Performance Evaluation of Spatial Modulation and QOSTBC for MIMO Systems †

Multiple-input multiple-output (MIMO) systems require simplified architectures that can maximize design parameters without sacrificing system performance. Such architectures may be used in a transmitter or a receiver. The most recent example with possible low cost architecture in the transmitter is spatial modulation (SM). In this study, we evaluate the SM and quasi-orthogonal space time block codes (QOSTBC) schemes for MIMO systems over a Rayleigh fading channel. QOSTBC enables STBC to be used in a four antenna design, for example. Standard QO-STBC techniques are limited in performance due to self-interference terms; here a QOSTBC scheme that eliminates these terms in its decoding matrix is explored. In addition, while most QOSTBC studies mainly explore performance improvements with different code structures, here we have implemented receiver diversity using maximal ratio combining (MRC). Results show that QOSTBC delivers better performance, at spectral efficiency comparable with SM.


Introduction
Since the demand for ever higher data rates is a major driver in research on telecommunications services for mobile devices, modern (and future) telecommunication standards are being proposed based on multiple-input multiple-output (MIMO) antenna configurations.In long-term evolution (LTE) and LTE-Advanced (LTE-A) for example, data rates on the Gigabit scale are being sought [1].The MIMO scheme exploits the probability that no two channel paths will have equally bad impairments, thus increasing the number of transmitting and/or receiver elements can improve the probability of correctly receiving transmitted information over a fading channel.For instance, if p is the probability that the instantaneous signal-to-noise ratio (SNR) falls below a critical value on each antenna branch (usually called the outage probability [2]), then is the probability that the instantaneous SNR is below the same critical value on all N R -receiver branches for independently faded channels [3].Since mobile receivers are generally compact, diversity techniques are best utilised in the transmitter.
MIMO schemes increase both transmitter and receiver diversity beyond one antenna.Many different transmitter diversity techniques have been studied by researchers [4].Maximal ratio combing (MRC) is an optimum combining method for flat fading channels with additive white Gaussian noise (AWGN) [4] and is applied in the receiver.
Examples of popular transmitter diversity techniques include spatial multiplexing and space-time block coding (STBC).Most recently, the spatial modulation (SM) technique [5,6] has been introduced.In SM, multiple antenna systems are designed with only one radio frequency (RF) chain in the transmitter.This technique is explored in this study and will be compared with QOSTBC alongside STBC.The standard QOSTBC scheme is limited by nonzero off-diagonal terms in its detection matrix.Here, a QOSTBC scheme that does not have this limitation is used.In this paper, this will be referred to as interference-free QOSTBC.
Spatial modulation (SM) is an attractive multi-antenna transceiver technique for MIMO system deployment.It improves spectral efficiency and has no inter-channel interference (ICI) at the receiver, provided the pulse shaping period does not overlap amongst antennas [7].SM reduces transmitter complexity and cost since only one transmitting antenna is activated during a transmission period, thus reducing the number of RF-chains to one.When SM was introduced, it was compared with STBC which uses up to 4receiver antennas [8].STBC improves power efficiency by maximizing spatial diversity, and improves capacity from diversity gain, which reduces error probabilities over the same spectral efficiency [6,8].QOSTBC thus improves signal quality reception and overall system performance consequent on this fact.
In this study, SM, STBC and QOSTBC will be compared in terms of their bit error ratios (BER).Each of these is Signal modulation involves mapping a fixed amount of information into one symbol.Each symbol represents a constellation point in the complex two dimensional signal plane [6].Extending this plane to three dimensions yields what has been referred to as spatial modulation [6,8], a three-dimensional signal mapping (modulation) scheme that activates only one transmitting antenna out of many at one time.
In signal modulation, for instance using M-PSK as explored in this study, the number of bits that can be transmitted is given by ) On the other hand, SM permits the mapping/transmission of more bits (n) as a consequence of the number of transmitting elements: where N T is the number of transmitting elements.By (2), the data rate is increased by ) ( log 2 T N .This is done by mapping the information in a q-vector of n bits into a new s-vector of N T bits in each timeslot such that only one element in the resulting vector is non-zero. The position of the element in the s-vector chooses the transmitting antenna element over which the symbol will be transmitted (or that can be made active) in a transmission timeslot.Let the active antenna be designated as s l ; notice that Since data are encoded in information symbol and antenna number (as in (3)), the estimation of antenna number is essential.For a noiseless system of the form y = hx, where h is the channel matrix, the estimate of the transmitted symbol can be expressed as [8] g(k) where h H is the Hermitian transpose of h.The antenna number can then be estimated as [6]     Then, based on the estimated antenna index, the estimate of the transmitted symbol is given by where D is the constellation demodulator function [7].The SM demodulator uses these two estimates to find the message respective to the antenna branch by performing an inverse mapping of the initial SM mapping table.

