1. Introduction
Cooperative communication enables multiple single-antenna terminals to create a “virtual” physical array in order to obtain spatial diversity [1,2]. However, many existing works are based on half-duplex (HD) operation. HD operation requires orthogonal channels (time slots, frequency bands, etc.) to forward the source data, resulting in the loss of spectral efficiency [3]. Many schemes has been proposed to overcome this problem, such as successive relaying [4], frame-level virtual FD relaying [5] and buffer-aided relaying [6]. All of these schemes are based on elaborated designed transmission protocols and more rigorous constraint and have not broken through the fundamental limitation of half-duplexing. In-band full-duplex (IBFD) communication provides the ability of simultaneous transmission and reception in the same frequency band, which gives relay systems the potential to improve spectral efficiency while retaining spatial diversity. An obvious drawback of FD operation is the self-interference (SI) introduced by the leakage of signal and energy from the output to the input of the same terminal. In recent years, a number of studies on self-interference cancellation (SIC) [7-10] have made the FD operation more feasible in practice. SIC aims to reduce the SI to an acceptable level that will not severely affect the detection of the intended signal.
1.1 Related Literature
Earlier researches [11-13] have not taken SI into account. However, the experimental results in have pointed out that the SI might not be completely mitigated even with advanced SIC methods. Hence, recent works consider the effect of residual SI on a variety of performance metrics. The authors in [14], [15] study the channel capacity of FD relay networks and stated that FD relaying outperforms its HD conterpart. In [16], it is shown that the unidirectional FD relaying is superior to the two-way HD relaying with physical-layer network coding. Kim et al [17] investigate the optimal power allocation problem in the context of cognitive radio. Rodríguez et al in [18] analysis the non-orthogonal FD amplify-and-forward (AF) relaying in terms of pairwise error probability (PEP). They conclude that even with the presence of residual SI, the FD relaying system can achieve full diversity. In the scenarios where the self-interference might worsen the quality of service, the system tends to operate in the traditional duplex modes. Feng et al. address the duplex mode selection problem in [19] and [20] along with the channel allocation and power control. In common sense, the system benefits more from weaker SI. However, the authors in [21] found that under some conditions, perfect SI cancellation may result in an unstable queue for message storage in a random access multiuser network, which brings a fresh point of view of relay-assisted system design.
Besides single relay networks, the multiple relay networks also draw attention. Some works focus on the parallel relaying systems, i.e., no inter-relay channel exists. In [22], Coso et al. investigate the achievable rates of different relaying protocols, namely: AF, DF, compress-and-forward and linear relaying, in the parallel FD relaying network. Zamani et al. [23] propose a hybrid decode-amplify forward relaying protocol and obtain its maximum throughput and expected-rate. In the scenarios of satellite-to-satellite communication [24], where isolation of relays can be achieved dute to absence of multipath propagation and the SI can be effectively supressed by directive antennas, significant gains can be achieved by FD operation comparing with HD relaying.
From the practical perspective, coexisted FD relay nodes may cause crosstalk due to essence of simultaneous transmission and reception. Yuksel et al. in [25] show that considering inter-relay communication and clustered network, the relay system can provide high spatial diversity. Without taking residual SI into account, the authors in [26] modeled the cooperative transmission with crosstalk as a partial distributed linear convolutional space-time coding (STC), which can achieve full asymchronous diversity with a minimum means square error and decision feedback equalizer (MMSE-DFE) receivers.
1.2 Contributions and Organization of This Paper
Inspired by the aforementioned studies, we investigate the error performance of uncoded full-duplex AF two-relay systems in the presence of residual SI. In the considered system, two FD relays assist the source node to forward information to the destination node, the direct link is assumed to endure heavy shadow and deep fading such that can be ignored. Crosstalk between the relays is considered as information exchange instead of interference to be mitigated. The main contributions of this paper are summarized as follows:
1) The closed-form expression of PEP is derived. Also we obtain the upper bound of PEP in the high transmit power region, which is useful to carry out the diversity and coding gain.
2) The diversity gain is discussed with the consideration of residual SI. Theoretical and numerical results show that full diversity is achievable if the variance of residual SI is not a function of the transmit power. Otherwise, the diversity gain is degraded. The scale of degradation is related to the quality of SIC.
3) The coding gain is also investigated. Higher coding gain can be obtained by increasing the frame length, suprassing more SI and proper power allocation.
The rest of this paper is organized as follows. The system model and transmission protocol are elaborated in Section 2. The closed-form expression and the upper bound of PEP are worked out in Section 3. Section 4 analyzes the error performance in terms of diversity and coding gain. Numerical results are shown and discussed in Section 5. Finally, conclusion is provided in Section 6.
