Optimal Power Allocation with Hybrid Relaying Based on the Channel Condition

This paper considers a hybrid relay network consisting of the source, the amplify-and-forward (AF) relay, the decode-and-forward (DF) relay, and the destination. In hybrid three-hop relay systems, the transmitted signal from source can be received at the destination after processing the signals through two relays. If the first relay amplifies and forwards the received signal, and the second relay decodes and forwards the received signal, the system model is considered to be an AF-DF relay system. The reverse case is considered for the DF-AF relay system. The AF-DF and DF-AF relay systems have different error rates and achievable throughput with respect to the channel conditions between two nodes. We propose optimal power allocation schemes for two different relays in order to maximize the achievable rate under a sum relay power constraint for given channel gains and transmit power from the source. By solving the optimization problem to maximize the achievable rate for each relay network, the transmit power values in closed form are derived. When the channel gains are the same, the optimal power allocation scheme for the AF-DF relay network proves that greater power should be allocated at the first relay to maximize the achievable rate. In the case of the DF-AF relay network, we derive an optimal power allocation scheme for the four possible cases. Under the same signal-to-noise ratio (SNR) condition, at the first hop we show that the achievable rate of the AF-DF relay network is greater than that of the DF-AF relay network when the channel gain between two relays is greater than that between the second relay and destination. Simulation results show that the proposed power allocation schemes provide a higher achievable rate than the equal power allocation scheme and the grid search schemes.

There are several relaying strategies: amplify-and-forward (AF) and decode-and-forward (DF).In the AF relaying scheme, a relay simply amplifies the received signals from the source and retransmits them to the destination without performing any signal regeneration, which may lead to the propagation of noise and interference.For the DF relaying scheme, a relay decodes the received signals and retransmits the recovered signals to the destination.Although the DF relaying scheme achieves extra coding gain, the error propagation is caused by decoding errors at the relay.
To get advantages of both the AF and the DF, hybrid relaying schemes were studied in [8,9].In [8], the authors analyzed the bit error probability for both the AF relaying and the DF relaying with respect to signal-to-noise ratio (SNR) and proposed a hybrid relaying scheme which changes the relaying scheme based on analyzed bit error probability.Like [8], [9] calculated the symbol error probability for both homogeneous relaying and hybrid relaying networks and simulated the symbol error rate (SER) according to the location of relay.The hybrid relaying schemes in [8,9] have higher bit error rate (BER) and SER performances than the simple homogeneous relaying schemes.These hybrid relaying networks obtain more gains than the homogeneous relaying schemes.
Recently, the power allocation problem in a cooperative system has also attracted lots of research attention.In [10] and [11], power allocation schemes have been proposed to maximize the capacity under a sum transmit power constraint for AF relay and DF relay, respectively.The optimal power allocation schemes for hybrid network have not been analyzed for a two-hop AF and DF cooperative relay system employing outage probability as the optimization criterion in [12].In [13], optimal power allocation based on average end-to-end symbol error probability (SEP) as the optimization criterion is performed for a two-hop DF cooperative relay system.In [14] and [15], the instantaneous received signal-to-noise ratio (SNR) and its approximate expression are exploited to obtain optimal power allocation for an AF multi-hop relaying system.The optimal power allocation based on outage probability in a DF multi-hop system is discussed in [16].In [17], the power allocation scheme that minimizes a bit error rate (BER) at the destination for uncoded AF with Rayleigh fading channel under a sum transmit power consumption was proposed.The optimal power allocation strategy is proposed in [18] to maximize achievable secrecy rates under an overall transmit power constraint assuming that a single relay is located at each individual hop.
In hybrid relaying networks, the error performance is mainly affected by received SNR which is changed by transmit power, channel power and noise power.This paper proposes optimal power allocation schemes for hybrid relay networks within limited total power.The proposed scheme has higher achievable rate than the equal power allocation scheme and can approach the maximum achievable rate with lower power than the equal power allocation scheme.Also, after applying our power allocation schemes, the achievable rates for two hybrid relay networks, i.e.AF-DF and DF-AF, are different according to channel state from a relay to another relay and from a relay to a destination.So, a proper relaying scheme can be selected adaptively based on estimated channel state.
In inter-cell communication system which is more common than intra-cell communication system, three-hop relaying transmission is sufficient to achieve optimal throughput and to find the optimal relay node [19].The simulation results in [19] show that three-hop relaying transmission has better throughput performance than two-hop and four-hop relaying transmission.For transmission with two-hop relaying network, the transmission range is short and it causes a decrease of achievable throughput in the inter-cell communication.Due to the short range, two-hop system does not select better relaying node in terms of throughput performance than three-hop system.Also, for transmission with four-hop or more relaying hops, the throughput is severely decreased because the routing with four-hop or more relaying hops increases forwarding delay and it causes many overheads and signal processing delay for overall systems.Therefore, the number of hops in the relay network is confined to three.The proposed schemes enable the achievable rate to maximize by adaptively allocating the power to the first and the second relay nodes.For adaptive power allocation of each relay node, we derive the transmit power values in closed-form for each relay network according to channel condition.Analytical solutions are derived, and the proposed power allocation schemes are compared with the equal power allocation scheme.In addition, we compare the achievable rates of the proposed power allocation schemes when SNR of the first hop is the same.The simulation results show that the proposed optimal power allocation scheme requires lower transmit power to achieve a specific achievable rate than the equal power allocation scheme.Therefore, this paper contributes to reduction for the lower limit of transmit power consumption to satisfy the achievable rate in a cooperative communication.Also, in the next generation system, green communication has attracted more and more attention.The reduction for the lower limit of transmit power consumption by the proposed scheme contributes to the implementation of green communication in next generation system.
The remainder of this paper is organized as follows.In Section II, the system model of three-hop relay networks is presented.In Section III, the optimal power allocation schemes are proposed for three-hop AF-DF and DF-AF relay networks, respectively.Section IV shows the simulation results, and the conclusion is drawn in Section V.

