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Layered spatial modulation
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 233 (2018)
Abstract
In this paper, we propose a layered spatial modulation (LSM) scheme that combines a multilayered coding scheme with the spatial modulation. At each antenna selection, the number of active transmit antennas equals the number of layers. The proposed LSM scheme maximizes the diversity order and coding gain at each antenna selection. The exact pairwise and upper bound bit error probabilities are derived. Several examples of different spatial modulation schemes and the proposed LSM scheme are simulated with the same number of transmit antennas and same spectral efficiency. As expected, the proposed LSM scheme satisfies significant increase in the performance compared to all existing spatial modulation schemes.
Introduction
Significant improvements in the bandwidth efficiency and notable reduction in the probability of error can be obtained by using multiple input multiple output (MIMO) technology [1, 2]. This technology uses multiple transmit and multiple receive antennas where space and time diversities and spatial multiplexing are applied. Several spacetime trellis and block coding schemes that improve the diversity and coding gains were proposed [3,4,5,6,7,8,9]. On the other hand, a number of coding schemes were introduced to improve the spatial multiplexing gain [10,11,12,13,14,15,16,17]. All the above schemes use more than one transmit antenna in the same time. Simultaneous activation of multiple transmit antennas causes the interchannel interference. To avoid this problem, spatial modulation (SM) scheme was presented [18, 19]. This scheme uses one transmit antenna at each time period depending on the first group of input bits. The second group of bits is used to determine which symbol is selected from the constellation set. Unfortunately, this scheme needs large number of transmit antennas to increase the spectral efficiency of wireless system. To reduce the number of transmit antennas, generalized spatial modulation scheme was proposed [20]. This scheme uses more than one transmit antenna at each time period. However, the transmit diversity is not exploited in this scheme which yields significant degradation in the performance. To overcome this problem, a scheme that combines spatial modulation and spacetime block coding (STBC) referred as STBCSM was proposed [21]. This scheme takes the advantages of both spatial modulation and spacetime block coding. Also, it maximizes the diversity order and multiplexing gain. This scheme suffers from degradation of the spectral efficiency offered by the spatial dimension compared with the conventional spatial modulation. To improve the spectral efficiency, highrate STBCSM (HSTBCSM) scheme and spatial modulation STBC scheme with cyclic structure (STBCCSM) were introduced [22, 23]. The number of codewords provided by these schemes is twice the number of codewords provided by the STBCSM in [21]. The transmit diversity order achieved by both schemes is equal to two. However, the HSTBCSM suffers from small degradation in the performance. At the same spectral efficiency and with transmit antennas less than those of the STBCSM scheme, the performance of STBCCSM approaches that of the STBCSM. The authors in [24] proposed the complex interleaved orthogonal design SM with the high degree of spatial modulation (CIODSMH). The spectral efficiency of this scheme is as that of the HSTBCSM scheme, and its performance is less than that of the STBCSM. To further increase the spectral efficiency and the performance, cyclic temporally and spatially modulated STBC (STBCTSM) scheme was proposed [25]. In this scheme, Alamouti STBC is rotated over four symbol periods to ensure a transmit diversity order of 2.
In this paper, we propose a layered spatial modulation (LSM) scheme that avoids large number of transmit antennas and provides high spectral efficiency. The proposed LSM scheme achieves significant improvement in the performance compared to that of the all above spatial modulation schemes. The proposed LSM scheme is based on a multilayered coding scheme combined with the spatial modulation scheme in [18, 19]. At each antenna selection, the proposed LSM scheme is used with transmit diversity order equals the number of active transmit antennas and maximum coding gain based on specific codeword matrices. The number of layers of the proposed scheme is equal to the number of active transmit antennas at each antenna selection.
