Abstract
Motivated to enhance the error performance (EP) of generalised complex quadrature spatial modulation (GCQSM) systems, this study proposes a scheme that builds on GCQSM and uses hexagonal quadrature amplitude modulation (HQAM) constellations which have the advantages of a maximised Euclidean distance with relatively low peaktoaverage power ratio, compared to conventional QAM (CQAM) systems. This in turn, leads to an enhancement of the EP of GCQSM schemes. The proposed scheme utilises a rotated hexagonal 8QAM (H8QAM) set. Thus, the proposed scheme is herein named; Generalised QSM using H8QAM (GQSMH8QAM). In this study, the EP of the proposed GQSMH8QAM scheme is investigated over Rayleigh frequency flatfading channels with additive white Gaussian noise. Additionally, a theoretical average bit error probability (ABEP) expression of the GQSMH8QAM scheme is formulated and validated using Monte Carlo simulations. Compared to simulation results, the ABEP proves to be increasingly tight at high signaltonoise ratio values. Obtained simulation results also show an improvement in the EP of the GQSMH8QAM scheme over various SM schemes like GCQSM, CQSM and conventionalgeneralized spatial modulation (CGSM), at the same spectral efficiency (SE). An improvement in the EP of 0.61 dB with SE of 8 bits/s/Hz is seen in \(4\times 4\) GQSMH8QAM over \(4\times 4\) GCQSM using C8QAM, 2.58 dB over \(4\times 4\) CQSMC64QAM and a gain of 4.85 dB over \(4\times 4\) CGSMC64QAM.
Introduction
Nowadays, the world has been forced to adapt to the the Covid19 living conditions. Thus, there has been a high demand and compulsory need for the human race to work at home/indoors with no/minimum physical interaction. This has resulted in a high dependence on the internet and wireless communications networks. Hence, the need for wireless communications with reliable link margins and high data rates. Multipleinput multipleoutput (MIMO) systems have been shown to improve/enhance spectral efficiency (SE) and /or improve the link reliability of wireless networks against multipath fading^{1}. The idea behind MIMO systems is to improve the error performance (EP) of wireless communication networks using spatial diversity and/or enhancing SE using spatial multiplexing^{1}.
Conventional spatial modulation (CSM) is one of many MIMO techniques that can improve SE by the utilization of both the spatial constellation (transmit antenna indices) and the signal constellation^{2}. In CSM, information bits are categorized into two; the bits for transmitting antenna indices and the symbols bits^{2}. Only a single transmit antenna per channel use is needed in CSM. Hence, compared to other transmission schemes like Alamouti spacetime block codes and vertical Bell layered spacetime scheme (VBLAST), CSM was found to have a better EP because it precludes interchannel interference ICI and interantenna synchronization^{3}. Also, CSM was found to be more bandwidth efficient when compared to conventional modulation with a single transmit antenna^{4}. This is because it’s specified transmit antenna per time slot, also conveys additional information^{4}. However, the number of physical antennas that can be employed at the transmission side limits the SE of CSM schemes, and thus, conventional generalized spatial modulation (CGSM) was proposed to alleviate this disadvantage^{5}. CGSM overcomes the CSM disadvantage by curbing the restriction of transmit antennas being a power of two in CSM schemes^{5}. Thus, in CGSM, the input data stream bits are mapped into two categories, the symbol signal bits and the spatial constellation bits (antenna pairs combination bits). In CGSM, a unique combination/grouping of active transmit antennas per time slot is represented by an index. This unique combination/grouping relies on the input random data stream^{5}. Thus, CGSM is different from CSM as two or more transmit antennas are activated per channel use as compared to one active transmit antenna per channel use in CSM^{5}. Hence, CGSM enhances the overall SE by \(\log _2(N_c)\); where \(N_c=\lfloor \log _{2}\begin{pmatrix} N_T \\ N_A\end{pmatrix} \rfloor _{2^p}\) pairs of permissible transmit antennas for transmission, with \(\lfloor \cdot \rfloor _{2^p}\) representing the largest integer that is less than or equal to \((\cdot )\), (that is, an integer p power of 2). Whereas, \(\begin{pmatrix} \cdot \\ \cdot \end{pmatrix}\) represents the binomial coefficient of the argument. \(N_T\) is the total number of transmit antennas for the CGSM scheme, and \(N_A\) has been discussed before in^{5}. The chosen value of use for \(N_A\) in this paper, is 2.
