This paper studies a wire-tap channel in which a source node wants to communicate securely to a destination node in the presence of an eavesdropper and under the aid of an amplify-and-forward (AF) relay operating in full-duplex (FD) mode. The residual self-interference due to FD transmission is explicitly taken into account. The secrecy capacity and the respective optimal power allocation schemes for this system are examined under both individual and joint power constraints. At first, the related optimization problems are shown to be quasi-concave. As such, the globally optimal solution exists and is unique. Due to the non-linearity of the derivative, we apply a simple bisection method for root finding and obtain a simple expression for the optimal power allocation scheme. To further provide some insight on the solutions, we apply the method of dominant balance to analyze the capacity and power allocations in different high power regions. It is then demonstrated that full relay power is only needed when the power at the relay is sufficiently small compared to the power at the source. Comparisons with half-duplex (HD) relaying also revealed that FD can achieve a significantly higher secrecy capacity. Finally, numerical results are presented to confirm the optimality of the solutions.

This paper investigates the error performance of a full-duplex (FD) amplify-and-forward (AF) single-relay system under the effect of residual self-interference whose variance is proportional to the λ-th power of the transmitted power (0≤λ ≤1). Our focus is on the cooperative FD linear relaying (LR) protocol that makes use of direct source-destination link. At first, a closed-form expression of the pairwise error probability (PEP) is derived for the considered system. This expression allows us to analyze the diversity behavior in high transmission power regions. Thanks to the use of the direct link, it is shown that the FD LR system can attain the same diversity function as its half-duplex (HD) counterpart as long as a suitable precoder is applied. Different from previous works that either ignore the direct link or treat it as a source of interference, a non-zero diversity order is thus achieved and the error floor behavior can be eliminated despite the existence of self-interference in FD. More interestingly, it is then demonstrated that transmitting a superposition of all symbols in HD mode maximizes the asymptotic coding gain. Although HD relaying is hence asymptotically optimal, simulations results reveal that FD is advantageous at practical bit error rate (BER) levels when λ is sufficiently small.

This paper investigates the error and diversity performances of full-duplex (FD) amplify-and-forward (AF) singlerelay systems under the effect of residual self-interference. The variance of this interference is assumed to be proportional to the λ-th power of the transmitted power (0 ≤ λ ≤ 1). The study considers the cooperative linear relaying protocol with direct source-destination link and the dual-hop scheme without direct link, both under uncoded and coded frameworks. At first, closed-form pairwise error probability expressions are derived for the uncoded systems, which are then used to obtain tight bounds to the bit error rate (BER) of the coded systems. To shed an insight on the diversity behavior, asymptotic expressions at high transmission powers are also presented. Different from previous works that treat the direct link as interference, this paper shows that FD linear relaying systems with a suitable precoder can attain the same diversity function as their half-duplex (HD) counterparts. However, further analysis shows that HD orthogonal AF using a superposition constellation is asymptotically optimal in terms of maximum coding gain. In addition, it is shown that the diversity of FD dual-hop systems is a decreasing function of λ and is equal to zero when λ = 1. Although HD relaying is asymptotically optimal under the considered protocols and interference model, illustrative results show that FD relaying is advantageous at practical BER levels when λ is sufficiently small.

In this paper, we analyze the capacity of the static half-duplex single-relay amplify-and-forward (AF) system under both per-node and joint power constraints. Different from multiple-input multiple-output systems, the channel matrix of the cooperative AF system is a function of the parameters to be optimized and hence water-filling over the square of the singular values is no longer optimal. Furthermore, given that the mutual information of the AF system is not a concave function, conventional optimization methods cannot be applied. Instead, by deriving and comparing all local solutions, we characterize the optimal input covariance matrix at the source and the optimal power allocation scheme at the relay that maximize the achievable rate. First, for the individual power constraint scenario, it is shown that the capacity of the AF system is achieved by either a direct transmission (DT) scheme, a non-orthogonal AF (NAF) beamforming (BF) protocol with a unit-rank covariance matrix, or a NAF system using a specific full-rank covariance matrix. Then, for the global power constraint scenario, it is shown that only a DT or a NAF-BF protocol can achieve the capacity. In both cases, orthogonal transmission is strictly suboptimal. The capacity of the AF system is finally analyzed for some concrete examples, such as under asymptotically high and low transmission powers and for several network models.

