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Differential spatial modulation for highrate transmission systems
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 6 (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.
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.
Motivated by the aforementioned challenge, in this paper, we propose a new DSM scheme based on the concept of spatial constellation (SC), SC codewords [20] and HRSM codeword [21], called HighRate Differential Spatial Modulation (HRDSM), for an arbitrary number of transmit antennas. In the proposed HRDSM scheme, transmit signal matrix (i.e., HRDSM codewords) are generated simply by multiplying SC codewords by signal symbols drawn from an MPSK constellation. The role of the SC matrices is to determine how the constellation symbols are weighted and which antenna combination is selected to transmit the HRDSM codewords. It is noted that the role of SC matrices is similar that of the dispersion matrices (DM) for SM in [7, 8]. However, the designs of these two matrices are based on different approaches. The advantages of the proposed HRDSM scheme can be elaborated as follows:

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.
The contributions of this paper can be summarized as follows:

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.
System model
Figure 1 shows the transmitter and receiver configuration of the proposed HRDSM with n_{ T } transmit antennas and n_{ R } receive antennas working over a Rayleigh flatfading channel. Assume that at time instant t, symbol s_{ t } is transmitted via the n_{ t }th transmit antenna, n_{ t }=1,2,⋯,n_{ T }. The transmitted signal vector is given by \({\mathbf {s}} = {\left [ {\begin {array}{*{20}{c}} {\begin {array}{*{20}{c}} 0& \cdots &0&{{s_{t}}} \end {array}}&0& \cdots &0 \end {array}} \right ]^{T}}\), where only the n_{ t }th element is nonzero. In our proposed HRDSM system, we collect the transmit signal vector s in time and form a n_{ T }×T actual transmit signal matrix S. As proved in [23], for noncoherent MIMO systems, for arbitrary block length T, number of receive antennas n_{ R } and signaltonoise ratio (SNR), the capacities obtained for n_{ T }>T and n_{ T }=T are equal. Therefore, we set T=n_{ T } in our proposed scheme. This means that each transmit signal matrix is sent during T=n_{ T } symbol durations. The actual transmit signal matrix S satisfies the following conditions:

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}.
The actual transmitted signal matrix S_{t+1} is computed via the following formula
Note that relation Eq. (1) is the fundamental differential transmission relation. Without loss of generality, we choose \(\phantom {\dot {i}\!}\mathbf {C}_{0} = \mathbf {I}_{n_{T}}\), where \(\phantom {\dot {i}\!}\mathbf {I}_{n_{T}}\) is the n_{ T }×n_{ T } identity matrix. Then, the chain of transmitted matrices is given by:
Let H_{ t } be the n_{ R }×n_{ T } fading matrix with the (i,j)th entry h_{ ij } denoting the normalized complex fading gain from transmit antenna j to receive antenna i. At the receiver side, the chain of received matrices, Y_{0},…,Y_{ t },Y_{t+1}…, is given by
and
where Y_{ t } is the n_{ R }×T received signal matrix and N_{ t } is the n_{ R }×T AWGN matrix. Using the differential transmission relation in Eqs. (1) and (2), we can rewrite Eq. (3) as
where N_{ t }^{′} is the n_{ R }×T AWGN matrix. Therefore, to estimate the information matrix, the optimal ML detector can be derived as
This is equivalent to
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 }.
For a given n_{ T }, with M_{1}−PSK constellation, we define the following n_{ T }×1 permutation vector \({{\mathbf {p}}_{p}} = \left [ {\begin {array}{*{20}{c}} {{p_{1}}}&{{p_{2}}}& \cdots &{{p_{{n_{T}}}}} \end {array}} \right ]\), (p=1, 2, ⋯,n_{ T }!). For the proposed HRDSM scheme, we define the basic form of the generator matrix G
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.