In the past decade, an effort to improve the EP and/or SE of SM systems, led to the derivation of conventionalquadrature spatial modulation (CQSM) in Mesleh et al.^{6}. CQSM extends the spatial constellation into two dimensions (the inphase (I) and quadrature (Q) dimensions). The I dimension is for transmitting the real part of a single amplitude/phase modulated symbol, and the Q dimension is for transmitting the imaginary part of the single amplitude/phase modulated symbol^{6,7}. The I components are modulated into cosine carriers and Q components are modulated into sine carriers, respectively. Hence with that, CQSM eliminates ICI, thereby enhancing the EP of SM systems^{8}. Additionally, CQSM enhances the SE (\(\mathfrak {m}\)) by \(\log _2(N_T)\) bits/s/Hz compared to the CSM, which achieves \(\mathfrak {m}=\log _2(M) + \log _2(N_T)\) bits/s/Hz; where M is the amplitude/phase modulation order^{6}. In Li et al.^{9}, CQSM is used where indexes of the designated receive antennas (\(N_R\)) are used to convey information. Thus, in Li et al.^{9}, an amplitude/phase modulated symbol is precoded so that only a single receive antenna is activated, thereby conveying more information to the receiver. Also, in Kim^{10}, CQSM has been used in antenna selection schemes to improve the EP and simultaneously reduce the detection complexity. However, despite the capability of improving the EP and SE of SM wireless communication networks, CQSM has a disadvantage of utilising many transmit antennas as compared to conventional spatial multiplexing (CSMux) techniques, thereby governing the improvement of the EP of SM systems^{11}.
Motivation
Motivated to improve the EP and/or SE of SM systems with a minimal number of transmit antennas, a generalized complex quadrature spatial modulation (GCQSM) scheme was proposed in Mohaisen et al.^{12}. The GCQSM scheme builds on CQSM by adding the attributes of CGSM to the CQSM scheme. It improves the SE by transmitting two amplitude/phase modulated symbols drawn from two different constellation sets at each channel use, using unique combinations of generalized I domain antennas and Q domain antennas. The two symbols could be transmitted from the same antenna at one point, and thus the total modulation set at the transmitter becomes the Minkowski sum of the original two constellation sets^{12}. Therefore, compared to CSM and CQSM, this leads to an increase in the size of the modulation set and a decrease in the minimum distance between transmitted symbols at the transmitter of GCQSM systems, which in turn results in the degradation of the EP of the GCQSM scheme^{13}. This results in a poor EP because the detector depends on the minimum Euclidean (MED) distance between the transmitted vector symbols^{14}.
To further enhance/improve the EP of SM systems, we propose to build on GCQSM by equipping it with hexagonal QAM (HQAM) constellations which have the benefits of a maximised MED (MMED) with a relatively low peaktoaverage (PA) power ratio^{15}. HQAM systems are QAM systems with a hexagonal structure and they are densely packed with a MMED. In^{15}, it was found that the structure of HQAM systems gave them an advantage of outperforming conventionalQAM (CQAM) systems by approximately 0.6 dB. SM and QSM systems with HQAM were introduced by Cogen et al. in^{16} and in^{17}. HQAM was found to be more energy efficient than square QAM, rectangular QAM and crossQAM. Hence, its use was found for many applications including multicarrier systems, MIMO systems, SM, QSM and advanced channel coding^{15} and^{17}. Cogen et al.^{17}, also solved the error floor degradation of the EP of QSM systems by rotating the HQAM symbols. Also, in^{12}, Mohaisen et al. investigated the rotation of symbols in GCQSM systems which also led to an improvement in the EP of GCQSM systems. In Naidoo et al.^{14}, and Singya et al.^{15}, the EP of SM systems was also found to depend on the MMED between two neighbouring constellation points and the average symbol energy. Hence, we propose to use HQAM systems as they have been proven to have a MMED and relatively low PA which improves the EP of SM schemes, as compared to CQAM systems.