In this paper, achievable rates and power allocation (PA) schemes are studied for a two-way (TW) amplify-and-forward (AF) relaying system over Rayleigh fading channels in high and low signal-to-noise ratio (SNR) regimes. We consider three different AF techniques: i) the channel inversion (CI) scheme where the relay has full channel side information (CSI) of the two incoming links; ii) the fixed-gain scheme where the relay has only channel distribution information (CDI); and iii) a mixed (MX) scheme where the relay has CSI of one link but only CDI of the other. First, focusing on high SNR regimes, approximations to the achievable rates of the three systems are derived. The approximations are shown to be tight and can be used to analyze the systems under consideration. In particular, it is demonstrated that the CI system provides the best sum rate performance, followed by the MX and CDI techniques. A suboptimal yet effective PA scheme to maximize the sum rate at high SNR is then proposed. In low SNR regimes, we first derive rate approximations for the considered AF systems. It is then shown that the three systems achieve a similar sum rate. An asymptotically optimal PA is then proposed to maximize the sum rate at low SNRs. Finally, the sum rate of the direct transmission (DT) scheme is compared to that of the TWAF system. Although the TWAF is proved to be inferior at low SNRs, it outperforms the DT at high SNRs in various network configurations.

In this paper, we establish the achievable sum-rate of a half-duplex single-relay two-way amplify-and-forward network in which the relay uses channel knowledge of the two incoming links to cooperate with the two sources. Specifically, by assuming that the relay can acquire full channel knowledge and that Gaussian codebooks are used at the source nodes, the optimal power adaptation scheme at the relay that maximizes the achievable sum-rate under a long-term average power constraint is derived. While the maximization of the sum-rate is shown to be a convex optimization problem, obtaining optimal solutions are challenging. By using the Karush-Kuhn-Tucker conditions, we first show that finding the optimal relay adaptation scheme is equivalent to finding the root of a quartic polynomial. The closed-form optimal solutions are then obtained. Important insights on the proposed adaptation scheme are also presented and discussed. Numerical results reveal that the derived relay adaptation technique outperforms the conventional fixed-gain and variable-gain amplification coefficients at low signal-to-noise ratio regimes, thanks to the benefit of dynamic power allocation.

In this paper, we develop optimal power adaptation schemes by means of power amplification coefficients at the relay for the half-duplex single-relay non-orthogonal amplify-and-forward (NAF) system with channel side information (CSI) available at the relay. In particular, assuming that the relay has full knowledge of the channel gains, optimal power adaptation schemes are established in closed-form using both the mutual information (MI) and pairwise error probability (PEP) criteria with Gaussian inputs at the source. The proposed solutions can be understood as a multi-dimensional (multi-D) distributed water-filling in time and space. Numerical results show that the proposed power amplification methods provide a significant improvement over conventional schemes using either channel distribution information (CDI) or channel inversion (CI). The gain is observed with regard to both the MI with Gaussian inputs and the bit error rate (BER) performance using finite constellations such as quadrature amplitude modulation (QAM).

