Asynchronous Two-Way Relaying Networks Using Distributed Differential Space-Time Coding

. A signal detection scheme is proposed for two-way relaying network (TWRN) using distributed differential space-time coding (DDSTC) under imperfect synchronization. Unlike most of existing work, which assumed perfect synchronization and channel state information (CSI) at all nodes, a more realistic scenario is investigated here by considering the signals transmitted from the twosourcenodesarrivingattherelaynotexactlyatthesametimeduetothedistributednatureofthenodes,andnoCSIisavailableat anynode.Theproposedsignaldetectionschemeisthendemonstratedtoremovetheimperfectsynchronizationeffectsignificantly throughsimulationresults.Furthermore,pairwiseerrorprobability(PEP)oftheasynchronousTWRNisanalyzedandderivedfor bothsourcenodes.BasedonthesimplifiedPEPexpression,anoptimumpowerallocation(OPA)schemeisthendeterminedto furtherimprovethewholesystemperformance,whenneitherthesourcenortherelayhasanyknowledgeoftheCSI.


Introduction
Cooperative communications have attracted much attention nowadays, by allowing nodes in the network to cooperate and form a virtual antenna array [1,2].Compared with one-way relaying, two-way relaying networks (TWRNs) [3] have the advantage of high spectral efficiency, where two source nodes exchange information via the help of the relay nodes located between them.Recently, distributed spacetime coding (DSTC) for TWRNs was extensively investigated [4][5][6] due to the diversity and multiplexing gain of multipleinput and multiple-output (MIMO) technology.Most of the existing studies on DSTC consider coherent detection by assuming that the channel state information (CSI) is known at the receiver.However, in fast-fading scenario, accurate CSI is hard to acquire, and training symbols required for channel estimation will decrease the spectrum efficiency and increase computation complexity, especially when there are multiple relays in wireless networks.Therefore, differential modulation has been considered to address this problem since it does not require the knowledge of CSI at either the transmitter or the receiver [7,8].
Similar to the coherent detection scenario [9,10], several protocols have been proposed for TWRNs using differential detection.One of the most commonly used protocols is the amplify-and-forward (AF) scheme [11,12].In this scheme, both source nodes transmit information to the relay node at the same time, the relay then amplifies the received superimposed signal and broadcasts to both sources.For multiple relay nodes, space-time coding is used before amplifying the signals.This AF based bidirectional relaying is also referred to as analog network coding (ANC), which is very useful in wireless networks since the wireless channel acts as a natural fulfillment of network coding by superimposing the wireless signals over the air.In [11], distributed differential space-time coding by AF was applied to TWRNs for the first time.However, the correctness of the currently detected symbol significantly affects the decoding of next symbols, resulting in severe error propagation.To solve this problem, Huo et al. [12] presented a differential space-time coding with distributed ANC (DDSTC-ANC) scheme for TWRNs with multiple relays.The DDSTC-ANC scheme has been proved to achieve the same diversity order as the coherent detection scheme, but the performance of which is 3 dB away compared 2 International Journal of Antennas and Propagation with that of the coherent detection due to the differential modulation.
So far, almost all work on DDSTC with TWRNs has assumed that the transmission is perfectly synchronized by assuming that the relay nodes receive the signals from both source nodes at the same time, which can be difficult to achieve in practical systems due to the distributed nature of the nodes, and the channels may become dispersive with imperfect synchronization even under flat fading [13][14][15][16].In [15], a signal detection scheme for differential bidirectional relaying with ANC under imperfect synchronization was put forward, but it only considers a single relay node.In [16], the authors proposed a simple detection scheme for distributed space-time block coding under imperfect synchronization for TWRNs.However, perfect CSI is required at all nodes.To the best of our knowledge, little has been reported for TWRNs with multiple relays using DSTC under imperfect synchronization, when neither the sources nor the relays have any knowledge of the CSI.
Therefore, a differential signal detection scheme for asynchronous TWRNs with multiple relays using DDSTC is proposed in this paper.Due to imperfect synchronization, the symbols that relays broadcast back to sources are not symmetrical, signal detection will not be the same at the two sources, which will be described in detail thereafter.Due to the importance of resource allocation for the TWRN system [17,18], the performance of the proposed detection schemes is analyzed and PEP for both sides is derived.Moreover, an optimum power allocation (OPA) scheme is presented to further improve the system performance, based on the simplified PEP expression.
The rest of this paper is organized as follows.Section 2 introduces the system model.In Section 3, the detection schemes of the different sides are proposed, respectively, by two subsections.Section 4 presents the performance analysis and OPA for the system.The simulation results and corresponding conclusions are provided in Section 5. Section 6 summarizes the paper.