Thus, to improve the EP of SM systems, we propose a GQSM scheme that builds on GCQSM^{13}, by combining it with rotated HQAM systems like Cogen et al.^{17}. The proposed scheme adopts rotated hexagonal8QAM (H8QAM) and utilises generalized combinations of the I and Q domain antennas. This is because H8QAM has a MMED between neighbouring symbols and a relatively low PA as compared to C8QAM^{18}. Also, the proposed scheme uses generalized antenna combinations because they help enhance and/or improve EP and SE^{5}. Thus, the benefits of GCQSM and rotated HQAM systems lead to an improved EP. The H8QAM concept could be extended to other various HQAM constellations like H32QAM and others, however as discussed in^{18}, this results in a poor EP as the size of the hexagonal lattice structure increases. Hence the proposed scheme utilises only H8QAM as it is the optimum H8QAM compared to others of the same family^{18}. A constellation comparison diagram between rotated H8QAM and C8QAM is included in "Appendix 1" of this paper. The creation of H8QAM symbols was discussed in^{18}. It was also well covered including the solution of the error floor in the EP of QSM systems by Cogen et al.^{17}. This paper adopts the same way of creation and rotation of H8QAM symbols like in^{17}.
In summary, this paper, we propose a GQSM scheme that improves the EP of SM systems by equipping GCQSM systems with rotated H8QAM^{17}. The proposed scheme uses the CQSM approach of splitting two rotated H8QAM symbols into their respective I and Q components. The split symbols are then transmitted by generalized combinations of the I and Q domain antennas like in CQSM. This helps with eliminating ICI and thereby further enhancing the EP of the proposed scheme, since the I and Q components of the CQSM scheme are modulated into cosine and sine carriers^{8}. Thus, the main core contributions of this paper are:

Proposal of a new GQSM scheme called GSM with HQAM (GQSMH8QAM), that improves the EP of GCQSM systems by equipping them with rotated H8QAM. The H8QAM systems have a lattice structure that has a MMED and relatively low PA compared to CQAM. This helps improve the EP of SM schemes.

Derivation of an upper bound average bit error rate (ABER) expression for the GQSMH8QAM scheme over independent and identically distributed (i.i.d) Rayleigh frequencyflat RFF fading channels.

Validation of the derived analytical bound using Monte Carlo simulation results.
Finally, the paper is organised as follows. We introduce our proposed GQSMH8QAM system in Section "System model". It is presented in the form of a MIMO \(N_T \times N_R\) H8QAM system shown in Figs. 1 and 2. For this system, there exists \(\mathfrak {n}=\begin{pmatrix}N_T \\ N_A\end{pmatrix}\) possible transmission antenna pairs of which only \(N_c\) pairs are allowed for transmission, with \(N_c\le \mathfrak {n}\). The receiver computational complexity analysis (CCA) based on real valued multiplications and additions of the GQSMH8QAM scheme is then presented in Section "MLD CCA of the proposed GQSMH8QAM schemebased on real valued multiplications and additions", followed by the performance analysis of the proposed scheme in Section "Proposed error performance analysis of GQSMH8QAM". Section "Simulation and numerical results analysis" then dwells on the results and performance comparisons. Finally, we conclude the paper in Section "Conclusion".
Notation
\((\cdot )^*\) represents the conjugate and \(\cdot \) denotes the Euclidean norm operator. We use bold lowercase letters to denote column vectors and uppercase letters for matrices, respectively. Scalar quantities are then represented by regular letters. For a set of \(S\times L\) complex matrices, we use \(\mathbb {C}^{S\times L}\). For complex arguments, we use \(\Re (\cdot )\) and \(\Im (\cdot )\) to represent the real and imaginary parts, respectively. \(Q(\cdot )\) is for representing the Gaussian Qfunction, \(E\{\cdot \}\) is for denoting the expectation operator, \(\underset{w}{{\text {argmin}}}\{{\cdot }\}\) and \(\underset{w}{{\text {argmax}}}\{{\cdot }\}\) represent the minimum or maximum value of an argument with respect to w, respectively. \([\cdot ]^T\) and \((\cdot )^H\) are the transpose and Hermitian operators, respectively. Finally, \(\left\Vert \cdot \right\Vert _F\) denotes the Frobenius norm operator.
