 Research
 Open Access
Differential spatial modulation for highrate transmission systems
 Thu Phuong Nguyen^{1}Email author,
 Xuan Nam Tran^{1},
 MinhTuan Le^{2} and
 Huan X. Nguyen^{3}
https://doi.org/10.1186/s1363801710131
© The Author(s) 2018
Received: 21 July 2017
Accepted: 17 December 2017
Published: 4 January 2018
Abstract
This paper introduces a new differential spatial modulation (DSM) scheme which subsumes both the previously introduced DSM and highrate spatial modulation (HRSM) for wireless multiple input multiple output (MIMO) transmission. By combining the codeword design method of the HRSM scheme with the encoding method of the DSM scheme, we develop a highrate differential spatial modulation (HRDSM) scheme equipped with an arbitrary number of transmit antennas that requires channel state information (CSI) neither at the transmitter nor at the receiver. The proposed approach can be applied to any equal energy signal constellations. The bit error rate (BER) performance of the proposed HRDSM schemes is evaluated by using both theoretical upper bound and computer simulations. It is shown that for the same spectral efficiency and antenna configuration, the proposed HRDSM outperforms the DSM in terms of bit error rate (BER) performance.
Keywords
 MIMO
 Differential spatial modulation
 Highrate differential spatial modulation
1 Introduction
In recent years, many transmission techniques for wireless multiple input multiple output (MIMO) communication systems have been proposed. A byword is the spatial modulation (SM) [1, 2], which has attracted increased research interest recently. By exploiting the difference in the channel impulse response of a wireless link from a transmit antenna element to a receive one, the transmit antenna indices are utilized as an additional means to carry information. Compared with the Vertical Bell Laboratories Layered SpaceTime (VBLAST) [3] and spacetime block codes (STBCs) [4, 5], SM has several advantages which are attained by the following three essential features: (1) unlike the VBLAST, in an SM scheme, only one transmit antenna is activated during transmission, thereby completely avoiding the problem of interchannel interference (ICI) among the transmit antennas; (2) since only one antenna is activated for signaling the SM, transmitter needs to use only a single radio frequency (RF) chain, which certainly helps to reduce the hardware cost as well as energy consumption compared with VBLAST and STBCs; (3) the spatial position of each transmit antenna is utilized to convey information, thus allowing SM to obtain a spectral efficiency that increases logarithmically with the number of transmit antennas. Coherent SpaceTime Shift Keying (CSTSK) [6–8] is possible of striking a flexible tradeoff between the obtainable diversity and multiplexing gain. This scheme was shown to exhibit a better performance than the SM and SSK schemes since it is possible to obtain both transmit and receive diversity, rather than only receive diversity as in SM. But unlike SM, CSTSK needs multiple RF chains at the transmitter. CSTSK is also inferior to SM in that it requires strict InterAntenna Synchronization (IAS) as VBLAST and STBC. Furthermore, lowcomplexity ML detection algorithms for SM systems with M−QAM, M−PSK modulation are proposed in [9] and [10], respectively. In these detection algorithms, the ML search complexity is independent of the constellation size. As a consequence, SM is a promising MIMO technique over the conventional MIMO schemes.
Until now, most investigations on SM assumed that the CSI is available at the receiver. This assumption is reasonable when the channel varies slowly compared with the symbol rate, and knowledge of the channel can be obtained via training. Unfortunately, obtaining channel knowledge is not always realizable when the channel changes quickly. Moreover, for MIMO channels, the number of channel coefficients to be measured is equal to the product of the number of transmit antennas and the number of receive antennas. The length of the training sequence grows proportionally with the number of transmit antennas [11]. This could result in a large decrease of the overall system throughput. In order to dispense with the CSI estimation, a blind detector have been recently proposed for SM [12]. However, this detector requires a large number of observed symbols and thus increased computational complexity for signal estimation. Therefore, solutions that do not require CSI and large observations, such as differential modulation, are very useful.
Differential signaling is a widely used approach to deal with the problem of highmobility wireless communications without requiring CSI. The differential transmission concept has been successfully implemented in many MIMO systems, e.g., the differential Alamouti scheme [13] and differential spatial multiplexing [14]. However, this approach could hardly be directly applied to SM or Space Shift Keying (SSK) because the channel in SM/SSK systems is the virtual modulation unit, which makes the design of differential SM/SSK unique and difficult. Very recently, a number of differential schemes for SM have been proposed [15–19], called differential SM (DSM), which can be applied to any equal energy signal constellations. Similar to SM, DSM activates only one antenna at a symbol instant. Therefore, ICI is avoided and the requirement of IAS is relaxed. However, the performance of DSM is restricted at a given spectral efficiency.

