# Guard Band-Based Resource Sharing for Device-to-Device Communications Underlying Cellular Network.

1. IntroductionDevice-to-device communication underlying the cellular network is a promising technology in future wireless networks to improve the resource utilization efficiency and extend coverage areas [1]. However, D2D communication in the cellular spectrum poses a host of challenges to the network. For example, new resource allocation schemes must be designed to mitigate or avoid interference between cellular and D2D links.

Resource allocation methods for D2D users can be categorized into two schemes: the orthogonal resource sharing and the nonorthogonal resource sharing. The former assigns dedicated resources to D2D users, and the latter requires them to share resources with cellular users. The nonorthogonal resource sharing scheme achieves better resource utilization efficiency [2-4]; however, additional manipulations, such as power control, distance limitation, and measures related to mode selection [2-7], are needed to suppress the interference between cellular users and D2D users.

In [2], a resource sharing algorithm is proposed to find the optimal transmission power for D2D communication without degrading the QoS of cellular users. Tang et al. present a series of distributed power control methods to avoid interference and to enhance radio resource utilization in cellular and D2D hybrid networks [3]. Further, Yu et al. employed different resource sharing modes to analyze optimal resource allocation and power control between cellular and D2D connections that share the same resources [4]. In [5-10], the authors considered the similar problems and presented a number of distance-dependent algorithms based on power optimization, uplink reuse allocation, and power management. However, all these solutions hinge on the condition that each D2D link and its paired cellular user operate in the same frequency band, which raises the issue that the D2D receiver may pick up total interference from its paired cellular user. Therefore, the D2D users may not achieve satisfactory SINR (Signal to Interference and Noise Ratio) performance when D2D users and cellular users are geographically located in a small area; as a result, the outage probability performance would be significantly degraded.

In communication systems, to reduce the impact of adjacent channel interference, nearly 10 percent of the system bandwidth at the edge of the allocated bandwidth is reserved as the guard band. For example, 3GPP TS 25.101 regulates that the adjacent carrier spacing is 5 MHz, but the actual occupied bandwidth is 1 + [mu] times the chip rate in WCDMA system, where [mu] is the roll-off factor of the root-raised cosine filter. This leaves the guard band (5 - 3.84(1 + [mu])) MHz. In LTE, when the allocated bandwidth is 20 MHz, its actual occupied bandwidth is 18 MHz, leaving the guard band 2 MHz. Based on the guard band, Chen et al. proposed a novel scheme of utilizing the guard band in LTE uplink, which tried to optimize the overall spectrum efficiency of the two systems adjacently deployed on frequency [11].

In contrast to existing works, this paper considers the low energy characteristics of WCDMA signals in the guard band and presents a novel resource sharing strategy to mitigate cellular user-induced interference at the D2D receiver. In addition, the D2D outage probability performances are discussed mathematically in both flat-fading and frequency-selective channels. Numerical results show that the proposed resource sharing strategy outperforms the conventional strategy.

2. System Model

In cellular networks, uplink resource sharing is a common technique, since uplink resources tend to be underutilized in frequency division duplexing (FDD) based cellular networks [12], especially in multimedia services. As a result, uplink resource sharing has been the subject of a number of D2D system designs.

This paper will explore resource sharing problems in one cell, leaving intercell interference out of discussion. Consider a scenario involving a fully loaded cellular network, where N active cellular users occupy N orthogonal channels in a cell, and there are no spare resources. Figure 1 depicts one BS (Base Station) and N orthogonal cellular users, with each user occupying a frequency band indexed by i = 1,2, ..., N. A total of N - 1 D2D pairs exist in this network. The goal of our proposed scheme is to decrease the D2D users' average outage probability with the proposed scheme.

In what follows, we use [DT.sub.i] and [DR.sub.i] to denote the D2D transmitter and receiver, respectively, and the cellular user is denoted by [CU.sub.i]. The following three assumptions are underlying the system model.

Assumption 1. The distance between [DT.sub.i] and [DR.sub.i] is assumed to be [[rho].sub.i], and the distances from BS to [CU.sub.i] and [DT.sub.i] are denoted by [r.sup.c.sub.i] and [r.sup.d.sub.i], respectively. For any [CU.sub.i], the probability density function (PDF) of its distance [r.sup.c.sub.i] from BS is f([r.sup.c.sub.i]) = 2[r.sup.c.sub.i]/[R.sup.2], [r.sup.c.sub.i] [member of] (0,R] [5], where R is the cell radius. Moreover, resource sharing is region constrained such that if the [DR.sub.i] angle is [[theta].sub.0], the cellular user's angle [[theta].sub.i] is uniformly distributed in [[[theta].sub.0] + [pi]/2, [[theta].sub.0] + 3[pi]/2). And [OMEGA] denotes the set of [CU.sub.i] locations that meet the region constraint. Since [[rho].sub.i] is much smaller than [r.sup.d.sub.i], the distance from BS to [DR.sub.i] also approaches [r.supd.sub.i]. Assuming that the relative angle between [DR.sub.i] and [CU.sub.i] is [theta], then the distance between [CU.sub.i] and [DR.sub.i] is given by