In this paper, we establish in closed-form the capacity and characterize the optimal input covariance matrix at the source and the optimal power allocation scheme between source and relay for a half-duplex single-relay amplify-and-forward (AF) system with static channel gains. Different from multiple-input multiple-output (MIMO) systems, the channel matrix of the AF system is a function of the parameters to be optimized and hence water filling over the square of the singular values of this matrix is no longer optimal. Furthermore, given that the mutual information of the AF system is not a convex function, conventional optimization methods cannot be applied to find the optimal input covariance, the power allocation, and the capacity. Instead, by analyzing all local maximizers, it is shown that the capacity of the AF system is achieved by either the direct transmission (DT) scheme, the orthogonal AF (OAF) protocol, or the non-orthogonal AF (NAF) protocol using a non-diagonal covariance matrix. The choice of protocol depends on the signal-to-noise ratio (SNR) and network configuration. By further analyzing the asymptotic mutual information, it is shown that the DT scheme is dominant in low and high SNR regimes. The capacity of the AF system is also provided for comparison among several network models. Specifically, it is demonstrated that in a general linear network model, the NAF protocol can provide significant gains over the other schemes at medium SNRs.

In this paper, we investigate the capacity and optimal power allocation (PA) scheme between the source and relay for a dual-hop amplify-and-forward (AF) system over non-symmetric Rayleigh fading channels with channel information available at the relay. At first, a closed-form expression of the mutual information (MI) between the input and output of the considered channel is obtained. Since only the exponential integral is involved, the derived expression is useful in finding the optimal PA to achieve the capacity. By further considering high and low signal-to-noise ratio (SNR) regimes, we present tight yet simple approximations to this MI, which can be used to show the advantage of knowing channel information at the relay. Then, focusing on the problem of optimal PA, we first derive a closed- form derivative of the MI. A simple bisection method is then proposed to find the optimal PA scheme. While uniform PA is shown to achieve the capacity at any SNR over the symmetric channel, its optimality can only be observed at low SNRs over a non- symmetric channel. In other SNR regimes, numerical results reveal that uniform PA experiences a significant loss. A comparison between the dual-hop and direct transmission scheme is also made, where we show that the dual-hop scheme using the optimal PA can provide impressive rate increases in medium SNR ranges in various network configurations.

In this paper, we investigate the achievable rate and ergodic capacity of a general non-symmetric half-duplex non-orthogonal amplify-and-forward (NAF) single-relay channel in Rayleigh fading environments, assuming that the channel state information is only available at the destination. The considered channel, which captures pathloss and shadowing effects over the transmission links, includes the symmetric one as a special case. At first, for a given power allocation scheme, simple and closed-form expressions of the upper and lower bounds on the achievable rate are derived. As shown by various numerical examples, the gap between the upper and lower bounds is small in the entire range of SNRs, which makes them useful in finding the optimal power allocation solution to achieve the capacity. Focusing on the two extreme cases of low and high SNRs, we then provide relatively tight approximations of the achievable rate. Using these approximations, it is then revealed that at both high and low SNR regimes, the ergodic capacity is achieved when the relay is inactive. Equivalently, NAF relaying does not yield any advantage over direct transmission at low and high SNRs. The results and observations in this paper therefore provide some further important insights on NAF relaying.

This paper proposes a precoding scheme over multiple cooperative frames to increase the diversity order of a bandwidth efficient Bit Interleaved Coded Modulation (BICM) system over a Non-orthogonal Amplify-and-Forward (NAF) half-duplex single-relay channel. By deriving a union bound on the bit error probability, it is shown that the diversity gain function of the considered system is (Nf· dH)-th power of that of uncoded cooperative systems, where Nfis the number of precoded cooperative frames and dHis the minimum Hamming distance of the outer code. An optimal class of precoders is then derived to optimize the asymptotic coding gain. It is then shown that the source should transmit a superposition of all symbols in the broadcasting phases, while being silent in all cooperative phases for best asymptotic performance. By further analyzing the first iteration performance, a design criterion is then developed to find optimal superposition angles for good convergence behavior. A pragmatic approach is then proposed to find good rotation angles. Analytical and simulation results show that the proposed scheme provides a significantly higher order of time and cooperative diversities and better coding gains than previous precoding schemes.

