Spatial Index Modulation With the Combination of Both Antenna Indexes and Two Constellations

In this paper, in order to further exploit the antenna index (AI) domain for increasing the spectral efficiency, we propose a new schematic of enhanced generalized spatial modulation with both three dimension constellation and QAM constellation (EGSM-3DQ) to enhance the active AI combinations with the construction of three dimension (3D) signal constellation. In the EGSM-3DQ, a transmit vector symbol (TVS) consists of two spatial vector symbols (SVS) with the aid of one vector index bit, one SVS of which is obtained by modulating three components of a 3D signal symbol on two or three active transmit antennas (TAs) through the 3D Signal Mapper and AI Vector Modulator, another SVS of which is obtained by modulating one or multiple mapped QAM symbols on one or multiple active TAs through the AI Vector Selector. In addition, by jointing the AIes and multiple constellations such as the 3D and QAM, PAM constellations, grouping EGSM (gEGSM) is proposed to develop the group index combinations to increase extra index information. Finally, the comparisons of squared minimum Euclidean distances (MED) and the average bit error probability are provided. Simulation results show that the proposed schemes improve the bit error rate performance as compared with conventional systems.


I. INTRODUCTION
In the past decade years, index modulation technologies, which transmit one or multiple transmit data symbol by activating one transmit antenna (TA) or multiple TAs in the total of TAs, are wisely researched for improving the spectral efficiency (SE) in wireless communication systems.

A. RELATED WORK
In spatial modulation (SM) [1] that only activates only one TA to convey data symbol (e.g. quadrature amplitude modulation/phase shift modulation (QAM/PSK)), the spatial index bits depend entirely on the number of TAs, which need to satisfy the power of two, i.e., log 2 N t , where N t is the number of TAs. Although the SM relaxes the problem of both the The associate editor coordinating the review of this manuscript and approving it for publication was Yiming Huo .
inter channel interference and inter antenna synchronization, the number of TAs increases exponentially by the power of two when increasing the spatial index bits. In generalized SM (GSM) [2], [3], [4], in order to relax the number of TAs and improve the spatial index bits, the combinations of multiple antenna indexes (AIes) are investigated to activates simultaneously several TAs for the transmission of different data symbols such as QAM/PSK, i.e., log 2 C n a N t , where n a is the number of active TAs and · is the floor operation. In order to break through the limitations of the single spatial domain, quadrature spatial modulation (QSM) [5] extends the spatial domain to the in-phase and quadrature spatial dimensions. Thus, QSM can achieve more log 2 C 1 N t information bits than SM in terms of spatial index bits. To further improve the SE while enhancing the bit error rate (BER) performance, a lot of works on generalization of QSM (GQSM) [6], [7], [8] have been developed. Although the GQSM systems can achieve 2n a log 2 C n a N t of spatial index bits by increasing the active AI combinations, the complexity of detection will be relatively increased with the increasing of active TAs in the in-phase and quadrature dimensions. In other words, the number of spatial index bits are increased at the cost of the detection complexity. Quadrature index modulation with three dimension (3D) constellation (QIM-TDC) [9] is proposed to transmit two components of a 3D symbol by using two active TAs in the in-phase dimension and to transmit the remaining one by using one active TA in the quadrature dimension, respectively. Consequently, log 2 C 2 N t + log 2 C 1 N t number of spatial index bits are achieved. In order to further improve the spatial index bits, with the aid of two states of both 1 and j, spatial modulation with spatial constellation (SM-SC) [10] develops the group index domain according to the D dimensions of the employed data symbol, thus it has log 2 C n a N t +D to be conveyed. However, for the above systems such as QSM, QIM-TDC and SM-SC, the squared minimum Euclidean distance (MED) is changed to 2 E av , where E av is the average energy per transmit vector symbol (TVS). From this, it will affect the the reliability of communications. In order to avoid this problem while increasing the spatial index bits, on the joint design of active TA(s) and multiple signal constellations, enhanced SM (ESM) [11], [12] is proposed to either transmit one primary symbol (e.g. M -ary QAM) by using one active TA or transmit two secondary symbols by using two active TAs. Thus, the spatial index bits carried by ESM is log 2 (C 1 N t + 2C 2 N t ). In order to not only further carry more spatial index bits but also keep the characteristic of the squared MED (i.e., 4 E av ), generalized spatial modulation with multi-index modulation (GSM-MIM) [13] has been developed to exploit multiple index domains with the aid of two distinct signal constellations (e.g., pulse amplitude modulation (PAM) and QAM) and the sign j. Based on the in-phase and quadrature dimensions, a lot of works [14], [15] on developing more spatial index bits have done and achieved the higher SE. In [14], improved QSM (IQSM) is proposed to activate two out of all TAs in the in-phase and quadrature dimensions for transmitting the real and imaginary parts of two signal symbols, respectively. Thus, the number of spatial index bits transmitted by the IQSM system are increased to 2 · log 2 In double GSM (DGSM) [15], all AI combinations of activating one or more than one TA are developed to convey the spatial index bits that is 2 · log 2 (C 1 N t + · · · + C N t/ 2+1 N t ) . In order to make the best use of the resource of the idle TAs in large scale TAs, all TAs are divided into multiple groups [16], [17], [18]. In [16], the spatial index bits carried by grouping GSM (gGSM) is G· log 2 (N t G) . However, grouping GSM only exploits the spatial index bits in one spatial domain, so as to limit the exploiting of the spatial index information. From this, the in-phase and quadrature dimensions of the spatial domain are considered. In both GQSM [17] and parallel QSM (PQSM) [18], each group of which performs the design principle of QSM, the spatial index bits the PQSM and GQSM can convey are 2G · log 2 N t G . However, the GQSM and PQSM schemes do not consider the design of signal constellations. In other words, the spatial gain from signal constellations and the spatial index information exploited by jointing design of both signal constellations and the active TAs need to be further developed.

