SONG Wei and WANG Wenzheng

1.School of Information and Electronics,Beijing Institute of Technology,Beijing 100081,China;2.School of Electronics Engineering and Computer Science,Peking University,Beijing 100871,China

Abstract: Media based modulation (MBM) is expected to be a prominent modulation scheme, which has access to the high data rate by using radio frequency (RF) mirrors and fewer transmit antennas. Associated with multiuser multiple input multiple output(MIMO),the MBM scheme achieves better performance than other conventional multiuser MIMO schemes. In this paper,the massive MIMO uplink is considered and a conjunctive MBM transmission scheme for each user is employed. This conjunctive MBM transmission scheme gathers aggregate MBM signals in multiple continuous time slots, which exploits the structured sparsity of these aggregate MBM signals. Under this kind of scenario, a multiuser detector with low complexity based on the compressive sensing(CS)theory to gain better detection performance is proposed.This detector is developed from the greedy sparse recovery technique compressive sampling matching pursuit (CoSaMP) and exploits not only the inherently distributed sparsity of MBM signals but also the structured sparsity of multiple aggregate MBM signals.By exploiting these sparsity, the proposed CoSaMP based multiuser detector achieves reliable detection with low complexity.Simulation results demonstrate that the proposed CoSaMP based multiuser detector achieves better detection performance compared with the conventional methods.

Keywords:media based modulation(MBM),radio frequency(RF)mirror, compressive sensing (CS), multiple input multiple output(MIMO), multiuser detector, compressive sampling matching pursuit(CoSaMP).

1.Introduction

The explosive growth of data rate requirements and user equipment (UE) will be involved in the next generation wireless communication systems [1]. To solve this issue,massive multiple input multiple output (MIMO) is considered as a promising technology to improve the spectral efficiency by employing hundreds of antenna elements(AEs) at the base station (BS). For the uplink transmission, multiple users transmit signals synchronously at the UE equipped with multiple antennas. However, one specific radio frequency(RF) chain is usually equipped with each AE,which leads to considerable power consumption and hardware cost.To mitigate the influences caused by the increased RF chains, innovative schemes such as spatial modulation(SM) [2–5], space shift keying(SSK) [6–8]and media-based modulation(MBM)[9–13]are proposed to improve the performance via using fewer RF chains than the traditional methods.

SM provides a low complexity spectral enhancing way for the MIMO system [14]. In the SM scheme system,each user is equipped with RF chains and multiple antennas at the transmitter.Only one transmit antenna is selected in a given channel use and the signal constellation symbol setΩwithM-ary modulation, e.g., 16 quadrature amplitude modulation (QAM), is transmitted by the selected antenna.Assuming that there areNttransmit antennas in the system.The achieved throughput of each user is log2(Nt)+log2Mbits per channel use(bpcu)in the case of one active RF chain.For instance,the spectral efficiency of four information bits can be achieved with the use of 8 phase shift keying (8PSK) and two transmit antennas. To improve the throughput,the number of the transmit antennas should be increased exponentially. However, there is no enough space for each user to own an excess of transmit AEs.

SSK is a modulation scheme based on SM and can be considered as a particular case of SM [15].In SSK transmission,only one RF chain is connected to multiple antennas and the antenna index is used for relaying information instead of transmitting symbols.In this way,the detection complexity of SSK is lower than SM. Forminformation bits,Nt=2mtransmit antennas are utilized to achieve the throughput of log2(Nt)bpcu.Similarly,only one transmit antenna is activated to convey the information bits in the SSK scheme.The spectral efficiency of SSK also improves with the exponential increase of transmit antennas,which is expensive from the perspectives of the hardware cost and energy consumption increase.