[L.sub.i] = [square root of ([([r.sup.c.sub.i]).sup.2] + [([r.sup.d.sub.i]).sup.2] - 2[r.sup.c.sub.i] [r.sup.d.sub.i] cos [theta]). (1)

Assumption 2. Let [h.sup.d.sub.i] and [g.sup.d.sub.i] denote the channel from [CU.sub.i] to [DR.sub.i] and [DT.sub.i] to [DR.sub.i], respectively. Further, the channel from [CU.sub.i] to BS is denoted by [h.sup.c.sub.i], and the channel from [DT.sub.i] to BS is given by [g.sup.c.sub.i]. If channel gain follows independent Rayleigh distribution, then, according to [13], [[absolute value of [g.sup.d.sub.i]].sup.2] and [[absolute value of [h.sup.d.sub.i]].sup.2] follow independent exponential distribution. In addition, the path loss factor a is set to be 4 [5].

Assumption 3. The system has a guard band between two adjacent channels to suppress interferences between them. The carrier frequency of the cellular user i is denoted as [f.sup.c.sub.i] and that of the D2D user i as [f.sup.d.sub.i].

3. Guard Band-Based Resource Sharing

3.1. Scheme Description. In this part, we will first present the conventional resource sharing technique and then propose a new one to improve the reliability of D2D communications. For convenience, the two strategies are denoted as MODE 1 and MODE2, respectively.

In MODE1, the D2D user's carrier frequency [f.sup.d.sub.i] is equal to its paired cellular user's center frequency [f.sup.c.sub.i]; that is, [f.sup.d.sub.i] = [f.sup.c.sub.i]. As shown in Figure 2, [B.sup.d.sub.L,i] is the lower band bound and [B.sup.d.sub.H,i] is the upper band bound of the [DT.sub.i]; [B.sup.c.sub.L,i] is the lower and [B.sup.c.sub.H,i] is the upper band bound of [CU.sub.i]. With [B.sub.p] denoting the passband, [B.sup.d.sub.L,i] = [f.sup.d.sub.i] - [B.sub.p]/2, [B.sup.d.sub.H,i] = [f.sup.d.sub.i] + [B.sub.p]/2, and the same applies to [B.sup.c.sub.L,i] and [B.supc.sub.H,i]. Interference received by the D2D link i is caused by [CU.sub.i], as seen in Figure 2. If the power of the cellular signal i is denoted by P, then the cellular interference from [CU.sub.i] is P.

In MODE2, [f.sup.d.sub.i] is chosen as the center of the guard band between two adjacent cellular channels; that is, [f.sup.d.sub.i] = ([f.sup.c.sub.i] + [f.sup.c.sub.i+1])/2. As Figure 3 shows, when the D2D receiver obtains the desired signal with a band-pass filter with passband [[B.sup.d.sub.L,i], [B.sup.d.sub.H,i]], it also picks up cellular interference from [CU.sub.i] and [CU.sub.i+1]. Denote the power of the interference coming from [CU.sub.i] as [P.sub.I] (which means integration over region I in Figure 3) and the power of the interference coming from [CU.sub.i+1] as [P.sub.IV]. Clearly, the cellular interference is [P.sub.I] + [P.sub.IV]. Define the interference fraction factor as [epsilon], [epsilon] = [P.sub.I]/P.

Proposition 4. If a guard band exists between two adjacent channels with center frequencies [f.sup.c.sub.i] and [f.sup.c.sub.i+1] and the D2D user's center frequency [f.sup.d.sub.i] is equal to ([f.sup.c.sub.i] + [f.sup.c.sub.i+1])/2, the definedcellular interference fraction factor 0 < [epsilon] < 0.5.

Proof. For convenience, we calculate the parameter e in the baseband, as shown in Figure 4. P(f) is the power spectrum, [B.sub.p] is the filter passband, and [B.sub.g] is the guard band. Denote

[mathematical expression not reproducible]. (2)

Let P = [P.sub.I] + [P.sub.II] + [P.sub.III]. Since [P.sub.I] = P - [P.sub.II] - [P.sub.III] and [P.sub.I] = [P.sub.II], [epsilon] can be expressed as

[epsilon] = [P.sub.I]/P = 1/2 (P - [P.sub.III])/P = 1/2 (1 - [P.sub.III]/P). (3)

According to (3), it is obvious that 0 < [P.sub.III]/P < 1. As a result, 0 < [epsilon] < 0.5.

Equation (3) shows that e is affected by the ratio of the interference power in the guard band [P.sub.III] to the total power P. For different CUs, the ratios are identical; that is, the interference fraction factor is also [epsilon] for [CU.sub.i+1].

