assumed. As a reference for the SU-MIMO case, eigenbeamforming (see Chapter 8) was used as this MIMO scheme showed the best performance for SU-MIMO (see Chapter 9). In order to compare the throughput results obtained for SU-MIMO eigenbeamforming with the bitrates obtained for MU-MIMO, the bitrates acquired for SU-MIMO have been divided by the number of users M with M = 2 in the inhome PLC scenario. This appears to be an acceptable estimation, considering that SU-MIMO needs to apply TDD (or FDD), using M time instants (or M times the bandwidth) to give the same service MU-MIMO would.

The MIMO PLC channels obtained in the ETSI measurement campaign (see Chapter 5) were used. Figure 19.7 shows an example of the measured channels in one home. In this example, the transfer functions (shown by the solid lines) between 5 outlets were recorded. An example of the MU scenario (refer also to Figure 19.3) is illustrated by the dashed lines, where the channels from outlet P4 to the outlets P2 and P3 are used. The SDD scenario (refer also to Figure 19.5) is illustrated by the dotted lines, where the channels from P1 to the P2 and from P3 to P5 are used. All possible combinations for each scenario (MU or SDD) for each home and all measurement sites were used for the simulations. This results in 357 different channel combinations for the MU scenario and 201 channel combinations for the SDD scenario. The noise is assumed to be additive white Gaussian noise (AWGN) and uncorrelated for the receive ports. The transmitters have N_T = 2 transmit ports and the receivers have N_R = 4 receive ports.

FIGURE 19.7
Depiction of channels measured in one house and possible MU (dashed lines) and SDD (dotted lines) setup.

19.4.2  MU Scenario

Figure 19.8a shows the bitrate depending on the transmit to noise power level. The sum bitrate of both streams was averaged over all 357 channels. The number of users is M = 2, that is, one transmitter communicates to two receivers. Figure 19.8b shows a zoomed plot of Figure 19.8a. There is only a marginal gain of MU-MIMO (with the NuSVD algorithm) compared to SU-MIMO with eigenbeamforming (BF). At a transmit to noise power level of 56 dB, the gain of MU-MIMO is 1.35 Mbit/s or 1.6% compared to SU-MIMO.

FIGURE 19.8
MU – throughput results averaged over 357 channels. (a) Averaged bitrate comparison for 357 channels. (b) Averaged bitrate comparison for 357 channels zoomed around 56 dB.

The main reason for the similar performance between MU-MIMO and SU-MIMO in the MU scenario is the high spatial correlation of the MIMO PLC channels (for details of the spatial correlation in MIMO PLC, refer to Chapters 4 and 5). Assume the following ideal case for MU-MIMO with two receivers. The MU-MIMO precoding vector of the first link is identical to the first column vector of the SU-MIMO eigenbeamforming matrix of this first link, that is, the precoding vector to cancel the interference to the other user is the same precoding vector which maximises the SINR of this spatial stream to the intended user. In addition, the MU-MIMO precoding vector of the second link is identical to the first column vector of the SU-MIMO eigenbeamforming matrix of the second link. If we further assume that the SINR conditions of the second spatial stream of the SU-MIMO eigenbeamforming allow no data transmission on the second spatial stream, then the total bitrate of MU-MIMO would be doubled compared to SU-MIMO eigenbeamforming. Expressed in simplified terms, the channels to the two receivers could be considered to be orthogonal and each link could be optimised independently. However, due to the high spatial correlation, this is not true, and the precoding to cancel out the interference to the other users is far from optimal for this link.

It was observed that the use of random channel matrices as MU-MIMO channels led to a much higher performance gain of MU-MIMO compared to SU-MIMO.

Figure 19.9 shows the cumulative distribution function (CDF) of the throughput for the 357 MU-MIMO channels. There is a small gain of 6.6 Mbit/s at the median point. For the high-throughput area (CDF values between 0.7 and 0.9), there is some gain of MU-MIMO compared to SU-MIMO which can be estimated in 9.6 and 12.1 Mbit/s, respectively, while SU-MIMO shows better results than MU-MIMO for low-throughput values (high coverage, low CDF values).