B. MOTIVATION AND CONTRIBUTION
Considering previous literatures, inspired by the design idea of GSM-MIM and DGSM, a new design of enhancing the active AI combinations using the 3D constellation, called as enhanced generalized spatial modulation with both 3D constellation and QAM constellation (EGSM-3DQ), is proposed to increase the SE. Then, in order to further exploit the index domain, by jointing the AI combinations and multiple constellations, the grouping EGSM scheme is proposed and develop the group index information. The main contributions of this paper can be summarized as follows: • In view of the resource of N t TAs and the utilization of three components in the 3D signal constellation, the EGSM-3DQ scheme expands the number of the active AI combinations with the 3D constellation. In the EGSM-3DQ, the total of N t TAs are divided into two groups of TX 1 and TX 2 TAs. Then, two or three out of the group of TX 1 TAs are used to modulate the 3D constellation point (CP) symbols 3D (m x , m y , m z ) for forming a spatial vector symbol (SVS) X. The active TA(s) from the group of TX 2 TAs is(are) used to modulate the mapped QAM symbol(s) for forming other one SVSX. With the Vector Combiner, using one extra bit to determine the form of TVS from the combinations of X andX.
• Specifically, in order to be capable of exploiting more spatial index bits by expanding the active AI combinations, two out of TX 1 TAs are activated to respectively modulate two components of the spatial signal point (SSP) that has four forms: {(m x , m y + jm z ), (jm x , m y + jm z ), (m y + jm z , m x ), (m y + jm z , jm x )}, which are obtained by converting a 3D symbol s 3D (m x , m y , m z ) with the aid of the 3D→2D Convertor. Thus, it has 4C 2 TX 1 AI vectors to be designed to four forms of SSP symbols. On the other hand, three out of TX 1 TAs are activated to respectively modulate three components of the SSP that has two forms: {(m x , m y , m z ), (jm x , jm y , jm z )}, which are obtained by one key controller controlling two states of {1, j}. In this case, we can design 2C 3 TX 1 AI vectors. Consequently, there exist 4C 2 TX 1 + 2C 3 TX 1 AI vectors that are candidate for being constructed into an AI vector set for carrying spatial index bits.
• In order to further exploit the index domain for increasing the SE, with the 3D constellation and the QAM and PAM constellations, grouping EGSM (gEGSM) is developed and exploits the group index combinations to increase extra index information.
• We analyze the squared MED between the TVSs, which verify the advantage of the proposed schemes as compared with GSM-MIM. Simulation results are conducted to demonstrated that the proposed schemes outperform other schemes in terms of BER performance.
The rest of the paper is organized as follows. In Section II, we introduce system model of EGSM-3DQ and an improved 3D signal constellation. In Section III, system model of gEGSM is introduced. In Section IV, the squared MED comparison and the average bit error probability are presented. Finally, simulation results are given and discussed in Sections V and our conclusions is given in SectionVI.
The abbreviations for various systems used throughout the paper are listed in TABLE 1

