1. Introduction
Cooperative communications have drawn much attention due to its capability of mitigating the fading effect of wireless channels and achieving spatial diversity [1-2]. A variety of cooperative schemes have been proved and their performances have been analyzed [3-12]. In comparison with other schemes, relay selection (RS) scheme usually achieves full diversity with less synchronization or overhear/feedback requirement [13], and thus it attracts considerable attention. Depending on the strategy that relays utilize for cooperation, the relay networks are generally classified as decode-and-forward (DF) and amplify-and-forward (AF). Among these two strategies, AF is very attractive in terms of having less computational burden at the relays.
Many studies of the relay selection schemes are based on frequency flat fading channel models [14-29]. However, in high data-rate wireless applications, the transmission bandwidth is larger than the coherence bandwidth of the channel which results in a frequency-selective channel [30]. For such high rate communication in cooperative relay networks, existing techniques for frequency flat fading channels need to be adapted, or new techniques need to be designed for frequency-selective channels. A widely used approach to overcome the degrading effects of frequency-selectivity is orthogonal frequency division multiplexing (OFDM) which has been recently applied to cooperative communications [31-34]. Nevertheless, OFDM is not a good option if low peak-to-average power ratios (PAPRs) are desired [35-37]. Single-carrier (SC) transmission can overcome the drawback of OFDM [38-39]. In the literature, research efforts on RS for SC transmission under frequency selective channel are limited, a brief review of which is provided in the following. A single relay selection (SRS) scheme for SC transmission is proposed in [40-43]. SRS is attractive because of its simplicity, but it may fail to meet the QoS performance required by users due to its limited diversity gain. To increase the diversity gain, multiple relay selection (MRS) should be considered. In [44], several MRS schemes are developed. All the schemes [40-44] are investigated under the assumption that full and perfect channel state information (CSI) of all the relayed paths at the receiver is known. In addition, although the MRS scheme in [44] achieves considerable performance gains, the BER performance of system may not be improved much as the number of cooperating relays exceeds a certain value, and more relays than necessary may be selected. The increase of relay nodes would degrade the spectral efficiency and energy saving comparing to single relay selection schemes. To achieve better tradeoff between BER performance and spectral efficiency and to save more energy, the MRS scheme should be able to adaptively adjust the number of nodes according to the current channel state.
In this paper, we propose an adjustable MRS scheme under frequency selective channel that uses orthogonal relay channels, distributed AF relays, and multi-branch joint detector (MJD) at the destination, and without the assumption that full and perfect CSI at the receiver is known. The key concept of adjustable MRS is the adjustment of cooperating relay node number. The proposed scheme is based on the min-max design criterion, which means to minimize the maximum number of selected relays participating in cooperation. According to the criterion, the MRS scheme can adaptively adjust the number of selected relays by comparing the steady-state mean square joint error (MSJE) for different number of relay, and obtain the optimum number No. Optimum means the minimum number of selected relay, No, achieving the minimum level of steady-state MSJE. Thus, better tradeoff between BER performance and spectral efficiency and to save more energy can be achieved.
2. System Model
We consider a multiple-relay assisted cooperative wireless communication system with a single source (S), N half-duplex relay terminals (Ri), i = 1, 2, ..., N, and a single destination (D), as depicted in Fig. 1. The source, destination, and all relays are equipped with single transmit and receive antenna. We assume the AF relaying and adopt the user cooperation proposed in [1]. Specifically, in the broadcasting phase, the source node transmits to the relay nodes. In the relaying phase, the relay nodes take turn to forward a scaled noisy version of the received signal to the destination node in different time slots. For simplicity, we assume that there is no direct link between the source and the destination, and all relays have the same average power constraint.
Fig. 1.A cooperative network with one source, multiple relays, and one destination
The channel impulse responses (CIRs) for S → Ri and Ri → D links for ith relay terminal are given by hSRi = [hSRi[0],⋯, hSRi[LSRi]] and hRiD = [hRiD[0],⋯, hRiD[LRiD]], respectively, where LSRi and LRiD denote the corresponding channel memory lengths. All channel links are assumed to experience frequency selective fading. The discrete-time channel impulse response are modeled as zero-mean Gaussian random variables with an exponential power delay profile [45-47]
where Lx ∈ {LSRi, LRiD} and σ characterizes the delay spread of channel, δ(•) is the Dirac delta function, PR is the average power of the multipath components.
