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

Authors: Minjie Qian; Yanliang Jin; Zhuo Wu; Tao Wang
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

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 two source nodes arriving at the relay not exactly at the same time due to the distributed nature of the nodes, and no CSI is available at any node. The proposed signal detection scheme is then demonstrated to remove the imperfect synchronization effect significantly through simulation results. Furthermore, pairwise error probability (PEP) of the asynchronous TWRN is analyzed and derived for both source nodes. Based on the simplified PEP expression, an optimum power allocation (OPA) scheme is then determined to further improve the whole system performance, when neither the source nor the relay has any knowledge of the CSI.


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## Body

## 1. 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 space-time coding (DSTC) for TWRNs was extensively investigated [4–6] due to the diversity and multiplexing gain of multiple-input 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 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 [13–16]. 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. Section2 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.Notation. Throughout this paper, capital and boldface lower-case letters denote matrices and vectors, respectively. ·*, ·T, ·H, and ·-1 stand for complex conjugate, transpose, conjugate transpose, and inverse, respectively, for both matrix and vector. E· denotes the expectation. diagx1,x2 represents 2×2 matrix whose ith diagonal entry is xi.


## 2. 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,S1 and S2, exchange information through relay nodes R1 and R2, 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 R1 and R2 simultaneously, while in the BC phase, the relays broadcast the amplified superimposed signal back to the source nodes. Let fi and gi (i=1,2) denote the fading coefficients of the channels S1-Ri and S2-Ri, respectively. In the MA phase, both relays receive a superposition of the signals transmitted from S1 and S2. 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 S1 and S2 can be represented as two-dimensional vectors sjt=sj1tsj2tT(j=1,2), normalized as Esj(t)sj(t)H=I. Considering that s1t and s2t 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 s1t and s2t at two relay nodes are different, and they are assumed as τ1 and τ2 corresponding to nodes R1 and R2, respectively. Since the effort of synchronization is always required, τ1 and τ2 are assumed no greater than the symbol period T. 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 S1 is perfectly synchronized to R1 and R2. The received signals at relay node Ri can then be expressed as(1)rit=P1fits1t+P21-αi2gits2t+P2αigit-1s2t-1+vit,i=1,2,
where P1 and P2 are the transmitted power of S1 and S2. vit=vi1tvi2tT represents the noise in the MA phase, which follows a zero-mean white Gaussian distribution, that is, vit~CN0,N0I. αi stands for the imperfect coefficient of channel fading between S2 and Ri, which reflects the effect