System Model
A TWRN with two source nodes and two relay nodes is considered in this paper, all equipped with a single antenna and working in the half-duplex mode.The source nodes,  1 and  2 , exchange information through relay nodes  1 and  2 , using two phases, the multiple access (MA) phase and the broadcast (BC) phase, as shown in Figure 1.In the MA phase, both sources transmit signals to  1 and  2 simultaneously, while in the BC phase, the relays broadcast the amplified superimposed signal back to the source nodes.Let   and   ( = 1, 2) denote the fading coefficients of the channels  1 −   and  2 −   , respectively.In the MA phase, both relays Second time slot First time slot receive a superposition of the signals transmitted from  1 and  2 .The number of symbols in a distributed differential space-time coding block is normally assumed to be equal to the number of relay nodes.Since two relay nodes are considered in this TWRN, the signals transmitted from  1 and  2 can be represented as two-dimensional vectors s  () = [ 1 ()  2 ()]  ( = 1, 2), normalized as {s  ()s  ()  } = I.Considering that s 1 () and s 2 () are imperfectly synchronized during the first phase, therefore they arrive at the relay nodes at different time with a relative time delay.In the distributed TWRNs, there are two nodes in the relay, the relative relay time of s 1 () and s 2 () at two relay nodes are different, and they are assumed as  1 and  2 corresponding to nodes  1 and  2 , respectively.Since the effort of synchronization is always required,  1 and  2 are assumed no greater than the symbol period .Such a relative time delay will still cause "intersymbol interference (ISI)" from neighboring symbols at the receiver.Without loss of generality, we assume that the signal from  1 is perfectly synchronized to  1 and  2 .The received signals at relay node   can then be expressed as where  1 and  2 are the transmitted power of  1 and  2 .k  () = [V 1 () V 2 ()]  represents the noise in the MA phase, which follows a zero-mean white Gaussian distribution, that is, k  () ∼ CN(0,  0 I).  stands for the imperfect coefficient of channel fading between  2 and   , which reflects the effect of timing delay   .Normally, we have   = 0 for   = 0, which means the synchronization situation, and   = √1/2 (  = √1 −   2 ) for   = 0.5, which means the power of delay signal is equal to that of current signal.The fading coefficients   () and   (), denoting the Rayleigh channel fading from source  1 to Relay   and source  2 to Relay   , that is,   () ∼ CN(0,  2   ) and   () ∼ CN(0,  2   ), are assumed to be constant over one frame and change independently from one frame to another for simplicity [15,17,18].So, let   () =   ,   () =   , and r  () can be expressed as International Journal of Antennas and Propagation 3 Since differential modulation is considered in this paper, a 2 × 2 unitary matrix () is used to encode the signal at nodes  1 and  2 .At time , it is encoded as s  () =   ()s  ( − 1) ( = 1, 2), where s  ( − 1) is the signal transmitted by   at time  − 1.For space-time coding, a block is often constructed for transmission [10], which satisfies where   and   are two 2 × 2 complex matrices.For simplicity, it is designed that either   is unitary,   = 0 (case I), or   is unitary,   = 0 (case II).
In the BC phase, the th relay node   utilizes r  () to generate a symbol vector x  () to satisfy the space-time coding scheme, which is a linear combination of r  () and its conjugate [12,19].Hence, Considering amplify-and-forward (AF) protocol in the relay nodes, the transmitted signal at the th relay can be represented as where   () is the scaling factor at   and specially given by where    is the transmitted power of   and it is assumed that    =   , so we have   () =  (constant).Then, the relay nodes   broadcast the coded symbol vector x  ().The signals received at two source nodes are expressed as follows, respectively.At node  1 , where w  () = [ 1 ()  2 ()]  ( = 1, 2) denotes the additive white Gaussian noise (AWGN) at  1 .  () ( = ,  − 1) is the space-time coding block which satisfies (7), and it is also a linear construction of   () and its conjugate [20]   () = ( Besides,   () is in the differential modulation with   () as follows: Let and ] + [  11 ()  12 () ]; then y 1 () can be abbreviated as It is easy to prove that {n 1 ()n 1 ()  } =  2 n 1 ()I, and where h 21 () =  [ and n 2 has the same property as n 1 ; that is,  2 n 2 () = (∑ 2 =1 || 2 |  | 2 + 1) 0 .For the first block, a known vector can be transmitted to both source nodes for differential modulation which satisfies {s  ()  s  ()} = 2 ( = 1, 2), for example, [1 1] .Here, let