MBM as a novel technology can be combined with conventional MIMO and even massive MIMO to solve the issues of the hardware cost and energy consumption in the SM or SSK scheme [14]. The basic idea of MBM is to generate various channel states between the transmitter and receiver by changing the RF properties near transmit antennas.In an MBM system,RF mirrors are placed around a transmit antenna.Each RF mirror represents an ON/OFF status, which is called ‘mirror activation pattern (MAP)’and a different combination of MAPs maps into different information bits. When the RF mirror is in ON status, the signals will pass through. On the other hand, the OFF status indicates that the RF mirror turns into a barrier and the signals will not pass through.Each combination of ON/OFF status makes the transmitter to receiver transmission path different, which results in a variety of different channel states.Such a design makes a total of 2NRFcombinations ofNRFRF mirrors and will conveyNRFbits of information.

Therefore, MBM can be considered as a type of index modulation [16] and RF mirrors serve as indexed transmit entities. In this way, MBM can possess the same signal noise ratio (SNR) result as conventional modulation schemes when conveying smaller sized modulation symbols. Additionally, the same bit error ratio (BER) performance can be achieved by using fewer antennas at the BS with multiuser MBM as that of multiuser SM and conventional modulation in the same massive MIMO settings[17].MBM achieves better performance in the case of uplink massive MIMO especially when a high number of receive antennas is considered.

Compared with the conventional modulation,the advantages of MBM are as follows[14].First,MBM is capable of enlarging the receiving constellation dimension without increasing energy consumption.Second,the noise components between multiple receiving antennas are independent of each other. By selecting a subset of channel configurations, MBM obtains better performance on spectral efficiency and energy saving. In order to achieve the same bit error performance, the SM scheme requires more receive antennas than the MBM scheme in the same massive MIMO circumstance[18].Therefore,the MBM scheme is an absorbing mean for massive MIMO systems.

Implementing a multiuser detector with low complexity in the uplink massive MBM MIMO scheme is intractable.The maximum likelihood(ML)detector is regarded as an optimal one, whereas it possesses an excess of complexity. The nonlinear sphere decoding detector also suffers from excessive complexity when the number of antennas is large [19]. The linear low complexity detection algorithm such as minimum mean square error (MMSE) performs well for conventional massive MIMO schemes[20],but it is inadaptable due to the exploding increased number of transmit AEs and reduced number of RF chains at BS. Compressive sensing (CS) based detection algorithms leveraged in [21–23] are efficient approaches for multiuser detection,but they are valid only in small scale MIMO systems.

In this paper,a conjunctive MBM transmission scheme for each user is employed and a multiuser detector with low complexity based on compressive sampling matching pursuit(CoSaMP)is proposed.The multiuser detectors of SM systems by using the CS theory were developed by some researchers. A CS based detector was proposed by exploiting the structured sparsity of SM[24].By utilizing the distributed sparsity in SM systems, a CS based multiuser detection algorithm was exploited in [25]. Similar as SM,MBM also possesses such kind of distributed sparsity and structured sparsity.This conjunctive MBM transmission scheme gathers aggregate MBM signals in multiple continuous time slots, which exploits the structured sparsity of these aggregate MBM signals. The transmission scheme and the proposed detector aim to utilize the inherently distributed sparsity of the MBM signal and the structured sparsity of multiple MBM signals.The simulation results demonstrate that the proposed CoSaMP based multiuser detector outperforms the conventional detectors.

The rest of this paper is organized as follows.Section 2 introduces the multiuser MBM system model. The proposed CoSaMP based multiuser detector is described in Section 3. Section 4 presents the simulation results. Finally,Section 5 concludes this paper.

NotationThe italic boldface lowercase and uppercase symbols denote vectors and matrices,respectively.Superscripts(·)T, (·)Hand(·)?denote the transpose,conjugate transpose,and Moore-Penrose inversion operators,respectively.InandOndenote then×nidentity and null matrices,respectively,while 0n.is the vector of the sizenwith all the elements being 0.[A]m,ndenotes themth row andnth column element ofA.The?0and?2norm operations are given by‖ · ‖0and‖ · ‖2, respectively. The support set of the vectorxis denoted by supp{x}.A|Γ. denotes the submatrix whose columns comprise the columns ofAdefined in the setΓ.x〉Λdenotes the sub-vector whose elements comprise the nonzero elements defined in the setΛ.Θ= max(a;n) stands for a set whose elements are the indices of the largestnelements of the vectora.|a|denotes a vector consisting of the modulus of the elements ofa.