During the broadcasting phase I, the received signal at the ith relay terminal is given by
where ∗ denotes convolution, s(k) is the signal transmitted by the source node, nRi(k) is additive white Gaussian noise (AWGN) with zero-mean, and its variance is N0,i. s(k) can be obtained by sampling the s(t) with analogue-to-digital converter (A/D), s(t) is given by
where d(j)’s are equiprobable uncorrelated binary phase-shift keying (BPSK) data and Tf is the symbol duration. w(t) denotes Gaussian pulse waveform, which is a real function, time-limited to the interval [0,Tf], and normalized, i.e.,
During the relaying phase, at the destination terminal, the received signal from ith relay Ri is given by
where NDi(k) is the corresponding AWGN with zero mean and variance N0,Di. βi is the gain of relay Ri, to remain within its power constraint, we follow [1] and set the gain of the relay Ri as
where P is the average energy per symbol used at each terminal, hSRi[k] is the kth channel coefficient between the source and the selected relay Ri, and N0,i is the variance of the zero mean additive white Gaussian noise corresponding to the received signal for ith relay Ri at the destination.
3. Proposed Adjustable Multiple Relay Selection Scheme
In this section, we propose an adjustable MRS scheme under frequency selective channel. We assume that the relay nodes use the same transmission bandwidth but employ time division so that the relays transmit on a channel orthogonal to the destination. Optimal MRS can be achieved through an exhaustive search over all the relays. However, its complexity is exponential in the network size. It is obvious that this search method is impracticable for the networks with a large number of relay nodes. To tackle with the problem, we follow [16] and adopt the relay ordering. The MRS can be solved with linear complexity and perform close to the optimal scheme. With such an ordering, we only need to find the number of cooperating relays. In [16], it is assumed that each relay knows its own channels and the receiver knows all channel values (Under frequency flat fading channel, all channels are generated as Gaussian random variables with zero-mean and unit-variance). In this paper, we do not assume that relays or receiver knows the channel values. The receiver adopts the multi-branch joint detector (MJD) proposed in our previous works [48]. We use the steady-state mean square error (MSE) of single-branch detector (SBD) as the measurement to order the relays. Then the adjustable MRS selects sequentially the cooperating relays according to the relay ordering and obtains the optimal number of cooperating relays by comparing steady-state mean square joint error (MSJE) for different number of cooperating relay.
3.1 Preparation
The MJD adopts fractionally spaced linear filter structure, as showed in Fig. 2, which is capable of performing the functions of the matched filter, T-spaced equalization of the conventional linear receiver [49], and the combining technique for multiple received signals via diverse paths. The detector will directly process the sampled signals after the analogue-to-digital converter (A/D). For the received signal rDi(t), the input signal of detector is rDi(k). In the detector, the signal vector for rDi(k) is given by
where Lj is the observation window length (OWL). The corresponding tap coefficient vector is as follows
where cj(n) is initialized with cj = [0,⋯,0], j ∈ {D1, D2⋯,DN}.
Fig. 2.Structure of MJD
For convenience, we introduce two column vectors
Tap coefficients of the MJD are adjusted using proposed adaptive filtering algorithms in [48].
where both µ and δ are positive constant, eN(n) is the joint error. Unlike the case in single-branch detector (only one relay participates in cooperation), in MJD, the input signals of all branches should be jointly processed.
3.2 Relay Ordering Based on Steady-State MSE
When N is 1, there is only one relay participating in the cooperation. The MJD will reduce to a single-branch detector (SBD). We will use the SBD to process the received each signal from different relays. For the received signal rDi(k), the output SNR of the SBD can be given by [50]
where Jmse,Di denotes the steady-state MSE, which can be obtained by averaging the mean square error signal after the convergence of the algorithm (10), as follows
where eDi(n) = d(n) - yDi(n), Nmse denotes the number used to calculate the Jmse,Di. Here, the Jmse,Di is obtained with Nmse training sequences after the algorithm in (10) has achieved the steady-state. The ergodicity of process is assumed, thus the ensemble averages can be transformed into time averages. It is clear from (13) that the smaller the Jmse,Di is, the larger the output SNR is. Therefore, the steady-state MSE can be used as the measurement to order the relays.