Maximal Ratio Combining (MRC)
MRC is an optimum receiver combining technique for a flat fading channel with additive white Gaussian noise (AWGN) [4], used to provide N R -receiver diversity order.A schematic example of a system with MRC is shown in Figure 1. .For a linear system, the received signal can be described in the form: y = hs + z (7) where y is a vector containing received symbols from each branch of the N R receiver elements, defined as: y multipath fading channel defined as ( [9]): [ , a i,l is the path gain, and l i,


is the l th phase of the multipath.Similarly, z the AWGN due to the i th receiver antenna.
If the channel coefficients are perfectly available in the receiver, the detector attains optimal maximum likelihood (ML) decoding as [10]


. The term 2 hs y  is the Euclidean distance metric for ML decoding [11].PSK symbols have equal energy, thus the bias energy term in (8a) is dropped [11] so that [10]: where   *  is the conjugate of    .The optimal decision rule linearly combines the received signals through different diversity branches after co-phasing and weighting them with their respective channel gains.If an equivalent channel is known, the MRC rule becomes [10,12] Although affected by * h , the noise terms are still Gaussian.
The effective instantaneous SNR with MRC is where  is the average SNR per antenna branch.It can be seen that the SNR of N R branch diversity with MRC is the sum of the instantaneous SNRs for each branch.In [2], it as presented in Figure 2 (BER is the bit error ratio).The plot in Figure 2 shows that an MRC system does not act as a single element with N R times greater SNR.Due to the fact that independent faded symbol copies are received on each antenna branch, the slope of the BER changes as N R increases (but falls off exponentially).
Thus from Figure 2, the MRC combining diversity technique provides significant improvement as the number of receiver antennas increases.

Orthogonal Space Time Block Codes
Space time block coding or STBC was introduced to improve the performance of multi-antenna systems over constrained bandwidth.It achieves full diversity and a full spatial rate (R s ) over two antenna spaces, for example [13], In (10), there are two antenna spaces (N T ) and two time slots (T) so that R s = N T /T = 1.The symbol s provides two antenna spaces, h 1 and h 2 .In the first time slot, s 1 and s 2 will be transmitted over h 1 and h 2 , respectively.Similarly, in the second timeslot * 2 s  and * 1 s will be transmitted over h 1 and h 2 , respectively.Thus, in the receiver, It is easier for the receiver to decouple the informationbearing symbol if an equivalent virtual channel matrix (EVCM) to the orthogonal codes of (10) can be constructed.Thus, we take the conjugate of the second row of signal received in the second timeslot in (11), In a linear form, (12) can be rewritten as , where and z In the receiver, the STBC code permits linear decoding as I is the power gain, where 2 I is a (2×2) identity matrix.Thus, the rows and columns of EVCM of the Alamouti code are orthogonal.The receiver decouples the transmitted signals s 1 and s 2 .Using EVCM simplifies the implementation of the STBC scheme and also reduces the receiver complexity.

