# Optimal Distributed Detection for Complex Field Network Coding Assisted Wireless Sensor Networks with Nonorthogonal Communications

Authors: Yulun Cheng; Longxiang Yang; Hongbo Zhu
Journal: Mobile Information Systems
 (2016)

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## Abstract

Complex field network coding (CFNC) enables signal transmission and information fusion to be operated over nonorthogonal channels, which essentially improves both the bandwidth efficiency and transmission reliability of the wireless sensor networks (WSNs). This paper investigates the power allocation for the CFNC assisted WSNs. By improving the accuracy of the approximation forQ function, the closed-form expression of the pairwise error probability bound for the considered network is derived, so as to improve the optimization efficiency of the power allocation. On the basis of the expression, the relay power coefficient is optimized under the fixed power constraint. Besides, the impacts of the topology parameters on the power allocation are also analyzed accordingly. Simulations demonstrate that the proposed power allocation outperforms the benchmark in terms of the detection performance. The obtained expression of the power coefficient and insights can provide useful suggestions for the power allocation design of the CFNC assisted WSNs.


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## Body

## 1. Introduction

Due to the economical deployment and wide range of applications [1], wireless sensor networks (WSNs) have drawn an upsurge of research interests in recent years. Generally, a WSN consists of low powered sensor nodes, which only own constrained processing and communication capabilities. This fact puts limits on the WSNs in both the bandwidth efficiency and transmission reliability in wireless fading channels [2].Distributed detection (DD) [3] is an efficient solution for the limitation of WSNs. In the DD, each sensor node transmits no raw data but the decision extracted from it, which help to reduce bandwidth expense. Besides, DD is also able to combat the adverse effects of wireless fading channels, because the same raw data are observed by the decisions of multiple sensor nodes and transmitted to the fusion center through multiple independent channels. For the above merits, DD has been extensively studied by the literatures. For example, towards the scenarios that the decisions of the sensors are conditionally independent [4, 5] and correlated [6, 7], the optimal DD and decision fusion have been analyzed, respectively. However, in these works, the communication is assumed to be error-free. References [8, 9] investigate the effects of the noise and wireless fading in the decision fusion, in which the channel state information (CSI) is assumed to be fully known at the receivers. Later, the case is also studied in [10] when only partial CSI is available.In these previous studies, to avoid multiaccess interference (MAI), all the sensor nodes transmit in the manner of orthogonal communications, where only one sensor node is permitted to transmit at a certain time. This manner leads to the inefficient utilization of the bandwidth and constrains the throughput of the network, especially for the large WSNs. Therefore, nonorthogonal communications- (NOC-) enabled DD [11–14] appears. With NOC, all the sensor nodes transmit simultaneously over the multiaccess channel, so the bandwidth efficiency is greatly improved.Interfered by the MAI, the transmission reliability is a key issue for the NOC-enabled DD. Recent work [15] employs complex field network coding (CFNC) [16] as a solution of NOC, in which the cooperative relay [17, 18] is also integrated. On the one hand, by using the cooperative relay, the diversity gain can be obtained to improve the transmission reliability. On the other hand, through power allocation, the influence of MAI is also mitigated. Hence, by combing CFNC and cooperative relay, [15] provides an efficient solution in both bandwidth and transmission reliability for WSNs.In the power allocation of [15], the critical step is to obtain a closed-form expression of the detection performance. However, to yield a traceable expression of the pairwise error probability (PEP) bound, the Chernoff bound [19] is utilized to approximate Q function in the computation [15], which is not very tight [20] and may cause degradation to the optimized result.This paper focuses on the optimization of the power allocation for the CFNC assisted WSNs, and the major contributions are summarized as follows:(i)
Contrary to the former work [15], we employ the result of [20] to improve the accuracy of approximation for Q function and derive the closed-form expression of the PEP bound for the considered network.(ii)
We utilize the obtained expression to optimize the relay power by considering fixed power budget. Moreover, the impacts of the topology parameters are also analyzed accordingly, through which the relationship between the power allocation and node location is revealed.(iii)
We examine the performance of the obtained result by comparing it with the one of [15].The rest of this paper is organized as follows. Section2 presents the system model of CFNC assisted DD. In Section 3, the PEP analysis and power optimization are presented. Simulation results are provided in Section 4. Finally, Section 5 concludes the paper.