Signal Detection
Due to the imperfect synchronization, detection methods at the two source nodes are not the same.They are proposed and presented as follows, respectively.

Detection at Node 𝑆 1
Theorem 1.If the relay matrices have the property: tr {     } = 2 for  = , tr {     } = 0 for  ̸ =  (where   stands for   or   ) [12], it can be elicited that So, h11 () can be approximated as where L is the frame length.Then let If detection of s 2 () is in the same way as in the perfect synchronization case, that is, ignoring ISI √ 2  2 ( − 1)h 13 (), there will be a severe error floor, which is the same at node  2 .To eliminate the error floor caused by imperfect synchronization, a detection scheme is proposed to remove the ISI as much as possible.When  = 1, the initial value is h 13 (0) = 0,  2 (−1) = 0; hence we have Then ỹ1 (0), ỹ1 (1) can be calculated as By using the least square (LS) decoder, the transmitted signal can be recovered as Ũ2 (1) = arg min ỹ1 (1) −   (1) ỹ1 (0)     .

Detection at
Node  2 .Using the same estimation method of h11 () in node  1 , h22 and h23 here can be approximated as So, ỹ(2) 2 () can be calculated as The transmitted signal can be detected by LS again as It is proved in the simulation results that the method of reestimating h 22 () and h 23 () can effectively eliminate the error floor and ensure the detection performance.

Constellation Rotation.
Note that the value of S2 ()  S2 (− 1) − S2 ( − 1)  S2 ( − 2) can be equal to zero, which may affect the accuracy of h13 ().This issue also exists in estimating h 23 ().To solve this problem, a rotation angle is required for the symbols modulated [21].For BPSK constellation, the effective rotation angle is in the interval [−/2, /2].To simplify, the rotation angle may be set as  = /2.Here, we give s 2 () as an example on how to achieve the constellation rotation.Set s 2 () as and it is easy to calculate that Then we can get which is impossible to be zero.This constellation rotation scheme is also applied to s 1 ().