II. PROPOSED EGSM-3DQ
A. SYSTEM MODEL Let us consider that the schematic framework of the EGSM-3DQ system with N t transmit antennas (TAs) and N r receiver antennas, as is shown in Fig. 1. Note that the number of TAs are consist of two groups, TX 1 and TX 2 TAs (i.e., we define N t = TX 1 + TX 2 ), TX 1 TAs of which for transmitting a 3D constellation point (CP) symbols 3D and TX 2 TAs of which for transmitting τ QAM CP symbols: s 1 , · · · , s τ , (1 ≤ τ < TX 2 ), respectively. Specifically, assume that a total of m = log 2 N +I SIB +τ ·log 2 M information bits, which are divided into three blocks: the mapping 3D CP bits (log 2 N ) with the N -ary 3D constellation, the spatial index bits (I SIB ) and the mapping QAM CP bits (τ · log 2 M ) with the QAM constellation, are entered the transmitter of the proposed EGSM-3DQ in each TVS duration, where N and M respectively denote the modulation orders of 3D constellation and QAM constellation, τ denotes the number of mapped QAM symbols. Furthermore, I SIB , denoting the number of spatial index bits, is divided into three groups: I A , I B and I v .
For both the group of I A and the block of mapping 3D CP bits (log 2 N ), the 3D Signal Mapper and AI Vector Modulator encode them into a SVS XD. The detailed design is as follows: As depicted in Fig. 1, the block of mapping 3D CP bits, containing log 2 N bits, is mapped into a 3D CP symbols 3D (m x , m y , m z ) from the N -3D constellation that will be designed in II-B. The group of I A , containing log 2 (N A ) , N A = 2 log 2 | | information bits, is converted into a decimal numberD, (1 ≤D ≤ N A ) by converting the I A into a decimal, where · denotes the floor operation. Note that | | denotes the number of all AI vectors in the set ) possible AI vectors for activating α, α ∈ {2, 3} out of TX 1 TAs, and will be introduced in II-C, where C x y denotes combination operation. After entering the group of bits into the Size Comparator, according to judgment of the Size Comparator, there exists two outputs: 1) IfD ≤ | 1 |, the Size Comparator outputs the AI number λ, λ ∈ {1, · · · , | 1 |}, which is fed into not only the Two-State Output-I but also the AI Vector Selector-I and the Ceil Algorithm Device. In this case, the parameter κ 1 is set to ''0'' (i.e., κ 1 = 0 ), which is used to switch the k1 up for feeding the 3D CP symbols 3D (m x , m y , m z ) into the 3D→2D Convertor. Then, according to the output δ (1 ≤ δ ≤ 4) of the Ceil Algorithm Device whose expression of the algorithm is δ= Ceil(λ C 2 TX 1 ), the 3D→2D Convertor convert the 3D symbols 3D In the AI Vector Selector-I, the AI number λ is employed to select a specific V λ , which containing two non-zero elements equaling to ''1'', from the AI vector set Note that, from the design of the set 1 are used to modulate the corresponding SSP symbol Q 2 δ . According to the selected AI vector V λ , two components of the corresponding SSP symbol Q 2 δ are modulated on two out of TX 1 TAs activated by the AI vector V λ , resulting in a SVS X λ . 