System model
For the proposed scheme in Figs. 1 and 2, the input data stream is divided into two categories, the spatial input bits (antenna pair indices) and two rotated H8QAM symbol bits. The first input category is the spatial information. Spatial input bits (\(\varvec{d}=\log _{2}N_c\)) are assigned to a unique generalized \(K{\rm th}\) antenna pair of transmit antennas (\(k_1\) and \(k_2\)), which are for the I dimension of the system. The same \(\varvec{d}\) bits are assigned to a second but different generalized \(K{\rm th}\) antenna pair of transmit antennas (\(k_3\) and \(k_4\)), which are for the Q dimension of the system. An example is where \(N_T=4\), two possible sets of antenna pairs \(\begin{bmatrix}(1,3); (1,4); (2,3); (2,4)\end{bmatrix}\) for the I dimension and \(\begin{bmatrix}(2,4); (2,3); (1,4); (1,3)\end{bmatrix}\) for the Q dimension are assigned the same bit indices \(\begin{bmatrix}00; 01; 10; 11\end{bmatrix}\). A mapping table of the grouped bits (I dimension transmit antenna pairs and Q dimension transmit antenna pair combinations), is given in Tables 1 and 2. The antenna mapping for the I and Q dimensions are different in order to avoid ICI or the use of the same pair of antennas to simultaneously transmit a mixed signal of the real and imaginary parts. This enhances the EP of the proposed system.
The second category of the input data stream is the symbols bit stream. A bit stream \(\varvec{b}=\begin{bmatrix} b_1 b_2 ... b_{2r}\end{bmatrix}\), with \(r=\)log\(_2\) M is fed into a mapper (\(\mathfrak {G_1}\)). In mapper (\(\mathfrak {G_1}\)), the 2r input bits are mapped onto a rotated H8QAM signal constellation set to yield two symbols \( S _{q}^{1}\) and \( S _{q}^{2}\), where \(q=1+\sum _{v=1}^{2r}2^{2rv}b_v\), \(q\in [1:M^2]\). The creation of the H8QAM symbol constellation and its rotation is well discussed by Cogen et al.^{17}. This paper adopts the same way of creating H8QAM systems. Also, Cogen et al.^{17}, discusses the degrading effect of error floor on the EP of QSM systems when using HQAM without rotation. Thus, Cogen et al.^{17}, then devised a solution of using optimum rotation angles on HQAMQSM systems to curb the error floor on the EP of HQAMQSM schemes. The optimum rotation angle for the chosen H8QAM schemes in this paper, is \(\frac{\pi }{4}\)^{17}. The mapping of the rotated H8QAM symbols to bits is provided in Table 3 and Fig. 10. The SE of the proposed GQSMH8QAM scheme is given as,
The modulated symbols (\( S _{q}^{1}\) and \( S _{q}^{2}\)) are split into their respective real parts and imaginary parts (\( S _{q}^{1I}+ \Im { S _{q}^{1Q}}\) and \( S _{q}^{2I}+\Im { S _{q}^{2Q}}\)). The real parts of the symbols are transmitted via the I dimension using the inphase transmit antenna pairs (\(k_1\) and \(k_2\)). The imaginary parts of the symbols are in turn transmitted through the Q dimension using the quadrature antenna pairs (\(k_3\) and \(k_4\)). Thus, the split symbols are transmitted by four transmit antennas concurrently per time slot, in the form of an \(N_T\times 1\) transmit vector \(\textbf{x}\) shown in Fig. 3. Vector \(\textbf{x}\) has four nonzero elements, which are the two real parts (\( S _{q}^{1I}\) and \( S _{q}^{2I}\)) and the two imaginary parts (\( S _{q}^{1Q}\) and \( S _{q}^{2Q}\)) of the symbols. Moreover, the positions of these four nonzero elements correspond to the indices of the four chosen active transmit antennas. \(\theta _k\) is the rotation angle applied to different transmit antenna pairs as shown in Tables 1 and 2. This is done in order to further minimise ICI that could be generated by overlapping antenna pairs^{14}. The discussion of the selection of the antenna pairs and the optimum \(\theta _k\) was discussed in^{14} and^{19}. Hence, Tables 1 and 2 show the chosen transmit antenna pairs and the corresponding optimum \(\theta _k\) values used in this paper according to^{14}.