Maximum Likelihood for STBC Detection
It is assumed that channel state information (CSI) is known to the receiver.Thus, the channel coefficients h 1 and h 2 can be used in the decoding for a ML detector [14].The detector is optimum if the ML detector can find the codewords   .In STBC, the Euclidean distance metric for ML decoding is [11,15]   The joint conditional PDF of y 1 and y 2 given the channels h 1 and h 2 over which the codewords s 1 and s 2 are transmitted can be expressed as [11]  where 2  is the variance of z 1 and z 2 ; these are equal for uncorrelated Gaussian random variables.To reduce the computational complexity in the detector, 2 2 2 1 y y  terms that are not required in the decision will be dropped.Then expanding (15), it is found that For PSK symbols, signal points in the constellation have equal energy.Thus, the bias energy terms ( ) are ignored; the optimum ML detection further simplifies to Only PSK symbols will be considered in this study.

MIMO-STBC
Suppose that there are N R receiver antennas.Thus, each of h 1 and h 2 can be treated respectively as a vector of the form: We know that if the equivalent channel can be derived, then the MRC when there are N R maximum receiving elements becomes [12]  r

H
is an identity matrix multiplied (as in the case N R = 1) by the channel gains such as . The noise term is rather amplified by . The degree of impact of H v j H on the noise term impacts the closeness of the Euclidean distance metric in the receiver; this depends on the fading of the channel.The complexity in the decoupling of the transmitted message in the receiver reduces to finding only .

Codes
A major limitation in the use of STBC is that N T > 2 is not supported.QOSTBC is a class of STBC that removes the two-transmit antenna limitation.

Standard QOSTBC
Standard QOSTBC achieves a full spatial rate but not full diversity [16].In [11,15,17,18], different methods for constructing QOSTBC have been described.QOSTBCs are STBCs with N T > 2 and timeslots T = 2, 4 and 8 with complex entries; STBC codes with R s = N T /T = 1 are said to attain a full rate [11,19].
An example of a full rate (R s = 1) STBC code with N T = 4 and T = 4 is given as [19,20] where Ω represents the standard Alamouti STBC [21], ) is an example of QOSTBC.The rate of full-diversity codes is 1  s R [22].Unfortunately, standard QOSTBC codes do not attain full diversity and also do not permit linear processing due to coupling terms which lie off the leading diagonal of the detection matrix [20,23].
As seen in (20) As with Alamouti STBC codes, EVCM can be formulated by taking the conjugates of the second and fourth rows in the received matrix, thus D, where D is a sparse matrix with ones on its leading diagonal and at least N 2 /2 zeros at its off-diagonal positions; its remaining (selfinterference) entries are bounded in magnitude by 1.
In the receiver, the EVCM can be used to simplify decoding.For instance, let the decoding method proceed as: On the other hand, β is the self-interfering term that limits full-diversity performance expected of this type of QOSTBC systems.Alternative channel estimation for linear receivers, such as zero-forcing (ZF) [24], compensates for the β-term except for the noise elements.This yields sub-optimal results.

2.4.2
Interference-Free QOSTBC Independently, two different researchers have proposed an interference-free QO-STBC; one method involves the use of Givens rotations [25] while the other uses eigenvalues [20,23].Both of these methods yield similar results.However, the eigenvalues approach is less complex and this will be reviewed in brief.D 4 M = V; this is the principle of diagonalizing a matrix [26].It follows that V contains the required diagonal terms of the diagonal matrix, v i I, with no interference terms.
Recall the decoding matrix of ( 25): its modal matrix from

h h h h h h h h h h h h h h h h h h h h h h h h h h h h h
Thus, assuming a linear system of ( ML detection, the receiver finds    To enable QOSTBC, EVCM is constructed as (27) for 4×1 transmission; for 3×1 design, h 4 is set to zero.Noise terms (AWGN) are generated and added to each receiver branch respective to the transmitting branch.In the receiver, the linear decoding shown in ( 28) is performed to estimate the transmitted signals.
It is assumed that the channel state information (CSI) is known; the ML detection method is said to be optimum to find the transmitted data as [14] where

He
is the Euclidean distance metric at the receiver and SNR is the ratio of transmitted and received powers per antenna.From the Chernoff upper bound [28], the pairwise error probability (PEP) approximates to