Similarly, we can use the method to calculate the steady-state MSE for all the received signals rDi(k) with equation (14). Based on the calculation results of steady-state MSE for each received signal rDi(k), the relay nodes can be ordered according to the size of steady-state MSE. If (1, 2,⋯, K) is a relay ordering for (R1, R2,⋯,RK), then the ordering of steady-state MSE should satisfy Jmse,D1 < Jmse,D2 < ⋯ < Jmse,DK.
3.3 Adjustable Multiple Relay Selection Scheme
The purpose of adjustable MRS is to achieve better performance, thus, the steady-state mean square joint error (MSJE) of MJD is used as a criterion to adjust the number of relays participating in the cooperation. Based on the relay ordering in 3.2, to solve the optimal MRS problem, we only need to find the smallest value among Jmje{1}, Jmje{1,2}, … , Jmje{1,...,K} · Jmje{1,...,K} represents the steady-state MSJE to which the algorithm (10) has converged, after the processing for received multiple signals from relays (R1, R2,⋯,RK) with the algorithm.
Research shows that as the number of cooperating relays exceeds a certain value, the performance gains would become smaller [51]. Therefore, there is an optimal value for the number of cooperating relays, which can achieve better tradeoff between BER performance and spectral efficiency and to save more energy. In this section, the optimal value of the number of cooperating relays is defined as the smallest integer No that satisfies:
where N denotes the number of relay nodes, N and No are positive integers, and ε is a predetermined value according to the system requirements. The meaning of (15) is to find the minimal No satisfying the constraint Jmje{1,...,N}−Jmje{1,...,N−1} ≤ ε. When the number of relay nodes participating in cooperation is larger than No, after the algorithm (10) has converged, any two steady-state MSJE corresponding to two successive number of cooperating relays can be regarded as same.
It is immediately clear from (15) that a cost function for searching No may be obtained as
which means that the minimum N that satisfies Jmje{1,...,N}−Jmje{1,...,N−1} ≤ ε. Idealy, we may have Jmje{1,...,N}−Jmje{1,...,N−1} < 0, but due to the effect of noise, the Jmje{1,...,N} is not always less than Jmje{1,...,N−1}. In practice, for parameter ε, it may not be chosen absolutely properly, because when N is large enough, all Jmje{1,...,N} are with a narrow range. In addition, although we can use (16) to obtain the optimum number No in the mean, the proposed algorithm may suffer from slow convergence. This is because, on average, the algorithm adjusts the number of cooperating relays by comparing Jmje{1,...,N} and Jmje{1,...,N−1}, which are the steady-state MSJE of algorithm (10). Therefore, we introduce MSJE as a criterion to adjust the number of cooperating relays. The proposed algorithm is described as follows.
The different outputs of MJD with different number of cooperating relays can be used to compute a corresponding joint error signal with (11). The distinct error signals can be squared and averaged to obtain an output mean square joint error measure for adjustable MRS at each number
where NTS denotes the number of trainning sequences used to compute MSJEN(n), eN(n) is the joint error, and can be calculated with (11).
At each number of cooperating relays, the MSJEN(n) is obtained with NTS training sequences. Ideally, each MSJEN(n) would be smaller than the previous one, MSJEN−1(n). In practice, this may not hold owing to errors introduced by the adaptive filtering algorithm in the form of excess MSJE (EMSJE). The aim of the number update algorithm is to detect the number of relays at which the MSJE level becomes insignificantly smaller or even larger than the previous number.
The process of adjusting the number of cooperating relays is as follows
If MSJEN(n) < MSJEN−1(n) ⇒ add a relay
where N(k) denotes the current number of cooperating relay at discrete time k.
If αMSJEN(n) > αMSJEN−1(n) ⇒ remove a relay
where 0 < α < 1. The function of α is to determine the amount of worsening necessary to force the proposed algorithm to decrease the number of cooperating relays. The larger α is, the more frequently the algorithm will decrease the number of cooperating relays. If the value of α is too large, due to the influence of noise, the number of relays will be adjusted frequently, which makes it impossible to converge to the optimal relay number (only converge to a number which is less than the optimal number). Especially when the number of relays reaches a certain value, with the increase of the number of relays, the difference of MSJE between successive two numbers of relays becomes smaller and smaller, the increase of a relay is not enough to make the MSJE significantly reduced. The α works like a threshold, which makes the number of relays would not be easily reduced. Only when the condition αMSJEN(n) > MSJEN−1(n) is met, the number of relays would be reduced. If α is too small, when reaching a certain number of relays, the condition αMSJEN(n) > MSJEN−1(n) is difficult to be met. Then, the number of relays would only have the increasing trend, which may lead to the algorithm converge to a larger number of relays (larger than the optimal number).