of timing delay τi. Normally, we have αi=0 for τi=0, which means the synchronization situation, and αi=1/2αi=1-αi2 for τi=0.5T, which means the power of delay signal is equal to that of current signal. The fading coefficients fit and git, denoting the Rayleigh channel fading from source S1 to Relay Ri and source S2 to Relay Ri, that is, fit~CN0,σfi2 and git~CN0,σgi2, are assumed to be constant over one frame and change independently from one frame to another for simplicity [15, 17, 18]. So, let fit=fi, git=gi, and rit can be expressed as
(2)rit=P1fis1t+P21-αi2gis2t+P2αigis2t-1+vit,i=1,2.Figure 1
Transmission model of DTWRN.Since differential modulation is considered in this paper, a2×2 unitary matrix Ut is used to encode the signal at nodes S1 and S2. At time t, it is encoded as sjt=Ujtsjt-1j=1,2, where sjt-1 is the signal transmitted by Si at time t-1. For space-time coding, a block is often constructed for transmission [10], which satisfies
(3)AkUjt=UjtAk,ifBk=0,BkUj*t=UjtBk,ifAk=0,
where Ak and Bk are two 2×2 complex matrices. For simplicity, it is designed that either Ak is unitary, Bk=0 (case I), or Bk is unitary, Ak=0 (case II).In the BC phase, theith relay node Ri utilizes rit to generate a symbol vector xit to satisfy the space-time coding scheme, which is a linear combination of rit and its conjugate [12, 19]. Hence, A1=1001, B2=01-10. Considering amplify-and-forward (AF) protocol in the relay nodes, the transmitted signal at the ith relay can be represented as
(4)xit=βit(Airit+Birit*),
where βit is the scaling factor at Ri and specially given by(5)βit=PRiσfi2P1+1-αi2σgi2P2+αi2σgi2P2+N0=PRiσfi2P1+σgi2P2+N0,
where PRi is the transmitted power of Ri and it is assumed that PRi=PR, so we have βit=β (constant). Then, the relay nodes Ri broadcast the coded symbol vector xit. The signals received at two source nodes are expressed as follows, respectively. At node S1,
(6)y1t=βf1A1r1t+βf2B2r2*t+w1t=P1βS1tf12f2f2*+P2βS2t1-α12f1g11-α22f2g2*+P2βS2t-1α1f1g1α2f2g2*+βf1v11-f2v22*f1v12f2v21*+w11tw12t,
where wit=wi1twi2tTi=1,2 denotes the additive white Gaussian noise (AWGN) at S1. Sjττ=t,t-1 is the space-time coding block which satisfies (7), and it is also a linear construction of sjτ and its conjugate [20]
(7)Sjτ=A1sjτ,B2sj*τ=sj1τ-sj2*τsj2τsj1*τ.
Besides, Sjt is in the differential modulation with Sjt as follows:
(8)Sjt=A1sjt,B2sj*t=A1Ujtsjt-1,B2Uj*sj*t-1=UjtA1sjt-1,UjtB2sj*t-1=Ujt·Sjt-1.
Let h11t=βf12f2f2*, h12t=β1-α12f1g11-α22f2g2*, h13t=βα1f1g1α2f2g2*, and n1t=βf1v11-f2v22*f1v12f2v21*+w11tw12t; then y1t can be abbreviated as
(9)y1t=P1S1th11t+P2S2th12t+P2S2t-1h13t+n1t.
It is easy to prove that En1tn1tH=σn12tI, and σn12t=∑i=12β2fi2+1N0. Similarly, at node S2,
(10)y2t=βg1A1r1t+βg2B2r2*t+w2t=P1S1th21t+P2S2th22t+P2S2t-1h23t+n2t,
where h21t=β[g1f1g2f2*], h22t=β1-α12g121-α22g2g2*, h23t=βα1g12α2g2g2*, n2t=βg1v11-g2v22*g1v12g2v21*+w21tw22t, and n2 has the same property as n1; that is, σn22t=∑i=12β2gi2+1N0.For the first block, a known vector can be transmitted to both source nodes for differential modulation which satisfiessjtHsjt=2j=1,2, for example, 11T. Here, let s10=11T, s20=11T as initial state; then S10=1-111, S20=1-111.


