1. Introduction

Since modern communication networks must deliver an ever increasing data rate in order to satisfy the users' traffic demand, the spectral efficiency (SE, as measured in bits/s/Hz/km2) of the wireless networks should be substantially improved. Apart from it, the radio coverage should also be extended for facilitating a seamless wireless access to the mobile users. Cooperative communications system comprising multiple relays, regarded as a promising solution for combating the shadowing effect, extending the radio coverage and improving the network throughput, facilitating a better immunity against signal fading and a more system-wide power saving, etc., has attracted a wide attention [1]-[4].

However, the spectrum-utilization penalty in classical resource-allocation policies may even deteriorate the benefits from using multiple relays, if an orthogonal channel allocation (e.g., in terms of carrier frequencies, time slots or codes) among relays is required. In order to address the above-mentioned problem, opportunistic relay selection relying on limited channel state information (CSI) feedback can be implemented [5]-[9]. The existed studies showed that the method of forwarding the source's data via the optimally chosen relay is effective in balancing the achievable diversity order and the attainable SE [10]-[12].

Many of the existing studies have been focused on half-duplex (HD) relaying mode [13], [14]. Unlike the HD mode, the full-duplex (FD) relaying mode allows for concurrent transmission and reception of a communication device in a single time/frequency channel [15]-[18], thus substantially improving the system's SE. However, as a downside, the FD mode may suffer from a performance erosion imposed by the self-interference1. A substantial SE-improvement than the conventional HD mode can be attained [19], [20] as long as the self-interference in the FD devices can be effectively suppressed.

Apart from it, power allocation would also play an important role in effectively suppressing the self-interference power imposed on the FD devices [21], [22]. Several works have been performed to deal with the power allocation in FD relays (see, e.g. [23] and the references therein). Generally, power allocation techniques in single-relay systems can be developed subject to one of the following two constraints:

Again, the spatial diversity gain can be substantially improved by employing multiple FD relays [24]. However, power allocation for FD based multiple-relay-selection algorithms is still left for further study.

In this paper, opportunistic decode-and-forward (DF) based FD relay selection with power allocation in cooperative communications systems is studied. The main contributions of this paper are emphasized as follows:

The remainder of this paper is organized as follows. Section 2 introduces the system model of opportunistic FD relay selection. Both the cumulative distribution function (CDF) and probability density function (PDF) of the received signal-to-noise ratio (SNR) at the destination are also derived in this section, followed by the closed-form expressions of outage probability subject to various power allocation policies, such as equal power allocation (EPA), optimal power allocation (OPA) under IPC and SPC, in Section 3. Section 4 gives out the numerical results. Finally, Section 5 concludes this paper.

Notation: FX(⋅) and fX(⋅) represent the CDF and PDF of the random variable (RV) X , respectively. Var(X)represents the varance of the RV X.

2. System Model

In this section, we consider a cooperative network comprising a source node S, N parallel FD relays operating at DF mode, and a destination D (please see Fig. 1). Note that the FD-based relay receives and transmits data simultaneously via the same frequency band, the residual self-interference (i.e., γLI) will always be non-zero even after performing self-interference cancellation2. Without loss of generality, the direct S → D link is assumed to suffer from deep fading and will be unavailable for signal transmission. Furthermore, we assume that S and Ri perform signal transmission/forward with power of PS and PR, respectively. Additionally, each node receives and transmits data with single antenna, respectively, and suffers from additive white Gaussian noise (AWGN) with zero mean and variance .

Fig. 1.A cooperative network comprising multiple FD relays.

In the following, we consider the DF rather than AF relaying mode to perform signal forwarding. The signal is transmitted with one phase according to the essential of FD relaying mode. Furthermore, assume S transmits x(t) at time t , the received signals for Ri and D can be given by

and

respectively, where hab denotes the circularly symmetric complex Gaussian channel gain, a,b ∈ {Ri,S,D} , nRi (t) and nD(t)(t) represent the AWGN received at Ri and D , respectively, and τ denotes the signal-processing time delay for FD relays.