MIMO-QOSTBC
The maximum achievable diversity level for an N T × N R MIMO system is N T N R [11], where N T is the number of transmit elements and N R is the number of receiver elements.In the ML detection case, the error matrix is  e   For an i.i.d Gaussian channel with many receiver elements (e.g.N R ), the average Chernoff bound simplifies to [27]

P
) ( A major performance degradation in this case arises from the error matrix or 2 l  in the estimated noise power which is amplified by H H . An advantage in this case (interference-free QOSTBC) accrues from the gain contributed by H H H  which amplifies the amplitude of the received signals.This varies depending on the eigenvalue ( , the error probability is bounded as [11]

P
) ( The idea of using the rank criterion is to attain maximum possible diversity, N T N R .The nonzero eigenvalue term, however, provides information about the coding gain.

Simulation Results and Discussion
In this section, the simulation results are discussed.The results involve comparisons of the spatial modulation scheme as a method for MIMO transmission with STBC and QOSTBC schemes.Assuming a Raleigh fading channel, the receiver is equipped with the full knowledge of the channel state information and the antennas are reasonably separated to avoid correlation.To investigate a 3×1 QOSTBC design, the fourth antenna element is nulled (i.e., h 4 = 0).The process for 4×1 QOSTBC implementation is then repeated for 3×1 QOSTBC.In Figure 4, it is observed that interference-free (Int-free) QOSTBC achieves about 2 dB gain with respect to the standard QOSTBC for both 4×1 QOSTBC and 3×1 QOSTBC.The gain results from the elimination of the interfering terms described above.

Simulation of STBC and QOSTBC
To simulate the MIMO-STBC case, some 7×10 5 random input symbols, a, are generated and mapped using QPSK.The resulting symbols are demultiplexed into 1 s , and 2 s so that they can be transmitted over antenna spaces 1 h and 2 h , as shown in Figure 5. Since there are N R receiver antennas, each of h 1 and h 2 is treated respectively as a vector as in (18).In the receiver, the received information on each branch is demodulated and the channel detection performed.The information received from respective branches is combined by MRC.The resulting symbol s ˆ is demapped using QPSK.For other mapping orders, such as 8-PSK, 16-PSK, 32-PSK and 64-PSK, only QPSK has been substituted in Figure 5.The results are first compared with the data in [21].The results in Figure 6 are comparable to those reported in Figure 4 of [21] for 2×1 and 2×2 MIMO systems.Similar to the STBC simulation case in Figure 5, QOSTBC is enabled after generating random symbols.The resulting symbols are demultiplexed into 1 s , 2 s , 3 s and 4 s so that they can be transmitted over antenna spaces 1 h , 2 h , 3 h and 4 h .For MIMO-QOSTBC, each 1 h , 2 h , 3 h and 4 h is treated respectively as a vector of up to N R receiver antennas.In the receiver, the channel is compensated.The information detected across each antenna branch is combined with another using MRC, then using QPSK, the received symbol estimate is demapped.

Simulation of SM
Similarly, to simulate the MIMO for the SM case, some n×10 5 random input symbols (where n is from (2)), a, are generated and mapped using QPSK to yield b; this is the two-dimensional signal modulation.SM is a third dimension added to the default two-dimensional signal modulation.SM maps the symbols into a table of  s is an N T ×L matrix with complex entries.N T here denotes that there are N T possible transmitting branches provided by the SM design.In each column of the matrix, only one element is nonzero which corresponds to the antenna index that can be activated at that time.The symbol corresponding to the selected branch is transmitted over the channel, and AWGN is added.The channel estimation proceeds and the antenna index that will be used to predict the transmitted information is derived.Performance Evaluation of Spatial Modulation and QOSTBC for MIMO Systems

Results and Discussions
The results are compared for all transmitter diversity schemes.For a fair comparison, the channel and noise terms are made similar.However, the mapping scheme orders vary but permit equally likely data rates.In [29], we discussed the cases of SM and STBC for up to an N R = 4 MIMO system and QOSTBC for only to N R = 2.In this study, we include QOSTBC for up to an N R = 4 MIMO system.