System model
Fig. 1 shows the system model consisting of a source s, the first relay r 1 , the second relay r 2 , and a destination d.The nodes operate in the half-duplex mode, i.e., they are not able to receive and transmit at the same time and same frequency.We assume that the channel gains are acquired from channel state information (CSI) of a system such as reference signal (RS) of 3GPP LTE and that relays know the used power for transmission of a source.In case of this three hop relay network, three RS should be allocated and destination feedbacks the CSI through reverse links of the relay network.For channel estimation, various schemes can be concerned [20,21].However, the perfect CSI is assumed to compare the maximum performance with other power allocation schemes.In addition, it is assumed that the total power for relaying is fixed.Since the relays also need power to transmit their own signal, this condition is needed.In case of equal power allocation scheme, the fixed total power is allocated to all relays equally.However, since channel condition is not concerned, the equal allocation scheme allocates the burden inefficiently.Therefore, optimization should be applied for power efficiency.This fixed total power can be normalized for comparison with other power allocation schemes [22,23].
An aim of hybrid relay network is that it achieves both high throughput performance and simple implementation.The different property between AF-DF and DF-AF relay network is that these two schemes have different performances with respect to achievable throughput and implementation according to channel condition.Generally, because the AF scheme severely amplifies the noise power, the error performance for the AF scheme is lower than the error performance for the DF scheme.The error performance for the AF scheme is almost the same as error performance for the DF scheme when the channel condition is good.However, because the DF scheme always know channel state information (CSI) to decode the received signals, one of main disadvantage for the DF scheme is that the real-time implementation is harder than the AF scheme in multi-hop transmission system.The AF scheme which simply amplifies the received signal does not require CSI.The AF-DF scheme is adequate when a communication link between the source and the first relay is good with simple implementation by amplifying the received signal and has high throughput performance when a communication link between the first relay and the second relay is not good.Also, the DF-AF scheme has high throughput performance when a communication link between the source and the first relay is not good and is adequate when a communication link between the first relay and the second relay is good with simple implementation.In this paper, both the system models of the AF-DF and DF-AF are represented for general analysis of hybrid relay network in various channel conditions.