2) IfD > | 1 |, the Size Comparator outputs the AI number β, β ∈ {1, · · · , | 2 |}, which is fed into not only these two Two-State Output-I, II but also the AI Vector Selector-II. In this case, the parameter κ 1 is set to ''1'' (i.e., κ 1 = 1 ) in the Two-State-Output-I, which is used to switch the k1 down for feeding directly the 3D CP symbols 3D (m x , m y , m z ) into the AI Vector Selector-II. Then, according to the AI number β, a specific V β , which containing three non-zero elements equaling to ''1'', is selected from the AI vector set 2 = V 1 , V 2 , · · · , V | 2 | , and used to activate three out of TX 1 TAs for respectively modulating three components of thes 3D (m x , m y , m z ), resulting in a SVS X. Due to producing the same SVS X, this will cause unrecoverable detection errors at the receiver. In order to avoid this problem. With the aid of the Two-State Output-II, if β ≤ C 3 N t , κ 2 is set to ''0'' (i.e., κ 2 = 0), and is used to switch the key k2 down. Thus ε = 1. Otherwise, ε = j. Consequently, according to the key k2 control, the expression of ε · X β may be given by Finally, we can obtain the SVS XD for TX 1 TAs, as follows (1) Furthermore, for the second group of I B , containing s τ ) obtained by mapping the τ ·log 2 M information bits, resulting in a spatial vector symbol X, where ∈ , denotes the set that contains all possible AI vectors that can activate τ out of TX 2 TAs. Note that, each vector in the set has τ non-elements equaling to ''1'' to activate τ TAs, and when M = 2, the 2-QAM constellation : To further achieve one more information bit, we utilize two vector symbols X,X to construct two different forms of TVS, i.e., {S 0 , S 1 }, one of which is determined by one bit. These two different TVS forms may be designed as follows where I v is one information bit that is used to select one of the two different forms of TVSs to be transmitted. Specifically, in the Vector Combiner of Fig. 1, the last group of I v , only containing one information bit, is fed into the Vector Combiner and then select a TVS from {S 0 , S 1 }. Finally, through the Space Mapper, the selected TVS S is mapped on N t TAs.
To further illustrate the working principle of the proposed EGSM-3DQ, examples of forming a EGSM-3DQ symbol are given in TABLE 2. Assumed that N t = 8, and QAM symbols ares 3D = (m x , m y , m z ) and one s 1 , respectively. Moreover, the AI vector set is provided by the example of (9). In Table 2, the AI vector VD with the AI num-berD is obtained by the information bits {b 0 , b 1 , b 2 , b 3    Based on the above design, the SE (bits/s/Hz) of the EGSM-3DQ can be calculated by { ± 2, ±4, · · · , ±2Z}. Thus, the design of the 3D constellation can be as follows: where Z denotes the integer operation. For example, Table 3 In this case, we can design 4C 2 TX 1 AI vectors, which are given by (6), as shown at the bottom of the next page, where the a, b, c positions are replaced by m x , m y , m z values of a 3D CP symbol, respectively. 2) When three out of TX 1 TAs are activated, each active TA is used to modulate each component m χ , χ ∈ {x, y, z} of the mapped 3D symbols 3D m x , m y , m z , respectively. Furthermore, with the aid of the k2 controller controlling two states of {1, j} in Fig. 1, the mapped 3D symbols 3D m x , m y , m z may be converted into the following symbol forms: Thus, we can design 2C 3 N t AI vectors for transmitting the three components of the mapped 3D symbol s 3D m x , m y , m z , which are given by where the a, b, c positions are replaced by the three components m x , m y , m z of a 3D CP symbol, respectively. Consequently, based on the above mentioned design, we can construct a set = { 1 , 2 } with C 2 TX 1 ×4+C 3 TX 1 ×2 AI vectors. For instance, assumed that TX 1 = 4, an AI vector set that has a total of C 2 4 × 4 + C 3 4 × 2 = 32 AI vectors is given by (9), as shown at the bottom of the next page. It can be obtained from (9), five extra index information bits can be achieved.