The exact pairwise and upper bound bit error probabilities of the proposed LSM scheme are derived. Simulation results are demonstrated with several examples of the proposed LSM scheme and recent spatial modulation schemes combined with STBC. All schemes are compared with the same number of transmit antennas and comparable spectral efficiency. These results show that the proposed LSM scheme significantly outperforms all other spatial modulation schemes.
This paper is organized as follows: In Section 2, the methods that we used are presented. Conventional spatial modulation scheme is presented in Section 3. In Section 4, we propose the LSM scheme. We derive the pairwise and bit error probabilities of the proposed LSM scheme in Section 5. In Section 6, numerical results of the proposed LSM and different spatial modulation schemes are demonstrated. Finally, the conclusions of this paper are inserted in Section 7.
Methods
The proposed LSM scheme is a combination of a multilayered coding scheme and the spatial modulation. The codeword of the multilayered coding scheme contains a number of layers with the same number of active transmit antennas. A number of phase shifts are inserted in the codeword among the layers and symbols to maximize the diversity and coding gains. When all transmit antennas are active, one codeword is transmitted over one codeword period. When the total number of transmit antennas is greater than or equals twice the number of active transmit antennas, two codewords are transmitted from different active transmit antennas over two different codeword periods at each antenna selection case. Several examples of the proposed LSM and the stateoftheart schemes are simulated using Matlab software. The channel is modeled by using the Monte Carlo method. The schemes are compared to each other with the same number of transmit and receive antennas and same spectral efficiency.
Conventional spatial modulation scheme
The SM scheme in [18, 19] is described in this section as shown in Fig. 1. This scheme uses two groups of bits. The first group of k bits is used to select one active transmit antenna i out of N_{all} antennas at each time period, and the second group of u bits is used to select the symbol c from the constellation set that has 2^{u} elements. This results in transmission rate of (k + u) bits per channel use. The advantage of this approach is that it avoids the problem of interchannel interference.
The signal at receive antenna j can be written as
where E_{s} is the average symbol energy, c_{i} is the symbol c transmitted from antenna i, and a_{ji} is the path gain between transmit antenna i and receive antenna j. It is assumed that the path gains are samples of a zero mean complex Gaussian random variable with variance of 0.5 per dimension. The additive noise v_{j} at the receive antenna j is assumed to be independent samples of a zero mean complex Gaussian random variable with variance of N_{0}/2 per dimension.
The exact pairwise error probability of transmitting c_{i} and deciding \( {\tilde{c}}_{\tilde{i}} \) is given by [20]
where \( {\tilde{N}}_0={N}_0+{E}_s{\leftc\right}^2+{E}_s{\left\tilde{c}\right}^2 \) and I_{x}(w_{1}, w_{2}) is the regularized beta function given by
with
The bit error probability of spatial modulation scheme is given by
where \( N\left({c}_i,{\tilde{c}}_{\tilde{i}}\right) \) is the number of bits in error between \( {c}_i\ \mathrm{and}\ {\tilde{c}}_{\tilde{i}} \).
As an example of this scheme is to apply 3 bits per channel use. The first 2 bits are used to select the active transmit antenna out of four antennas, and the third bit is used to select the symbol from the binary phase shift keying (BPSK) constellation set. The transmission and encoding of this example is shown in Table 1.
Proposed LSM scheme
Proposed multilayered coding scheme without spatial modulation
In this subsection, we present a multilayered coding scheme that maximizes the diversity order and spatial multiplexing gain as shown in Fig. 2.
In this scheme, multi layers with the same number of transmit antennas are applied. Each layer contains a number of symbols. The received signal matrix is given as
where V is N_{r} × N_{t} noise matrix at the receive antennas whose entries are independent samples of a zero mean complex Gaussian random variable with variance of N_{0}/2 per dimension. A is N_{r} × N_{t} channel matrix whose entry a_{ji} is the path gain between transmit antenna i and receive antenna j, and it is given as