In the proposed HRDSM scheme, information bits are carried by both SC codewords and constellation symbols. Therefore, a substantial increase in spectral efficiency is achieved as compared to other differential modulation schemes.

In order to improve the spectral efficiency, it is possible to increase either the signal space Ω_{HR−DSM} or the M−PSK constellation. The scheme is thus more flexible than the DSM [16].

Compared with the DSM system, the proposed scheme is more advantageous under the spaceconstraint situations. For the DSM, in order to increase the spectral efficiency, more transmit antennas need to be used. The proposed HRDSM scheme, however, can keep the number of transmit antenna within the limit and increase the M−PSK constellation size.

The proposed scheme is superior to the DSM system as it has two separate constellations. In the DSM system, increasing the spectral efficiency reduces distance between the signal points in the DSM constellation. Thanks to having two separate constellations, the proposed HRDSM can keep the signal space Ω_{HR−DSM} at a required level and increase the M−PSK signal constellation.

A systematic approach is presented for the design of SC codewords for an arbitrary number of transmit antennas, assuming n_{ T }≥2.

A theoretical union bound on the bit error rate of the HRDSM scheme is derived in the closed form. The derived bound can be used as a means to evaluate the BER performance of HRDSM when the signaltonoise power ratio (SNR) is sufficiently high.

Computer simulation results, supported by the theoretical upper bound, are provided to benchmark the BER performance of the propose HRDSM scheme with those of related differential transmission schemes, such as the differential Alamouti [13], the singleantenna differential scheme (DPSK), the differential scheme of quasiorthogonal spacetime block code (DQOSTBC) [22], and the DSM.
The rest of paper is organized as follows. System model of the proposed HRDSM is introduced in Section 2. The design of the SC codewords for the HRDSM is presented in Section 3. In Section 4, a closedform of pairwise error probability (PEP) and a theoretical upper bound are derived. Section 5 presents simulation results and analysis. Finally, conclusions are drawn in Section 6.
Notation: Throughout the paper, we use the following mathematical notations. (·)^{ H }, (·)^{ T }, and ∥·∥_{ F } denote the Hermitian transpose, transpose, and Frobenius norm of a vector or matrix, respectively. Re(·) denotes the real part of a complex number.
2 System model

Only one antenna remains active at each time instant. This means that only one entry in any column of S is nonzero.