A. Comparison of Analytical and Simulation Results
To validate our results we show first a comparison of analytical and simulation results with the standard QO-STBC in Figure 7.The simulation results shown in Figure 7 are for a 4×1 QOSTBC diversity scheme.It can be found that the simulation result of eQOSTBC and the analytical result of eQOSTBC reasonably agree up to about 10 -4.5 BER, where both results perfectly aligned.The variation at the start of the plot can be attributed to the parameter setting.Meanwhile, the interference terms in the detection matrix of the standard QOSTBC degrade its performance.Hence, the eQOSTBC outperforms the standard QOSTBC both analytically and in simulations.Both results of eQOSTBC outperform the standard QOSTBC scheme by about 2 dB, for instance, at 10 -4.5 BER.

B. Two Bits Transmission
For further evaluations, we transmit two bits using SM, STBC and QOSTBC.The two bits from the SM scheme are provided by two transmit antennas and the BPSK scheme of (2) with four receiving elements.On the other hand, the two bits for STBC and QOSTBC are consequent on (1).The results are shown in Figure 8.At relatively low transmission power, it is found from Figure 8 that transmitting with 2 antennas and receiving with 4 antennas for the STBC scheme is better by some 1 dB at 10 -2 BER than transmitting with 4 antennas and receiving by 4 antennas using the QOSTBC scheme.Increasing the transmission power thus improves the performance of QOSTBC up to 7.5 dB where both schemes performed equally; further increase in power however shows better performance for QOSTBC than for STBC.The interfering terms (β) discussed in (25) and then eliminated in (28b) impact the signal amplitude as they affect the gain h  .The results here show that results can be improved by simply increasing the transmission power.However, comparing the results of QOSTBC and SM schemes, it is found that the QOSTBC scheme outperforms the SM by about only 9 dB at 10 -3 BER and then by about 11 dB at 10 -5 BER.This would increase for increased symbol lengths.

C. Three Bits Transmission
In Figure 9, we compare the results of the SM, STBC and QOSTBC schemes for three bits transmission.The three bits transmission investigation is typical of those reported in Figure 3 of [6].As in [6], it can be seen that transmitting on 2 antennas and receiving on 4 antennas using the SM scheme for QPSK is better than transmitting on 4 antennas and receiving on 4 antennas using BPSK by about 1 dB.On the other hand, comparing the STBC and QOSTBC schemes, it is found that 4×4 QOSTBC performs better than 2×4 of the 8PSK scheme by about 3 dB at 10 -5 BER.Then comparing them with the SM scheme, it is found that both STBC outperform SM scheme (2×4-QPSK) by about 6 dB at 10 -5 BER while QOSTBC outperforms SM scheme (2×4-QPSK) by about 8 dB respectively.The benefit of the discussed QOSTBC scheme is its ability to attain full diversity and a full spatial rate resulting from the elimination of the coupling terms.

D. Four Bits Transmission
Again, we compare the performance of STBC and QOSTBC with the SM scheme when four bits are transmitted.The results are shown in Figure 10.Both 4×4-QPSK and 2×4-8PSK SM schemes perform similarly with 2×4-16PSK of STBC at 10 -2 BER although with a performance marginally better than STBC.On the other hand, 4×4-16PSK QOSTBC clearly outperforms the other schemes at 10 -3 BER by about 4 dB, for example.

E. Six Bits Transmission
Finally, transmitting six bits using SM is investigated as in [6] and [8].We then compare the results with that of transmitting six bits with the STBC and QOSTBC schemes.
The results are shown in Figure 11.
In agreement with references [6] and [8], Figure 11 clearly shows that using the SM schemes to transmit six bits is more economical, in terms of transmission power, than using the STBC scheme.However, SM shows improved performance as the signal modulation order increases.The SM scheme transmission of six bits using 4×4-16PSK outperforms 2×4-32PSK progressively.On the other hand, comparing the QOSTBC and SM schemes, the 4×4-64PSK of the QOSTBC scheme outperforms all other SM schemes and the STBC scheme.Specifically, the 4×4-64PSK QOSTBC scheme outperforms the best performing SM (16PSK that uses 4×4 antennas) among all SM schemes of six bits transmission, by about 3 dB at 10 -4 BER.