AF and DF Relay Network
In this subsection, we assume that the first relay r 1 considers AF protocol and the second relay r 2 considers DF protocol.
In the first time slot, the source transmits the signal x ad,s with transmit power P s to the first relay.The received signal y ad,1 at the first relay can be expressed as where h 1 is the channel coefficient from source to the first relay and n 1 is the zero-mean additive white Gaussian noise (AWGN) with unit variance at the first relay.
In the second time slot, the first relay transmits the signal x ad,1 with transmit power P ad,1 to the second relay.The transmitted signal x ad,1 at the first relay is where β ad is the amplification factor for AF relay and it is given by The received signal y ad,2 at the second relay can be expressed as where h 2 is the channel coefficient from the first relay to the second relay and n 2 is the zero-mean AWGN with unit variance at the second relay.
The second relay decodes the received signal.In the third time slot, the second relay transmits the signal x ad,2 with transmit power P ad,2 to the destination.The received signal y ad,d at destination can be expressed as where h 3 is the channel coefficient from the second relay to destination and n d is the zero-mean AWGN 120 with unit variance at the destination.The achievable rate is given by where γ ad is the SNR for AF-DF relay network and it is given by

DF and AF Relay Network
In this subsection, we assume that the first relay r 1 considers DF protocol and the second relay r 2 considers AF protocol.In the first time slot, the source transmits the signal x da,s with transmit power P s to the first relay.The received signal y da,1 at first relay can be expressed as where h 1 is the channel coefficient from source to the first relay and n 1 is the zero-mean additive white Gaussian noise (AWGN) with unit variance at the first relay.The first relay decodes the received signal.In the second time slot, the first relay transmits the signal x da,1 with transmit power P da,1 to the second relay.The received signal y da,2 at the second relay can be expressed as where h 2 is the channel coefficient from the first relay to the second relay and n 2 is the zero-mean 125 AWGN with unit variance at the second relay.
In the third time slot, the second relay transmits the signal x da,2 with transmit power P da,2 to the destination.The transmitted signal x da,2 at the second relay is where β da is the amplification factor for AF relay and it is given by The received signal y da,d at the destination can be expressed as where h 3 is the channel coefficient from the second relay to destination and n d is the zero-mean AWGN with unit variance at the destination.The achievable rate is given by where γ da is the SNR for DF-AF relay network and it is given by

Optimal Power allocation schemes
We propose the optimal power allocation schemes for hybrid three-hop relay networks which 130 maximize the achievable rate under a sum relay power constraint for given channel gains and transmit power from source.

AF and DF relay network
In this subsection, we propose the optimal power allocation scheme for three-hop AF and DF relay network.

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The optimization problem to maximize the achievable rate under a sum relay power constraint can be written as max We define the ratio of P ad,1 to P as α ad .By using α ad , the optimization problem is rewritten as max where γ ad,1 (α ad ) and γ ad,2 (α ad ) are given, respectively, as As α ad increases, γ ad,1 (α ad ) increases and γ ad,2 (α ad ) decreases.Therefore, by solving the equation γ ad,1 (α ad ) = γ ad,2 (α ad ), the optimal α ad is obtained as where a, b, and c are given, respectively, as When the channel gains are the same, α ad in ( 19) is represented as where and λ ad is given by From ( 21), we can know that α ad is greater than 1/2.In other words, we should allocate more power to the first relay than the second relay to maximize the achievable rate when the channel gains are the same. When the SNR for AF-DF relay network in ( 7) is rewritten as where γ ad,s2 and γ ad,2d are given, respectively, as Because γ ad,s2 has a similar form of the harmonic mean of P s |h ad | 2 and P ad,1 |h ad | 2 , the increment of γ ad,s2 is less than that of P ad,1 as P ad,1 increases.On the other hand, the increment of γ ad,2d is equal to that of P ad,2 as P ad,2 increases.Therefore, the increment of γ ad,s2 is less than that of γ ad,2d when the increments of P ad,1 and P ad,2 are the same.To maximize the minimum value between γ ad,s2 and γ ad,2d in (23), it is necessary to further increase γ ad,s2 which does not increase as much as γ ad,2d .In addition, to increase γ ad,s2 more than γ ad,2d , we should allocate more power at the first relay than the second relay.