Each antenna is activated only once in the n_{ T } successive time instants of the transmit signal matrix. This means that only one entry in any row of S is nonzero.
An example of the transmit signal matrix for n_{ T }=3 is given by \({\mathbf {S}} = \left [\begin {array}{ccc} 0 & s_{12} & 0\\ s_{21} & 0 & 0\\ 0 & 0 & s_{33} \end {array} \right ]\), where \(s_{{n_{t}}t}\) denotes the transmit symbol over the n_{ t }th antenna at time t. This matrix determines that at time instant 1, the symbol s_{21} is transmitted over transmit antenna 2 while transmit antennas 1 and 3 remain idle. Similarly, at time time instants 2 and 3, the symbols s_{12} and s_{33} are transmitted over antennas 1 and 3, respectively, while the other two transmit antennas remain idle. According to this design, each antenna is activated only once during each block, making differential operation possible so long as the wireless channel remains unchanged over two successive HRDSM blocks.
The proposed HRDSM transmitter works as follows: At time t+1, (l+m) data bits enter the HRDSM transmitter, among which l bits are mapped into a n_{ T }×T matrix X_{t+1}, out of K matrices in the signal space Ω_{HR−DSM} with basic elements drawn from M_{1}−PSK constellation, while remaining m=log2M bits are mapped into a M−PSK constellation symbol x_{t+1}. The resulting n_{ T }×T HRDSM codeword C is generated by multiplying X by x, i.e., C_{t+1}=X_{t+1}·x_{t+1}.
3 HRDSM codeword design
As HRDSM codewords C are obtained simply by multiplying matrices X in the signal space Ω_{HR−DSM} by the constellation symbol x, our objective is to design suitable matrix X for a given n_{ T }.
where column vector \({{\mathbf {x}}_{n}} = {\left [ {\begin {array}{*{20}{c}} {{x_{1n}}}&{{x_{2n}}}& \cdots &{{x_{{n_{T}}n}}} \end {array}} \right ]^{T}}\), (n=1, 2, ⋯, n_{ T }), is a n_{ T }×1 vector with complex valued entries; \(\Gamma = \left \ {{{\mathbf {x}}_{n}}} \right \ = \sqrt {{\sum \nolimits }_{{n_{t}} = 1}^{{n_{T}}} {{{\left  {{x_{{n_{t}}n}}} \right }^{2}}}} \) is the magnitude of x, which is used to normalize the transmit power.
 1.
Assign the first permutation vector to \({{\mathbf {p}}_{1}} = \left [ {\begin {array}{cccc} 1&2& \cdots &{{n_{T}}} \end {array}} \right ]\).
 2.
Built (n_{ T }!−1) permutation vectors p_{ p } of p_{1}:
\({{\mathbf {p}}_{2}} = \left [ {\begin {array}{*{20}{c}} 2&1& \cdots &{{n_{T}}} \end {array}} \right ], \cdots, {{\mathbf {p}}_{{n_{T}}!}} = \left [ {\begin {array}{cccc} {{n_{T}}}&{{n_{T  1}}}& \cdots &1 \end {array}} \right ]\).
 3.Arrange the G(p_{ p }) matrices corresponding to permutation vectors p_{ p } as follows:
 (a)
Fix the p_{1}th element of vector column x_{1} to 1 and 0s elsewhere.
 (b)
Let the p_{ n }th elements of remaining column vectors x_{ n } be selected from M_{1}−PSK constellation symbols and 0s elsewhere.
 (a)
 4.
Generate the corresponding matrix X: X_{ q }=G(p_{ p },M_{1}), \(\left ({q = 1,\,2,\, \cdots,\,{n_{T}}!M_{1}^{{n_{T}}  1}} \right)\).
The motivation behind assigning the only nonzero element of vector column x_{1} to 1 is to guarantee that the proposed HRDSM scheme obtains high performance. Thanks to this assignment, for the case X_{ i }≠X_{ j } and x_{ i }≠x_{ j }, we get C_{ i }≠C_{ j }. Without this assignment, there possibly exist X_{ i }≠X_{ j } and x_{ i }≠x_{ j } such that C_{ i }=C_{ j }, leading to a wrong detection at the receiver.
is maximized for all pairs of distinct codewords C_{ i }≠C_{ j } and for all combinations of (M,M_{1}).
According to the design procedure, for a given n_{ T } and with M_{1}−PSK constellation, a total of \(Q = {n_{T}}!M_{1}^{{n_{T}}  1}\) matrices X in the signal space Ω_{HR−DSM} can be obtained. Therefore, one matrix X is able to carry \(l{\mathrm {= }}\left \lfloor {{\text {lo}}{{\mathrm {g}}_{2}}\left ({{n_{T}}!M_{1}^{{n_{T}}  1}} \right)} \right \rfloor \) information bits. In addition, one M−PSK constellation symbol corresponds to m=log_{2}M information bits. Both of them are transmitted within n_{ T } symbol periods. Consequently, the spectral efficiency of the proposed HRDSM scheme is equal to \(\frac {1}{{{n_{T}}}}\left ({l + m} \right) = \frac {1}{{{n_{T}}}}\left [ {\left \lfloor {{\text {lo}}{{\mathrm {g}}_{2}}\left ({{n_{T}}!M_{1}^{{n_{T}}  1}} \right)} \right \rfloor + {{\log }_{2}}M} \right ]\) bpcu. Clearly, the additional spectral efficiency offered by our proposed scheme is substantially higher than that of DSM.
4 Theoretical upper bound of HRDSM
