Multiple access spatial modulation
 Nikola Serafimovski^{1}Email author,
 Sinan Sinanović^{1},
 Marco Di Renzo^{2} and
 Harald Haas^{1}
DOI: 10.1186/168714992012299
© Serafimovski et al.; licensee Springer. 2012
Received: 26 March 2012
Accepted: 4 September 2012
Published: 19 September 2012
Abstract
In this study, we seek to characterise the behaviour of Spatial modulation (SM) in the multiple access scenario. By only activating a single transmit antenna for any transmission, SM entirely avoids interchannel interference, requires no synchronisation between the transmit antennas and a single radio frequency chain at the transmitter. Most contributions thus far have only addressed aspects of SM for a pointtopoint communication system. We propose a maximumlikelihood (ML) detector which can successfully decode incoming data from multiple simultaneous transmissions and does not suffer from the nearfar problem. We analyse the performance of the interferenceunaware and interferenceaware detectors. We look at the behaviour of SM as the signaltointerferenceplusnoise ratio goes to infinity and compare it to the complexity and cost equivalent singleinputmultipleoutput (SIMO) system. Two systems are considered to be equivalent in terms of complexity if their respective detection algorithms are of the same order in$\mathcal{O}(\xb7)$notation. Simulation results show that the interferenceaware SM detector performs better than the complexity equivalent multiuser MLSIMO detector by at least 3 dB at an average biterrorratio of 10^{−3}.
Introduction
Multipleantenna systems are fast becoming a key technology for modern wireless systems. They offer improved error performance and higher data rates, at the expense of increased complexity and power consumption[1]. Spatial modulation (SM) is a recently proposed approach to multipleinputmultipleoutput (MIMO) systems which entirely avoids interchannel interference, requires no synchronisation between the transmit antennas and achieves a spatial multiplexing gain[2]. This is performed by mapping a block of information bits into a constellation point in the signal and spatial domains[3]. In SM, the number of information bits, ℓ, that are encoded in the spatial domain can be related to the number of transmit antennas N_{ t } as N_{ t }= 2^{ ℓ }. This means that the number of transmit antennas must be a power of two unless fractional bit encoding[4] or generalised SM[5] are used. SM offers an intrinsic flexibility to trade off the number of transmit antennas with the modulation order in the signal domain to meet the desired data rate. It should be noted that SM is shown to outperform other pointtopoint MIMO schemes in terms of average biterrorratio (ABER)[3].
In the single user scenario, only a single transmit antenna is active at any instance, this avoids the need for complicated interference cancellation algorithms at the SM receiver. In addition, unlike other MIMO schemes, the number of receive antennas is independent of the number of transmit antennas. Several articles are available in literature which are aimed at understanding and improving the performance of SM in various scenarios, e.g.,[6–8]. The study in[6] seeks to improve the ABER performance of SM by introducing trellis coding on the transmitting antennas. The study in[9] shows that the detector complexity of SM is independent of the number of transmit antennas. The optimal detector is known with and without channel state information at the receiver in[10–12]. The optimal power allocation problem for a twotransmit with one receive antenna system is solved in closed form in[13] and the performance of SM in correlated fading channels is considered in[14, 15]. Recent work has also shown that SM can be combined with space–time block codes to attain spectral efficiency gains[16] by exploiting transmitside diversity. At this point, it is worth noting that if we choose to use only the spatial constellation of SM to transmit information, then SM is reduced to spaceshiftkeying (SSK) as proposed in[17]. To this extent, we note that all presented work can be extended to SSK without loss of generality.
MIMO techniques can also be used in relaying networks to improve the diversity, provide multiplexing gains and aid in interference cancellation. To this extent, the orthogonal decode and forward (DF) algorithm decodes the received signal at the relay, then reencodes and retransmits this information, establishing a regenerative system. Outage probabilities, mutual information calculations and transmit diversity bounds for orthogonal amplify and forward (AF) and DF relaying are derived in[18] with the endtoend performance being considered in[19] where DF is shown to perform better in terms of the ABER when compared to AF. However, the ABER of regenerative systems depends on the ABER on the individual links. In particular, since SM is shown to outperform other spatial multiplexing techniques on a single link, the application of SM to relaying systems is also shown to provide significant signaltonoise ratio (SNR) gains when compared to orthogonal DF[20]. Nonetheless, these results are only applicable in a noiselimited relaying system. The deployment of relaying systems around the cell edges, however, may result in interferencelimited systems. Therefore, to enable the deployment of SM in a relaying scenario, the ABER performance of SM on a single link must also be assessed in the interferencelimited environment.