2.System model

Consider an MBM based massive MIMO multiuser uplink system, as shown in Fig. 1, where BS employsNBSreceive antennas to serveKUEs with each UE equipped with a single transmit antenna andNRFRF mirrors.These RF mirrors generateS= 2NRFpossible MAPs with each MAP involving ON/OFF status associated with one RF mirror, where the UE can transmit one of the MAPs mapped to a bit sequence that containsNRFinformation bits for a given channel use[26].To illustrate the mapping relationship between the bit sequences and MAPs,Table 1 shows a mapping example forNRF=3 bits.

Fig.1 Block diagram of MBM based multiuser detection for massive MIMO uplink systems

Table 1 Mapping relationship between information bit sequences and MAPs for NRF =3

The uplink multiuser detection lastsTtime slots and the MBM signal vector of thekth UE in thetth time slotxk,t ∈CS(1 ≤k≤K,1 ≤t≤T) can be expressed asxk,t=ek,tsk,t,whereek,t ∈CSis the selected MAP index vector corresponding to a specified information bit sequence andsk,t ∈C denotes the signal constellation symbol transmitted by a UE.Due to only one ofSMAPs selected by each UE,only one entry ofek,tis set to 1,and the residual entries inek,tare 0.Therefore,the support set ofek,tcan be determined as

whereΠ={1,2,...,S}is the MAP index set. On the other hand, the transmit signal constellation symbolsk,tcomes from the signal constellation symbol setΩwithM-ary modulation(e.g.,M-QAM),i.e.,sk,t ∈Ω.Therefore, the overall throughout in the uplink for an MBM based massive MIMO system havingKUEs isK(NRF+log2M) bpcu. For example, the total throughput of an MBM based multiuser system withK= 4,NRF= 4 andM=16 is 32 bpcu.

For multiuser detection in the uplink, the signal vectoryt ∈CNBSreceived by BS in thetth time slot is given by

whereis the MBM channel gain matrix of thekth UE withbeing the channel gain vector corresponding to theith MAP for 1 ≤i≤S,andzt ∈CNBSis the complex additive white Gaussian noise(AWGN)vector with the covariance matrix

By collecting an allKchannel matrix associated withKUEs,we have the aggregated channel gain matrixH=. Furthermore, due to the close placement of RF mirrors at the UEs, the channels associated with different MAPs exhibit typical spatial correlation, and the similar spatial correlation can be also established among the receive antennas equipped at the BS. Hence,His the correlated flat Rayleigh fading MIMO channel gain matrix, which can be expressed by using the Kronecker model[27],i.e.,

whereRk ∈CS×Sis the RF mirror correlation matrix of thekth UE that reveals the spatial correlation among the channels corresponding to different MAPs.For thekth RF mirror correlation matrix, given the correlation valueρt,Rk(1 ≤k≤K)is given by

In terms of the receive correlation matrixRBS, its element follows the exponentially decaying correlation model, i.e., given the correlation valueρr, [RBS]m,n=ThusRBScan be expressed as

According to the aggregated channel gain matrixH,(2)can be rewritten as

whereis the aggregated MBM signal vector andHin continuous time slots can be regarded as quasi staic.Finally,by aggregating the received signal vectorsytof(7)over theTtime slots intoY=[y1y2··· yT]∈CNBS×T,we have

whereandZ=are the aggregated MBM signal and noise matrices,respectively.