III. PROPOSED GROUPING EGSM SCHEME
In this section, on the basis of the design structure of the EGSM-3DQ, we propose the gEGSM scheme, in which not only the spatial index domain is further exploited to increase the SE but also the squared MED is further increased for enhancing the reliability of wireless communications.
The system model for the gEGSM equipped with N t = TX 1 + TX 2 + TX 3 TAs is depicted in Fig. 2, where the Vector Group Generator is designed for exploiting the extra information I G . In the Vector Group Generator, the resulted SVSs: X,X, X are fed into the three vector blocks (e.g. S 1 , S 2 , S 3 ) through the controlling of three Switches with the information bits I G . Specifically, as shown in the left of Fig. 2, the TX bits m are divided into five data sub-streams: log 2 N , I A , I G , I B and I C . Furthermore, according to the design of the 3D Signal Mapper and AI Vector Modulator in the EGSM-3DQ scheme, the two sub-streams of both log 2 N and I A containing log 2 N A bits are modulated into a SVS X to be transmitted on TX 1 TAs. For the sub-stream of I B , which are further divided into log 2 N B and ϑ log 2 M B bits, ϑ log 2 M B bits by the PAM Modulator are modulated into ϑ M B -PAM symbols:sn1, · · · ,snϑ from the constellation set = ±2, · · · , ±2µ , ±2j, · · · , ±2μ j , µ ,μ ∈ N ,Ñ denotes the natural number. Then, log 2 N B , N B = 2 log 2 C ϑ TX 2 bits by the AI Vector Selector B are used to select a specific AI vector B x , x ∈ {1, · · · , N B } from the AI vector set B = B 1 ,B 2 , · · · ,B N B to modulate ϑ PAM symbols, resulting in a SVSX, where B ∈¯ ,¯ denotes the set that contains all possible AI vectors that can activate ϑ out of TX 2 TAs. Similarly, the sub-stream of I C are further divided into log 2 N C and τ log 2 M C bits. τ log 2 M C bits by the QAM Modulator are modulated into τ M C -QAM symbols: s n 1 , · · · , s n τ . Then, in the AI Vector Selector C, s n 1 , · · · , s n τ symbols are modulated by the specific AI vector C y , y ∈ {1, · · · , N C } with the log 2 N C , N C = 2 log 2 (C τ TX 3 ) bits on τ active TAs of TX 3 TAs, resulting in a SVS X , where C y is from the AI vector set C = C 1 ,C 2 , · · · ,C N C and C ∈ , denotes the set that contains all possible AI vectors that can activate τ out of TX 3 TAs.
Furthermore, in order to exploit more extra index bits, the Vector Group Generator is designed. Specifically, with the aid of the Group Index Selector, the Vector Group Generator feeds the three SVSs X,X, X into the corresponding vector blocks (e.g. S 1 , S 2 , S 3 ). Since the three SVSs X,X, X are different, according to the permutation of them, they can be formed into different group index patterns for achieving extra index bits. Hence, in the Group Index Selector, the set of group index patterns can be designed as where three components [G ς (1), G ς (2), G ς (3)] of the specific pattern G ς with the group index ς that is the decimal representation of I G bits, are used to control three Switches: k1, k2, k3 for feeding three SVSs into the specific vector blocks (e.g. S 1 , S 2 , S 3 ), respectively. For the sake of transmitting the integer multiple of one bit, only four group index patterns of G are considered as legitimate, the remaining is not considered. Without loss of generality, the group index set may be designed as G = {G 1 , G 2 , G 3 , G 4 }. To further illustrate the design principle of the Group Index Selector, an example is provided. Assumed that I G = {1 0}, it has ς = 3, and then G 3 = [G 3 (1), G 3 (2), G 3 (3)] = [2, 1, 3] is selected. In this case, since G 3 (1) = 2, the SVS X is fed into the 2-nd vector block S 2 through the control of Switch k1. Similarly, since G 3 (2) = 1 and G 3 (3) = 3, the two SVS X, X , under the control of two switches of both k2 and k3, are fed into the 1-st and 3-rd vector blocks S 2 , S 3 , respectively. Finally, a TVS symbol is obtained as S = [X, X, X ] T . Based on the above mentioned design and analysis, the SE of gEGSM can be expressed as