Then, a general procedure for designing the matrix X for n_{ T } transmit antennas is summarized as follows:

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}), \(\left ({q = 1,\,2,\, \cdots,\,{n_{T}}!M_{1}^{{n_{T}}  1}} \right)\).
For example, for n_{ T }=3, with BPSK constellation, we can construct a signal set Ω_{HR−DSM} which includes \(q = {{n_{T}}!M_{1}^{{n_{T}}  1}}=3!2^{2}=24\) matrices X as follows:
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.
In order for the proposed HRDSM system to achieve high performance, the transmitted signal space Ω_{ C } needs to be designed such that the minimum Euclidean distance between two arbitrary transmission signal matrices C_{ i }, C_{ j }∈Ω_{ C } is maximized [24]. Define the difference between two matrices C_{ i } and C_{ j } as d_{ i,j }=C_{ i }−C_{ j }, then the Euclidean distance between two transmission signal matrices C_{ i } and C_{ j } can be expressed as
Then, for a given spectral efficiency, M and M_{1} are selected such that the following minimum Euclidean distance
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.
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 \({{\mathbf {Q}}_{t}}{\mathbf {Q}}_{t  1}^{H} = {{\mathbf {E}}_{t}}\).
Let η_{ c } and η_{ e } be the decision variables for transmission matrices C and E, respectively. Besides, let P(C→EH) be the pairwise error probability of deciding E when C is transmitted for a given channel realization H. Then, P(C→EH) can be expressed as
where
and
Substituting Eq. (2) into Eqs. (10) and (11), we have
Note that the secondorder noise terms in Eq. (12) are ignored since they are quite small compared to other noise terms when SNR is large enough. Let
where ρ is defined as
and
For given transmission matrices C and E, Q_{t−1},Q_{ t },S_{t−1} and S_{ t }, and ρ can be considered as deterministic quantities. Therefore, we can easily show that E[trace{Re(Θ)}]=0. Taking the expectation of both sides of Eq. (13), we have:
The computation of the variance of Δ is more complicated, because some terms in Eq. (15) are correlated, although most of the terms are assumed to be mutually independent. It is proved in the “Appendix” section that the variance of Δ is given by:
which can be simplified to
From Eqs. (9), (13), (16), and (18), it follows that
where Q denotes the Gaussian tail function, and γ=E_{ s }/N_{0} is the SNR per symbol. Defining the instantaneous SNR as
and using the alternative form of the Gaussian Qfunction [25], we can write
Averaging Eq. (21) over all realizations of the channel matrix H, we obtain the PEP as
where p(γ_{ b }) is the PDF of γ_{ b } given in [26].
Let u represent a sequence with q information bits and \({\hat {\mathbf {u}}}\) denotes an error sequence with the same number of information bits. The bit error probability P_{ b } of the proposed HRDSM scheme is unionbounded by [27, 28]:
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).
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.
Comparison between theoretical and simulation results
In Fig. 2, the theoretical and simulation BER performances versus the signaltonoise power ratio (SNR) are plotted. The theoretical and simulation results match well in the high SNR regions for all cases. This implies that the bound given by Eq. (23) can be used as a tool to evaluate BER performances of not only the proposed HRDSM scheme but also the DSM scheme when SNR is sufficiently high.
Comparison between the differential detection HRDSM and the coherent detection SM
In order to support our analysis in Section 4, Fig. 3 presents an example to compare the differential detection scheme of HRDSM with the coherentdetection SM scheme. The simulations are realized with n_{ R }=4 and n_{ T }=2, 4. The BER performances of the differential and the coherent schemes are compared under the same spectral efficiency of 3 bpcu. Note that when SM and HRDSM use the same signal constellation, their spectral efficiencies are not equal.
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.
BER performance comparison of proposed HRDSM and other transmission schemes.
In Fig. 4, we compare the BER performance of HRDSM and DSM with the same spectral efficiency of 2.5 bpcu. To obtain spectral efficiency of 2.5 bpcu, the HRDSM utilizes BPSK and 8PSK constellations, while DSM adopts the QPSK constellation. The simulations are carried out with n_{ T }=2 and n_{ R }=1, 2, and 4. It can be seen from the figure that the proposed HRDSM outperforms DSM.