Therefore, the PDF of SNR for a → b link can be derived as , where denote the average SNR and instantaneous SNR of the a → b link, respectively, where a,b ∈ {S,D}∪Φ and Φ = {R1,R2,....,RN} , with hab denoting the circularly symmetric complex Gaussian channel gain.

The max−min relay selection scheme activates the relay having the best end-to-end link, while considering the impact of the residual self-interference, as given by [23, Eq. 3]

where γD denotes the received SNR of D, and γRi represents the received SNR of the i -th relay. By taking into account the impact of self-interference at the FD relays, the equivalent received SNR at Ri and D can be formulated as [25]

where γLI denotes the residual self-interference-to-noise ratio at the relay, and denotes the channel power to noise ratio at each Ri → D link. Consequently, the equivalent SNR of the activated link can be represented as γeq = min{γRk , PRγRkD} .

The above-mentioned relay selection method is assumed to be controlled by a CU device, which collects all the information regarding the instantaneous CSI and then feeds back the link-selection decision to the relays. The CU thus activates the relay that satisfies (1). Hence, the CU requires the CSI knowledge of all the ∀S → Ri and Ri → D channels as well as self-interference level. It is worth mentioning that the CU based control requires an additional power consumption and bandwidth reservation. Without loss of generality, the backhaul cost of CU is neglected in this paper. Furthermore, the self-interference power at the relays can be substantially reduced by employing some existed cancellation techniques operating at both the analog and digital domains, making the residual self-interference channels (hLI) be modelled as Rayleigh fading parameters with a reasonable validity.

In order to evaluate the link quality of the proposed systems, both the PDF and CDF of the equivalent SNR should be derived. From Appendix I (25), the CDF of γeq can be derived as

where , denotes the variance of residual self-interference-to-noise ratio, and represent the average SNR of S → Rk and Rk → D link, respectively. Furthermore, the PDF of the received SNR can be derived as

As compared to the HD relay selection schemes, the FD relay selection schemes cannot be optimized by simply increasing the transmit power of relays (i.e., for improving the received SNR of Ri → D link) owing to the performance erosion imposed by the enhanced residual self-interference power. Therefore, performing a proper power control method would play a critical role in optimizing the FD based opportunistic relay-selection systems.

3. Outage Probability Analysis

In this section, the closed-form expressions of the outage probability for both FD (with EPA and OPA) and HD modes are derived. Without loss of generality, normalized transmission power is assumed at both the source and relay nodes, whereas the total transmission power is normalized to 2 units.

3.1 Outage Probability Subject to EPA Rule

From (5), for a given pre-set threshold3 γth, the outage probability of the opportunistic relay selection scheme can be derived as

where we have assumed that all the activated nodes transmit signals at their respective maximum power level. The optimization problem is a convex-optimization problem in terms of (PS, PR). Furthermore, The optimum power for outage probability can be solved by using sub-gradient method.

3.2 Outage Probability Subject to OPA Rule

In this mode, outage probability can be formulated as

Note that (8) can be further simplified as

where , with .

Evidently, (9) is a monotonically increasing function of PS. Furthermore, the first factor of (the second factor of f(PS, PR), i.e. ) is a monotonically deceasing (increasing) function of PR. The outage probability can thus be minimized by using the OPA rule under policies of IPC and SPC, respectively.

1) Outage Probability under IPC Policy: The OPA based outage probability under IPC policy can be formulated as

Furthermore, (10) is not jointly convex subject to PS and PR. Hence, the optimal power allocation of the source and relay (OPA under IPC) can thus be derived as

leading to

When = 0, (12) reduces to , corresponding to the HD mode.