Most contributions thus far, however, have only addressed SM aspects for pointtopoint communication systems, i.e. the single user scenario. Notable exceptions are given in[21, 22], where the authors focus their analysis on scenarios employing SSK. The aim of this study is to characterise the behaviour of SM in the multiuser, interferencelimited scenario and compare it to the complexity and cost equivalent multiuser MIMO system. We emphasise that SM requires only a single radio frequency (RF) chain at the transmit side since only one is active at any particular instance. Requiring only a single RF chain at the transmitter means that multiuser SM is not comparable in terms of cost to the more complicated spatialmultiplexing multiuser systems analysed in[23–26].
Furthermore, the study in[27] shows that the most energy consuming part of a wireless base station is the power amplifiers and consequently RF chains associated with each transmitter. The study in[28] demonstrates that the power requirements of a base station increase linearly with the number of RF chains added. In addition to higher power consumption, multiple RF chains imply higher manufacturing costs and interantenna synchronisation problems. To this extent, SM is an optimal system for utilising the advantages of multiple transmit antennas while still maintaining a single RF chain for Green communications. The aggregate power usage in a system employing SM is significantly lower than a system employing classical MIMO techniques. Furthermore, the lower detection complexity for SM reduces mobile station power usage, enabling a longer battery life for the mobile terminal[9]. Understanding the performance of SM in a multiuser system is necessary to assess its suitability for practical deployment scenarios. In this context, it is of interest if the particular structure of the SM encoding scheme can be exploited to devise novel multiuser detection techniques.
In this study, we first characterise the performance of a single user detector as applied in an interferencelimited scenario, i.e. we analyse a ML interferenceunaware optimal receiver. We then propose an ML detector which can successfully decode incoming data in the multiuser scenario and is not interference limited, i.e. an interferenceaware detector which can successfully decode data from several nodes. For each detector, we develop an analytical framework to support simulation results and closed form solutions are provided to compute the ABER over identical and independently distributed (i.i.d.) Rayleigh fading channels.
The remainder of this article is organised as follows. In the “System model” section, the system and channel models are introduced. In the “Analytical modelling and receiver design” section, the performance of SM in the multiple access scenario is characterised and the analytical modelling for the multiuser detector is proposed. The “Simulation results and discussion” section provides simulation results to substantiate the accuracy of the developed analytical framework. In the “Summary and conclusions” section, we summarise and conclude this study.
System model
The basic idea of SM is to map blocks of information bits onto two information carrying units[3]: (i) a symbol, chosen from a complex signalconstellation diagram, and (ii) a unique transmitantenna, chosen from the set of transmitantennas in an antennaarray, i.e. the spatialconstellation. Jointly, the spatial and signal constellation symbols form a single SM constellation symbol. If, for example, we wish to transmit a total of 4 bits/s/Hz using SM with four available transmit antennas; then the first 2 bits would define the spatialconstellation point identifying the active antenna, while the remaining 2 bits would determine the signalconstellation point that will be transmitted.
where E_{ m } is the average transmit energy per symbol,${N}_{t}^{\left(u\right)}$is the index of the active transmit antenna from a total of${N}_{t}^{\left(u\right)}$available on node u, r is the index of the receive antenna from a total of N_{ r } available on the receiving node,${\alpha}_{\left(u\right)}^{2}$is the power of the channel attenuation coefficients between all receive antennas and all transmit antennas on the link between node u and the receiver,${h}_{{n}_{t}^{\left(u\right)},r}$is the fast fading channel coefficient between the active transmit antenna n_{ t } on node u and the receiving antenna r, x^{(u)}is the signal constellation symbol transmitted from the set of all possible signal constellation points,${\mathcal{X}}^{\left(u\right)}$, for node u and η_{ r }, is additive white Gaussian noise (AWGN), defined as a complex normally random variable with zero mean and variance N_{ o },$\mathcal{C}\mathcal{N}\left(0,{N}_{o}\right)$.
Throughout the study E_{ X }[x^{2}] = 1, meaning the average power in the signal constellation${\mathcal{X}}^{\left(u\right)}$is normalised to one for all u. To avoid repetitive definitions of symbols, we note that symbols denoted with$\widehat{\xb7}$are simply an element of the same set as the symbol without$\widehat{\xb7}$, i.e.$\widehat{x}$ comes from the same set as x. Furthermore, all bold letters are vectors. If we set the signal constellation to be only a single constellation point, where N_{ t } is chosen such that log_{2}(N_{ t }) equals the spectral efficiency, then all presented work can directly be applied to any system employing SSK by simply replacing x^{(u)}= 1.
Analytical modelling and receiver design
We analyse the ML detector for use in the multiple access scenario. The detector computes the Euclidean distance between the received vector signal, y, and the set of all possible received signals, selecting the closest one.
Interferenceunaware detection
where${N}_{t}^{\left(\xi \right)}$is the number of available transmit antennas on node ξ, ·_{F} is the Frobenius norm and${\mathcal{X}}^{\left(\xi \right)}$has a total of M^{(ξ)} constellation points. We note that u represents the index of a general node in the system, and ξ is the index of the desired node whose data stream is being decoded.