## 3. 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.


### 3.1. Detection at NodeS1

Theorem 1.
If the relay matrices have the property:tr⁡OiOjH=2 for i=j, tr⁡OiOjH=0 for i≠j (where Oj stands for Aj or Bj) [12], it can be elicited that
(11)ES1tHy1t=2P1h11t.
So, h~11t can be approximated as
(12)h~11t=12L1P1∑l=1L‍S1t-lHy1t-l,
where L is the frame length. Then let
(13)y~1t=y1t-P1S1t·h~11t=P2S2th12t+P2S2t-1h13t+n1t.If detection ofs2t is in the same way as in the perfect synchronization case, that is, ignoring ISI P2S2t-1h13t, there will be a severe error floor, which is the same at node S2. To eliminate the error floor caused by imperfect synchronization, a detection scheme is proposed to remove the ISI as much as possible. When t=1, the initial value is h130=0, S2-1=0; hence we have
(14)y10=P1S10h110+P1S20h120+n10.
Then y~10, y~11 can be calculated as
(15)y~10=P2S20h120+n10,y~11=P2S21h121+P2S20h131+n11.
By using the least square (LS) decoder, the transmitted signal can be recovered as
(16)U~21=argmin⁡Uk1y~11-Uk1y~10.
Since S20 is initialized as S20=1-111, set S~20=S20=1-111, so S~21=U~21S~20, and then h131 can be estimated as
(17)h~131=1P2S~21HS~20-1·S~21Hy~11-S~20Hy~10.
When t⩾2, use Eh~13 instead of h~13t to increase the accuracy with
(18)h~13avet=1t∑n=1t‍h~13t.
Then the ISI part can be removed using h~13avet, let y~1′t denote the remaining signal, and it can be calculated as
(19)y~1′t-1=y~1t-1-P2S2t-2h~13avet-1≈P2S2t-1h12t-1+n1t-1,(20)y~1′t=y~1t-P2S2t-1h~13avet-1≈P2S2th12t+n1t=P2U2tS2t-1h12t+n1t=U2ty~1′t-1+n~1t,
where n~1t=n1t-U2tn1t-1. The transmitted signal is then detected by LS as
(21)U~2t=argmin⁡Ukty~1′t-Ukty~1′t-1,
and then S~2t=U~2tS~2t-1, and estimate h13t again using
(22)h~13t=1P2S~2tHS~2t-1-S~2t-1HS~2t-2-1·S~2tHy~1t-1-S~2t-1Hy~1t.
The detection process is then repeated as described in steps (18)~(22), which improves the accuracy of h~13avet as t increases.


### 3.2. Detection at NodeS2

Using the same estimation method ofh~11t in node S1, h~22 and h~23 here can be approximated as
(23)h~22t=12L1P2∑l=1L‍S2t-lHy2t-l,h~23t=12L-11P2∑l=1L‍S2t-1-lHy2t-l.
Then set y~2t as
(24)y~2t=y2t-P2S2th~22t-P2S2t-1h~23t=P1S1th21t+n2t=U1ty~2t-1+n~2t,
where n~2t=n2t-U1tn2t-1. The transmitted signal from S1 can be detected by LS as
(25)U~1t=argmin⁡Ukty~2t-Ukty~2t-1.
However, h~22t and h~23t estimated above are not accurate enough, which will lead to error floor in the detection. The reason is that when estimating h22t, ES2tHy2t includes part of ES2tHS2t-1h23t. S2t and S2t-1 are not completely independent in statistical terms, so ES2tHS2t-1h23t is not 0 while ES1tHS2t-1h13t=0 for node S1. The same problem is also existing in the estimation of h23t. To eliminate the inaccuracy of h~22t and h~23t, a method is proposed as follows. Though h~22t is not accurate enough, it can be used. Firstly, rewrite h~22t and h~23t as h~221t and h~231t. Then define y^2t as
(26)y^2t=y2t-P2S2th~221t=P1S1th21t+P2S2t-1h23t+n2t.
Use the similar estimation method of h13t to estimate h23t; denote the result as h~23ave2t. The only difference is that S2t-1 is already known at node S2, which leads to a more accurate value. Set y^2′t=y2t-P2S2t-1h~23ave2t; then h23t can be reestimated as
(27)h~222t=12L1P2∑l=1L‍S2t-lHy^2′t-l.
So, y~22t can be calculated as
(28)y~22t=y2t-P2S2th~222t-P2S2t-1h~23ave2t.
The transmitted signal can be detected by LS again as
(29)U~12t=argmin⁡Ukty~22t-Ukty~22t-1.
It is proved in the simulation results that the method of reestimating h22t and h23t can effectively eliminate the error floor and ensure the detection performance.


### 3.3. Constellation Rotation

Note that the value ofS~2tHS~2t-1-S~2t-1HS~2t-2 can be equal to zero, which may affect the accuracy of h~13t. This issue also exists in estimating h23t. 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 -π/2,π/2. To simplify, the rotation angle may be set as θ=π/2. Here, we give s2t as an example on how to achieve the constellation rotation. Set s2t as
(30)s2t∈±1,±1T,t=4m-3,4m-2±i,±iT,t=4m-1,4m00000000000000.0000000m=1,2,…,
and it is easy to calculate that
(31)S2t∈±1-111,±11-11,t=4m-3,4m-2±i-iii,±ii-ii,t=4m-1,4m000000000000000000000000000000000000(m=1,2,…).
Then we can get
(32)S2tHS2t-1-S2t-1HS2t-2∈±2±2i002±2i,±2±2i∓2i2,0000000±±2i2-2±2i,02±2i-2∓2i0,
which is impossible to be zero. This constellation rotation scheme is also applied to s1t.