DF and AF relay network
In this subsection, we propose the optimal power allocation scheme for three-hop DF and AF relay network.
The optimization problem to maximize the achievable rate under a sum relay power constraint can be written as max R da (P da,1 , P da,2 ), s.t.P da,1 + P da,2 = P.
We define the ratio of P da,1 to P as α da .By using α da , the optimization problem is rewritten as max where γ da (α da ) is given as Firstly, we consider the Case a(1) when By using (30), the optimization problem in ( 27) is rewritten as where f da,1 (α da ) and f da,2 (α da ) are given by , where if ω da,1 ≤ ω da,2 then 6: Taking partial derivative of (32) with respect to α da and equating it to zero, we can obtain ω da,2 which maximizes f da,1 (α da ) as where The f da,1 (α da ) increases for 0 < α da ≤ ω da,2 and decreases for ω da,2 ≤ α da < ω da,1 .Since is less than zero, f da,2 (α da ) decreases as α da increases for ω da,1 < α da < 1.As a result, the optimal α da is 155 ω da,2 when ω da,1 ≥ ω da,2 and ω da,1 when ω da,1 ≤ ω da,2 .Secondly, we consider the Case a(2) when By using (30) and where f da,3 (α da ) and f da,4 (α da ) are given by where f da,5 (α da ) is given by By solving the equation ∂ f da,5 (α da ) ∂α da = 0, the optimal α da is obtained as where From (34), we can know that α da is the same as ω da,2 .Secondly, we consider the Case b(2) when By using where f da,6 (α da ) is given by By solving the equation = 0, the optimal α da is obtained as 1/2.From ( 29) and (34), we can know that ω da,1 does not depend on |h 3 | 2 and ω da,2 does not depend on P s and |h 1 | 2 .
The Algorithm 1 explains the procedure to find α da for DF and AF relay network.

Simulation Results
This section presents the achievable rates of the proposed and equal power allocation schemes for hybrid three-hop relay networks.For the equal power allocation schemes, α ad and α da are fixed to 1/2.Fig. 3(a) and Fig. 3(b) show the achievable rates and α ad for AF-DF relay network when P s = 10 dB.From Fig. 3(a), it is observed that the achievable rate of the optimal power allocation scheme is greater than that of the equal power allocation scheme regardless of channel gains.1.Among the results in Fig. 3(a), the achievable data rate expressed by dashed line of optimal power allocation and equal power allocation converges after the power constraint of 10dB.This can be understood from (17) and (18).After the power constraint of 10dB, γ ad of the two allocation schemes is determined  by (17) and is hardly subject to α ad .From Fig. 3(b), we can know that α ad is greater than 1/2 when As mentioned, the SNR at destination γ ad is determined by minumum value between SNR at the second relay γ ad,s2 and SNR of third hop γ ad,2d .In addition, the increment of γ ad,s2 is less than that of γ ad,2d when increments of the transmit power from each relay are the same.To maximize γ ad , we need to further increase γ ad,s2 which does not increase as much as γ ad,2d .Therefore, we should allocate more power at the first relay than the second relay to increase γ ad,s2 more than γ ad,2d .4(a), it is observed that the optimal power allocation schemes for Case a(1) and b(1) provide a higher achievable rate than the equal power allocation scheme.The α da decreases for Case a(1) and increases for Case b(1) as P increases.As mentioned in Case a(2), α da is 1/2 for (a) Achievable rates for the power allocation schemes.for P ≥ 13.0103 dB.The achievable rate of the optimal power allocation scheme for Case a(2) is the same as that of the equal power allocation scheme when P ≤ 13.0103 dB.Then, the optimal power allocation scheme for Case a(2) provides a higher achievable rate than the equal power allocation scheme when P ≥ 13.0103 dB.
Fig. 5(a) and Fig. 5(b) show the achievable rates of the proposed power allocation schemes for hybrid three-hop relay networks when SNR of the first hop γ s1 is 0 dB, −3 dB and −6 dB.When |h 3 | 2 has a greater value than |h 2 | 2 , it is observed that the achievable rate for DF-AF relay network is greater than that of AF-DF relay networks regardless of γ s1 .On the other hand, the achievable rate for AF-DF relay network is greater than that of DF-AF relay networks when |h 2 | 2 has a greater value than |h 3 | 2 .As mentioned, the SNR at destination γ ad for AF-DF relay network is determined as the minimum value between the SNR at the second relay γ ad,s2 and the SNR of the third relay γ ad,2d .Because γ ad,s2 has a form similar to harmonic mean between γ s1 and SNR of the second hop, the increment of γ ad,s2 is less than that of γ ad,2d when the increments in SNR of each hop are the same.Therefore, to maximize γ ad , we need to further increase γ ad,s2 which does not increase as much as γ ad,2d .For a given γ s1 , γ ad,s2 can be increased by increasing |h 2 | 2 .Unlike the AF-DF relay network, the SNR at destination γ da   for DF-AF relay network has a similar form of the harmonic mean between γ s1 and the SNR of the third hop when γ s1 is less than the SNR of the second hop.For a given γ s1 , γ da can be increased by increasing |h 3 | 2 .Therefore, γ ad has a greater value than γ da when |h 2 | 2 is sufficiently larger than |h 3 | 2 .On the other hand, γ da has a greater value than γ ad when |h 3 | 2 is sufficiently larger than |h 2 | 2 .
Fig. 6(a) and Fig. 6(b) show the achievable rates of the equal power allocation schemes for hybrid three-hop relay networks when SNR of the first hop γ s1 is 0 dB, −3 dB and −6 dB.As shown in Fig. 5(a) and Fig. 5(b), the achievable rate of DF-AF relay network is greater than that of AF-DF relay networks when |h 3 | 2 has a greater value than |h 2 | 2 and vice versa.It is observed that the achievable rates of the equal power allocation schemes are lower than that of the proposed power allocation schemes.
From Fig. 3 to Fig. 6, it is noticed that proposed optimal allocation uses less P than equal power allocation for keeping same achievable rate.Also, the results consider achievable rate per unit bandwidth.Therefore, the advantage increases linearly according to the bandwidth of systems.Since bandwidth of recent communication systems has been increased continuously to accommodate   future data traffic, proposed optimal allocation scheme can contribute to power efficiency of the recent wideband systems.
The simulation results in Fig. 5 and 6 show that the appropriate hybrid relaying according to the channel condition has significant performance improvement compared with power allocation scheme.For further comparisons, the power allocation schemes of [22] and [23] are referenced.Since the system model for each paper is different, we refer to the simulation results of each paper.In [22], the proposed power allocation is used to improve diversity gain by cooperative transmission in the hybrid decode-amplify-forward cooperative communication system.Simulation results for the achievable rates according to power usage showed about 3dB performance improvement compared with equal power allocation.Also, in [23], the proposed power allocation was used to improve diversity gain by cooperative transmission with relay selection in the cooperative communication system with multiple relays.Simulation results for the achievable rates according to power usage showed the performance improvement of less than 1dB.However, from the simulation results of Fig. 5 and 6, according to channel condition, we confirm the performance improvement over 5dB at in range of 5 ∼ 10dB of used power by using appropriate hybrid relaying compared with other types of hybrid relaying.Therefore, the analysis of hybrid relaying and power allocation according to the channel condition yields meaningful results.