FIGURE 19.9
MU – CDF over 357 channels computed for a Tx/noise level of 56 dB.

FIGURE 19.10
SDD – averaged bitrate comparison for 201 channels.

19.4.3  SDD Scenario

Following the model of the MU scenario, the throughput results obtained are averaged over 201 real measured channels. Figure 19.10 shows the comparison between SU-MIMO (with eigenbeamforming) and MU-MIMO (NuSVD). The results presented here belong to a scenario with two transmitter–receiver pairs.

Several differences can be observed in comparison with the results introduced for the MU scenario. Figure 19.10 shows a performance gain of MOSDM (NuSVD SDD) compared to SU-MIMO with eigenbeamforming. This BD method, which uses only one spatial stream between each transmitter–receiver pair, shows a throughput increase of approximately 20% in comparison to SU-MIMO, for a transmit to noise power level of 56 dB. This gain is also observed in Figure 19.11, which shows the CDF of the throughput.

FIGURE 19.11
SDD – CDF over 201 channels computed for a Tx/noise level of 56 dB.

19.5  Conclusions

This chapter investigated the adaptation of MU-MIMO algorithms to MIMO PLC. The aim of this work was to analyse the suitability of MU-MIMO algorithms to MIMO PLC systems and study the benefits and improvements that they could bring to the PLC system. The adaptation of wireless MU-MIMO strategies to MIMO PLC faces some challenges, namely, the limited number of ports at the transmitter (only two) and the relatively high spatial correlation.

Two possible scenarios introduced in Section 19.2 have been analysed during this work. The MU scenario with one transmitter and several receivers presents a high degree of similarity to the wireless environment, where one BS offers service to several users located on one cell. On the other hand, the SDD scenario was introduced which involves at least two pairs of transmitters and receivers operating at the same time and on the same frequency band.

In particular, an algorithm called MOSDM [21] was applied to PLC. This algorithm turned out to be a good solution for MIMO PLC [12]. The algorithm applies precoding at the transmitter in order to cancel out inference to other users.

Simulations were performed to evaluate and compare MU-MIMO performance to SU-MIMO systems. The SU-MIMO system reference was based on eigenbeamforming. The channels obtained in the ETSI MIMO PLC measurement campaign (see Chapter 5) were used. A total set of 357 channels were used in the MU scenario, while 201 of them were used for the SDD scenario. The performance in terms of throughput of MU-MIMO was increased by 1.5% and 20% compared to SU-MIMO with eigenbeamforming, for the MU and SDD scenario, respectively. Clearly, only a small improvement is achieved for the MU scenario. However, 20% of throughput increase is reached when two transmitters are simultaneously working, that is, in the SDD scenario.

In the SDD scenario, a synchronisation between the transmitters is required in order for the receivers to estimate and cancel out the interference from the interfering transmitters. In the MU scenario, this synchronisation is not needed since only one transmitter is used. Possible directions for future research in MU-MIMO for the PLC field could include the following:

•  The investigation of the use of bands above 30 MHz. It could increase the efficiency of MU-MIMO since the spatial correlation above 30 MHz is reduced, compared to lower frequencies (see also Chapter 5).

•  Finding a parameter or set of features that would detect, in advance, when MU-MIMO precoding techniques will yield better throughput results than SU-MIMO precoding techniques or vice versa. In this case, a hybrid algorithm could be used which toggles between MU-MIMO and SU-MIMO.

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*  In linear algebra, the null space (or kernel) of a m × n matrix A is the set of all vectors x for which Ax = 0: ker$\left(A\right)=\left\{x\in {ℂ}^n:Ax=0\right\}$ where 0 is the m × 1 zero vector. The kernel of a matrix is a linear subspace of the n-dimensional Euclidean space. The dimension of the null space of A is called the nullity of A [22].