According to(8),the optimal ML detection criterion for the MBM signal matrixXis given by

Although the ML signal detector in (9) is optimal, its computational complexity exhibits exponential increase with the amount of users since the size of exhaustive search is equal to(S ·M)KT[28].The excessively high computational complexity can be unaffordable in practice for the multiuser MIMO systems where the number of the UEKtends to be very large. As the near-optimal solutions, the sphere decoding detectors [29] are indeed capable of reducing the computational complexity, but they may still suffer from prohibitive complexity cost when the values ofS,M,KandTare very large. Moreover, some suboptimal low complexity linear algorithms,such as matched filter (MF), zero forcing (ZF) and MMSE, which are applied in traditional MIMO systems to solve the overdetermined signal detection problem [30], are not suitable to addressing the large-scale underdetermined(e.g.,NBS

3.Proposed CoSaMP based multiuser detector for MBM MIMO uplink

In this section,a successive MBM signal design employed at the UEs is first proposed to present the common sparse support set in(8).Then,by exploiting the sparsity of MBM signals,a multiuser detection algorithm is proposed based on the CS theory. Furthermore, the computational complexity of the proposed multiuser detection algorithm is evaluated.Specifically,for an MBM signal vector,there is only one non-zero element inSelements,which means the MBM signal vectors are inherently sparse and their sparsity factor is 1/S.According to(7),the aggregated MBM signal vectorxtin thetth time slot consists ofKsubvectors with sparsity level one,and thus its sparsity factor isK/(KS) = 1/S. This inherent sparse property in the CS theory can be utilized to improve the accuracy of signal detection,and then a greedy sparse recovery techniques based algorithm is proposed to solve the CS problem.

3.1 Successive MBM signal design

The successiveTtime slots at the data transmission stage are considered as a group to design MBM signals. To effectively reduce the switching frequency of RF mirrors at the UEs, the same selected MAP index vector is adopted in different time slots, i.e.,ek,t=ek,t=···=ek,T(1 ≤k≤K),but the signal constellation symbols are different for every time slot.Therefore,the MBM signal vectors in different time slots share the same common sparse support set,i.e.,

Due to the same common sparse support set sharing by different time slots for all UEs, the aggregated MBM signal vectors also share the same common sparse support set,i.e.,

According to(8)and(11),we can observe that different columns of the aggregated MBM signal matrixXhave the same common sparse support set.By exploiting this sparsity, a modified CoSaMP algorithm is proposed, named partitioned CoSaMP(PCoSaMP)algorithm and detailed in the next subsection,to improve the performance of signal detection significantly.

3.2 Proposed PCoSaMP algorithm

Based on the analysis above,all the columns of the aggregated received signal matrixYcan be addressed simultaneously to solve the following optimization problem:

To solve the optimization problem (12) above, a PCoSaMP multiuser detection algorithm is proposed by exploiting the common sparse support set among measurements of multiple time slots and the priori sparse information of the MBM signal vector.By utilizing the proposed PCoSaMP multiuser detection algorithm,the MBM signal vectors ofKusers inTsuccessive time slots can be estimated, i.e.,(1 ≤k≤K,1 ≤t≤T),whereandare the estimated selected MAP index vector and signal constellation symbol,respectively.

The proposed PCoSaMP algorithm is summarized in Algorithm 1.Specifically,the residual vectorrt(1 ≤t≤T)and the support setΞ(0)can be initialized in Step 2. According to the priori sparse information of the constraint condition in(12),i.e.,Step 3 means that the total number of iterations is equal to that of UEsK.Step 5 performs the correlation calculation between the channel matrices and the residual in previous iteration,and Step 7 attempts to obtain the most likely preliminary support set based on the correlation values of Step 5.Step 9 acquires the union set by combining the support sets in the current and previous iterations, and this union set is used to solve the least squares(LS)problem in Step 11,which contributes to discarding the incorrect indices and update the most likely support set of the current iteration in Step 13,i.e.,Ξ(i).Step 15 calculates the MBM signal estimation in the current iteration by using the LS criterion,and Step 17 updates the residue estimates based on this estimation and the refined support setΞ(i).Then,the ultimate MBM signal estimationand the support setΞ(K)are in Step 20. Finally, because(1 ≤k≤K,1 ≤t≤T), the support setΞ(K)can directly determine the estimated selected MAP index vectors and its information bit sequences,and the estimated signal constellation symbols can be obtained by minimizing the Euclidean distance betweenand the legitimate signal constellation symbols in the setΩ.