Conclusion
Under a sum relay power constraint, this paper proposed the optimal power allocation schemes to maximize the achievable rates for hybrid three-hop relay networks when the channel gains and the transmit power from source are given.By solving the optimization problem, we derived the transmit power value from the first relay in closed-form for AF-DF and DF-AF relay networks.When the channel gains are the same for the AF-DF relay network, we showed that more power should be allocated at the first relay than the second relay to maximize the achievable rate.In addition, we derived the optimal power allocation scheme for DF-AF relay network for the possible four cases.When the SNR of the first hop is the same, it is shown that the optimal power allocation scheme for AF-DF relay network provides a higher achievable rate than that for DF-AF relay network when the channel gain between two relays is higher than that between the second relay and destination.On the contrary, the achievable rate of DF-AF relay network is greater than that of AF-DF relay network when the channel gain between the second relay and destination is higher than that between two relays.We can choose the optimal power allocation scheme which can provide the best performace in a given environment.Both the analytical solutions and simulation results have shown that the proposed power allocation schemes outperform the equal power allocation schemes.

Figure 3 .
Figure 3. Performance for AF-DF relay network when P s = 10 dB.

Fig. 4 (
Fig. 4(a) and Fig. 4(b) show the achievable rates and α da for DF-AF relay network when P s = 10 dB.The Case a(1) an a(2) are described when |h 2 | 2 = |h 3 | 2 and |h 2 | 2 = |h 3 | 2 , respectively.Then, the Case b(1) is described when |h 2 | 2 = |h 3 | 2 .From Fig.4(a), it is observed that the optimal power allocation schemes for Case a(1) and b(1) provide a higher achievable rate than the equal power allocation scheme.The α da decreases for Case a(1) and increases for Case b(1) as P increases.As α da for the optimal power allocation scheme.

Figure 4 .
Figure 4. Performance for DF-AF relay network when P s = 10 dB.

Figure 5 .
Figure 5. Achievable rates for the proposed power allocation schemes.

Figure 6 .
Figure 6.Achievable rates for the equal power allocation schemes.