2) Outage Probability under SPC Policy: In this case, the OPA under SPC policy can be formulated as

The optimal transmit power allocation can thus be achieved by solving

with λ denoting the Lagrangian multiplier associated under the SPC policy.

After Simplifying (14), we obtain

where ζ = 2a-b+c+ab-ac, η=4a-6b+2c+4ab, and ε=12b-4ab.

By solving the root of the cubic polynomial of (15) using some mathematical software (e.g., Mathmatic or Matlab), we can obtain three root of PR as

where , , and . When ∃0 ≤ ωi ≤ 2,i ∈ {1,2,3} , the optimal transmit power of relay can be derived as

When = 0 and = , the power of Rk and S can be given by and , respectively.

4. Numerical Results

In this section, we evaluate the proposed scheme in terms of the outage probability over i.i.d. Rayleigh fading channels by usingMonte Carlo simulation. After performing self-interference cancellation at the FD relays, the residual self-interference power () will be proportional to the power of Rk → D link ().

In Fig. 2, outage probability is evaluated by considering various number of relays (i.e. N), α=2bps/Hz, and . When N = 1 is considered (i.e., without performing relay selection), the lowest spatial diversity order is obtained. Particularly, the spatial diversity order in the FD based relaying systems can be improved by employing more relays.

Fig. 2.Outage probability for the FD scheme with EPA and the OPA under IPC versus the average SNR of the S → R links for different N with

As shown in Fig. 3, in the FD mode, OPA scheme under SPC policy outperforms that under IPC policy in terms of coding gain by about 2 dB. As shown in Fig. 3, the EPA have appeared the floor effect by the residual interference powers and noise for outage probability at the high SNR. However, when the residual interference powers had been suppressed by power allocation, the OPA hasn’t appeared the floor effect for outage probability at the high SNR. Besides, the outage performance of OPA scheme is superior to that of EPA scheme.

Fig. 3.Outage probability for the FD scheme with EPA and the OPA under SPC versus the average SNR of the S → R links for different N with

Fig. 4 analyzes the outage probability with = 23 dB for different values, while keeping N = 3 unchanged. It is shown that the curve of HD under OPA policy is identical to that under EPA policy. Meanwhile, it is also shown that the spatial diversity order in the FD relay cooperative systems becomes independent of , since the slopes of curves keep unchanged for different . If < 10 dB is satisfied, the FD mode outperforms the HD mode in terms of outage probability. Otherwise, the HD mode may obtain an advantage over the FD mode. Furthermore, it is also shown that the optimal power allocation in FD schemes under IPC policies outperform the HD mode in terms of coding gain by more than 4 dB, provided that the self-interference can be successfully suppressed to the level that is below the noise power. The floor effect for outage probability is observed in the DF-mode cooperative communications systems. As shown in Fig. 4, increasing implies obtaining a lower outage probability at low-SNR regime. However, if we further increase the SNR (when approaches 25 dB in this simulation), the outage probability cannot be further decreased, because the selected relay has successfully decoded the symbol at the high-SNR link, and further increasing SNR may not contribute to the reduction of outage probability. Consequently, a floor effect appears at the outage probability.

Fig. 4.Outage probability for the FD scheme with EPA, the OPA under IPC and the HD mode versus the average SNR of the S → R links for different

5. Conclusion

The performance of the FD based relay selection scheme under DF relaying mode was evaluated by deriving the closed-form expressions of the CDF, the PDF and the outage probability of the activated link. The proposed scheme with OPA scheme (under IPC and SPC policies) was also validated by using simulations. Particularly, the theoretical analysis was shown to match the corresponding numerical results well. Furthermore, it was also shown in the numerical results that the other parameters, including the number of relays, the residual self-interference, and the SNR of relaying links, etc., all substantially impact the performance of the multi-relay systems. Finally, simulation results showed that the FD based mode could outperform the HD based mode in terms of coding gain by more than 4 dB, provided that the self-interference at the FD relays can be sufficiently suppressed.