define the symbols at the receive antenna r. With this, we can pose y = A + B + η.
represents half of the signaltointerferenceplusnoise ratio (SINR) between node ξ and the receiver. Throughout the study, we average only across the fast fading channel statistics. As (11) shows, γ_{ I }is still dependent on the magnitude of the modulated signal symbols of the interfering nodes, x^{(u)}. This means that all expressions using γ_{ I }maintain their conditioning on the modulated signal symbols of the interfering nodes.
where the u th summation from the (N_{ u }−1) summations above is defined for all${x}^{\left(u\right)}\in {\mathcal{X}}^{\left(u\right)}$and u ≠ ξ with M^{(ξ)} being the cardinality of${\mathcal{X}}^{\left(\xi \right)}$. The symbol$\sum _{\begin{array}{l}{x}^{\left(\xi \right)},{n}_{t}^{\left(\xi \right)},\\ {\widehat{x}}^{\left(\xi \right)},{\widehat{n}}_{t}^{\left(\xi \right)}\end{array}}^{{M}^{\left(\xi \right)}{N}_{t}^{\left(\xi \right)}}$is defined as a fourfold summation, two for all$\phantom{\rule{0.3em}{0ex}}{x}^{\left(\xi \right)},\phantom{\rule{0.3em}{0ex}}{\widehat{x}}^{\left(\xi \right)}\in {\mathcal{X}}^{\left(\xi \right)}$and two for the indices${n}_{t}^{\left(\xi \right)},\phantom{\rule{0.3em}{0ex}}{\widehat{n}}_{t}^{\left(\xi \right)}\in \left[1,\dots ,{N}_{t}^{\left(\xi \right)}\right]$. In addition,${d}_{\xi}(b,\widehat{b})={d}_{\xi}({n}_{t},{\widehat{n}}_{t})+{d}_{\xi}(x,\widehat{x})$, where${d}_{\xi}(\xb7,\widehat{\xb7})$denotes the Hamming distance between the binary representations any two symbols coming from the same set for node ξ.
The average power carried by any signalsymbol constellation in (12) is 1. Nonetheless, a variable amplitude modulation scheme means that the instantaneous SINR changes. The instantaneous SINR must strictly be defined to study the asymptotic behaviour of the system. This is necessary because the instantaneous SINR is an argument of the PEP which is defined using the Qfunction. To obtain the ABER, the PEP must be averaged across all channel realisations and all signalsymbol constellations. A closedform expression for the asymptotic behaviour of (12) and (13) is therefore difficult to obtain. However, we note that (22) and (23) are special cases of (12) and (13) which enable a simpler theoretical analysis of the asymptotic behaviour of the system as the SINR grows to infinity. Simulation results in the “Simulation results and discussion” section show that the asymptotic bounds derived for SM using a constant amplitude modulation scheme are also valid for SM using 4QAM. In addition, it is shown that for the pointtopoint single user scenario, SM using PSK may have a better ABER performance than SM using QAM depending on the number of available transmit antennas[31]. We now proceed with the asymptotic analysis of this system.
Asymptotic analysis of an interferenceunaware detector
In this section, we investigate some asymptotic cases to highlight trends in SM at high SNR. Simulations in the “Simulation results and discussion” section show that the presented results are asymptotically tight in the high SNR region. We first define${\text{SNR}}_{\xi}={E}_{m}{\alpha}_{\left(\xi \right)}^{2}/\left(2{N}_{o}\right)$and${\text{SIR}}_{\xi}=\frac{{\alpha}_{\left(\xi \right)}^{2}}{\sum _{u\ne \xi =1}^{{N}_{u}}{\alpha}_{\left(u\right)}^{2}\phantom{\rule{0.3em}{0ex}}{x}^{\left(u\right)}{}^{2}}$. With these definitions, we look at three systems, the asymptotic performance of SM and SIMO in the noiselimited scenario and the asymptotic performance of SM in the interferencelimited scenario.
SNR_{ ξ }≫1 and SINR≈SNR (noiselimited scenario)
is the expectation of${\left(\frac{{\sigma}_{z}^{2}}{2}\right)}^{{N}_{r}}$across the various possibilities of${\sigma}_{z}^{2}$for$x,{n}_{t},\widehat{x}$ and${\widehat{n}}_{t}$. We are not aware of a closed form solution to the more generic expression for (30) given a variableamplitude modulation. However, we can upper bound (30) by setting${\sigma}_{z}^{2}=min\left(\underset{z}{\overset{2}{\sigma}},2\right)$. In particular, the general form of${\sigma}_{z}^{2}$is defined by the underlying SM signalsymbol constellation size, M^{(ξ)}. To this extent, expressions for${\sigma}_{z}^{2}$are defined using the upper bound for square QAM constellation sizes. A summary of the derivation of (32) is provided in Appendix.
In general, it can be shown that as N_{ r }→∞, the inverse of (35) tends to zero and is always less than 1. Since the inverse of (35) is always less than 1, the addition of an extra receive antenna implies a smaller ratio, which means a lower ABER and hence coding gains for the system.
SIMO system utilising QAM (noiselimited scenario)
where${\gamma}_{\text{QAM}}=\frac{3}{2(\stackrel{~}{M}1)}\gamma $. The interested reader is invited to look at the work in[32] for more details in obtaining (36). We can now pose${\text{ASER}}_{\text{QAM}}/{log}_{2}\left(\stackrel{~}{M}\right)\approx {\text{ABER}}_{\text{QAM}}$provided Gray mapping is used ([32], eq. 8.7), where$\stackrel{~}{M}={M}^{\left(\xi \right)}{N}_{t}^{\left(\xi \right)}$.
Relative coding gains of SM using 4QAM compared to SIMO using $\stackrel{\mathbf{~}}{\mathit{M}}$ QAM
N _{ r }  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{1}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{2}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{3}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{4}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{5}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{6}}$ 