Algorithm 1Proposed PCoSaMP algorithm

Input:Received signal matrixY= [y1y2··· yT],channel matrixH,and sparsity levelK

1%Initialization%

2Ξ(0)={?},i=1,rt=yt,=0KSfor 1 ≤t≤T

3whilei≤Kdo

4 %Correlation%

5ct=HHrt,?t

6 %Preliminary support set%

8 %Combine support set%

9

10 %First least squares%

11

12 %Update support set%

14 %Second least squares%

15

16 %Update residual%

17

18i=i+1

19End while

Output:Estimated MBM signals···vectorfor 1 ≤t≤T.

It should be pointed out that the main difference between the proposed PCoSaMP in Algorithm 1 and the traditional CoSaMP algorithm is that the PCoSaMP algorithm jointly detects measurements of multiple time slots by exploiting the common sparse support set while the CoSaMP algorithm recovers only one sparse signal.Therefore, compared with the traditional CoSaMP algorithm,the multiuser signal detection performance of the proposed PCoSaMP algorithm can be improved significantly.For the special case ofT= 1, that is without utilizing the common sparse support set among measurements of multiple time slots,the proposed PCoSaMP algorithm is equivalent to the traditional CoSaMP algorithm, which can be illustrated in simulations.

3.3 Computational complexity evaluation

In terms of optimal ML detector in (9), its computational complexity is O((MNRF)K), which is excessively high due to the exponential order.Therefore,the ML detection algorithm is not suitable for the multiuser MBM system in this paper.By contrast,the MMSE based detector for massive MIMO and the CS based SM detector[21]for general MIMO have low computational complexity,and their computational complexities are O(NBS(NRFK)2+(NRFK)3)and O(2NBS(K)2+(K)3), respectively.As for the proposed PCoSaMP multiuser signal detector, its computational requirements mainly come from the LS operations,whose computational complexity is O(T(2NBS(K)2+(K)3)). By comparing the computational complexity of these multiuser signal detectors mentioned above,it should be pointed out that the orders of magnitude of the proposed PCoSaMP based detector are lower than those of the traditional MMSE or CS based detector.

4.Simulation results

To verify our proposed algorithm,the bit error rate(BER)performance of the proposed detector is carried out to be compared with that of the other multiuser systems by using the traditional MMSE based, MF based and CoSaMP based signal detector for the massive MIMO uplink system. The BER performance influenced by different arguments’values of the transmitter and receiver is also compared.In addition,the proposed algorithm is implemented by using Matlab 2017a.

Fig.2 describes the BER performance achieved by different signal detectors as a function of the SNR under the MBM massive MIMO uplink system forK=8,M=64,NRF= 3,ρt= 0,ρr= 0.5, andNBS= 32. The traditional MMSE and MF based detectors suffer from a BER performance floor when the SNR value is greater than 10 dB,but the BER performance of the MF based detector works inferior than the MMSE based detector. The BER performance of the CoSaMP based detector works superior than the MF based detector when the SNR value is greater than 10 dB and works superior than the MMSE based detector when the SNR value is greater than 15 dB.WhenT= 4, the advantage of the proposed detector becomes clearer.The performance gap between the situation ofT= 2 andT= 4 becomes larger when the SNR value is greater than 4 dB. In conclusion, relative to using the traditional signal detector,the proposed PCoSAMP based multiuser detector only suffers from little BER loss in the massive MIMO uplink system. It can be observed from Fig.2 that the proposed PCoSAMP based multiuser detector achieves better BER performance than the other conventional CS based detectors mentioned in Fig.2 and has near optimal performance compared with the ML detector.