## 3.1. Detection at NodeS1

Theorem 1.
If the relay matrices have the property:tr⁡OiOjH=2 for i=j, tr⁡OiOjH=0 for i≠j (where Oj stands for Aj or Bj) [12], it can be elicited that
(11)ES1tHy1t=2P1h11t.
So, h~11t can be approximated as
(12)h~11t=12L1P1∑l=1L‍S1t-lHy1t-l,
where L is the frame length. Then let
(13)y~1t=y1t-P1S1t·h~11t=P2S2th12t+P2S2t-1h13t+n1t.If detection ofs2t is in the same way as in the perfect synchronization case, that is, ignoring ISI P2S2t-1h13t, there will be a severe error floor, which is the same at node S2. To eliminate the error floor caused by imperfect synchronization, a detection scheme is proposed to remove the ISI as much as possible. When t=1, the initial value is h130=0, S2-1=0; hence we have
(14)y10=P1S10h110+P1S20h120+n10.
Then y~10, y~11 can be calculated as
(15)y~10=P2S20h120+n10,y~11=P2S21h121+P2S20h131+n11.
By using the least square (LS) decoder, the transmitted signal can be recovered as
(16)U~21=argmin⁡Uk1y~11-Uk1y~10.
Since S20 is initialized as S20=1-111, set S~20=S20=1-111, so S~21=U~21S~20, and then h131 can be estimated as
(17)h~131=1P2S~21HS~20-1·S~21Hy~11-S~20Hy~10.
When t⩾2, use Eh~13 instead of h~13t to increase the accuracy with
(18)h~13avet=1t∑n=1t‍h~13t.
Then the ISI part can be removed using h~13avet, let y~1′t denote the remaining signal, and it can be calculated as
(19)y~1′t-1=y~1t-1-P2S2t-2h~13avet-1≈P2S2t-1h12t-1+n1t-1,(20)y~1′t=y~1t-P2S2t-1h~13avet-1≈P2S2th12t+n1t=P2U2tS2t-1h12t+n1t=U2ty~1′t-1+n~1t,
where n~1t=n1t-U2tn1t-1. The transmitted signal is then detected by LS as
(21)U~2t=argmin⁡Ukty~1′t-Ukty~1′t-1,
and then S~2t=U~2tS~2t-1, and estimate h13t again using
(22)h~13t=1P2S~2tHS~2t-1-S~2t-1HS~2t-2-1·S~2tHy~1t-1-S~2t-1Hy~1t.
The detection process is then repeated as described in steps (18)~(22), which improves the accuracy of h~13avet as t increases.