## 2. System Model of CFNC Assisted DD

We consider the CFNC assisted DD in a parallel WSN, which is utilized to monitor the region of interest (ROI) through binary hypothesesH1 and H0. The network is consisted of N sensor nodes, one fusion center (FC), and one relay node R. As is shown in Figure 1, each sensor node Sk firstly acquires raw measurement from the ROI to generate its decision uk(1or0), which is then modulated to produce its signal xk(-1or1). Here, the sensing accuracy of Sk is described by the probability of detection PDk and probability of false alarm PFk, which are expressed as(1)PDk=Pr⁡xk=1∣H1,PFk=Pr⁡xk=1∣H0.After that, all N signals are transmitted to FC, which are distorted by the fading and noise. Finally, FC combines all the received signals and casts a final decision.Figure 1
System model of CFNC assisted DD.With orthogonal communications, it costsN time slots (TS) to transmit all N signals. In contrast, with CFNC assisted NOC, only two TSs are needed. According to the scheme, each Sk owns a unique signature θk. In TS 1, each Sk produces its CFNC signal by multiplying θk and xk and then transmits θkxk simultaneously. Hence, N CFNC signals overlap at both R and FC, which can be expressed as(2)ysr=∑k=1Nrkhkrθkxk+zr,ysd=∑k=1Ngkhkdθkxk+zd,where ysr and ysd are the received signals at R and FC, respectively. rk and gk denote the path loss coefficient of Sk-to-R and Sk-to-FC links, respectively, while hkr and hkd capture the corresponding fading coefficients. Throughout the manuscript, we assume that the channel state information (CSI) is available at all the receivers. zr and zd denote the background noise at R and FC, respectively, which are modeled as additive white Gaussian noise (AWGN) with zero mean and variance of N0. After that, R decodes the received signals by maximum likelihood (ML) detection, which is(3)x^1,…,x^N=argx1,…,xN⁡min⁡ysr-∑k=1Nrkhkrθkxk2.Then, R encodes the obtained signals as(4)xr=∑k=1Nθkx^kand transmits xr to FC in TS 2. So the received signal at FC can be written as(5)yrd=arrhrdxr+zd,where coefficient a denotes the power allocated to R.Finally, FC combines all the received signals in TS 1 and 2. With the optimal fusion rule in Bayesian sense, the corresponding likelihood ratio is computed as(6)Λ^ysd,yrd=∑Xfysd∣H1,X∑X^fyrd∣H1,X^PX^∣XPX∣H1∑Xfysd∣H0,X∑X^fyrd∣H0,X^PX^∣XPX∣H0,where X=x1,…,xNT, X^=x^1,…,x^NT. To simplify the analysis, we follow the assumption utilized in [15]; that is, X^=X. Thus, (6) can be simplified as(7)Λysd,yrd=∑Xfysd∣H1,Xfyrd∣H1,XPX∣H1∑Xfysd∣H0,Xfyrd∣H0,XPX∣H0.When X is given, the conditional probability of ysd(yrd) is independent of the hypothesis, so (7) can be simplified as(8)Λysd,yrd=∑Xfysd∣Xfyrd∣XPX∣H1∑Xfysd∣Xfyrd∣XPX∣H0.According to the optimal likelihood ratio based fusion rule [9], the condition distributions and probabilities in (8) can be given as(9)fysd∣X=12πσ2exp⁡-ysd-∑k=1Ngkhkdθkxk22σ2,fyrd∣X=12πσ2exp⁡-yrd-arrhrdxr22σ2,PX∣H1=∏k=1NPDkuk1-PDk1-uk,PX∣H0=∏k=1NPFkuk1-PFk1-uk.And then, FC generates the final decision as(10)H1is  ture,ifΛysd,yrd≥t,H0is  ture,ifΛysd,yrd<t,where t is the detection threshold. By means of minimizing the error detection probability, the optimal value of t can be obtained as(11)t=PH0PH1,where P(H1) and P(H0) area priori probabilities of H0 and H1, respectively. Thus, the detection metrics of the scheme can be expressed as(12)PF=PΛysd,yrd≥t∣H0,PD=PΛysd,yrd≥t∣H1,where PF and PD are the false alarm probability and detection probability, respectively. Thus, the average error probability can be expressed as(13)Pe=PFPH0+1-PDPH1.


