A Precoding Aided Space Domain Index Modulation Scheme for Downlink Multiuser MIMO Systems

In this correspondence, we propose a space domain index modulation (IM) scheme for the downlink of multiuser multiple-input multiple-output (MU-MIMO) systems. Instead of the most common approach where spatial bits select active receiver antennas, in the presented scheme the spatial information is mapped onto the transmitter side. This allows IM to better exploit large dimensional antenna settings which are typically easier to deploy at the base station. In order to mitigate inter-user interference and allow single user detection, a precoder is adopted at the BS. Furthermore two alternative enhanced signal construction methods are proposed for minimizing the transmitted power or enable an implementation with a reduced number of RF chains. Simulation results for different scenarios show that the proposed approach can be an attractive alternative to conventional precoded MU-MIMO.


I. INTRODUCTION
The ever-growing demand for faster and more energy efficient communications has been driving research towards finding newer and more effective solutions for the physical layer of beyond 5G (B5G) networks. Amongst the different possible approaches, index modulation (IM) schemes [1] have been captivating a lot of research interest in the wireless community. IM schemes rely on the indices of the building blocks to convey additional information bits and are able to offer interesting tradeoffs in terms of error performance, complexity, and spectral efficiency (SE), making them potential candidates for B5G [2]. Well-known examples of IM are spatial modulations (SM) [3] and generalized SMs (GSM) [4], which are multiple-input multiple-output (MIMO) schemes where part of the information bits are used for selecting one or more active antennas, which then transmit M-ary modulated symbols. A lot of research work has been done on SM/GSM, mostly focusing on single-user (SU) [5]- [7], with a few works also addressing uplink [8]- [10] and downlink [11]- [16] multiuser (MU) scenarios. In the case of downlink, most of the proposed space domain IM schemes typically fit into two main types of approaches. The first one [11]- [13] is often referred to as receive SM (R-SM), and is based on the idea of using part of the information bits for selecting a subset of receiver antennas that will receive the intended signals free of inter-channel interference. This approach resorts to a specially designed precoder and can reduce the complexity of the receivers. Even though only a small subset of the antennas receives symbols, they must all be active at all times. Furthermore, the small number of antennas that can typically be deployed at the mobile equipment limits the SE gains. The second approach [14]- [16] avoids this limitation by mapping the spatial bits to the transmitter side. In [14] the authors proposed a precoder that allows the MU system to be decomposed into independent SU-SM systems. Only simple SM is considered, with each user's bits mapped to the active transmit antenna index and an M-QAM constellation symbol. At the receiver, single stream maximum likelihood (ML) detection is applied. In [15], the authors assume single antenna receivers in a MU-SM system and adopt a similar approach, with a precoder designed for eliminating MU interference (MUI), enabling SU ML detection. In [16], block diagonalization (BD) precoding is combined with GSM. Spatial bits are broadcast to all users and only the conventional modulated symbols carry unicast messages. BD is applied to remove MUI while a ML detector searches amongst all the different possibilities of active antenna combinations.
Motivated by the work above, in this correspondence we consider the downlink of a MU-MIMO system where a base station (BS) transmits precoded space domain IM symbols, similarly to a virtual GSM. To enable an efficient implementation of the system, two alternative enhanced signal construction methods are presented. The main contributions of this letter are summarized as follows:  A space domain IM scheme which we refer to as precodingaided transmitter side space domain index modulation (PTSDIM) is designed for the downlink of a MU-MIMO system. The approach uses a precoder that mitigates MUI and allows SU GSM detection. Instead of the usual approach where spatial information selects active receiver antennas, in PTSDIM spatial information is mapped onto the transmitter. The advantage lies on the possibility to work with large dimensional antenna settings which are usually much easier to deploy at the BS than at the users, thus enabling higher SE.  To exploit the implicit sparsity of PTSDIM signals and the availability of large numbers of transmit antennas, we propose the transmission of an equivalent signal with reduced power that produces the same received signal at all the receivers. The approach is transparent to the receivers and, therefore, does not require additional complexity.  We also propose a signal construction method which allows the use of a reduced number of active antennas, simplifying the hardware implementation at the BS. Similarly to the other approach, this method is also transparent the receivers.