At the receiver, the signal is given by \(\textbf{y}\in \mathbb {C}^{N_R\times 1}\) (2), as
where the fading channel matrix is represented by \(\textbf{H}\in \mathbb {C}^{N_R\times N_T}\) and \(\textbf{n}\in \mathbb {C}^{N_R\times 1}\) is the AWGN noise vector. \(\textbf{h}_{k_l}(1\le l\le N_T)\) is the lth column vector of the channel gain matrix \(\textbf{H}=\begin{bmatrix}\textbf{h}_{k_1}&\textbf{h}_{k_2}&\textbf{h}_{k_3}&...&\textbf{h}_{k_{N_T}}\end{bmatrix}\) and \(\textbf{h}_{k_l}=\begin{bmatrix}\textbf{h}_{1k_l}&\textbf{h}_{2k_l}&\textbf{h}_{3k_l}&...&\textbf{h}_{N_Rk_l}\end{bmatrix}^T\). The elements of both \(\textbf{n}\) and \(\textbf{H}\) are assumed to be i.i.d Gaussian random variables (GRVs) with distribution \(CN (0,1)\), respectively. Finally, \(\frac{\rho }{4}\) is the average signaltonoise ratio (SNR) per receive antenna.
The optimum maximum likelihood detection (MLD) for the proposed scheme, can be written as in (3) and it entails a joint estimation of the transmission antenna pairs indices (\(\hat{K}\)) and the sent symbols (\( S _{\hat{q}}^{1}\) and \( S _{\hat{q}}^{2}\))^{11} and^{12}, as follows,
where \(\mathbf {\mathfrak {g}}=\sqrt{\frac{\rho }{4}}\bigg (\left( \textbf{h}_{k_1} S _{q}^{1I} + \textbf{h}_{k_2} S _{q}^{2I}\bigg )e^{j\theta _k} + \Im \bigg (\textbf{h}_{k_3} S _{q}^{1Q} + \textbf{h}_{k_4} S _{q}^{2Q}\bigg )e^{j\theta _k} \right) \) and \(\hat{\textbf{x}}\) is the estimated received vector shown in Fig. 4, respectively.
MLD CCA of the proposed GQSMH8QAM schemebased on real valued multiplications and additions
The CCA of the GQSMH8QAM scheme is similar to that discussed in Holoubi et al.^{20}. It is based on real value multiplications and additions. In^{20}, it is given as \((20N_R1)2^{\mathfrak {m}}\). In this section, all the incorporated CCAs of various schemes in this paper (including enhanced spectral efficiency GSM (ESEGSM), GSM multiplexing two symbols(MIMOGSM) and constellation reassigned GSM (GSMCR)), have been discussed before in^{6} and^{21}. As seen in Tables 4 and 5, the GQSMH8QAM scheme has the highest CC as compared to all the schemes included in this paper. GQSMH8QAM has an optimal MLD at the expense of a high CC. However, it is essential to also consider that the proposed scheme uses fewer transmit antennas as compared to various MIMOSM schemes under the same conditions. An example is shown in Tables 9 and 10 found in "Appendix 1". Table 10 shows that the proposed scheme requires 4 transmit antennas to achieve a SE of \(\mathfrak {m}=8\) bits/s/Hz as compared to CSM and CGSM with \(N_T=32\) and 10, to achieve the same SE under the same conditions, respectively. Hence, the disadvantage of a high CC in the proposed scheme, is compensated by the advantage of using a less number of transmit antennas as compared to other various MIMOSM schemes incorporated in this paper. Due to a high CC shown by the proposed scheme, as upcoming work, we intend to formulate low complexity detection algorithms to overcome this disadvantage.