## 3. PEP Analysis and Power Allocation

In this section, we determine the sensor signatureθ=θ1,…,θNT and the relay power coefficient a by optimizing the ML bound on the PEP of the considered network. Since each sensor detection is binary, X has 2N possibilities, and we denote Xi as ith possible detection vector, while its corresponding CFNC symbol can be expressed as Ci=θTXi. Thus, the PEP at FC can be written as(14)PEPF≜PCi⟶CiatR∣CiPCi⟶Cjat  FC∣Ci⟶CiatR,Ci+PCi⟶CjatR∣Ci1-PCi⟶Ciat  FC∣Ci⟶CjatR,Ci,where PCi→CiatR∣Ci and PCi→CjatR∣Ci are the correctly decoding probability and PEP at R when Ci is given, respectively. PCi→Cjat  FC∣Ci→CiatR,Ci is the PEP at FC given that Ci is sent and R decodes correctly. PCi→Ciat  FC∣Ci→CjatR,Ci is the correctly decoding probability at FC given that Ci is sent and R decodes incorrectly.When the CSIs are known atR, its PEP can be written as(15)PCi⟶CjatR∣Ci=P∑k=1NrkhkrθkXik-Xjk+zr2≤zr2=Pzr∗∑k=1Nrkhkrθkdijk+zr∑k=1Nrkhkrθkdijk∗≥∑k=1Nrkhkrθkdijk2,where “∗” stands for complex conjugate; Xi(k) and dij(k) are kth component of Xi and (Xi-Xj), respectively. Thus, zr∗(∑k=1Nrkhkrθkdij(k))+zr(∑k=1Nrkhkrθkdij(k))∗ follow the Gaussian distribution with zero mean and 4σ2∑k=1Nrkhkrθkdijk2 variance, so (15) can be expressed as(16)PCi⟶CjatR∣Ci=Q∑k=1Nrkhkrθkdijk2σ.Besides, the PEP at FC given that R correctly decodes Ci can be expressed as(17)PCi⟶Cjat  FC∣Ci⟶CiatR,Ci=P∑k=1NgkhkdθkXik-Xjk+zd2+arrhrd∑k=1NθkXik-Xjk+zd2≤2zd2.By a similar analysis as in (15), (17) can be expressed as(18)PCi⟶CjatFC∣Ci⟶CiatR,Ci=Q∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk22σ.Also, the correct decoding probability at FC given, that R decodes erroneously, can be obtained as(19)PCi⟶CiatFC∣Ci⟶CjatR,Ci=Qarrhrd∑k=1Nθkdijk2-∑k=1Ngkhkdθkdijk22σ∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk2.By substituting (16), (18), and (19) into (14), the instantaneous PEP at FC can be obtained as(20)PEPF=Q∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk22σ×1-Q∑k=1Nrkhkrθkdijk2σ+Q∑k=1Nrkhkrθkdijk2σ×1-Qarrhrd∑k=1Nθkdijk2-∑k=1Ngkhkdθkdijk22σ∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk2.To obtain the traceable relationship between PEPF and the power coefficient a, (20) is bounded as(21)PEPF≤Q∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk22σ+Q∑k=1Nrkhkrθkdijk2σ.A closed-form expression ofQ(x) is required to further simplify (21). Different from the previous work [15] in which the Chernoff bound [19] is utilized, we employ (1), (2) and (14) in [20] to approximate Q(x) as(22)Qx≈1/6exp⁡-x2/2+1/2exp⁡-2x2/32.We utilize (22) on the purpose that the accuracy of the approximation for the derived closed-form expression can be improved, which is vital for the final optimization of a. Here, the accuracy comparison of the approximations are depicted in Figure 2, where Q(x) is proposed as the benchmark. It shows that the gap between (22) and Q(x) is narrower, which indicates that (22) is superior to the Chernoff bound in the accuracy of approximation.Figure 2