Notations: In this paper, scalars are denoted with regular lower case letters. Vectors and matrices are denoted with bold lower and upper case letters. We denote AH and AT as the Hermitian transpose and regualr transpose of the matrix A, respectively. det(⋅) stands for the determinant operator. And E[⋅] is the expectation operator. IN represents the N×N identity matrix. Cov[⋅] is the covariance operator.
2. System Model
Consider a cooperative system which consists of four nodes: one HD source node S, two FD relay nodes R1 and R2, and one HD destination node D, as shown in Fig. 1. All nodes are equipped with single antenna. Since the nodes operate in the same frequency band, the two FD relays may cause crosstalk. We denote the channel from S to Ri (S → Ri) as fi, from Ri to D (Ri → D) as gi and the crosstalk channel from Ri to Rj as hij, i, j ∈ {1, 2}. By the reciprocity of wireless channels, we have h12 = h21 = h. The Rayleigh block fading is considered, which means that the channel gains remain constant during the transmission period. All the channel gains are assumed to be independent and identically distributed (i.i.d.) cyclic symmetric Gaussian random variables (RV) with zero mean and unit variance. We make the following assumption of CSI: the source has no knowledge of CSI; the relays know the second-order statistics of S → R and the crosstalk channels; the destination has global CSI.
Fig. 1.Two FD relaying system with crosstalk and residual SI
2.1 Transmission Protocol
The source data is transmitted in frame, s = [s(1), s(2),⋯, s(T)]T. The frame length T is an even number (T ≥ 2). The transmit signal s has an expected power equal to one, i.e., E[sHs] = 1. In the t-th time slot, S broadcasts s(t) to the relays. Meanwhile, the relays receive the crosstalk from each other. The received signal at Ri can be written as
where ci(t) is the residual SI after SIC, and ni(t) is the additive white Gaussian noise (AWGN) with zero mean and unit variance. Ps is the transmit power at the source. xi(t) is the transmit signal from Ri. The relays amplify and forward the signal which is received in the previous time slot, i.e.,
where b is the amplification coefficient which will discussed later. Substituting (2) into (1) we get
where vi(t) = ci(t) + ni(t). Recursively substituting into (3), it can be rewritten as
where k = i when (t − l) is even, otherwise k = j, i, j∈{1,2}, i ≠ j. Grouping ri(t) into a vector, the received signal at Ri in T symbol periods can be written as
where ri = [ri(1), ri(1),⋯ri(T)]T, fij = [fi fj]T, ui = B1vi + B2vj is the equivalent residual SI plus noise at Ri, vi = [vi(1), vi(1),⋯, vi(T)]T and vj = [vj(1), vj(1),⋯vj(T)]T, i, j ∈{1, 2}, i ≠ j, and
where and are the element at the m-th row and n-th column of B1 and B2, respectively.
The received signal at D can be written as
where M = [B1s B2s]. nd = [nd(1), nd(2),⋯nd(T)]T is the AWGN at D, w = b[u1 u2]g + nd is the equivalent residual SI plus noise at D, f = [f1 f2]T, g = [g1 g2]T, .
It can be observed from (6) that the uncoded source data s is transformed into the space-time coded signal M. The coding matrices B1 and B2 are not priorly designed and distributed, but involuntarily constructed during the information exchange through the crosstalk channel [26]. We refer to this particular distributed space-time coding (DSTC) scheme as the crosstalk self-coding (CSC). The formation of CSC is built on two preconditions: 1) the bidirectional crosstalk channel between the FD relays; and 2) the non-regenerative (AF) protocol. When operating in HD mode, the relays cannot exchange information simultaneously. Similarly, if the decode-and-forward protocol is employed, the transmit signal from Ri will be treated as interference at Rj, which leads to the failure of information exchange.
It should be noted that we consider the frame length of T ≥ 2 . When T =1, the considered system would retrogress to a HD relaying system, in which the CSC matrices cannot be constructed. The proposed transmission protocol provides no profit for such a system.
2.2 Statistical Model of Residual Self-Interference
To the best of the authors’ knowledge, the accurate relationship between the transmit signal and the residual SI is still unknown in practice. In this paper, we follow the previous works [19,27] and model the residual SI as an additive Gaussian RV,
where Pr is the transmit power at each relay, and the values of β (β > 0 ) and λ ( 0 ≤ λ ≤1 ) represent the quality of SIC. Smaller β and λ reflect a better SIC performance. In addition, we assume that the residual SI is independent of the signal and noise.