It is assumed that the path gains in A are samples of a zero mean complex Gaussian random variable with variance of 0.5 per dimension. In (6), C is the codeword matrix given by
where c_{j, i} is the ith symbol of the jth layer. The symbols c_{j, i}^{'}s are selected by the bits b_{1}b_{2} … b_{u}. In (9) and (10), \( {\alpha}_1,{\alpha}_2,\cdots, {\alpha}_{N_t} \) and \( {\beta}_1,{\beta}_2,\cdots, {\beta}_{N_t} \) are the phase shifts among transmitted symbols and layers, respectively. It is worthy of noting that this scheme provides spectral efficiency of N_{t} symbols per channel use.
The upper bound of the average probability of codeword matrix C_{m} and the decoder decided C_{n} is given in [13] as
where R = (C_{m} − C_{n})(C_{m} − C_{n})^{H} and the superscript H denotes Hermitian transpose. This result was derived by using the Chernoff’s upper bound [26].
In the case of high signal to noise ratio, (11) can be written as
where r and λ_{1}, λ_{2}, ⋯, λ_{r} are the rank and the eigenvalues of matrix R, respectively.
The angles \( {\alpha}_1,\cdots, {\alpha}_{N_t},{\beta}_1,\cdots, {\beta}_{N_t} \) have to be chosen to increase the upper bound of (11). This can be obtained by maximizing the rank r to become N_{t} and the product of eigenvalues (λ_{1}λ_{2}⋯λ_{r}) of R.
It is worthy of noting that this coding scheme achieves diversity order of N_{t} × N_{r} and maximum coding gain \( {\left({\lambda}_1{\lambda}_2\cdots {\lambda}_{N_t}\right)}^{\frac{1}{N_t}} \) under specific values of \( {\alpha}_1,\cdots, {\alpha}_{N_t},{\beta}_1,\cdots, {\beta}_{N_t} \).
Proposed LSM scheme
In this section, we combine the proposed layered coding scheme presented in Section 4.1 with the spatial modulation scheme presented in [18, 19] to form a layered spatial modulation (LSM) scheme. Assume that the total number of antennas = N_{all} and the number of active transmit antennas at each antenna selection = N_{t}. In Section 4.1, N_{all} = N_{t} and one codeword matrix is transmitted over N_{t} symbol periods (one codeword period) with diversity order equals N_{t} × N_{r}. In this case, there is no antenna selection and the proposed scheme uses \( {N}_t^2 \) symbols per N_{t} channel use. For N_{all} ≥ 2N_{t}, two different codewords are transmitted from different active transmit antennas over two different codeword periods at each antenna selection case. Depending on the antenna selection case, each codeword is transmitted circularly from N_{t} active antennas along the total number of transmit antennas. This produces \( N=\frac{2{N}_{\mathrm{all}}!}{\left({N}_{\mathrm{all}}{N}_t\right)!{N}_t!}+{N}_{\mathrm{all}} \) antenna selections and the number of bits related to these selections = k = log_{2}N. The proposed LSM scheme achieves diversity order of N_{t} × N_{r} and maximum coding gain at each case of the antenna selections. Since each codeword contains \( {N}_t^2 \) symbols, the proposed LSM scheme uses \( \left(k\ \mathrm{bits}+2{N}_t^2\ \mathrm{symbols}\right) \) per 2N_{t} channel use which is equivalent to \( \left(\frac{k}{2{N}_t}\ \mathrm{bits}+{N}_t\ \mathrm{symbols}\right) \) per channel use. Note that N_{t} symbols = N_{t}log_{2}M bits. This yields \( \left(\frac{k}{2{N}_t}+{N}_t{\log}_2M\ \right)\ \mathrm{bits} \) per channel use. For the proposed LSM scheme, the received signal matrix over the two codeword periods is sent to the maximum likelihood decoder to estimate the k bits (b_{1}, b_{2},…, b_{k}) related to antenna selections and the \( 2{N}_t^2{\log}_2M \) bits related to the symbols in the two codeword matrices. Two different examples are introduced to explain the concept of the LSM scheme.
Example 1 We consider the proposed scheme with N_{all} = N_{t} = 2. In this case, there is no antenna selection. Therefore, one codeword with two layers is transmitted from two antennas over one codeword period of time. Assume that the symbols c_{11} and c_{12} are related to layer 1 and the symbols c_{21} and c_{22} are related to layer 2. According to (8), (9), and (10), the transmitted codeword matrix C of this example is given by