## 3.2. Detection at NodeS2

Using the same estimation method ofh~11t in node S1, h~22 and h~23 here can be approximated as
(23)h~22t=12L1P2∑l=1L‍S2t-lHy2t-l,h~23t=12L-11P2∑l=1L‍S2t-1-lHy2t-l.
Then set y~2t as
(24)y~2t=y2t-P2S2th~22t-P2S2t-1h~23t=P1S1th21t+n2t=U1ty~2t-1+n~2t,
where n~2t=n2t-U1tn2t-1. The transmitted signal from S1 can be detected by LS as
(25)U~1t=argmin⁡Ukty~2t-Ukty~2t-1.
However, h~22t and h~23t estimated above are not accurate enough, which will lead to error floor in the detection. The reason is that when estimating h22t, ES2tHy2t includes part of ES2tHS2t-1h23t. S2t and S2t-1 are not completely independent in statistical terms, so ES2tHS2t-1h23t is not 0 while ES1tHS2t-1h13t=0 for node S1. The same problem is also existing in the estimation of h23t. To eliminate the inaccuracy of h~22t and h~23t, a method is proposed as follows. Though h~22t is not accurate enough, it can be used. Firstly, rewrite h~22t and h~23t as h~221t and h~231t. Then define y^2t as
(26)y^2t=y2t-P2S2th~221t=P1S1th21t+P2S2t-1h23t+n2t.
Use the similar estimation method of h13t to estimate h23t; denote the result as h~23ave2t. The only difference is that S2t-1 is already known at node S2, which leads to a more accurate value. Set y^2′t=y2t-P2S2t-1h~23ave2t; then h23t can be reestimated as
(27)h~222t=12L1P2∑l=1L‍S2t-lHy^2′t-l.
So, y~22t can be calculated as
(28)y~22t=y2t-P2S2th~222t-P2S2t-1h~23ave2t.
The transmitted signal can be detected by LS again as
(29)U~12t=argmin⁡Ukty~22t-Ukty~22t-1.
It is proved in the simulation results that the method of reestimating h22t and h23t can effectively eliminate the error floor and ensure the detection performance.


## 3.3. Constellation Rotation

Note that the value ofS~2tHS~2t-1-S~2t-1HS~2t-2 can be equal to zero, which may affect the accuracy of h~13t. This issue also exists in estimating h23t. 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 -π/2,π/2. To simplify, the rotation angle may be set as θ=π/2. Here, we give s2t as an example on how to achieve the constellation rotation. Set s2t as
(30)s2t∈±1,±1T,t=4m-3,4m-2±i,±iT,t=4m-1,4m00000000000000.0000000m=1,2,…,
and it is easy to calculate that
(31)S2t∈±1-111,±11-11,t=4m-3,4m-2±i-iii,±ii-ii,t=4m-1,4m000000000000000000000000000000000000(m=1,2,…).
Then we can get
(32)S2tHS2t-1-S2t-1HS2t-2∈±2±2i002±2i,±2±2i∓2i2,0000000±±2i2-2±2i,02±2i-2∓2i0,
which is impossible to be zero. This constellation rotation scheme is also applied to s1t.


## 4. 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.


### 4.1. PEP of NodeS1

In Section3, the differential detection expression at S1 is derived as y~1t=Vty~1t-1+n~1t. Define VΔ,kjt=Vkt-Vjt and S2,Δ,kjt=S2,kt-S2,jt. The PEP of mistaking the kth STC block by the jth STC block can be evaluated by averaging the conditional PEP over the channel statistics [12] as
(33)PkjS1γ=Efi,giQVΔ,kjty~1t-122σn~12t,
where γ=P/N0 is the signal-to-noise ratio (SNR), P is the total transmitted power in the TWRN, and Qx is the Gaussian Q-function. Since it is very difficult to analyze y~1t-1 directly, we use y~1t-1≈P2S2t-1h12t-1 as in (20) instead in the following analysis. In Section 2, fit and git are assumed to be constant over one frame, so h12t is constant; that is, h12t=h12t-1. Based on (8), S2,Δ,kjt=VΔ,kjtS2t-1, so PkjS1γ can be simplified as
(34)PkjS1γ≈Efi,giQP2S2,Δ,kjth12t22σn~12t.
As derived in Section 2, h12t=β1-α12f1g11-α22f2g2*T. Define h12t=βFtg^t, where g^t=1-α12g1t1-α22g2*tT and Ft=diagf1t,f2t. Then, according to [12], PkjS1γ can be derived as
(35)PkjS1γ=1π∫0π/2Efi∏i=12‍1+lθ,tλifit2-1dθ,
where lθ,t=P2β2σg21-αi2/8∑i=12β2fit2+1N0sin2θ, and λii∈1,2 denotes the singular value of S2,Δ,kjtHS2,Δ,kj(t). The mean of fit2 is σf2, so the term ∑i=12fit2 in lθ,t can be approximated as ∑i=12fit2≈2σf2. Hence,
(36)lθ,t≈l′θ=P2β2σg21-αi282β2σf2+1N0sin2θ.PkjS1γ can then be expressed as
(37)PkjS1γ=1π∫0π/2Efi∏i=12‍1+l′θλifit2-1dθ.
It can be observed from (37) that the influence factor of PkjS1γ is the same as in synchronization case except the term 1-αi2. However, 1-αi2 is a constant during a frame. So the PEP expression of node S1 can be simplified at high SNR as
(38)PkjS1γ≈123‼4‼∏i=12‍1Miln⁡Mi=316∏i=12‍1Miln⁡Mi,
where Mi=(P2β2σf2σg21-αi2/82β2σf2+1N0)λi. Since the total transmission power is P, P=P1+P2+2PR. Denote P1=μ1P, P2=μ2P, and γ=P/N0. So Mi at high SNR can be expressed as Mi=C1iλiγ, where C1i=μ21-μ1-μ2σf2σg21-αi2/161-μ1-μ2σf2+μ1σf2+μ2σg2. Thus, the simplified PEP at high SNR can be rewritten as
(39)PkjS1γ=3161∏i=12‍C1iλiγ-2∏i=12‍ln⁡⁡C1iλi+ln⁡γ≈3161∏i=12‍C1iλiγ-2ln⁡γ2.