2.3 Amplifying Coefficient
For simplicity, we use the fixed-gain amplification [28]. Since the relays have the same average reception and transmission power, their amplification coefficients can be identical. Note that the consumed power at S and D is Ps and Pr, respectively, the amplifying coefficient can be calculated as
In the proposed system, a special case should be considered: At time slot t = 1, the relays keep silence, therefore no SI exists. At time slot t = 2, the power of received signal at each relay is higher than that in the last time slot because of the appearance of SI. By parity of reasoning, the average transmit power consumed by the relays at each time slot will increase with t. As shown in (7), the variance of residual SI is related to Pr. Hence, the accumulation of Pr will worsen the average signal to interference plus noise ratio (SINR) at D. We propose a simple solution to simultaneously fix the amplifying coefficient and transmit power: At time slot t = 1, each relay transmits a deterministic symbol s0 with the power of Pr. Then, the power of received signal at time slot t = 2 equals to that at time slot t = 1, and so on. When T is large, the loss of power efficiency at the relays can be neglected. Since D has global CSI, the corresponding terms of s0 can be directly removed from the received signal and will not affect the error performance. In the rest of this paper, the terms of s0 will be omitted for clarity.
3. Pairwise Error Probability
3.1 Maximum Likelihood Detection
Given that D has global CSI, w is a Gaussian RV with zero mean and the variance of Rww = E[wwH]. Assume that a message sp is transmitted from S, then the conditional received signal y|sp is also a Gaussian random vector with the mean of and the variance of Rww. The probability density function of y|sp can be written as
where Mp = [B1sp B2sp]. Thus, the ML detection can be straightforward given as
3.2 Pairwise Error Probability
Based on the ML detector in (10), the PEP of mistaking sp by sq, under certain channel realization, can be written as
where Mq = [B1sq B2sq]. Equation (11) provides the exact PEP of the considered system. However, the calculation of (11) is very hard and gives little insights. The rest of this section will focus on the upper bound of PEP which is more analyzable and tractable.
As a first step, we upper bound the PEP in (11) by the Chernoff bound. The main result is presented in the following theorem.
Theorem 1: With the ML detector in (10), the PEP of the considered system has the Chernoff bound of
where .
Proof: As in [29] and [30], the PEP in (9) has the Chernoff bound of
By (9), the exponent on the right-hand side of (13) can be written as
Substituting (14) into (13) yields the following equation after some simplification
Let , we have -λ(1 - λ) = -1/4, which minimizes the exponent at the right-hand side of (15). Due to the accumulation of residual SI and noise, w is non-white and non-stationary, which complicates the calculation of . In this paper, we use the approximation of the worst-case stationary residual SI and noise to upper bound (15), i.e., we use w′ = [w(T), w(T),⋯, w(T)]T to substitute w in (6), where w(T) is the T-th entry of w. Therefore, we have
where . By now, (15) can be upper bounded by (12) and the proof ends here.
Averaging (12) over f, we have
where . In the last step of (17), we use the fact that E[e-uHKu] = det-1(I + RK), for a given Gaussian random vector u subject to CN(0, R) and a Hermitian matrix K. Also we use the fact that det(I + AB) = det(I + BA) where B and A have proper dimensions.
Substituting (8) into (17), we have
where . In the high transmit power regime, i.e., Ps → ∞ and Pr → ∞, . Since is Hermitian, we can perform the eigendecomposition on ,
where Q is a unitary matrix, and Γ is a diagonal matrix that contains the eigenvalues of . Combining Q and QH with , the upper bound of PEP in (16) can be rewritten as
where . Denote z1 =g1 + g2, z2 = g1 - g2, and . The determinant on the right-hand side of (20) can be trivially rewritten as
Recall that is the Gram matrix of two linear independent vectors B1ŝ and B2ŝ, where ŝ = sp - sq. So we have γ11≥0 and γ22≥0. The determinant on the right-hand side of (21) can be lower bounded as
By (20), the upper bound of PEP in (20) can be updated as
3.3 Discussion on Correlation between Ri → D links
The diagonal elements of Γ is the equivalent power gains of the Ri → D links. Due to the information exchange between the relays, the original channel coefficients g1 and g2 are now transformed to the compounded channel coefficients g1 + g2 and g1 - g2. From the perspective of conventional multiple-input-multiple-output or space-time code analysis, the correlation among different channel coefficients directly affects the achievable diversity. If the channels are statistically independent, the system has potential to achieve full diversity; otherwise the diversity order is degraded. To that end, we need to explore the relationship between the two compounded channel coefficients. By introducing the following two useful lemmas [31], we can prove that g1 + g2 and g1 - g2 are statistically independent.