Else ⇒ the number of cooperating relay remains stable.
With the current adjusted number of cooperating relay, the tap coefficient vectors of all the branches are then jointly updated by the adaptive algorithm in (10).
The proposed MRS algorithm continually computes and compares the successive two MSJE for different number of cooperating relays at each iteration, then evaluates the impact the last added relay has on the MSJE level. If MSJEN(n) is much smaller than MSJEN−1(n), then most likely another relay node will improve the steady-state MSJE, therefore an extra relay node is added. If αMSJEN(n) is larger than MSJEN−1(n), then the last added relay is not contributing significantly to steady-state MSJE of the MJD and can be removed. Otherwise, the number of relay nodes remains stable.
4. Simulation Results
In this section, we give simulation results to justify our analysis and to evaluate the performance of the proposed adjustable MRS scheme. Binary phase-shift keying (BPSK) is used as the modulation scheme. All nodes are assumed to have the same power. In the simulations, we consider a cooperative network with one transmit-and-receive pair and 20 relays. All channel links are assumed to be quasi-static frequency selective channels. The channel impulse response coefficients are modeled with (1), the parameters are set as PR = 1, LSRi = LRiD = 10Tf/Ts,. σSRi = σRiD = 10Tf. and Ts = Tf / 5, Ts is sampling period of A/D. The OWL Lj of all the branch is set as 10Tf/Ts in (6).
4.1 Validation of Method of Relay Ordering
We first verify the method of relay ordering. The relay ordering is based on the comparison of steady-state MSE of SBD. Thus, we only need to verify the consistency of relay ordering based on SNR and steady-state MSE. We consider a communication system with one relay. For each calculated steady-state MSE Jmse,Di with (14), a data packet is transmitted, which has 500 symbols, these symbols are used as training sequences. All calculated Jmse,Di were obtained by averaging the MSE after the convergence of algorithm with (14).
In Fig. 3, we compare the estimated SNR γDi with (13) and actual SNR. It is observed that the calculation with (13) for SNR is not very accurate, which deviate from the actual SNR. However, it is noted that the estimated SNR remain the same trend as the actual SNR, that is to say, the size of estimated SNR keep the same ordering with that of actual SNR. According to the simulation results, it is proved that the method of relay ordering is feasible, we can use the steady-state MSE as the measurement to order the relay node.
Fig. 3.Consistency validation of relay ordering
4.2 Validation of Adjustable Multiple Relay Selection Scheme
Next, we would evaluate the adjustment ability of the proposed adjustable MRS scheme for the number of cooperating relays. We assume that the relay ordering (R1, R2,⋯,R20) has been obtained and SNRDi = SNRD1− 0.4(i−1), i = 2,⋯,20. To validate that the proposed adjustable MRS scheme can adaptively adjust the number of cooperating relays, at first, we would obtain the optimal number No by simulation.
Fig. 4 shows the BER performance when using different number of cooperating relays, the horizontal axis indicates the SNR SNRD1. In the simulation, a frequency selective channel is generated with (1). For the same frequency selective channel, a data packet of 10000 symbols is transmitted, 500 of these symbols are used as training sequences. The same received signal corresponding to the data packet is processed 200 times by the MJD. The point on BER curve is obtained by averaging the BER on each time. It is seen that the BER performance is close when the number of cooperating relays is set as 12-20. In order to achieve better tradeoff between BER performance and spectral efficiency and to save more energy, the optimal number No of cooperating relays is defined as the minimum number of relays which make the cooperative communication system approximate the optimal BER performance. It is observed from Fig. 4 that the optimum number No is about 12.