F. Evaluation of QO-STBC for N R Receivers
The interference-free QOSTBC provides information into the performance of the scheme with an increasing number of receivers, as shown in Figure 12.
Figure 12: Evaluation of receiver diversity order for N R QOSTBC system While the SM performance improvement stems largely from the mapping scheme rather than on the number of transmitting antennas, the QO-STBC scheme exploits the diversity gain of the transmitting antennas spaces when compared to the STBC scheme.For instance, in Figure 12, Performance Evaluation of Spatial Modulation and QOSTBC for MIMO Systems it is observed that as the number of the receiving antennas increases, the diversity gain increasingly diminishes for both mapping schemes shown (QPSK and 16PSK).The selfinterfering terms (β) in ( 25) even though eliminated in (28) further impact the true gain h  .
However, mobile nodes are not suited (in terms of size and battery life) in supporting a large number of antennas, for example, and the slope of the 4×2 MIMO BER plot reflects appreciable space and time coding gain better (in terms of difference between N R = 2 and 1 compared to N R = 3 and 2) than the rest N R = 3 and 4. Thus, interference-free QOSTBC is an excellent technique for MIMO configuration in modern and future wireless communication applications.It transfers the complexity of the multiple antenna design algorithm from the receiver to the transmitter; transmitters, such as base stations, are more flexible in supporting complex algorithms than the receivers (such as mobile devices).By EVCM, the QOSTBC scheme studied simplifies the decoupling of the transmitted information in the receiver.

Conclusion
In this study, three different diversity schemes for MIMO systems were studied.These schemes include spatial modulation, STBC and QOSTBC.In the study, we explored the performance of these diversity techniques at low and relatively high spectral efficiencies.Most QOSTBC studies emphasise performance improvement with different code structures; here we implemented the scheme to include receiver diversity using MRC.Up to 4×N R can be implemented and only 4×4 QOSTBC with MRC has been investigated.The study showed that over all, the MIMO QOSTBC (for instance 4×4) scheme performed better than SM while the MIMO-STBC technique performed worst at relatively high spectral efficiency but best at low spectral efficiency.From the study, it can be said that the strength of the SM diversity scheme is in the signal modulation order, that is, the performance of the SM diversity scheme improves as the signal modulation order increases up to four bits transmission (as investigated above).At relatively low spectral efficiency (e.g. up to four bits transmission), the modulation order (with lower transmitter antenna diversity) mostly impacted the performance of the SM scheme while at relatively high spectral efficiency (e.g. at six bits transmission), lower modulation (against higher modulation order) was improved by more transmitter antenna diversity.Notwithstanding, using the QOSTBC scheme with four receiver antennas, the performance of the SM diversity scheme is poorer.Finally, where a higher signal modulation order is required, then a higher number of transmitting elements must be preferred.

Figure 1 :
Figure 1: Schematic example of an MRC scheme In the transmitter, randomly generated symbols, a, are mapped using BPSK.The resulting symbols are transmitted over R N h h , , 1  multipath channels.In the receiver, some AWGN respective to each receiver branch is added, namely R N z z , , 1 .For a linear system, the received signal can be

Figure 2 :
Figure 2: BER performance of BPSK system for MRC with an increasing number of antennas.

D 4 4
is a Grammian matrix with h  in the leading diagonal of the D

κ
and its nonzero eigenvalues are   l  .QOSTBC systems attain full diversity, when κ = N T .As in the STBC case, if there are N R maximum receiver elements, then

Figure 7 :
Figure 7: Performance comparison of analytical and simulation results

Figure 8 :
Figure 8: Comparison of two bits transmissions using SM, STBC and QOSTBC

Figure 9 :
Figure 9: Comparison of three bits transmissions using SM, STBC and QOSTBC

Figure 10 :
Figure 10: Comparison of four bits transmissions using SM, STBC and QOSTBC