Lemma 1: Joint Gaussian RVs arise from nonsingular linear transformations on independent normal RVs.
Lemma 2: If X1, X2,⋯, Xn are jointly Gaussian and uncorrelated, then the Xi are independent.
Given that z1 and z2 are Gaussian distributed, we have
The above equation points out that z1 and z2 are joint Gaussian according to Lemma 1. The covariance of z1 and z2 can be calculated as
which suggests that z1 and z2 are also uncorrelated. So we can conclude that z1 and z2 are statistically independent by Lemma 2, and the considered system has the potential to achieve full diversity.
4. Derivation of Diversity and Coding Gain
In this section, we will derive the diversity and coding gain of the considered system based on the derived upper bound of PEP, and provide some remarks on the theoretical results.
Note that zi, i ∈ {1, 2}, is a linear combination of two independent Gaussian RVs, so |zi|2 has the exponential distribution of
Assume that the total consumed power of the considered system is P and Ps = αP, we have , 0<α<1. The upper bound in (23) can be calculated as
The diversity and coding gain can be worked out through (27), the main results are summarized in the following theorem.
Theorem 2: The PEP of the considered system can be upper bounded by
Proof: Set x = 1/P in (28). When P ≫ 1, we have 1/P → 0 which corresponds to the assumption of x → 0. Also, we have -Ei(-x) = -Ei(-1/P) = lnP+O(1). By omitting the lower-order terms of P, we obtain the expression in (28).
Remarks on Theorem 2:
1) The proposed CSC scheme achieves the diversity of 2(1 - λ) - ln lnP/lnP, which is jointly determined by the transmit power P and the quality of SI suppression λ. When λ = 0, the system achieves full diversity. The considered system needs efficient SIC to guarantee an acceptable diversity order, because the diversity order decays linearly with λ. In Section 2, we use two parameters, λ and β, to characterize the variance of SI. However, the diversity order does not involve β. Therefore, full diversity is achievable in the presence of residual SI if and only if the variance of residual SI is not a function of the transmit power at the relays.
2) The achievability of full diversity is also built on the full rank of , or essentially . In the studies of STC, is usually referred to as the coding gain distance (CGD) matrix. If drops rank, the diversity order decreases. The detailed expression of is
From (29) we know that if B1 ≠ B2, has full rank. When h ≠ 0, B1 ≠ B2 holds according to the definition of B1 and B2 in (5). When h = 0, B1 becomes an identity matrix, B2 becomes a zero matrix, and would be rank-one. In Section 2, we assume that h is subject to complex Gaussian distribution with zero mean and unit variance. The probability of h = 0 is zero. Hence, is full rank with probability one.
3) The crosstalk channel coefficient h should be treated as a deterministic and known complex at D. Note that each non-zero elements in B2 contains an odd-order moment of h. If we consider h as an RV, B2 will vanish since all odd-order moments of a Gaussian RV are zero, and the structure of CSC will be completely destroyed. Therefore, in the calculation of error probability, h should not be averaged over [26].
4) The term -ln lnP/lnP in the diversity order can also be observed in other relay systems, such as the HD DSTC and the FD NAF relaying. The reason is that the fading R → D links make the equivalent noise at the destination not Gaussian distributed. Comparing with the FD NAF system whose diversity order is 2 - ln lnP/lnP, the considered system in this paper has an additional reduction of -2λ. This is because that in the NAF systems, the signal conveyed in the direct link does not suffer from SI. However, in our work, all signals received from the relays are corrupted by residual SI at the relays.
5) Theorem 2 also shows the coding gain, which is
Equation (30) reveals that the coding gain is dominated by the frame length T, the power contrl factor α, and the quality of SIC β and λ. We do not involve det and , because in a given realization of the inter-relay channel, det and are determined by T and α. Due to the complicated relationship among the parameters in (30), we cannot precisely delineate the behavior of the coding gain. Alternatively, characters of the coding gain are demonstrated by numerical simulation in the next section.
5. Numerical Results
In this section, we present the numerical simulations of the proposed CSC scheme to verify the theoretical results. The simulations are divided into 2 parts: 1) the comparisons of the CSC scheme with the HD Alamouti STC [32], and existing full-duplex relay-assisted transmission schemes, such as the FD-NAF in [19] and the PDLC-STC in [26]; and 2) the illustration of the proposed CSC scheme with different quality of SIC λ, and frame length T.