Proposed error performance analysis of GQSMH8QAM
In this section, the ABEP of GQSMH8QAM is formulated. The detection discussed in section "System model" follows the same approach as^{6,11} and^{12}. Hence the ABEP of GQSMH8QAM is upperbounded by
where \(N\left( q,\hat{q}\right) \) is the total number of bit errors for the associated pairwise error probability (PEP) event \(P(\textbf{x}\rightarrow \hat{\textbf{x}})\) between the transmitted vector \(\textbf{x}\) and the received vector \(\hat{\textbf{x}}\). The PEP \(P(\textbf{x}\rightarrow \hat{\textbf{x}})\) is formulated as
where \(\omega _g=\frac{\rho }{8}\left\Vert \left( \textbf{h}_{k_1}d_1 + \textbf{h}_{k_2}d_2+ \textbf{h}_{k_3}d_3 + \textbf{h}_{k_4}d_4\right) \right\Vert _F^2=\frac{\rho }{8}\left\Vert \textbf{H}_k\textbf{G}_k\right\Vert _F^2= \frac{\rho }{8}\left\Vert \textbf{H}_k\right\Vert _F^2\left\Vert \textbf{G}_k\right\Vert _F^2= \frac{\rho }{8}\left\Vert \textbf{H}_k\right\Vert _F^2\left( (d_1)^2+(d_2)^2+(d_3)^2+(d_4)^2\right) \), with \(\textbf{H}_k=\begin{bmatrix} \textbf{h}_{k_1}&\textbf{h}_{k_2}&\textbf{h}_{k_3}&\textbf{h}_{k_4}\end{bmatrix}\), \(\textbf{G}_k=\begin{bmatrix}d_1&d_2&d_3&d_4\end{bmatrix}^T\), \(d_i=\left( S _q^{1I} S _{\hat{q}}^{1I}\right) e^{j\theta _k}\) and \(d_{(l+2)}=\left( S _q^{lQ} S _{\hat{q}}^{lQ}\right) e^{j\theta _k}\), \(\{i,l\}\in [1:2]\). Similarly, based on (11) in Koc et al.^{22}, \(\omega _g\) are chisquared RVs with \(2N_R\) degrees of freedom defined as \(\omega _g=\sum _{t=1}^{2N_R} \alpha _{\omega _{g,t}}^2\) with \(\alpha _{\omega _{g,t}}^2\sim N(0,\sigma _{\omega _g}^2)\) and \(\sigma _{\omega _g}^2=\frac{\rho }{8}\big ((d_1)^2+(d_2)^2+(d_3)^2+(d_4)^2\big )\). The derivation of the PEP is found in "Appendix 2".
Upon further simplification by taking the expected value of \(\omega _g\), the average PEP assuming \(N_R\) receive antennas is adopted and modified from^{6,11} and^{12}, to give
where \(\gamma =\frac{1}{2}\left( 1\sqrt{\frac{\sigma _{\omega _g}^2}{1+\sigma _{\omega _g}^2}}\right) \).
Hence, taking the Taylor series of (6) and using the same approach as^{6} and^{20} at high SNRs, (6) leads to (7) when higherorder terms are neglected.
The overall diversity gain (\(G_c\)) attained by a MIMOSM scheme has been defined in^{23}, as
where \(P_b\) denotes overall probability of error as a function of SNR. Hence given \(P_b(SNR)=P(\textbf{x}\rightarrow \hat{\textbf{x}})\) as given in (7), then (8) becomes
Hence substituting (7) into (9), leads to (10).
where \(\beta =\bigg (\frac{2^{2N_R1}\Gamma (N_R+0.5)}{\sqrt{\pi (N_R)!}}\bigg )\bigg (\frac{1}{8}\bigg )^{N_R}\bigg ((d_1)^2+(d_2)^2+(d_3)^2+(d_4)^2\bigg )^{N_R}\).
Since \(\lim _{\rho \rightarrow \infty } \bigg (\frac{\log (\beta )}{\log (\rho )}\bigg )=0\), therefore \(G_c=N_R\). Hence the diversity gain of the proposed system is \(N_R\).
Simulation and numerical results analysis
This section presents simulation results for the proposed GQSMH8QAM system with a different number of transmit antennas and comparisons with CSM, CGSM, CQSM, MIMOGSM and ESEGSM. The theoretical ABEP for CSM schemes was based on (9) in Naidoo et al.^{4}, that of CGSM schemes was based on (5) in Pillay et al.^{19}, and that of ESEGSM was based on (5) and (12) in Pillay et al.^{19}. The theoretical ABEP of CQSM was based on (3) and (4) in Oladoyinbo et al.^{8}, and that of MIMOGSM was based on (16) in Wang et al.^{21}. Another aim of this section is to validate the theoretical performance bound derived in (4). The average bit error rate BER of the proposed scheme was evaluated using Monte Carlo simulations over i.i.d RFF fading channels with AWGN. The BER performance was assessed for various SEs (\(\mathfrak {m}=8\) bits/s/Hz, \(\mathfrak {m}=9\) bits/s/Hz and \(\mathfrak {m}=10\) bits/s/Hz) as a function of the average SNR per receive antenna (\(\rho /4\)). These three SEs considered, are shown in Figs. 5, 6, 7, 9. The comparisons were made under the same conditions, which are; four receive antennas assumed in all scenarios, identical SE, optimal MLD method, and at a BER value of \(1\times 10^{5}\) for comparison. The formulae for evaluating the SEs of all the various scheme in Figs. 5, 6, 7, 9 are found in "Appendix 1", Table 8.