Suppose that the message C_{ t } is sent at each block. Since errors occur during transmission of the actual transmitted signal matrix S_{ t } due to channel fading and noise, after differential decoding, assume that the message E_{ t } in each block is detected. It follows that \({{\mathbf {E}}_{t}}{\mathbf {E}}_{t}^{H} = {{\mathbf {I}}_{{n_{T}}}}\), where I_{ n } is the n_{ T }×n_{ T } identity matrix. In order to measure the difference between C_{ t } and D_{ t }, we define \({{\mathbf {D}}_{t}} = {{\mathbf {E}}_{t}}{\mathbf {C}}_{t}^{H}\). So the matrix distance between C_{ t } and E_{ t } can be expressed as \(\text {trace}\left \{ {{\text {Re}} \left ({{{\mathbf {I}}_{{n_{T}}}}  {{\mathbf {D}}_{t}}} \right)} \right \}\). When no error occurs, \(\phantom {\dot {i}\!}{{\mathbf {D}}_{t}} = {{\mathbf {I}}_{{n_{T}}}}\), so \(\text {trace}\left \{ {{\text {Re}} \left ({{{\mathbf {I}}_{{n_{T}}}}  {{\mathbf {D}}_{t}}} \right)} \right \} = 0\). Since \({{\mathbf {D}}_{t}}{\mathbf {D}}_{t}^{H} = {{\mathbf {I}}_{{n_{T}}}}\), it follows that matrix D_{ t } has the same orthogonal property as the message matrix C_{ t } and the actual transmitted signal matrix S_{ t }.
Recall that the HRDSM transmitter transmits the matrices S_{ t } and S_{t−1} instead of directly transmitting the message matrices C_{ t }. Due to the influence of fading and noise, suppose that while S_{ t } and S_{t−1} are transmitted, Q_{ t } and Q_{t−1} are actually received which causes that the differentially decoded message matrices C_{ t } to become the error message matrices E_{ t }. Obviously, Q_{ t }=E_{ t }Q_{t−1}=E_{ t }C_{t−1} and \({{\mathbf {Q}}_{t}}{\mathbf {Q}}_{t  1}^{H} = {{\mathbf {E}}_{t}}\).
where p(γ_{ b }) is the PDF of γ_{ b } given in [26].
where \(w\left ({{\mathbf {u}},{\hat {\mathbf {u}}}} \right)\) is the Hamming distance between sequences u and \({\hat {\mathbf {u}}}\). The PEP Pr(C→E) is given by Eq. (22).
5 Performance evaluation
In this section, Monte Carlo simulations and the theoretical upper bound are used to study the BER performance of the proposed HRDSM scheme for different antenna arrangements, as well as to compare them against different MIMO systems, such as coherent detection SM [1] and DSM [15], the differential scheme of the well known Alamouti scheme [13], the DPSK scheme, DQOSTBC scheme [22]. Simulations are carried out over the quasistatic Rayleigh fading channel. We assume that the channel state information is perfectly known at the receiver of an SM system. In addition, ML detection is applied to all systems under consideration.
5.1 Comparison between theoretical and simulation results
5.2 Comparison between the differential detection HRDSM and the coherent detection SM
It is observed from the figure that compared to the SM, HRDSM suffers from a performance degradation of less than 3 dB in the SNR, particularly in the high SNR region.
5.3 BER performance comparison of proposed HRDSM and other transmission schemes.
In addition, it can be observed that in all antenna configurations, DA and DQOSTBC benefit more from increasing the SNR than does DSM and HRDSM. This is because the diversity order of HRDSM, as well as DSM, approaches that of single antenna system which is n_{ R } as detailed in [2]. In contrast, the Alamouti scheme with n_{ T }=2 provides full diversity, i.e., 2n_{ R } and QOSTBC with n_{ T }=4 provides half of the maximum achievable diversity. Thus, at high SNR, DA and DQOSTBC outperform HRDSM and DSM.
5.4 HRDSM with multiple transmit antennas
Another observation is that for the same number of receive antennas, HRDSM configurations with n_{ R }=1 will provide almost same performance. For n_{ R }=2, 4, HRDSM with n_{ T }=4 outperforms HRDSM with n_{ T }=2, 3. The likelihood of erroneous antenna detection decreases with an increasing number of receive antennas. This observation can be used as a guideline for selecting the signal constellation and the spatial configuration. To get the full potential of HRDSM, the system should be equipped with sufficient receive antennas.
6 Conclusions
In this paper, a new differential space time modulation scheme based on the HRSM codeword is proposed for MIMO systems. In addition, a theoretical upper bound is derived for the evaluation of the BER of the proposed HRDSM scheme. Simulation results show that this scheme outperforms the DSM scheme in terms of BER performance. The performance of the proposed scheme can be improved by selecting the set of transmission matrices having the largest minimum Euclidean distance. Undoubtedly, the proposed method paves a new way to design differential SM schemes with more transmit antennas.
7 Appendix
7.1 Evaluation of the variance of trace{Re(Θ)}
From Eq. (26), we can see that the variance of trace{Re(Θ)} consists of three parts: the variance of trace{Re(Θ_{1})}, the variance of trace{Re(Θ_{2})}, and the crosscorrelation between trace{Re(Θ_{1})} and trace{Re(Θ_{2})}. These quantities will be evaluated in the following subsections.
7.2 Evaluation of Var[trace{Re(Θ_{1})}], Var[trace{Re(Θ_{2})}]
7.3 Evaluation of Cov(trace{Re(Θ_{1})}, trace{Re(Θ_{2})})
Proof
Finally, substituting Eqs. (29) and (30) into Eq. (26), we obtain Eq. (17). □
Declarations
Acknowledgements
This work was supported by the Vietnam National Foundation for Science and Technology Development under Grant 102.022015.23.
Authors’ contributions
The authors have contributed jointly to all parts on the preparation of this manuscript, and all authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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