All channel gains are realized as i.i.d. complex Gaussian RVs with zero mean and unit variance, and keep constant during the transmission period. And all noises at the relays and the destination are also modeled as AWGN with zero mean and unit variance. The residual SI is realized as AWGN with zero mean, and the variance shown in (7). In all simulations, the transmit power P at the source and relays is normalized by the noise power and presented in decibel. Other involved parameters are listed in the following table.
Simulation A: This simulation illustrates the BER performance of the proposed CSC scheme, along with the NAF scheme, PDLC-STC scheme, and the HD Alamouti STC, against the total consumed power P. The configuration of the simulation scenario is shown in Table. 1. For all the FD scheme, we set λ=0.1. The results are shown in Fig. 2.
Table 1.Configuration of the simulations
Fig. 2.BER performance of the proposed CSC scheme against the total consumed power P
Due to the SI-free direct link between the source and the destination, the FD-NAF system with the diversity order of 2 (full diversity) has the best error performance. The CSC scheme with the diversity order of 1 - λ = 1.8 achieves a little higher BER, since the CSC scheme is built on a dual-hop system where all signal is corrupted by SI. On the other hand, thanks to the inherent advantage of ML decoder over the MMSE-DFE decoder, the CSC scheme outperforms the PDLC-STC scheme when P>15dB. The MMSE-DFE decoder performs symbol-by-symbol decoding and may cause error propagation, instead the ML decoder makes considerable improvement by jointly decoding all received symbols.
A conter-intuitive phenomenon occurs in the comparison of the CSC scheme and the HD Alamouti STC. For P<37dB, the CSC scheme outperforms the HD Alamouti STC; otherwise the Alamouti STC is superior. By common sense, the HD systems do not suffer from SI and should have better error performance than its FD counterparts. This contradiction can be explained in two aspect: 1) The HD Alamouti STC scheme employs higher-order modulation to achieve the same rate as the CSC scheme; 2) The CSC scheme provides coding gain which is absent in the Alamouti STC scheme. These extra improvements compensate the loss of error performance caused by SI. In a word, for a certain rate, the FD system with CSC is more reliable in practical transmit power regime. [16] has the similar conclusion for the FD-NAF systems.
Simulation B: Fig. 3 shows the BER performance of the proposed CSC scheme under different λ. Other parameters in this simulation are identical to Simulation A. The theoretical diversity orders under different λ are listed in Table 2.
Fig. 3.BER performance of the proposed CSC scheme under different quality of SIC
Table 2.Diversity orders under different λ
Clearly, the BER performance is degraded rapidly with the increase of λ, and an error floor can be observed when λ=1, which is in agreement of the diversity analysis in Remark 1) on Theorem 2. Fig. 3 reveals that the diversity gain is highly sensitive to λ, which means that a well-designed SIC method significantly helps the considered system to attain lower BER level.
Simulation C: Effects on BER performance of the frame length T is illustrated in Fig. 4. We can see that at the same transmit power level, the BER decreases with T, which indicates that the lengthening the frame increases the coding gain as proved in Theorem 2. However, it can be observed that increasing T from 2 to 5 provides more BER decreasement than increasing T from 5 to 10. The reason is that larger T results more accumulation of SI and noise, i.e., larger in (30). And a larger decreases the coding gain in reverse. Generally speaking, increasing T improves the BER, but excessive increasement provides less benefits.
Fig. 4.BER performance of the proposed CSC scheme under different frame length
Simulation D: The BER performance of CSC is demonstrated in Fig. 5. Comparing with the frame length, the power control scheme makes less contribution to the BER improvement. Fig. 5 shows a interesting result that there is a tradeoff in allocation power between the source and the relays. This can be easily explained as following: When α=1/4, the signals at the destination will be severely interfered due to the higher power at the relays; when α=3/4, the power of intended signal is small at the destination. And setting α=1/2 may reach a rough equilibrium between the SINR at the relays and that at the destination.
Fig. 5.BER performance of the proposed CSC scheme under different power allocation
6. Conclusion
This paper investigated the error performance of uncoded full-duplex AF two-relay cooperative systems in the presence of residual self-interference. Crosstalk between relays introduced by the FD operation was regarded as self-coding rather than interference and utilized to improve spatial diversity. A closed-form expression of the PEP was derived. The upper bound of PEP in high transmit power region was obtained to discuss the diversity and coding gain. It was shown that full cooperative diversity can be achieved when the variance of the residual SI is not a function of the transmit power. Otherwise, the diversity gain decreases linearly with the quality of SIC λ. In addition, the coding gain is principally determined by the frame length, and the power control factor. Generally speaking, λ plays a critical role in error performance of the considered system, and the validity of CSC is built on the ML decoder, efficient SIC, sufficient long frame and proper power control scheme.
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