Fig.2 BER performance of multiuser detectors versus SNRs using MMSE,MF,ZF,ML,CoSaMP and PCoSaMP algorithms

To observe the effect of different terms in our algorithm,we provide the BER performance comparison with different experimental parameters of the proposed algorithm.In Fig. 3, comparing the BER performance achieved by the proposed PCoSaMP multiuser detector with different numbers of users and RF mirrors, whereT= 2,M= 48,ρr= 0.4,ρt= 0 andNBS= 64 are considered.Considering the situation ofK=8 orK=16 andNRF=3 orNRF= 4, the result indicates that the BER performance ofK= 8 approaches better BER performance. The different values ofKhave a prominent impact on BER performance.The problem that we challenge is a large scale underdetermined problem and the values ofKhave an influence on the dimension of the transmit signals. When the value ofKincreases, the dimension of transmit signals also increases. This will make the underdetermined problem more serious and obtain worse BER performance.With the SNR increasing,the gap of the BER performance between theK=8 andK=16 situations becomes wider.We can find that the different values ofNRFhardly have any effect on the BER performance. It indicates that the proposed PCoSaMP multiuser detector possesses robustness.

Fig.3 BER performance of proposed PCoSaMP multiuser detector versus SNRs with different values of K and NRF

Fig.4 shows the BER performance achieved by the proposed PCoSaMP based multiuser detector for different arguments’values at the receiver,whereK= 8,M= 48,ρr= 0.4 andNRF= 3. Whenρr= 0 andρt= 0, and the channels become uncorrelated Rayleigh fading MIMO channels.The improvement of performance is achieved at the expense of reduced uplink throughput and the antenna number increment of the receiver will alleviate this situation. The different values ofNBSaffect the dimension of received signals and further influence the calculating results of the large scale underdetermined problem to be solved.For instance,when the value ofNBSdereases from 48 to 32,the BER performance degrades.Meanwhile,the different values of the correlation coefficientρrmake the receive correlation matrixRBSchange and finally have an effect on the channel matrix.The decreasedρrhas a positive effect on the BER performance. In the situation ofNBS= 48,the BER performance degrades a lot when the value ofρrchanges from 0.8 to 0.4. However, the BER performance degrades a little when the value ofρrchanges from 0.4 to 0.Meanwhile,the decreasedρrhas a positive effect on the BER performance. In Fig. 4, the best BER performance is the situation ofNBS= 48 transmitted by the uncorrelated Rayleigh fading MIMO channels.

Fig. 4 BER performance of proposed PCoSaMP multiuser dectector versus SNRs with different values of ρr and NBS

Finally, comparing the BER performance achieved by the proposed PCoSaMP based multiuser detector in Fig.5 with different values ofρtandM,whereK= 8,T= 2,NRF=3,ρr=0 andNBS=48 are considered.The probability of each signal constellation points inM-ary modulation is 1/Mand this indicates the BER performance will be worse when the value ofMbecomes larger.Comparing the BER performance of 16QAM and 64QAM,the results show that the BER performance of 16QAM is better than 64QAM.Similarly,the different values of the correlation coefficientρtmake the receive correlation matrixRTXchange and finally have an effect on the channel matrix.The simulation results show that the better BER performance can be gained when the values ofMandρtare both lower.

Fig. 5 BER performance of proposed PCoSaMP multiuser dectector versus SNRs with different values of ρt and M

5.Conclusions

MBM is considered as a potential modulation scheme as it reduces RF hardware consumption and offers performance advantages.In this paper,a signal detector with low complexity for MBM based MIMO uplink transmission is proposed. The joint MBM transmission scheme is leveraged by the users to introduce the structured sparsity of multiple aggregate MBM signals. The proposed method based on CoSaMP obtains reliable multiuser signal detection performance by exploiting the inherently distributed sparsity of MBM signals and the structured sparsity of multiple MBM signals. The simulation results demonstrate that the proposed CoSaMP based multiuser detector outperforms other traditional methods.