In this paper, we design a capacity-approaching coded modulation scheme for half-duplex non-orthogonal amplify-and-forward (NAF) single-relay channel. We apply the idea of multi-dimensional (multi-D) mapping employed in precoded multiple cooperative frames, concatenated with a simple outer binary code. Using union bounding techniques, it is first shown that for any unitary and full-diversity rotation G, the optimal multi-D labeling for NAF relaying shall maximize the average Euclidean distance between all pairs in the multi-D rotated constellation whose labels differ in only one bit. The extrinsic information transfer (EXIT) charts are then used to match the outer code, the multi-D mapping, and the precoder for near-capacity performance. It is demonstrated that the proposed scheme is promising for NAF relaying, in the sense that it can operate below the achievable rate obtained by using a conventional modulation scheme, i.e., the constrained capacity. In particular, for various spectral efficiencies, we obtain the bit error rate (BER) of 10-5or lower at a signal-to-noise ratio (SNR) that is 0.45dB-1dB below the traditional achievable rate and within 1.55dB-1.95dB from the ergodic capacity with Gaussian inputs.

ABSTRACT This paper proposes the idea of precoding over multiple cooperative frames with multi‐dimensional (multi‐D) mapping for a bit interleaved coded modulation system over an ergodic non‐orthogonal amplify‐and‐forward (NAF) half‐duplex single‐relay channel. The benefits of multiple‐frame precoding and multi‐D labeling are analyzed in two different regions: the error‐floor region to exploit diversity and the turbo pinch‐off region for near‐capacity performance. In the error‐floor area, it is shown that the diversity gain function of the considered system is th power of that of uncoded cooperative systems, where Nf is the number of precoded cooperative frames and dH is the minimum Hamming distance of the outer code. To optimize the asymptotic coding gain, it is then shown that the source and relay must transmit orthogonally for best asymptotic performance. In the turbo pinch‐off region, we demonstrate that the proposed system concatenated with a simple outer binary code can be employed to achieve near‐capacity performance. Using union bounding techniques, we show that the optimal multi‐D labeling for NAF relaying shall maximize the average Euclidean distance between all pairs in the multi‐D rotated constellation whose labels differ in only 1 bit. The extrinsic information transfer charts are then used to match the outer code, the multi‐D mapping, and the precoder. It is demonstrated that the proposed system is also promising for NAF relaying in the turbo pinch‐off region, in the sense that it can operate below the achievable rate achieved by a conventional coded modulation scheme. Copyright © 2011 John Wiley & Sons, Ltd.

This paper considers the precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) over non-orthogonal amplify-and-forward (NAF) single-relay channels with an arbitrary length of cooperative frame 2N. Based on the tight union bound on the bit error probability (BEP), we first derive an asymptotic design criterion with regard to a general 2N × 2N rotation matrix. This expression allows us to develop a class of precoder that not only achieves full cooperative diversity but also optimizes the asymptotic error performance. Interestingly, the developed class of optimal precoder indicates that the source should be kept silent in the cooperative phase. In the broadcasting phase, it is shown that power is distributed equally to 2N information symbols at the source. By further examining the structure of the optimal class of 2N × 2N precoders, we then reveal that precoding over a group of at least 2 information symbols is sufficient to fully exploit diversity and coding advantages. Such precoding technique, which is referred to as symbol grouping, therefore significantly reduces the system complexity without degrading the error performance.

This paper investigates the error performance and precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) system over a non-orthogonal amplify-and-forward (NAF)half-duplex single-relay channel. A tight union bound on the bit error probability (BEP) is first derived for an arbitrary block length of 2N using a 2N × 2N precoder. This bound provides an useful tool to predict the error performance. Attention is then devoted to the system using 2×2 precoder, where a closed-form expression of the bound is obtained. Based on this expression, an optimal 2×2 precoder applicable to any modulation scheme is developed. Different from the optimal precoders designed for uncoded NAF systems, the derived precoder indicates that the source only needs to send the superposition of signals in the first time slot and being silent in the second time slot in order to achieve the best asymptotic performance. Analytical and simulation results show that the proposed precoder not only exploits full cooperative diversity but also offers a significant coding gain over optimal precoders for uncoded NAF systems.