Fig. 4.BER performance for different number of relay nodes at SNRD1 = 4dB
Then, we would evaluate the adjustment ability for the number of cooperating relays with the proposed adjustable MRS scheme. In the simulation, for the same frequency selective channel, a training sequence packet of 1000 symbols is transmitted. The same received signal corresponding to the packet is processed 20 times by the proposed adjustable MRS. The convergence curve is obtained by averaging the curve on each times. In the adjustable MRS scheme, the α is set as α = 0.979 . A full study of the effect of the parameter on performance is not reported here. Fig. 5 shows the evolution curve of the number of cooperating relays, the horizontal axis indicates the SNR SNRD1. It is observed that the proposed adjustable MRS scheme can adaptively adjust the number of cooperating relays to about 12. The obtained number of cooperating relays approximates the optimal (minimum) number No in Fig. 4.
Fig. 5.Evolution curve of number of cooperating relays at SNRD1 = 4dB
In addition, the comparison of evolution curve with different values of α is also given in the Fig. 5. It is observed that when α is set as 0.979, the proposed algorithm can converge to optimal number of relays as showed in Fig. 4. However, when α is set as 0.999 or 0.6, the number of relays can not be adjusted to optimal value No. This is because the α is an important parameter on adjustment of number of relays, by setting the size of α, the amount of adjusting to decrease the number of relays can be controlled.
In view of the aforementioned results, it is testified that the proposed adjustable MRS can adjust adaptively the number of cooperating relays.
4.3 Comparison of BER Performance
To compare the BER performance of different RS schemes, a Monte-Carlo simulation is set up based on the channel model (1). For each channel realization, a data packet of 10000 symbols is transmitted, 500 and 1000 of these symbols are used as training sequences for conventional RS and the proposed scheme, respectively. For comparison purposes, 500 of 10000 symbols used as the training sequences are also simulated for the proposed scheme.
In Fig. 6, we check the BER performance of optimal SRS [40-43], optimal MRS [44] and the proposed adjustable MRS scheme, the horizontal axis indicates the SNR SNRD1. Both optimal SRS and MRS schemes assume that full and perfect CSI of all the relayed paths at the receiver is known, while the proposed adjustable MRS scheme doesn’t make the assumption. It is observed that the proposed adjustable MRS and MRS in [44] have evident BER performance improvement than optimal SRS scheme. This is because diversity gain of SRS scheme is limited. The more the number of cooperating relays is, the more diversity gain can be achieved. To meet the QoS performance required by users, multiple relays should be considered to participate in the cooperation. It is also noted that the proposed adjustable MRS with 1000 training sequences and MRS scheme in [44] achieve similar BER performance. However, compared with the MRS in [44], the proposed MRS scheme doesn’t need to know the CSI at the destination node, and use less cooperating relays. For example, it is observed from Fig. 4 that the optimal number of cooperating relays is 16 according to the MRS in [44], whereas the optimal (minimum) number is about 12 by the proposed adjustable MRS scheme. The increase of cooperating relays would degrade the spectral efficiency and energy saving. Thus, the proposed adjustable MRS scheme can achieve better tradeoff between BER performance and spectral efficiency and to save more energy. Moreover, the proposed scheme is able to adaptively adjust the number of cooperating relays according to the current CSI. Hence, the proposed scheme is more suitable for practical cooperative communication systems. In addition, it is also seen that the BER performance with 1000 training sequences is better than that with 500 training sequences. This is mainly because the proposed algorithm needs enough length of training sequences to achieve convergence, as showed in Fig. 5.
Fig. 6.BER performance of optimal SRS, MRS based on minimizing BER and the proposed adjustable MRS scheme
5. Conclusion
In this paper, an adjustable MRS scheme under frequency selective channel is proposed. The proposed scheme takes into account better tradeoffs among spectral efficiency, BER performance, and energy saving in cooperative relay networks. The proposed scheme can adaptively adjust the number of cooperating relays according to the comparison of steady-state MSJE for different number of cooperating relays, and obtain the optimal number of cooperating relays (means the minimum number of relay nodes, No, achieving the minimum level of steady-state MSJE).
To evaluate the performance of the proposed scheme, simulations were run using frequency selective channel model. The simulation results for the proposed scheme were compared with those of the optimal SRS and optimal MRS scheme based on minimizing BER. The BER performance of proposed MRS scheme outperforms the optimal SRS schemes, and is similar to MRS based on minimizing BER. Simulation results also show the proposed scheme can adaptively adjust the number of relay nodes according to the steady-state MSJE. Therefore, the proposed scheme provides flexibility to adapt in fading, and enables better tradeoffs between BER performance and spectral efficiency and to save more energy in cooperative relay networks.
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