II. SYSTEM MODEL
Let us consider the downlink of a MU-MIMO system where a BS equipped with Ntx antennas transmits to Nu users, each with Nrx antennas. We consider that the signal is represented as bits transmitted on a PTSDIM symbol. We consider the availability of channel state information at the transmitter (CSIT) which is used for preprocessing the symbols with a linear precoder The transmitted signal can thus be written as 1 1 After propagating through a flat fading channel, the baseband signal received at the k th user can be represented using  is the channel matrix between the BS and user k and is the noise vector containing independent zero-mean circularly symmetric Gaussian samples with covariance (3) is MUI which can be eliminated using a BD method [17]. In this case, the resulting H F matrix, with  [17]. Therefore, we assume a simple BD precoder without any power loading optimization (even though one can be employed for accomplishing power control between users). Each precoder matrix k F is designed in order to enforce that i k  H F 0 for all i k  and can be directly obtained from the null space of matrix An orthonormal basis can be found from the singular value decomposition (SVD)  (5) is basically a SU GSM received signal model, detection can be accomplished using a conventional GSM receiver such as the ordered block minimum mean-squared error (OB-MMSE) detector [18].

III. ENHANCED SIGNAL GENERATION
The direct application of the proposed BD precoder does not exploit the sparsity of s. In this section we describe two different approaches for implementing the transmission. Both methods are completely transparent to a receiver, which does not need to know if the BS transmitted the original precoded signal or the alternative one. Denoting the active AIC for user k as  

A. Transmitted Power Minimization
In the first approach, we design the transmitted signal with minimal power while enforcing the received signals arriving at the receivers to match the ones resulting from direct BD precoding at that time instance. Formally this can be written as This formulation represents an equality constrained least-norm problem whose solution for a "fat" H (i.e., with tx

B. Active Antenna Reduction
In the second approach, the transmitted signal is redesigned in order to reduce the number of active antennas, on N , while keeping the received signals close to the BD precoded ones.
This problem can be formulated as where λ>0 is a parameter commonly used in   -penalized least squares problems [19]. While a general interior point method can be applied to (10), the computational complexity is too high for the envisioned large problem settings. Therefore, we propose a proximal based iterative algorithm. First, we apply a projected gradient step [20] to the partial minimization problem which results in where q is the iteration number,  is the step size, denotes the Euclidean projection onto  . An estimate for the solution of the complete problem can then be obtained by finding an   q x that minimizes the   -norm while still remaining close to the projected gradient step   q x  . This is equivalent to solving which, by definition, corresponds to the proximal operator . This proximal operator is given component-wise by the (complex) soft threshold function as   To improve the typical slow convergence rate of gradient based methods [23], we can apply the acceleration described in [21] which results in (12) being replaced by two steps In this section, we evaluate the performance of the proposed PTSDIM schemes. It is assumed that all users experience the same path-loss with all the elements of H being independently drawn according to a complex Gaussian distribution    Fig. 1 shows the bit error rate (BER) as a function of the signal to noise ratio (SNR) per user for different configurations of PTSDIM in a MU scenario with Nu=15, Ntx=105, Ns=7, Na=2, Nrx=4 and 64-QAM. This corresponds to a SE of 16 bits per channel use (bpcu) and per user. An OB-MMSE detector [18] is employed at each receiver. As expected, the PTSDIM with minimum transmission power achieves the best performance with a gain over the simple PTSDIM of about 11dB at a BER of 10 -4 . Regarding the curves with active antenna reduction (AAR), we can observe that it is possible to improve the performance of the simple PTSDIM using fewer active antennas. In fact, with only Non=80 (a reduction of 23%), the performance becomes very close to the minimum power curve. It is important to note that when the number of active antennas becomes too low, the intended signal approximation degrades and an irreducible BER floor can arise (case of Non=60). It can also be observed that the AAR algorithm achieves better performance when using the proposed acceleration and polishing steps.
transmit antennas scales accordingly, only one curve is shown. The number of total active antennas in PTSDIM (Non) is always kept the same as in BD MU-MIMO. It can be seen that achieving a higher SE through additional antennas (in this case only one extra per user) while keeping the number of active antennas the same, can be advantageous in terms of SNR when compared with the use of a higher modulation order. It can also be observed that when Nu increases, the set of antennas available at the BS also enlarges and the gains of PTSDIM improve, reaching a value of 3dB for the scenario of 30 users.
V. CONCLUSIONS In this paper we described a MU-MIMO system where a base station transmits precoded space domain IM symbols. The spatial bits are mapped at the transmitter side and combined with a precoder which allows the proposed MU-MIMO scheme to become equivalent to several independent SU GSM-MIMO transmissions. Two different signal construction methods were proposed enabling a more efficient system implementation, with potential performance gains over conventional precoded MU-MIMO.