1  1.6  1.9  2.3  2.6  2.9  3.2 
2  0.67  0.38  0.22  0.12  0.068  0.038 
3  0.28  0.075  0.021  0.0057  0.0016  4.4(10^{−4}) 
Relative coding gains of SM using 16QAM compared to SIMO using $\stackrel{\mathbf{~}}{\mathit{M}}$ QAM
N _{ r }  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{1}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{2}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{3}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{4}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{5}}$  ${\mathit{N}}_{\mathit{t}}^{\mathbf{\left(}\mathit{\xi}\mathbf{\right)}}\mathbf{=}{\mathbf{2}}^{\mathbf{6}}$ 

1  3.8359  3.8724  4.0859  4.4031  4.7832  5.2024 
2  0.2551  0.1121  0.0542  0.0278  0.0147  0.0079 
3  0.0245  0.0050  0.0011  0.0003  0.0001  2.0(10^{ −5 }) 
Relative coding gains of SM using PSK compared to SIMO using $\stackrel{\mathbf{~}}{\mathit{M}}$ QAM
N _{ r }  $\stackrel{\mathbf{~}}{\mathit{M}}\mathbf{=}{\mathbf{2}}^{\mathbf{2}}$  $\stackrel{\mathbf{~}}{\mathit{M}}\mathbf{=}{\mathbf{2}}^{\mathbf{3}}$  $\stackrel{\mathbf{~}}{\mathit{M}}\mathbf{=}{\mathbf{2}}^{\mathbf{4}}$  $\stackrel{\mathbf{~}}{\mathit{M}}\mathbf{=}{\mathbf{2}}^{\mathbf{5}}$  $\stackrel{\mathbf{~}}{\mathit{M}}\mathbf{=}{\mathbf{2}}^{\mathbf{6}}$  $\stackrel{\mathbf{~}}{\mathit{M}}\mathbf{=}{\mathbf{2}}^{\mathbf{7}}$ 