### 4.2. PEP of NodeS2

Similarly to the derivation of PEP at nodeS1, PEP of node S2 can be expressed as
(40)PkjS2γ≈Efi,giQP1S1,Δ,kjth21t22σn~22t,
where σn~22t=2∑i=12βit2git2+1N0 and S1,Δ,kjt=S1,kt-S1,jt. If we define h21t=βGtf^t, where f^t=f1tf2*tT and Gt=diagg1t,g2t, it is easy to find that the elements in PkjS2γ have no relationship to the imperfect synchronization coefficient αi; that is, it is identical to the synchronization situation [12]. So PkjS2γ can be derived as
(41)PkjS2γ≈3161∏i=12‍C2iλiγ-2ln⁡γ2,
where C2i=μ21-μ1-μ2σf2σg2/161-μ1-μ2σg2+μ1σf2+μ2σg2.


### 4.3. Optimum Power Allocation

In order to analyze the overall performance of the system, the total PEP is considered. It can be calculated as(42)PkjS1γ+PkjS2γ≈3161∏i=12‍λi∏i=12‍C1i-1+∏i=12‍C2i-1γ-2ln⁡γ2=3161∏i=12‍λiC11-1C12-1+C21-1C22-1γ-2ln⁡γ2.
It is obvious that, to minimize the PEP at high SNR, C11-1C12-1+C21-1C22-1 should be minimized. For simplification, the source nodes and the relay nodes are assumed to have the same power, which is to say P1+P2=2PR=(1/2)P; that is, μ1+μ2=1/2; then C1i, C2i can be rewritten as C1i=μ2σf2σg21-αi2/321-μ2σf2+μ2σg2, C2i=0.5-μ2σf2σg2/320.5+μ2σg2+0.5-μ2σf2. Two cases are considered for further performance analysis. For case I, σf2=σg2=σ2. For case II, σf2=10σ2, σg2=σ2. In case I, C11-1C12-1+C21-1C22-1 can be simplified as
(43)C11-1C12-1+C21-1C22-1=32μ21-α12σ232μ21-α22σ2+320.5-μ2σ22=322σ41-α121-α22×2-α12-α22+α12α22μ22-μ2+0.25μ220.5-μ22.
Denote yμ2=(2-α12-α22+α12α22μ22-μ2+0.25)/μ220.5-μ22; obviously, when C11-1C12-1+C21-1C22-1 obtain the minimum value, yμ2 is minimum. This minimum value can be calculated by mathematical tools on computer easily, and the corresponding value of μ2 leads to the optimum power allocation of this system. Similarly, in case II, C11-1C12-1+C21-1C22-1 can be simplified as
(44)C11-1C12-1+C21-1C22-1=1610-9μ25μ21-α12σ21610-9μ25μ21-α22σ2+165.5-9μ250.5-μ2σ22=25625σ4Φ162+Φμ24+-261-99Φμ230000000000000+300.25+30.25Φμ22-145μ2+25×μ220.5-μ22-1,
where Φ=1-α121-α22. The OPA method also referred to the value of μ2 when C11-1C12-1+C21-1C22-1 is minimum.