IV. ML DETECTION AND PERFORMANCE ANALYSIS
In this section, the maximum-likelihood (ML) detection, the average bit error probability (BEP) and the squared MED comparison are introduced.

A. ML DETECTTION
At the receiver, the received signal vector of the proposed schemes may be given by where y ∈ C N r ×1 , H ∈ C N r ×N t denotes the Rayleigh fading MIMO channel matrix, whose expression can be given by each entry h n r ,µ of which is an independent and identically distributed (i.i.d) complex Gaussian random variable with CN (0, 1), where h n r ,µ , 1 ≤ n r ≤ N r , 1 ≤ µ ≤ N t denotes the n r -th row and µ-th column channel coefficient of the channel matrix H. n ∈ C N r ×1 is the complex additive white Gaussian noise (AWGN) vector with CN (0, σ 2 I N r ).
Assuming the channel state information is perfectly known, for the better detection of the original information bits, the ML detection is employed for the proposed schemes. For the EGSM-3DQ, the spatial index bits that include the AI bits (e.g. I A , I B and I v ) and the symbol index bits (log 2 N and τ · log 2 M ) are jointly detected and the detective algorithm may be expressed as where · 2 denotes the Frobenius norm,Î v denotes the detected vector combining bits.D,ξ denote the detected AI vector index.N ,M 1 , · · · ,M τ denotes the modulation order of the detected symbols. Furthermore, for the gEGSM, the detective algorithm may be expressed as ς ,D,N , x,n 1 , · · · ,n ϑ , y, n 1 , · · · , n τ = arg min ς,D,N ,x,n 1 ,··· ,n ϑ ,y,n 1 ,··· ,n τ

B. AVERAGE BEP ANALYSIS
With the assumption that channel state information is perfectly known. Based on the union bound technique [16], the average BEP of the proposed schemes are calculated out. According to (14) and (15), the average BEP of EGSM-3DQ and gEGSM can be respectively given by (16) and (17), as shown at the bottom of the next page, where e S →Ŝ is the total number of erroneous bits for the case of S = S,P(S →Ŝ) denotes the average pairwise error probability (PEP). Through taking the expectation of PEP on the channel matrix H using a moment generating function (MGF) from [16], the average PEP can be calculated by (18), as shown at the bottom of the next page, where Q (·) denotes the Gaussian Q function, h n r ,µ denotes the n r -th row and µ-th column channel coefficient of the channel matrix H,d 2

S,min
denotes the normalized squared MED between the TVSs. S(µ) andŜ(µ) are the µ-th row values of the TVS S and detectedŜ, respectively.

C. MINIMUM EUCLIDEAN DISTANCE COMPARISON
According to the (16) and (18), as is known, the bigger the squared MEDd 2 S,min is, the smaller the value of the average BEP is. Moreover, in the high signal noise ratio (SNR) region, the asymptotic performance is mainly determined by the squared MED. Hence, in order to outstanding the advantage of the proposed schemes in terms of the squared MEDs, the squared MEDs of the proposed schemes is analyzed in the subsection. After tedious but straightforward calculations for the squared MED of the proposed schemes, the normalized squared MED between the TVSs can be expressed as where E av denotes the average energy of each TVS. According to (19)  }. Note that a-b denotes a-QAM and b-QAM, which are simultaneously employed in these schemes. From the Table 4, the advantage of the proposed schemes in terms of the squared MED is verified and are bigger than that of GSM-MIM, which is provided in Table 4.