1  2  1.9889  2.1333  2.3511  2.6122  2.9022 
2  0.5000  0.2131  0.1067  0.0569  0.0311  0.0171 
3  0.1250  0.0228  0.0053  0.0014  3.7(10^{ −4 })  1.0(10^{ −4 }) 
From Tables1,2, and3, we can conclude that a singleinputsingleoutput system using QAM performs better than SM using QAM or PSK, i.e. the values in the first row of each table are greater than 1. These results are applicable only to SM which means that at least 1 bit must be sent via the signalsymbol.
SIR_{ ξ }≫ 1 and SINR ≈ SIR_{ ξ }(interferencelimited scenario)
in the interferencelimited scenario. We see that the limit of the ABER tends to (34) with a slight, but very important distinction: the system reaches an error floor. This is to be expected when the receiver is interferenceunaware. If γ_{ I } or${\gamma}_{\mathbf{I}}^{\mathrm{PSK}}$ are not large enough, i.e. the channel attenuations of the various nodes are similar, then this study presents an upper bound for the ABER of the system since (34) still defines the behaviour of the system in the limit. There are three consequences that should be considered similar to the noiselimited scenario analysed above: (i) the system error performance improves when more receive antennas are added at the receiver, (ii) the system error performance worsens as more SM constellation points are added, as either${N}_{t}^{\left(\xi \right)}$or M^{(ξ)} is increased and (iii) the detector will fail to decode any data emitted from a node whose desired signal is weaker than the interfering signal. Although analytical work for SM using QAM becomes intractable, numerical results demonstrate that SM using a variableamplitude modulation performs in a similar fashion to SM using PSK and leads to the same conclusions. In the remainder of this study, we show that the near–far problem is completely mitigated by applying a jointly optimal ML detector for SM in an interferencelimited scenario. In other words, all incoming streams will be decoded and, in particular, the error performance of the system will tend to zero as AWGN approaches zero, despite any interference.
Interferenceaware detection
The pairs,$\left({x}^{\left(\Omega \right)},{n}_{t}^{\left(\Omega \right)}\right)$and$\left({\widehat{x}}^{\left(\Omega \right)},{\widehat{n}}_{t}^{\left(\mathrm{\Omega}\right)}\right)$, come from the set of all possible symbolantenna pairings for all nodes, i.e. they independently take values from the set of all possible spatial and signal constellation points, Ω. We define$\text{PEP}\left({x}^{\left(\mathrm{\Omega}\right)},\phantom{\rule{0.3em}{0ex}}{n}_{t}^{\left(\mathrm{\Omega}\right)},\phantom{\rule{0.3em}{0ex}}{\widehat{x}}^{\left(\mathrm{\Omega}\right)},\phantom{\rule{0.3em}{0ex}}{\widehat{n}}_{t}^{\left(\mathrm{\Omega}\right)}\right)$to be the PEP between the symbol x^{(Ω)} emitted from antenna${n}_{t}^{\left(\mathrm{\Omega}\right)}$being detected as symbol${\widehat{x}}^{\left(\mathrm{\Omega}\right)}$emitted by antenna${\widehat{n}}_{t}^{\left(\mathrm{\Omega}\right)}$.
Note that (42) presents an analytical treatment of the most general case of SM using variable amplitude modulation for the signal symbol. Given this, the system using the interferenceaware detector behaves in a similar fashion to the noiselimited system, in that for an arbitrarily high SNR, each user can achieve an arbitrarily low ABER. It should be pointed out that due to the simultaneous detection process, the users with the best SNR will not be able to achieve their singleuserlowerbound (SULB). The exact effect of the additional nodes/users is further discussed in the “Simulation results and discussion” section.
Simulation results and discussion
In this section, we aim to show the performance of the interferenceunaware and interferenceaware detectors proposed in (2) and (41). In particular, we aim to show that (41) can successfully decode the incoming streams for all nodes. Numerical results demonstrate that (12) and (42) provide tight upper bounds for the ABER of the detectors at high SNR in the interferencelimited scenario. Furthermore, we demonstrate that the interferenceaware detector for SM performs better than the ML detector for a multiuser SIMO system using QAM.
The proposed interferenceaware detector is jointly optimal for all nodes and does not suffer from the near–far problem, but it needs full channel state information (CSI) from all possible transmitting antennas to each receiving antenna. In addition, finding the optimal solution is an exponentially complex problem. Assuming each node has the same number of transmit antennas, N_{ t }, and uses the same signal constellation with M points, then the interferenceaware ML detector proposed has$\mathcal{O}\left({\left(M{N}_{t}\right)}^{{N}_{u}}\right)$computational complexity which is proven to be NPcomplete[33].
To justify our$\mathcal{O}(\xb7)$complexity, we point to the key difference between SM and other multiuser MIMO schemes, the signal and spatial domains combine to form a single SM symbol. The constellation size, i.e. the spectral efficiency of any SM system, depends on the multiplication of the number of available transmit antennas and the signalsymbol constellation used, MN_{ t }. This is in stark contrast to other MIMO systems where each spatial branch is used to increase the diversity or multiplexing gains. In such a system, if each transmit antenna is used for multiplexing gains, the system has a maximum spectral efficiency of M^{ N }_{ t }. From here, the detection complexity of a single user SM system is given by$\mathcal{O}\left(M{N}_{t}\right)$, while the detection complexity of a single user MIMO system used for multiplexing gains is given by$\mathcal{O}\left({M}^{{N}_{t}}\right)$. In this case, the two systems have different spectral efficiencies. Alternatively, if the two systems operate at the same spectral efficiency, then their complexities will be of the same$\mathcal{O}$ order, but the cost, in terms of RF chains and power consumption would not be. The aim of this study is to characterise the behaviour of SM in the multiuser, interferencelimited scenario and compare it to the complexity and cost equivalent multiuser MIMO system. As discussed in the “Introduction” section and given the complexity expressions for the single user MIMO system and the single user SM system, it is apparent that the only valid complexity and cost equivalent comparison is to compare multiuser SM with multiuser SIMO. The optimal ML detector for the interferenceaware SIMO system has$\mathcal{O}\left({\left(\stackrel{~}{M}\right)}^{{N}_{u}}\right)$computational complexity, where$\stackrel{~}{M}=M{N}_{t}$from the “SIMO system utilising QAM (noiselimited scenario)” section, making it comparable to the interferenceaware SM detector. Recent work on sphere detection algorithms may be used to alleviate this computational cost[9]. Despite the generality of our results, we restrict our simulation results to two and three node scenarios for the sake of conciseness.