Performance Analysis
In this section, the Pairwise Error Probability (PEP) of the asynchronous TWRNs using DDSTC is derived.Due to the effect of imperfect synchronization, performance at the two source nodes is also asymmetric, which will be analyzed as follows, respectively.Total PEP and optimum power allocation method are also discussed in this section.The PEP of mistaking the kth STC block by the th STC block can be evaluated by averaging the conditional PEP over the channel statistics [12] as where  = / 0 is the signal-to-noise ratio (SNR),  is the total transmitted power in the TWRN, and () is the Gaussian Q-function.Since it is very difficult to analyze ỹ1 (− 1) directly, we use ỹ1 ( − 1) ≈ √ 2  2 ( − 1)h 12 ( − 1) as in (20) instead in the following analysis.In Section 2,   () and   () are assumed to be constant over one frame, so () can be simplified as As derived in Section 2, Then, according to [12],   1  () can be derived as where International Journal of Antennas and Propagation   1  () can then be expressed as It can be observed from (37) that the influence factor of   1  () is the same as in synchronization case except the term (1− 2  ).However, (1 −  2  ) is a constant during a frame.So the PEP expression of node  1 can be simplified at high SNR as where ).Thus, the simplified PEP at high SNR can be rewritten as where and () = diag { 1 (),  2 ()}, it is easy to find that the elements in   2  () have no relationship to the imperfect synchronization coefficient   ; that is, it is identical to the synchronization situation [12].So   2  () can be derived as where ) can be simplified as where Φ = (1 −   Bit error rate Synchronization in [12]  is random

Simulation Results
In this section, simulation results of the BER performance on both sides using the proposed signal detection and the OPA scheme are presented.Rayleigh fading channel is used as the channel model in the simulations.Transmitted power of the relay nodes is assumed as    =  1,2 = 1, that is, equal power allocation (EPA), if not specially pointed out.BPSK modulation is used, and the frame length is  = 100.
Figure 2 shows the performance of TWRN under imperfect synchronization using the existing differential detection scheme in [12].Set  = / √ 1 −  2 , the normalized imperfect synchronization coefficients, and take its values as 1, 0.5, 0.3, 0.2, and 0.1 for the simulations.For comparison, the performance of the TWRN under perfect synchronization is also presented [12].It can be concluded easily that, with  increasing, the detection error floor becomes higher.But in the real system,  is generated randomly since  is a random value, ranging from 0 to 1.The result is also provided in Figure 2; in this case, the error floor is almost the same as the case that  = 0.5.
Figure 3 shows the detection performance of the two source nodes using the proposed differential detection schemes for the two sides.It can be observed that the detection schemes proposed for both nodes  1 and  2 remove the high error floor caused by imperfect synchronization.The detection method on node  2 eliminates the error floor at high SNR after reestimating h 23 (), providing a BER performance approaching the synchronization situation.The BER of node  1 is 4 dB less than that of node  2 .The reason is that element  2 ( − 1) in the interference part is known to node  2 but unknown to node  1 , which has been mentioned in Section 3.  In Figure 4, it shows the BER performance of the proposed differential detection and power allocation scheme.It can be observed that, in both case I and case II, the BER of node  1 decreased while that of node  2 increased compared to equal power allocation (EPA), and the total BER of node  1 and node  2 is decreased for about 1 dB.So, it is obvious that OPA can balance the asymmetric performance of the signal detection at the two sources caused by imperfect synchronization, while the performance of the whole system can also be improved.

Conclusion
In this paper, we have proposed a signal detection scheme for TWRN under imperfect synchronization when neither the sources nor the relays have any knowledge of CSI.Due to the effect of imperfect synchronization, detection schemes and performance are different for both sources.Simulation results indicate that the proposed algorithms on both sides perform well, with the imperfect synchronization effect greatly removed.Furthermore, we derived the simplified PEP of the TWRN and determined the optimum power allocation scheme, which improves the performance of the whole system and leads to a symmetrical detection performance for both sides even though imperfect synchronization exists.
floor without eliminating imperfect synchronization Synchronization in[12] Detection at S1 Detection at S2 before re-estimation Detection at S2 aer re-estimation

Figure 3 :
Figure 3: BER performance of detection scheme on nodes  1 and  2 .

1 Figure 4 :
Figure 4: BER performance of the proposed detection scheme with optimum power allocation.
ln ( 1   ) + ln ()) PEP of Node  2 .Similarly to the derivation of PEP at node  1 , PEP of node  2 can be expressed as should be minimized.For simplification, the source nodes and the relay nodes are assumed to have the same power, which is to say  1 + 2 = 2  = (1/2); that is,  1 + 2 = 1/2; then  1 ,  2 can be rewritten as  1 =  2  2