Accuracy comparison of approximation forQ function.Therefore, by utilizing (22), (21) can be simplified as(23)PEPF≤exp⁡-∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk2/8σ2+exp⁡-∑k=1Nrkhkrθkdijk2/8σ212+exp⁡-∑k=1Nrkhkrθkdijk2/6σ2+exp⁡-∑k=1Ngkhkdθkdijk2+arrhrd∑k=1Nθkdijk2/6σ24.We denote w1=∑k=1Ngkhkdθkdijk2, w2=arrhrd∑k=1Nθkdijk2, w3=∑k=1Nrkhkrθkdijk2. Since hkd, hrd, and hkr follow complex Gaussian distribution with zero mean and unit variance, w1, w2, and w3 are exponent variables with a mean of λ1=∑k=1Ngkθk2dijk2, λ2=arr∑k=1Nθkdijk2, λ3=∑k=1Nrkθk2dijk2, respectively. Thus, the average PEP at FC can be bounded by averaging (23) over fading coefficients as(24)PEPF¯≤∫0∞∫0∞∫0∞exp⁡-w1+w2/8σ2+exp⁡-w3/8σ212+exp⁡-w1+w2/6σ2+exp⁡-w3/6σ24×∏i=13λi×exp⁡-∑i=13λiwidw1dw2dw3≤∏i=121/1+λi/8σ2+1/1+λ3/8σ212+∏i=121/1+λi/6σ2+1/1+λ3/6σ24.To make a fair comparison with [15], we similarly express the sensor signature in the polar form as θi=Piejφi, where Pi and φi denote the transmit power and phase of ith sensor signature, respectively. φi is also determined by the heuristic method in [15]. Thus, the total power constraint can be written as(25)∑i=1NPi+a∑i=1NPi=Pt,where Pt is the fixed power budget of the network.On the other hand, by considering the worst case, (24) can be bounded as(26)PEPF¯≤1/1+g-∑k=1NPkdmin2/8σ21/1+arr∑k=1NPkdmin2/8σ2+1/1+r-∑k=1NPkdmin2/8σ212+1/1+g-∑k=1NPkdmin2/6σ21/1+arr∑k=1NPkdmin2/6σ2+1/1+r-∑k=1NPkdmin2/6σ24,where g-=1/N∑k=1Ngk, r-=1/N∑k=1Nrk. dmin is the minimum distance of the CFNC constellation, which can be expressed as(27)dmin=mini≠j⁡ejφ1,…,ejφNT·Xi-Xj.By substituting (25) into (26), PEPF¯ can be simply expressed as(28)PEPF¯≤1/1+g-Pt-a∑k=1NPkdmin2/8σ21/1+arr∑k=1NPkdmin2/8σ2+1/1+r-Pt-a∑k=1NPkdmin2/8σ212+1/1+g-Pt-a∑k=1NPkdmin2/6σ21/1+arr∑k=1NPkdmin2/6σ2+1/1+r-Pt-a∑k=1NPkdmin2/6σ24.Using the approximation 1/(1+x)→1/x when x is in high SNR region, (28) can be simplified as(29)PEPF¯≤64σ4/g-rraPt-a∑k=1NPk∑k=1NPkdmin4+8σ2/r-Pt-a∑k=1NPkdmin212+36σ4/g-rraPt-a∑k=1NPk∑k=1NPkdmin4+6σ2/r-Pt-a∑k=1NPkdmin24≤433σ4g-rraPt-a∑k=1NPk∑k=1NPkdmin4+136σ2r-Pt-a∑k=1NPkdmin2.We employ (29) as the closed expression of PEPF¯, and the optimization of the power coefficient a can be obtained by solving the following equation as(30)∂PEPF¯∂a=-433σ4Pt-2a∑k=1NPkg-rra2Pt-a∑k=1NPk2∑k=1NPkdmin4+136σ2∑k=1NPkr-Pt-a∑k=1NPk2dmin2=0.Multiplying both sides of (30) by a2(Pt-a∑k=1NPk)2 leads to(31)86σ4Pt-2a∑k=1NPkg-rrdmin4-13σa∑k=1NPk2r-dmin2=0.We denote C=13/86 and ∑k=1NPk=Pt1; thus (31) can be simplified as(32)CPt12g-rra2+2r-Pt1aσ2-r-rdmin2Pt=0.The roots of (32) with respect to a can be expressed as(33)a=±r-2σ4+Cg-rrr-σ/dmin2Pt-r-σ2CPt1g-rr.By abandoning the negative root, the optimal value of a is obtained. Table 1 compares this result with the one in [15], and it shows that our expression is more straightforward to reveal the relationship between a and the topology parameters.Table 1