Simulation setup
Results for interferenceunaware detection
Figure4 shows that the increase in diversity resulting from the addition of only a single receive antenna significantly influences the system performance. The addition of the receive antenna increases the Euclidean distance between the received and hypothesis vectors, which results in a lower ABER. Comparing Figure2 with Figure3 and similarly Figure3 with Figure4 where the number of receive antennas is increased in each figure, showing how the addition of a single receive antenna is equivalent to lowering the interference$\left({\alpha}_{\left(u\right)}^{2}\right)$by more than 10 dB. In fact, the effect of each receive antenna is more pronounced as the imbalance between the desired and interfering links increases. By looking at the results for${\alpha}_{\left(2\right)}^{2}=1{0}^{2}$in Figure2, the results for${\alpha}_{\left(2\right)}^{2}=1{0}^{2}$in Figure3, and the results for${\alpha}_{\left(2\right)}^{2}=1{0}^{2}$in Figure4 at an SNR of 40 dB, we see the error rate of the simulation and analytical prediction moving from 2 × 10^{ −1 } in Figure2, to 4 × 10^{−3} in Figure3, to 9 × 10^{−5}in Figure4. This ABER decrease shows how the number of receive antennas dominates the performance of SM in general, and particularly in an interference limited scenario. Figure5demonstrates that the findings can be extended even in the presence of two interfering nodes.
From the presented results it is clear that when the interferenceunaware detector is used, the system ABER plateaus at the derived limits, irrespective of the transmit power being used. As discussed in the “Interferenceunaware detection” section, and as work in[20] has shown, the ABER improves when the number of transmit or receive antennas is increased, i.e. the system achieves coding gains.
Results for interferenceaware detection
To understand this, we can think of the multiuser ML detector as employing interference cancellation for the node with the worse channel attenuation. If the interfering user is sufficiently powerful, then the primary source of errors for the weakest node is the background AWGN rather than the randomness caused by the interfering signal[29]. All users that have good channel conditions can be considered as strong interferers, so when they are removed, the weakest nodes obtain performance closer to their SULB, i.e. the interferenceaware detector is akin to strong interference cancellation for the weakest node. On the contrary, for the nodes with better channel conditions, the primary source of errors is the randomness caused by the interfering signal rather than the background AWGN, which is why the nodes with better channel conditions never perform near their SULB.
Summary and conclusions
In this study, the performance of SM in the multiple access, interferencelimited scenario was investigated. Two ML detectors for use with SM were discussed.
The interferenceunaware detector was defined and studied in the limit as the SNR approached infinity. Its performance over uncorrelated Rayleigh fading channels was studied and a closed form solution for the upper bound of the system was provided. It was shown that this detector inevitably reaches an error floor which is dependent on the system SINR. The exact level was defined as a function and concrete examples were provided. It was shown that the increase in the number of receive antennas has a greater impact on the asymptotic performance of the system compared to reducing the interference in the system. The addition of a single receive antenna resulted in greater SNR gains than reducing the interference,${\alpha}_{\left(u\right)}^{2}$, by more than 10 dB at high SNR. This implies that the number of receive antennas dominates the performance of SM in general, and particularly in an interferencelimited scenario.
The interferenceaware ML detector for SM was proposed. As with the interferenceunaware detector, its performance over uncorrelated Rayleigh fading channels was studied and a closed form solution for the upper bound of the system was provided. In addition to avoiding the error floor present in the interferenceunaware detector, the jointly optimal detector results in a noiselimited scenario for the detection of all transmitted streams, i.e. an arbitrarily small ABER can be obtained by any user for a sufficiently high SNR. On the one hand, for the same spectral efficiency, increasing the number of transmit antennas at each of the nodes from 2 to 4 resulted in SNR gains of around 2 dB. This measure did not, however, have any effect on the coding gain difference between the ABER curves. On the other hand, increasing the number of receive antennas increased the diversity of the system. This, increased the coding gain difference between the ABER curves of the nodes because the receiver could distinguish the channels more easily and better mitigate interference. The impact on the diversity and coding gains further shows the importance of the number of receive antennas in any SM system. A limiting factor, as with all ML detectors, is the complexity. In addition, the receiver must have channel knowledge from all transmitting nodes. These two limitations constrain the application of this detector to the uplink scenario. The interferenceaware detector enabled SM to perform better in terms of ABER than the complexity and cost equivalent multiuser SIMO system in an interferencelimited environment.
This study demonstrated that in order to effectively apply SM in an interferencelimited scenario, the number of receive antennas should be maximised. Although more computationally complex than the interferenceunaware detector, the interferenceaware detector can guarantee that the system does not reach an error floor.
Appendix
Derivation of (32)
where ε ∈ {(1,1),(2,1),(3,1)} and d_{ ε }are the distances defined in Figure12 and${g}_{\text{QAM}}=\frac{3}{2\left({M}^{\left(\xi \right)}1\right)}$. We introduce the normalising factor g_{ QAM }, given in[32], to maintain unity power in our constellation. We see that there are only four possible combinations that satisfy (46). In particular