Firstly, Fig. 5 presents the BER performance curves of the GQSMH8QAM scheme with \(N_T=4, 6\) and 8, respectively. In Fig. 5, the analytical results are validated by simulation results and it is seen that they are tighter as the SNR values increase. They are tight at high SNR values because of the upperbounded ABER performance in all the configurations of the GQSMH8QAM scheme.
Secondly, Figs. 6 and 7 with Table 6 show the simulation results and the calculated EP gains of the proposed \(4\times 4\) GQSMH8QAM scheme over \(4\times 4\) various schemes of the same SE (\(\mathfrak {m}=8\) bits/s/Hz). As seen from Figs. 6 and 7 together with Table 6, the proposed scheme has an EP of 0.61 dB over \(4\times 4\) GCQSMC8QAM, 0.93 dB over \(4\times 4\) GSMCR64QAM scheme and 0.75 dB over \(4\times 4\) MIMOGSMC8QAM scheme. Also, Wang et al.^{21}, discussed MIMOGSM and it was shown to outperform CSM and CGSM schemes of the same SE. In^{14}, it is also seen that GSMCR outperforms CSM and CGSM schemes. Thus, the proposed scheme has performance gain over all the schemes included in this paper, as it outperformed the best of those discussed in literature^{14,21}. This might be because of the use of generalised antennas and H8QAM with a MMED that improved the EP of the proposed scheme.
Thirdly, Fig. 8 shows the proposed GQSMH8QAM scheme performance compared with the same schemes in Figs. 6 and 7 at a SE of \(\mathfrak {m}=9\) bits/s/Hz, \(N_R=4\) but under different number of transmit antennas. Fourthly, Fig. 9 shows the proposed GQSMH8QAM scheme performance compared with the same schemes in Fig. 6 but under different SEs and different number of transmit antennas. In Fig. 9, the \(8\times 4\) GQSMH8QAM scheme is compared with other various \(8\times 4\) schemes with a SE of \(\mathfrak {m}=10\) bits/s/Hz. Figure 9 together with Table 7 also exhibits the same behavior as Figs. 6 and 7. The proposed scheme outperformed all the various schemes that were compared to it (Fig. 10).
It is also important, to note that the proposed scheme (GQSMH8QAM), has another advantage of using a less number of transmit antennas under conditions of fixed \(N_R\) and fixed M at a certain SE (\(\mathfrak {m}\)) as compared to the other systems included in this paper. An example is shown in Tables 9 and 10 found in "Appendix 1", which show the number of \(N_T\)s required to achieve a certain SE at \(N_R=4\). Table 9 shows that the GQSMH8QAM scheme requires 4 transmit antennas to achieve a SE of \(\mathfrak {m}=8\) bits/s/Hz using H8QAM as compared to CSM which requires \(N_T=32\), CGSM which requires \(N_T=10\) and ESEGSM which requires \(N_T=8\) to achieve the same SE under the same settings. This shows that the GQSMH8QAM scheme has an advantage of less hardware costs as it requires a less number of transmit antennas as compared to the other systems. The only exception seen in Tables 9 and 10, is the MIMOGSM scheme which requires the same number of transmit antennas as the proposed scheme to achieve the same SE under identical settings as the proposed scheme. However, it is outperformed by the proposed scheme as seen from the results in Figs. 6, 7, 9.