Comparisons of the optimal power coefficients.
Work Closed-form expression ofa [15] a=PtPt+8σ2/dmin22r--4g-rrr-Pt8σ2/dmin2+4g-28σ2/dmin22/2g-rr-1 Proposed a = r - 2 σ 4 + C g - r r r - σ / d m i n 2 P t - r - σ 2 C ∑ k = 1 N P k g - r rTherefore, we utilize the obtained expression ofa to make the insightful analysis of the topology parameters and determine the proper power allocation. The results are presented as follows.(1) Whenσ2/dmin2→0, the proposed expression tends to(34)a=r-2σ4-r-σ2C∑k=1NPkg-rr=0.Equation (34) reveals that when the network works in the high SNR region, the optimal policy is to allocate the power budget to all the sensor nodes without employing the relay node, which helps to avoid error detection.(2) The derivative of the proposed expression with respect tog- can be obtained as(35)∂a∂g-=Crrr-σ/dmin2Ptg-/2r-2σ4+Cg-rrr-σ/dmin2Pt-r-2σ4+Cg-rrr-σ/dmin2Pt-r-σ2C∑k=1NPkg-2rr.By using relationshipx2+b-x=b/(x2+b+x), there are(36)r-2σ4+Cg-rrr-σdmin2Pt-r-σ2=Cg-rrr-σ/dmin2Ptr-2σ4+Cg-rrr-σ/dmin2Pt+r-σ2.Substituting (38) into (37), it leads to(37)∂a∂g-=r-σ/dmin2Pt1/2r-2σ4+Cg-rrr-σ/dmin2Pt-1/r-2σ4+Cg-rrr-σ/dmin2Pt+r-σ2∑k=1NPkg-<0.Equation (37) demonstrates that a is the monotone decreasing function given that g- grows. This result indicates that when the sensor nodes locate close to FC, more power should be allocated to them, because the direct links between the sensor nodes and FC are in good quality. On the contrary, when the sensor nodes locate far from FC, the quality of the direct links are poor, so more power should be allocated to the cooperative relay for reliable transmission.(3) The derivative of the obtained expression with respect torr can be similarly obtained as(38)∂a∂rr=r-σ/dmin2Pt1/2r-2σ4+Cg-rrr-σ/dmin2Pt-1/r-2σ4+Cg-rrr-σ/dmin2Pt+r-σ2∑k=1NPkrr<0.The result shows that a is the monotone decreasing function when rr grows, and it reveals that when the relay locates close to FC, more power should be allocated to the sensor nodes. This insight is reasonable, because in that scenario the link between the relay and FC is in good quality, so the end-to-end transmission error rate is dominated by the error detection at relay. If more power is allocated to the sensor nodes, the received SNR at relay can be improved, which is helpful to mitigate error propagation.(4) The insightful relationship betweena and r- is obtained as(39)∂a∂r-=2r-σ4+Cg-rrσ2/dmin2Pt/2r-2σ4+Cg-rrr-σ/dmin2Pt-σ2C∑k=1NPkg-rr.We denote w=Cg-rrPt/r-σ2dmin2, and (39) can be simplified as(40)∂a∂r-=Cg-rrPt2/r-2σ2dmin44∑k=1NPk1+w1+0.5w+1+w>0.The above result shows that a is the monotone increasing function with respect to r-. In fact, when the relay node locates close to the sensor nodes, error detection is slice, so the optimal policy is to fully utilize the cooperative relay by allocating more power to it.