$\left({d}_{(1,1)}^{2}+{d}_{(1,1)}^{2}\right){g}_{\text{QAM}}=0.4$, occurs (^{ M }(ξ)^{)2}/8 = 32 times,

$\left({d}_{(1,1)}^{2}+{d}_{(2,1)}^{2}\right){g}_{\text{QAM}}=1.2$, occurs (M^{(ξ)})^{2}/4 = 64 times,

$\left({d}_{(1,1)}^{2}+{d}_{(3,1)}^{2}\right){g}_{\text{QAM}}=2$, occurs (M^{ξ)})^{2}/8 = 32 times and

$\left({d}_{(2,1)}^{2}+{d}_{(2,1)}^{2}\right){g}_{\text{QAM}}=2$, occurs (M^{(ξ)})^{2}/2 = 128 times.
Having established the derivation of$\stackrel{~}{{\psi}_{1}}$, we turn our attention to$\stackrel{~}{{\psi}_{2}}$which corresponds to${\sigma}_{z}^{2}={x}^{\left(\xi \right)}{\widehat{x}}^{\left(\xi \right)}{}^{2}$. If we look at Figure12, we see there are additional combinations aside from the ones denoted. To this extent, our bound,${\sigma}_{z}^{2}=min\{\underset{z}{\overset{2}{\sigma}},2\}$, serves to simplify the counting and set many of them to 2. In particular, we see that when${\sigma}_{z}^{2}={d}_{(5,2)}^{2}$, then${\sigma}_{z}^{2}>2$. In this case, we bound${\sigma}_{z}^{2}$to 2 and realise that d_{(4,2)} is the largest distance that we must account for in our expectation analysis. We note that our approach is applicable to larger constellation sizes and, after some careful counting, we see that in general for a given constellation size of M^{(ξ)}we have that

d_{(1,2)} occurs a total of${D}^{\left(1\right)}=4\sqrt{{M}^{\left(\xi \right)}}\left(\sqrt{{M}^{\left(\xi \right)}}1\right)$times,

d_{(2,2)} occurs a total of${D}^{\left(2\right)}=4\sqrt{{M}^{\left(\xi \right)}}\left(\sqrt{{M}^{\left(\xi \right)}}2\right)$times,

d_{(3,2)} occurs a total of${D}^{\left(3\right)}=4\left(\sqrt{{M}^{\left(\xi \right)}}1\right)\left(\sqrt{{M}^{\left(\xi \right)}}1\right)$times and,

d_{(4,2)} occurs a total of${D}^{\left(4\right)}=8\left(\sqrt{{M}^{\left(\xi \right)}}1\right)\left(\sqrt{{M}^{\left(\xi \right)}}2\right)$times.
By looking at Figure12, we can easily see that for M^{(ξ)}= 16:

${d}_{(1,2)}^{2}\phantom{\rule{0.3em}{0ex}}{g}_{\text{QAM}}=0.4$, occurs exactly 48 times,

${d}_{(2,2)}^{2}\phantom{\rule{0.3em}{0ex}}{g}_{\text{QAM}}=1.6$, occurs exactly 32 times,

${d}_{(3,2)}^{2}\phantom{\rule{0.3em}{0ex}}{g}_{\mathrm{QAM}}=0.8$, occurs exactly 36 times and