## 4.1. PEP of NodeS1

In Section3, the differential detection expression at S1 is derived as y~1t=Vty~1t-1+n~1t. Define VΔ,kjt=Vkt-Vjt and S2,Δ,kjt=S2,kt-S2,jt. The PEP of mistaking the kth STC block by the jth STC block can be evaluated by averaging the conditional PEP over the channel statistics [12] as
(33)PkjS1γ=Efi,giQVΔ,kjty~1t-122σn~12t,
where γ=P/N0 is the signal-to-noise ratio (SNR), P is the total transmitted power in the TWRN, and Qx is the Gaussian Q-function. Since it is very difficult to analyze y~1t-1 directly, we use y~1t-1≈P2S2t-1h12t-1 as in (20) instead in the following analysis. In Section 2, fit and git are assumed to be constant over one frame, so h12t is constant; that is, h12t=h12t-1. Based on (8), S2,Δ,kjt=VΔ,kjtS2t-1, so PkjS1γ can be simplified as
(34)PkjS1γ≈Efi,giQP2S2,Δ,kjth12t22σn~12t.
As derived in Section 2, h12t=β1-α12f1g11-α22f2g2*T. Define h12t=βFtg^t, where g^t=1-α12g1t1-α22g2*tT and Ft=diagf1t,f2t. Then, according to [12], PkjS1γ can be derived as
(35)PkjS1γ=1π∫0π/2Efi∏i=12‍1+lθ,tλifit2-1dθ,
where lθ,t=P2β2σg21-αi2/8∑i=12β2fit2+1N0sin2θ, and λii∈1,2 denotes the singular value of S2,Δ,kjtHS2,Δ,kj(t). The mean of fit2 is σf2, so the term ∑i=12fit2 in lθ,t can be approximated as ∑i=12fit2≈2σf2. Hence,
(36)lθ,t≈l′θ=P2β2σg21-αi282β2σf2+1N0sin2θ.PkjS1γ can then be expressed as
(37)PkjS1γ=1π∫0π/2Efi∏i=12‍1+l′θλifit2-1dθ.
It can be observed from (37) that the influence factor of PkjS1γ is the same as in synchronization case except the term 1-αi2. However, 1-αi2 is a constant during a frame. So the PEP expression of node S1 can be simplified at high SNR as
(38)PkjS1γ≈123‼4‼∏i=12‍1Miln⁡Mi=316∏i=12‍1Miln⁡Mi,
where Mi=(P2β2σf2σg21-αi2/82β2σf2+1N0)λi. Since the total transmission power is P, P=P1+P2+2PR. Denote P1=μ1P, P2=μ2P, and γ=P/N0. So Mi at high SNR can be expressed as Mi=C1iλiγ, where C1i=μ21-μ1-μ2σf2σg21-αi2/161-μ1-μ2σf2+μ1σf2+μ2σg2. Thus, the simplified PEP at high SNR can be rewritten as
(39)PkjS1γ=3161∏i=12‍C1iλiγ-2∏i=12‍ln⁡⁡C1iλi+ln⁡γ≈3161∏i=12‍C1iλiγ-2ln⁡γ2.