## 4. Simulation Results

In this section, we examine the detection performance of the obtained power coefficient by employing the one of [15] as the benchmark. Moreover, the theoretical analysis on the topology parameters presented in Section 3 is also evaluated. MATLAB is employed to build simulation platform. The probability of error and receiver operating characteristics (ROC) curves are simulated as the performance metrics through the Monte-Carlo method. In the simulation of probability of error, it is assumed that P(H0)=P(H1)=0.5 in (13), while, in the simulation of ROC curves, P(H0) and P(H1) are changed for each simulated point. Simulation length is set as 100 million times per point. Each sensor node Si has identical false alarm and detection probabilities as PDk=0.5 and PFk=0.05. Also, we assume that the power budget Pt=2 per time slot. In the simulations, the signal-to-noise (SNR) is defined as (41)SNRin  dB=10log10⁡PtT02σ2,in which T0=0.5 is the time slot duration. Since Pt and T0 are fixed, we supply different SNRs by changing σ2 in the simulation. The network topologies are modeled by the path loss coefficients, through which both the symmetric and asymmetric scenarios are considered. Here, we assume the distance between S1 and FC is 1, and g1=1 is as the reference distance. The other path loss coefficients are given as gains or loss relative to it, that is, r(indB)=10log10⁡(r/g1).Figure3 shows the probabilities of error for the two compared schemes. The metric is simulated under a symmetric topology with g1=g2=0dB, r1=r2=10.45dB, rr=3.10dB, and N=2, in which the distance from the relay node to each sensor node is the same, as well as the one from FC to each sensor node. It can be observed that as SNR grows, the gap between the two schemes reduces; however, the proposed scheme is always superior to the benchmark in the same SNR region. This result verifies the effectiveness of the proposed optimization in Section 3 and shows that the accuracy improvement of approximation for Q function is able to yield better transmission reliability.Figure 3
Probability of error comparisons forN=2, g1=g2=0dB, r1=r2=10.45dB, rr=3.10dB.Figure4 shows the ROC curves of the two compared schemes in the same topology with different SNR values. It can be seen that the proposed scheme outperforms the benchmark in each SNR. In the other words, the proposed scheme has higher probability of detection than the benchmark under the same probability of false alarm. The results demonstrate that the proposed power allocation is helpful to improve the detection performance of the network.Figure 4
ROC curve comparison forN=2, g1=g2=0dB, r1=r2=10.45dB, rr=3.10dB.Besides symmetric topology, the asymmetric topology is also employed to evaluate the scheme. Figure5 depicts the probability of error when the relay node is closer to S1, and the distance from S2 to FC varies. It can be seen that although both compared schemes suffer from the decrease of g2, the proposed scheme is superior to the benchmark. In fact, the relay node locates farer from S2, so the error detection at the relay is dominated by the link between them. Meanwhile, when S2 locates farther from FC, the S2-to-FC link also demonstrates the detection of FC. Hence, the variety of g2 influences the performance of the network largely. However, since the proposed scheme can offer more accurate power allocation to deal with the topology variety, it yields better performance. In the figure, the performance gap is more obvious when S2 locates far from FC, which also verifies the accuracy improvement of the power allocation by the proposed optimization. Similarly, Figure 6 shows the corresponding curves of the asymmetric topology when SNR = 0 dB. The result also shows that the proposed scheme outperforms the benchmark.Figure 5
Probability of error comparison under variousg2 for N=2, g1=0dB, r1=10.45dB, r2=rr=3.10dB.Figure 6