Conclusion
In this paper, GQSMH8QAM scheme was proposed to improve the EP of MIMOSM schemes. The scheme builds on GCQSM and uses energy efficient H8QAM instead of the C8QAM, to improve the EP of MIMOSM schemes. Also, in this paper, an upper bound analytical EP formula for the proposed scheme in i.i.d (RFF) fading channels was derived. This analytical bound was validated using Monte Carlo simulations. The results showed that the derived analytical bound is increasingly tight with increasing SNR values. Furthermore, the proposed scheme outperformed GCQSM, CSM, CQSM, MIMOGSM, CGSM and other various MIMOSM schemes incorporated in this paper. Hence, from the results obtained, it can be concluded that the proposed scheme is a viable option of improving the EP of MIMOSM schemes. However, despite its capability of improving the EP of MIMOSM schemes, the proposed scheme was found to have a high complexity MLD. Hence, for future work the authors intend to work on low complexity detection methods for the proposed scheme. In addition, the authors also intend to build on the proposed scheme by fusing it with uncoded spacetime labelling diversity and golden codewords to improve/enhance the SE of MIMOSM systems.
Data availability
All data generated or analysed during this study is included in this published article [and its supplementary information files].
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All authors had equal work on this paper. N.S. wrote the Paper. H.X. proof read the paper and gave guidance on paper. N.P. also proof read and gave guidance on paper. All data generated or analysed during this study are included in this published article [and its supplementary information files].
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Appendices
Appendices
H8QAM constellation set
For the following Tables 9 and 10, where other schemes are excluded, means that they were not compatible for the conditions under comparison.
Derivation of the PEP
Based on the equivalent system model in (3), the conditional PEP, \(P(\textbf{x}\rightarrow \hat{\textbf{x}})\) can be formulated as
Substituting (2) into (A.11) leads to
where
\(d_i=\left( S _q^{1I} S _{\hat{q}}^{1I}\right) e^{j\theta _k}\) and \(d_{(l+2)}=\left( S _q^{lQ} S _{\hat{q}}^{lQ}\right) e^{j\theta _k}\), \(\{i,l\}\in [1:2]\).
Equation (A.12) can also be rewritten as;
where
and
Simplifying Equation (A.13) leads to (A.14).
Since \(\textbf{n}^H\in \mathbb {C}^{1\times N_R}\) and each entry of \(\textbf{n}^H=\begin{bmatrix} n_1^*&n_2^*&n_3^*&...&n_{N_R}^*\end{bmatrix}\) is Gaussian distributed, it therefore means that each entry of \(\Re \bigg \{\sqrt{\frac{\rho }{4}}\textbf{n}^H\textbf{h}_{k_z}d_z\bigg \}\), \(z\in [1:4]\) is also a GRV with distribution \(N\left( 0,\frac{\rho }{4}\left d_z\right ^2\sum _{l=1}^{N_R}\left h_{k_z}^l\right ^2\right) \). Hence, this implies that our decision variable is also Gaussian distributed as \(\Re \bigg \{\sqrt{\frac{\rho }{4}}\textbf{n}^H\bigg (\textbf{h}_{k_1}d_1 + \textbf{h}_{k_2}d_2+ \textbf{h}_{k_3}d_3 + \textbf{h}_{k_4}d_4\bigg )\bigg \}\sim N\left( 0,\frac{\rho }{8}\left\Vert \textbf{h}_{k_1}d_1 + \textbf{h}_{k_2}d_2+ \textbf{h}_{k_3}d_3 + \textbf{h}_{k_4}d_4\right\Vert _F^2\right) \) and it leads to
where \(\omega _g=\frac{1}{2}\textbf{C}=\frac{\rho }{8}\left\Vert \textbf{H}_k\textbf{G}_k\right\Vert _F^2=\frac{\rho }{8}\left\Vert \textbf{H}_k\right\Vert _F^2\left\Vert \textbf{G}_k\right\Vert _F^2=\frac{\rho }{8}\left\Vert \textbf{H}_k\right\Vert _F^2\left( (d_1)^2+(d_2)^2+(d_3)^2+(d_4)^2\right) \),with \(\textbf{H}_k=\begin{bmatrix} \textbf{h}_{k_1}&\textbf{h}_{k_2}&\textbf{h}_{k_3}&\textbf{h}_{k_4}\end{bmatrix}\) and \(\textbf{G}_k=\begin{bmatrix}d_1&d_2&d_3&d_4\end{bmatrix}^T\).
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Sibanda, N., Xu, H. & Pillay, N. Error performance analysis of generalized quadrature spatial modulation using H8QAM. Sci Rep 12, 20634 (2022). https://doi.org/10.1038/s41598022249508
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DOI: https://doi.org/10.1038/s41598022249508
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