The spectral efficiency of this case is equal to 2 symbols/s/Hz = (2 log_{2}M) bits/s/Hz. For the case of BPSK modulation, we assume that α_{1} = β_{1} = 0 as reference. The phase shifts α_{2} and β_{2} have to be chosen to increase the upper bound of (11). This can be obtained by maximizing the rank r and the product of eigenvalues (λ_{1}λ_{2}) of R. A rank r of 2 and maximum product of eigenvalues (λ_{1}λ_{2}) occur when \( {\alpha}_2={\beta}_2=\frac{\pi }{2} \). For the case of quadrature phase shift keying (QPSK) modulation, α_{1} = β_{1} = 0, \( {\alpha}_2=\frac{\pi }{6} \), and \( {\beta}_2=\frac{\pi }{4} \) and the spectral efficiency = 4 bits/s/Hz. For the case of 8QAM modulation, α_{1} = β_{1} = 0, α_{2} = 0.21π, and \( {\beta}_2=\frac{3\pi }{16} \) and the spectral efficiency = 6 bits/s/Hz. For the case of 16QAM modulation, α_{1} = β_{1} = 0, α_{2} = 0.1π, and \( {\beta}_2=\frac{\pi }{4} \) and the spectral efficiency = 8 bits/s/Hz.
Example 2 We consider the proposed LSM scheme with N_{all} = 4, N_{t} = 2 and 2 transmitted codewords C and E. The transmission approach of this example is demonstrated in Table 2. Note that the two codewords are transmitted from different active transmit antennas over two different codeword periods at each antenna selection case. The rotation of the two codeword matrices is different than that in [21,22,23,24,25]. The number of antenna selections \( N=\frac{2\times 4!}{\left(42\right)!2!}+4=16 \) as shown in Table 2 and k = 4. For BPSK, QPSK, 8QAM, and 16QAM modulation, the angles α_{1}, β_{1}, α_{2}, and β_{2} are the same as those in Example 1 whereas the angles θ_{1}, θ_{2}, θ_{3}, θ_{4}, and θ_{5} are selected to maximize the diversity order to 2N_{r} and multiplexing gain and they are equal to \( \frac{\pi }{12},\frac{2\pi }{12},\frac{4\pi }{12},\frac{5\pi }{12},\mathrm{and}\ \frac{3\pi }{12} \), respectively. The spectral efficiency of this example = \( \left(\frac{k}{2{N}_t}\ \mathrm{bits}+{N}_t\ \mathrm{symbols}\right) \) per channel use = (1 + 2log_{2}M) bits/s/Hz.
Performance analysis of the proposed LSM scheme
The performance of the proposed LSM scheme is derived in this section. At antenna selection related to b_{1}, b_{2},…, b_{k}, the received signal matrices Y_{m} and Y_{p} related to codeword matrices C_{m} and C_{p} transmitted from two different active antenna combinations over different symbol periods through two codeword periods are given by
where the active channel matrices A_{l} and A_{f} and the angles θ_{l} and θ_{f} are related to active antenna combinations l and f which are different in all antenna elements.
Define matrices G(b_{1}, b_{2}, …, b_{k}, C_{m}) and G(b_{1}, b_{2}, …, b_{k}, C_{p}) as
where C_{n} and C_{q} are different than C_{m} and C_{p}. The pairwise error probability of (b_{1}, b_{2}, …, b_{k}, C_{m}, C_{p}) and the decoder decided \( \left({\tilde{b}}_1,{\tilde{b}}_2,\dots, {\tilde{b}}_k,{\mathbf{C}}_n,{\mathbf{C}}_q\right) \) is given by
where the subscript F denotes the Frobenius norm of the matrix.
The mean and variance matrices of G(b_{1}, b_{2}, …, b_{k}, C_{m}) are given by
where \( {\mathbf{0}}_{N_r\times {N}_t} \) is N_{r} × N_{t} zero matrix and E is the expectation.
In a similar manner,
The mean and variance matrices of \( \mathbf{G}\left({\tilde{b}}_1,{\tilde{b}}_2,\dots, {\tilde{b}}_k,{\mathbf{C}}_n\right) \) are given by
V_{1} is independent of \( {\mathbf{A}}_l{\mathbf{C}}_m{e}^{j{\theta}_l} \) and \( {\mathbf{A}}_{\tilde{l}}{\mathbf{C}}_n{e}^{j{\theta}_{\tilde{l}}} \), then
In a similar manner,
\( \overline{\mathbf{G}}\left({\tilde{b}}_1,{\tilde{b}}_2,\dots, {\tilde{b}}_k,{\mathbf{C}}_q\right)={\mathbf{0}}_{N_r\times {N}_t} \) (28)
Let,
where the operation ./ is Hadamard division (element by element of the numerator and denominator matrices).
\( {\left\Vert \mathbf{D}\left({b}_1,{b}_2,\dots, {b}_k,{\mathbf{C}}_m\right)\right\Vert}_F^2 \), \( {\left\Vert \mathbf{D}\left({b}_1,{b}_2,\dots, {b}_k,{\mathbf{C}}_p\right)\right\Vert}_F^2 \), \( {\left\Vert \mathbf{D}\left({\tilde{b}}_1,{\tilde{b}}_2,\dots, {\tilde{b}}_k,{\mathbf{C}}_n\right)\right\Vert}_F^2 \), and \( {\left\Vert \mathbf{D}\left({\tilde{b}}_1,{\tilde{b}}_2,\dots, {\tilde{b}}_k,{\mathbf{C}}_q\right)\right\Vert}_F^2 \) are central chisquare random variables with 2N_{r} degrees of freedom. By using the approach in [20], the pairwise error probability can be obtained as
In (34), z_{1} and z_{2} are given by
By using the upper bound presented in [27], the average probability of bit error is given by
where \( v=2{N}_t^2\ {\log}_2(M) \) and \( {N}_e\left({b}_1,\dots, {b}_k,{\mathbf{C}}_m,{\mathbf{C}}_p,{\tilde{b}}_1,\dots, {\tilde{b}}_k,{\mathbf{C}}_n,{\mathbf{C}}_q\right) \) is the number of bits in error between b_{1}, …, b_{k}, C_{m}, C_{p} and \( {\tilde{b}}_1,\dots, {\tilde{b}}_k,{\mathbf{C}}_n,{\mathbf{C}}_q \). By substituting (34) in (37),
Numerical results and comparisons