## 4.2. PEP of NodeS2

Similarly to the derivation of PEP at nodeS1, PEP of node S2 can be expressed as
(40)PkjS2γ≈Efi,giQP1S1,Δ,kjth21t22σn~22t,
where σn~22t=2∑i=12βit2git2+1N0 and S1,Δ,kjt=S1,kt-S1,jt. If we define h21t=βGtf^t, where f^t=f1tf2*tT and Gt=diagg1t,g2t, it is easy to find that the elements in PkjS2γ have no relationship to the imperfect synchronization coefficient αi; that is, it is identical to the synchronization situation [12]. So PkjS2γ can be derived as
(41)PkjS2γ≈3161∏i=12‍C2iλiγ-2ln⁡γ2,
where C2i=μ21-μ1-μ2σf2σg2/161-μ1-μ2σg2+μ1σf2+μ2σg2.


## 4.3. Optimum Power Allocation

In order to analyze the overall performance of the system, the total PEP is considered. It can be calculated as(42)PkjS1γ+PkjS2γ≈3161∏i=12‍λi∏i=12‍C1i-1+∏i=12‍C2i-1γ-2ln⁡γ2=3161∏i=12‍λiC11-1C12-1+C21-1C22-1γ-2ln⁡γ2.
It is obvious that, to minimize the PEP at high SNR, C11-1C12-1+C21-1C22-1 should be minimized. For simplification, the source nodes and the relay nodes are assumed to have the same power, which is to say P1+P2=2PR=(1/2)P; that is, μ1+μ2=1/2; then C1i, C2i can be rewritten as C1i=μ2σf2σg21-αi2/321-μ2σf2+μ2σg2, C2i=0.5-μ2σf2σg2/320.5+μ2σg2+0.5-μ2σf2. Two cases are considered for further performance analysis. For case I, σf2=σg2=σ2. For case II, σf2=10σ2, σg2=σ2. In case I, C11-1C12-1+C21-1C22-1 can be simplified as
(43)C11-1C12-1+C21-1C22-1=32μ21-α12σ232μ21-α22σ2+320.5-μ2σ22=322σ41-α121-α22×2-α12-α22+α12α22μ22-μ2+0.25μ220.5-μ22.
Denote yμ2=(2-α12-α22+α12α22μ22-μ2+0.25)/μ220.5-μ22; obviously, when C11-1C12-1+C21-1C22-1 obtain the minimum value, yμ2 is minimum. This minimum value can be calculated by mathematical tools on computer easily, and the corresponding value of μ2 leads to the optimum power allocation of this system. Similarly, in case II, C11-1C12-1+C21-1C22-1 can be simplified as
(44)C11-1C12-1+C21-1C22-1=1610-9μ25μ21-α12σ21610-9μ25μ21-α22σ2+165.5-9μ250.5-μ2σ22=25625σ4Φ162+Φμ24+-261-99Φμ230000000000000+300.25+30.25Φμ22-145μ2+25×μ220.5-μ22-1,
where Φ=1-α121-α22. The OPA method also referred to the value of μ2 when C11-1C12-1+C21-1C22-1 is minimum.


## 5. 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 asPRi=P1,2=1, that is, equal power allocation (EPA), if not specially pointed out. BPSK modulation is used, and the frame length is L=100.Figure2 shows the performance of TWRN under imperfect synchronization using the existing differential detection scheme in [12]. Set ϕ=α/1-α2, 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 α increasing, the detection error floor becomes higher. But in the real system, ϕ is generated randomly since α 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 ϕ=0.5.Figure 2
Effect of different imperfect synchronization coefficientϕ.Figure3 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 S1 and S2 remove the high error floor caused by imperfect synchronization. The detection method on node S2 eliminates the error floor at high SNR after reestimating h23t, providing a BER performance approaching the synchronization situation. The BER of node S1 is 4 dB less than that of node S2. The reason is that element S2t-1 in the interference part is known to node S2 but unknown to node S1, which has been mentioned in Section 3.Figure 3
BER performance of detection scheme on nodesS1 and S2.In Figure4, 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 S1 decreased while that of node S2 increased compared to equal power allocation (EPA), and the total BER of node S1 and node S2 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.BER performance of the proposed detection scheme with optimum power allocation.
(a)
Case I:σf2=σg2=1
(b)
Case II:σf2=10,σg2=1


## 6. 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.



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*Source: 563737-2015-03-01.xml*