ROC curve comparison under variousg2 for N=2, g1=0dB, r1=10.45dB, r2=rr=3.10dB, and SNR = 0 dB.We also evaluate the proposed scheme in a more complex scenario in Figure7, in which N=4 and the distance from each sensor node to the relay node is identical. Similar to the situation in Figure 5, the decrease of g4 also deteriorates the performance of both schemes. Consistently, the proposed scheme has a better transmission reliability than the benchmark for each g4. To conclude, the above results show that the proposed optimization is constructive to improve the accuracy of power allocation, as well as the detection performance of the considered WSN.Figure 7
Probability of error comparison under variousg4 for N=4, g1=0dB, g2=g3=rr=0.91dB, r1=r2=r3=r4=13.98dB.Next, we examine the theoretical analysis presented in Section3. Figure 8 depicts the impact of the high SNR on the proposed scheme. The employed simulation parameters are the same as the ones in Figure 3. Here, we use direct transmission with CFNC as the benchmark, in which the power budget is equally allocated to the sensor node without considering the relay node. The figure shows that, in low and medium SNR regions, the proposed scheme outperforms direct transmission with considerable gap. However, in high SNR region, the gap is negligible. This result demonstrates that, in the high SNR region, the optimal power allocation is close to direct transmission, which corresponds well with the analysis in Section 3. So the simulation and analysis provide an insight that, in high SNR region, direct transmission is a simple yet efficient policy comparing with power allocation.Figure 8
The impact of the high SNR on the probability of error with symmetric topology andN=2.Figures9, 10, and 11 show the impacts of parameters g-, rr, and r- on the proposed scheme, respectively. In these simulations, SNR = 5 dB, and the other parameters are the same as the one in Figure 3. In Figure 9, three power coefficients are compared, in which aop is calculated according to the proposed expression for in Table 1 for each simulated value of g-, while amax and amin are by the expression when g-=0 and 10 dB, respectively. In other words, aop varies for each g- and falls into region [amin,amax]. It can be observed that, compared with the two fixed values, aop obtains better transmission reliability. This result shows that when the sensor nodes move towards FC, the proper transmission policy is to allocate more power to the sensor nodes.Figure 9
The impact of parameterg- on the probability of error with symmetric topology and N=2, SNR = 5 dB.Figure 10
The impact of parameterrr on the probability of error with symmetric topology and N=2, SNR = 5 dB.Figure 11
The impact of parameterr- on the probability of error with symmetric topology and N=2, SNR = 5 dB.Figure10 also demonstrates that aop is superior to amax and amin. This result reveals that power adjustment is constructive for the transmission performance when the relay moves towards FC. Besides, it shows that, in this case, more power should be allocated to the sensor nodes. Figure 11 depicts the comparisons when the relay node moves towards the sensor nodes. Similarly, aop yields optimal performance, while it also shows that more power should be allocated to the relay in this case. The above results verify the corresponding analysis in Section 3 and show that the proposed expression of a is effective in analyzing the impacts of the topology parameters and providing proper power allocation for each scenario.


## 5. Conclusion

This paper has investigated the power allocation for the CFNC-based DD in cooperative relay assisted WSN. In order to improve the accuracy of optimization, the closed expression of the PEP bound for the considered network is derived by utilizing a high accuracy approximation ofQ function. According to the obtained expression, the power allocation coefficient of the relay is optimized, while some useful insights on the relationship between the power allocation and topology parameters are also provided. Simulations demonstrate that the proposed optimization outperforms the benchmark [15] in terms of detection performance. Moreover, the impacts of the topology parameters on the power allocation are also verified. Therefore, the obtained expression of the power coefficient and the insights are helpful for the power allocation design in the considered WSN.



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