In this section, we compare the performance of the proposed LSM scheme with the spatial modulation schemes: SM [18, 19], STBCSM [21], STBCCSM [23], and STBCTSM [25] with the same number of antennas and same spectral efficiency.
Figure 3 shows the bit error probability of the proposed LSM scheme with two layers compared to SM scheme [18, 19] as a function of E_{b}/N_{0} where E_{b} is the bit energy. The proposed LSM and SM schemes are equipped with two transmit and two receive antennas. For the proposed LSM scheme N_{all} = N_{t} = 2 and the codeword is transmitted over one codeword period without any antenna selection. QPSK and 8QAM modulation are used for the proposed LSM and SM schemes, respectively. This yields spectral efficiency of 4 bits/s/Hz for both of them. Example 1 in Section 4.2 explains the proposed scheme for this case. As can be observed from the figure, the proposed scheme significantly outperforms the SM scheme.
In Figs. 4 and 5, the performance of the proposed LSM, STBCSM, STBCCSM and STBCTSM schemes is presented for N_{r} = 1 and N_{r} = 2. All these schemes except STBCCSM are equipped with four transmit antennas, and the SMCSM scheme is equipped with five transmit antennas. The number of active transmit antennas of all these schemes is equal to 2. To achieve spectral efficiency of 3 bits/s/Hz for all schemes, BPSK modulation is used for the proposed LSM, STBCCSM, and STBCTSM schemes and QPSK is used for STBCSM scheme. The proposed LSM scheme in this case is explained by example 2 of Section 4.2. In both figures, significant improvement in the performance of the proposed LSM scheme is achieved compared to that of the all other schemes.
Figure 6 further compares the performance of the proposed LSM scheme with SM scheme. Both schemes use eight transmit and two receive antennas. The proposed LSM scheme uses two active transmit antennas and QPSK modulation. The total number of antenna selections \( N=\frac{2\times 8!}{\left(82\right)!2!}+8=64 \), but in this example, we use only 16 antenna selections to produce k = 4 which yields spectral efficiency of 5 bits/s/Hz. The SM scheme uses QPSK modulation to satisfy the same spectral efficiency of 5 bits/s/Hz. As expected, similar behavior as in the case of Fig. 3 is observed.
Note that the generalized spatial modulation scheme in [20] uses more than one active transmit antenna at each time period to increase the spectral efficiency. The performance of this scheme is slightly smaller than the performance of the spatial modulation scheme in [18, 19] with the same spectral efficiency as explained in [20]. Therefore, no need to compare this scheme with our scheme since our scheme significantly outperforms the spatial modulation scheme in [18, 19]. Also, no need to compare our scheme with the HSTBCSM [22] and CIODSMH [24] schemes because their performance is less than that of the STBCCSM [23] and STBCTSM [25] schemes with the same transmit antennas and same spectral efficiency as can be shown in [23, 25].
Conclusions
We have proposed LSM scheme with high performance compared to all existing spatial modulation schemes. The proposed LSM scheme uses a number of layers with the same number of active transmit antennas. At each antenna selection, the proposed LSM scheme maximizes the diversity and coding gains. The performance of the proposed LSM scheme was analyzed. Numerical results of the proposed LSM and different spatial modulation schemes were presented. Significant improvement in the performance of the proposed LSM scheme is achieved compared to that of the other schemes.
Abbreviations
 BPSK:

Binary phase shift keying
 CIODSMH:

Complex interleaved orthogonal design SM with the high degree of spatial modulation
 E _{ b } /N _{0} :

Bit energy to noise ratio
 HSTBCSM:

Highrate spacetime block coded spatial modulation
 LSM:

Layered spatial modulation
 MIMO:

Multiple input multiple output
 MQAM:

Mary quadrature amplitude modulation
 QPSK:

Quadrature phase shift keying
 SM:

Spatial modulation
 STBC:

Spacetime block coding
 STBCCSM:

Spatial modulation spacetime block coding scheme with cyclic structure
 STBCSM:

Spacetime block coded spatial modulation
 STBCTSM:

Cyclic temporally and spatially modulated spacetime block coding
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Alkhawaldeh, S.A. Layered spatial modulation. J Wireless Com Network 2018, 233 (2018). https://doi.org/10.1186/s136380181250y
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Keywords
 MIMO
 Diversity
 Spatial multiplexing
 Layered spatial modulation
 STBC
 SM
 LSM