Figure 5 shows the BER curves of HRDSM and other transmission schemes with the spectral efficiency of 3 bpcu. The first one is the differential scheme of the wellknown Alamouti scheme (DA) with n_{ T }=2 [13]. The second is the singleantenna differential scheme, DPSK. The third is the differential scheme of quasiorthogonal spacetime block code [22] (DQOSTBC) with n_{ T }=4. And the last is the DSM. Simulation results indicate that, when n_{ R }=1, 2, the proposed HRDSM exhibits the worse performance than DA and DQOSTBC. However, when n_{ R }=4, HRDSM outperforms all the remaining schemes. This is because when n_{ R }=1, 2, the erroneous antenna detection of HRDSM is so significant that it redeems its coding gain. Increase in n_{ R } reduces the erroneous antenna detection of HRDSM considerably. Therefore, when n_{ R }=4, the coding gain of HRDSM due to the use of lower order modulation dominates the result and thus HRDSM outperforms the others.
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.
HRDSM with multiple transmit antennas
Figure 6 compares the BER performance of HRDSM with multiple transmit antennas with the same spectral efficiency of 3 bpcu. To obtain spectral efficiency of 3 bpcu, all antenna configurations utilize QPSK and 8PSK constellations, i.e., (M_{1},M)=(4,8). It can be seen from the figure that although all antenna configurations utilize the same modulation order, the HRDSM system with n_{ T }=4 provides the best BER performance, the HRDSM system with n_{ T }=3 has better BER performance compared with the HRDSM system having n_{ T }=2. In other words, the more transmit antennas are used, the less the performance loss of HRDSM is as compared to SM. This is because in the configuration with n_{ T }=3, 4, the best set of the signal space Ω_{HR−DSM} is selected in order to obtain improved performance.
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.
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.
Appendix
Evaluation of the variance of trace{Re(Θ)}
First, we can write
where
Now, the variance of trace{Re(Θ)} is given by:
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.
Evaluation of Var[trace{Re(Θ_{1})}], Var[trace{Re(Θ_{2})}]
trace{Re(Θ_{1})} can be evaluated as follows:
It could be observed from Eq. (25) that trace{Re(Θ_{1})} and trace{Re(Θ_{2})} have the same structure. Therefore, we can similarly get:
Finally, we can write
Evaluation of Cov(trace{Re(Θ_{1})}, trace{Re(Θ_{2})})
The crosscorrelation Cov(trace{Re(Θ_{1})}, trace{Re(Θ_{2})}) can be expressed as
where
and
Proof
Since S_{ t },H_{ t } and Q_{ t } are matrices with deterministic entries, the crosscorrelation Ψ is related to the 2 noise matrices N_{ t },N_{t−1} only. Moreover, these noise matrices are mutually independent of each other, and any crosscorrelation between two terms with different noise matrices is zero. Hence Ψ only consists of the crosscorrelation between the terms with the same noise matrix. For N_{ t } only, there is one term with noise matrix N_{ t }, so
hence, the crosscorrelation for N_{ t } is
For N_{t−1}
The crosscorrelation for N_{t−1} is
Therefore,
Finally, substituting Eqs. (29) and (30) into Eq. (26), we obtain Eq. (17). □
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Acknowledgements
This work was supported by the Vietnam National Foundation for Science and Technology Development under Grant 102.022015.23.
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Nguyen, T.P., Tran, X.N., Le, MT. et al. Differential spatial modulation for highrate transmission systems. J Wireless Com Network 2018, 6 (2018). https://doi.org/10.1186/s1363801710131
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DOI: https://doi.org/10.1186/s1363801710131
Keywords
 MIMO
 Differential spatial modulation
 Highrate differential spatial modulation