${d}_{(4,2)}^{2}\phantom{\rule{0.3em}{0ex}}{g}_{\text{QAM}}=2$, occurs exactly 48 times.
which when simplified becomes (32).
Reaching (38) from (34) with (31) and (37)
Derivation of (43)
such that$f\left(\stackrel{~}{\beta}\right)=\frac{1}{2}\left(1\sqrt{\frac{\stackrel{~}{\beta}}{1+\stackrel{~}{\beta}}}\right)$and$\stackrel{~}{\beta}=\frac{{E}_{m}}{4{N}_{o}}$$\sum _{u=1}^{{N}_{u}}{\alpha}_{\left(u\right)}^{2}{\vartheta}_{\left(u\right)}$.
Declarations
Acknowledgements
We gratefully acknowledge the support from the Engineering and Physical Sciences Research Council (EP/G011788/1) in the United Kingdom for this study.
Authors’ Affiliations
References
 Mietzner J, Schober R, Lampe L, Gerstacker WH, Hoeher PA: Multipleantenna techniques for wireless communications—a comprehensive literature survey. IEEE Commun. Surv. Tutor 2009, 11(2):87105.View Article
 Mesleh R, Haas H, Lee Y, Yun S: Interchannel interference avoidance in MIMO transmission by exploiting spatial information. In Proc. of the 16th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), vol. 1. (Berlin Germany, 11–14 September 2005), pp. 141–145;
 Mesleh R, Haas H, Sinanović S, Ahn CW, Yun S: Spatial modulation. IEEE Trans. Veh. Technol 2008, 57(4):22282241.View Article
 Serafimovski N, Sinanović S, Mesleh RY, Haas H, Di Renzo M: Fractional bit encoded spatial modulation (FBE–SM). IEEE Commun. Lett 2010, 14(5):429431.View Article
 Younis A, Serafimovski N, Mesleh R, Haas H: Generalized spatial modulation. In Asilomar Conference on Signals, Systems, and Computers. (Pacific Grove, CA, USA, November 2010);
 Mesleh R, Di Renzo M, Haas H, Grant PM: Trellis coded spatial modulation. IEEE Trans. Wirel. Commun 2010, 9(7):23492361.View Article
 Younis A, Mesleh R, Haas H, Grant PM: Reduced complexity sphere decoder for spatial modulation detection receivers. In 2010 IEEE Global Telecommunications Conference (GLOBECOM 2010). (Miami, Florida, USA, December 2010), pp. 1–5;
 Renzo MD, Haas H: Performance analysis of spatial modulation. In International ICST Conference on Communications and Networking in China (CHINACOM). (Beijing, China, August 2010, pp. 1–7);
 Younis A, Mesleh R, Haas H, Renzo M: Sphere decoding for spatial modulation. In Proc. of IEEE International Conference on Communications (IEEE ICC 2011). (Kyoto, Japan, 5–9 June 2011), pp. 1–6;
 Jeganathan J, Ghrayeb A, Szczecinski L: Spatial modulation: optimal detection and performance analysis. IEEE Commun. Lett 2008, 12(8):545547.View Article
 Hwang SU, Jeon S, Lee S, Seo J: Softoutput ML detector for spatial modulation OFDM systems. IEICE Electron Exp 2009, 6(19):14261431. 10.1587/elex.6.1426View Article
 Di Renzo M, Haas H: Spatial modulation with partialCSI at the receiver: optimal detector and performance evaluation. In Proceedings of the 33rd IEEE conference on Sarnoff. (IEEE Press, Piscataway, NJ, USA, 2010), pp. 58–63; http://portal.acm.org/citation.cfm?id=1843486.1843498
 Di Renzo M, Haas H: Improving the performance of Ssace shift keying (SSK) modulation via opportunistic power allocation. IEEE Commun. Lett 2010, 14(6):500502.View Article
 Handte T, Muller A, Speidel J: BER analysis and optimization of generalized spatial modulation in correlated fading channels. In Vehicular Technology Conference Fall (VTC Fall2009). Anchorage Alaska, USA, IEEE, (September 2009), pp. 1–5;
 Renzo MD, Haas H: Bit error probability of SMMIMO over generalized fading channels. IEEE Trans. Veh. Technol 2012, 61(3):11241144.View Article
 Basar E, Aygolu U, Panayirci E, Poor VH: Spacetime block coded spatial modulation. IEEE Trans. Commun 2011, 59(3):823832.View ArticleMATH
 Jeganathan J, Ghrayeb A, Szczecinski L, Ceron A: Space shift keying modulation for MIMO channels. IEEE Trans. Wirel. Commun 2009, 8(7):36923703.View Article
 Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior. IEEE Trans. Inf. Theory 2004, 50(12):30623080. 10.1109/TIT.2004.838089MathSciNetView ArticleMATH
 Hasna MO, Alouini MS: Endtoend performance of transmission systems with relays over Rayleighfading channels. IEEE Trans. Wirel. Commun 2003, 2(6):11261131. 10.1109/TWC.2003.819030View Article
 Serafimovski N, Sinanovic S, Di Renzo M, Haas H: Dualhop spatial modulation (DhSM). In Proc. of the Vehicular Technology Conference (VTC Spring). Budapest, Hungary, IEEE, 15–18, May 2011), pp. 1–5;
 Di Renzo M, Haas H: On the performance of SSK modulation over multipleaccess Rayleigh fading channels. In IEEE Global Telecommunications Conference (GLOBECOM). (Miami, Florida, USA, December 2010), pp. 1–6;
 Di Renzo M, Haas H: Bit error probability of spaceshift keying MIMO over multipleaccess independent fading channels. IEEE Trans. Veh. Technol 2011, 60(8):36943711.View Article
 Duplicy J, Badic B, Balraj R, Ghaffar R, Horvth P, Knopp FKR, Kovacs IZ, Nguyen HT, Tandur D, Vivier G: MUMIMO in LTE systems. EURASIP J. Wirel. Commun. Netw 2011, 2011: 13. 10.1186/16871499201113View Article
 Andrews JG, Choi W, Heath Jr RW: Overcoming interference in spatial multiplexing MIMO cellular networks. IEEE Wirel. Commun. Mag 2007, 14(6):95104.View Article
 Gesbert D, Kountouris M, Heath RW, byoung Chae C, Salzer T: From single user to multiuser communications: shifting the MIMO paradigm. IEEE Signal Process. Mag 2007, 24: 3646.View Article
 Tan CW, Calderbank AR: Multiuser detection of alamouti signals. IEEE Trans. Commun 2009, 57: 20802089. http://dl.acm.org/citation.cfm?id=1651065.1651094View Article
 Auer G, Gianni V, Godor I, Skillermark P, Olsson M, Imran M, Gonzalez M, Desset C, Blume O, Fehske A: How much energy is needed to run a wireless network? IEEE Wirel. Commun 2011, 11: 4049.View Article
 Desset C, Debaillie B, Giannini V, Fehske A, Auer G, Holtkamp H, Wajda W, Sabella D, Richter F, Gonzalez MJ, Klessig H, Godor I, Olsson M, Imran MA, Ambrosy A, Blume O: Flexible power modeling of LTE base stations. In IEEE Wireless Communications and Networking Conference (WCNC). (Paris, France, April 2012), pp.2858–2862;
 Verdu S: Multiuser Detection. Cambridge University Press, Cambridge,MA; 1998.MATH
 Alouini MS, Goldsmith A: A unified approach for calculating error rates of linearly modulated signals over generalized fading channels. IEEE Trans. Commun 1999, 47(9):13241334. 10.1109/26.789668View Article
 Di Renzo M, Haas H: Bit error probability of spatial modulation (SM) MIMO over generalized fading channels. IEEE Trans Veh. Technol 2012, 61: 11241144.View Article
 Simon MK, Alouini M: Digital Communication over Fading Channels. John Wiley & Sons, Inc, NEw York; 2005. ISBN: 9780471649533
 Verdu S: Computational complexity of optimum multiuser detection. Algorithmica 1989, 4: 303312. http://dx.doi.org/10.1007/BF01553893 10.1007/BF01553893MathSciNetView ArticleMATH
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.