V. SIMULATION RESULTS AND DISCUSSIONS
In this section, Simulation results using Monte Carlo method are presented and discussed for the BER performances of the proposed schemes, which are also compared with other IM schemes (e.g. GSM and GSM-MIM) at the same SEs. To make a fair comparison, all IM systems are equipped with the same number of TAs N t and receiver antennas N r , and detected by the ML detector to retrieve the original information bits. Moreover, the TVSs are transmitted over the Rayleigh fading MIMO channels, whose channel state information is perfectly known at the receiver. Note that, in our simulation results, we define the EGSM-3DQ scheme with (α, τ, N , M ) and GSM-MIM with (α,τ ,Ñ ,M ) for (TX 1 , }, a-b denotes a-QAM and b-QAM that are simultaneously employed in these schemes. In Fig. 3 and Fig. 4, simulation results is provided for the EGSM-3DQ scheme at the SE of 13, 14 bits/s/Hz and (N t , N r ) = (8,8), and made comparisons with the theoretical results. It can be observed that the simulation results match well with the theoretical results in the high SNR region for the effectiveness of the EGSM-3DQ scheme. Meanwhile, the simulation results for the EGSM-3DQ scheme with (1, 1, 8-3D, 4) and (1, 1, 8-3D, 8) are compared with that of other systems such as GSM-MIM with (1,2,2,8) and (1,2,2,(8)(9)(10)(11)(12)(13)(14)(15)(16), GSM with (n a = 3, 4-8QAM) and (n a = 3, 8QAM), and VBLAST with (N t = 3, 16-32QAM) and (N t = 3, 16-32QAM), respectively. It can be observed that the EGSM-3DQ scheme outperforms other systems in terms of BER performance at the SEs of 13, 14 bits/s/Hz.
In Fig. 5, we show the comparisons of the BER performance of the EGSM-3DQ scheme with that of the GSM-MIM system at the SE of 15, 16 bits/s/Hz and (N t , N r ) = (8,8).
According to the analysis of TABLE 4, the proposed gEGSM has bigger squared MED than other schemes such as EGSM-3DQ, GSM-MIM at the same SEs, it means that the gEGSM will reduce the BER values in comparison with other systems. Fig. 7 and Fig. 8 with the SEs of 17 and 18 bits/s/Hz are depicted and verify the BER performances of the gEGSM, and compared with that of the IQSM, PQSM, GSM-MIM and EGSM-3DQ schemes. For the SE of 17 bits/s/Hz in Fig. 7, due to the bigger squared MED provided in 4, the gEGSM with (8-3D, 1, 4QAM, 1, 4PAM) outperforms the IQSM with 8-16QAM, PQSM with 32QAM, GSM-MIM with (1,2,2,16) and EGSM-3DQ with (1, 2, 8-3D, 4-8) in terms of BER performance. For instance, approximately 0.6 dB, 2 dB, 3.5 dB, 6.5 dB SNR gain over the EGSM-3DQ, GSM-MIM, IQSM, PQSM are achieved at the BER value of 10 −3 , respectively. Also, for the SE of 18 bits/s/Hz in Fig. 7 with the change of the modulation order, compared with other schemes, the gEGSM still has better BER performance, which verifies the advantage of the gEGSM scheme.

VI. CONCLUSION
In this paper, in order to increase the SE and enhance the reliability of communication link, the EGSM-3DQ is proposed to further exploit the index domain by combining active AI combinations with the design of the improved 3D signal constellation. Then, the 3D Signal Mapper and AI vector Modulator is designed in detail for extending the active AI combinations. Furthermore, the gEGSM is proposed to further improve the SE and BER performance. Finally, the comparisons for the squared MEDs between the proposed schemes and GSM-MIM are provided and the average BEP is analyzed. Numerical results show that the proposed schemes outperform other schemes such as GSM, GSM-MIM, IQSM, PQSM at the same transmit rate in terms of BER performance.