3.2. Outage Probability Analysis in a Flat-Fading Channel. Let [x.sup.c.sub.j](t) represent the transmitted signal of [CU.sub.j] and [x.sup.d.sub.i](t) denote the signal of [DT.sub.i]. In a flat-fading channel, the received D2D signal at [DT.sub.i] is

[mathematical expression not reproducible], (4)

where [n.sup.d.sub.i](t) is additive white Gaussian noise with variance [[sigma].sup.d.sub.i]. Assuming that [p.sub.j] = E [[absolute value of [x.sup.c.sub.j]].sup.2] and [q.sub.i] = E[[absolute value of [x.sup.d.sub.i]].sup.2], in this paper, a widely used power control scheme for cellular user equipment ([CU.sub.j]) is considered, known as the target SNR power control scheme (TSPC) [14]. In this scheme, the cellular user's power is selected to reach a fixed SNR (Signal to Noise Ratio) target [xi], as shown in (5), where [[sigma].sup.c.sub.j] is the variance of the zero-mean Gaussian noise for [CU.sub.j]. In addition, the cellular power [p.sub.j] should satisfy its power constraint [p.sub.j] [less than or equal to] [P.sub.c max] :

[xi] = [p.sub.j] [([r.sup.c.sub.j]).sup.-[alpha]]/[[sigma].sup.c.sub.j]. (5)

The TSPC scheme is also applied to the D2D link. Two lemmas are utilized to derive the D2D outage probability.

Lemma 5. Let [[absolute value of [g.sup.d.sub.i]].sup.2] and [[absolute value of [h.sup.d.sub.i]].sup.2] follow independent exponential distribution with unit mean, x = [[absolute value of [g.sup.d.sub.i]].sup.2], and y = [[absolute value of [h.sup.d.sub.i]].sup.2], (a > 0). Then, the cumulative distribution function (CDF) of z = x/y is

[F.sub.Z] (z) = 1 - 1/1 + az. (6)

Proof. The proof is presented in Appendix A.

Lemma 6. Let [[absolute value of [g.sup.d.sub.i]].sup.2], [[absolute value of [h.sup.d.sub.i]].sup.2], and [[absolute value of [h.sup.d.sub.i+1]].sup.2] follow independent exponential distribution with unit mean, and x = [[absolute value of [g.sup.d.sub.i]].sup.2] and y = a [[absolute value of [h.sup.d.sub.i]].sup.2] + b[[absolute value of [h.sup.d.sub.i+1]].sup.2], (a > 0, b > 0). Then, the CDF of z = x/y is

[F.sub.Z] (z) = 1 - 1/(1 + az) (1 + bz). (7)

Proof. The proof is presented in Appendix B.

The D2D conditional outage probability is given by (8), where [[gamma].sub.0] is the D2D SINR threshold and [[gamma].sub.i,d] is the SINR for [DR.sub.i], calculated by (9). [mathematical expression not reproducible] is the probability density function of [[gamma].sub.i,d]. In (9), [P.sub.s] is the power of the desired signal at receiver [DR.sub.i], and [I.sup.c.sub.i] is the cellular interference:

[mathematical expression not reproducible] (8)

[[gamma].sujb.i,d] = [P.sub.s]/[[sigma].sup.d.sub.i] + [I.sup.c.sub.i]. (9)

The difference between MODE1 and MODE2 lies in the cellular interference [I.sup.c.sub.i], so we use a high SNR approximation; that is, of [[sigma].sup.d.sub.i] [approximately equal to] 0.

3.2.1. The Outage Probability of MQDE1. In MODE1, [P.sub.s] = [[absolute value of [g.sup.d.sub.i]].sup.2] [[rho].sup.-[alpha].sub.i] [q.sub.i], and [I.sup.c.sub.i] = [[absolute value of [h.sup.d.sub.i]].sup.2] [L.sup.-[alpha].sub.i] [p.sub.i]; then,

[mathematical expression not reproducible], (10)

where [[bar.p].sub.i] = [p.sub.i][L.sup.-[alpha].sub.i]/[q.sub.i][[rho].sup.-[alpha].sub.i]. According to Lemma 5, the conditional outage probability of MQDE1 can be written as

[mathematical expression not reproducible]. (11)

The outage probability of MQDE1 can be obtained by averaging over the positions of the cellular user [CU.sub.i]:

[mathematical expression not reproducible]. (12)

3.2.2. The Outage Probability of MODE2. In MODE2, [I.sup.c.sub.i] = [epsilon] [[summation].sub.j=i, i + 1] [[absolute value of [h.sup.d.sub.j]].sup.2] [L.sup.-[alpha].sub.j][p.sub.j]; then, the SINR can be expressed as

[mathematical expression not reproducible], (13)

where [mathematical expression not reproducible]. The conditional outage probability of MODE2 can be evaluated with Lemma 6, which is shown by

[mathematical expression not reproducible]. (14)

Similarly, when averaging over the positions of [CU.sub.i] and [CU.sub.i+1], the outage probability of MODE2 can be obtained:

[mathematical expression not reproducible]. (15)

3.3. Outage Probability Analysis in a Frequency-Selective Channel. When the system's communication bandwidth is much larger than the coherence bandwidth, the channel has a frequency-selective characteristic. In this case, the channel is modelled as a multipath fading channel, and the received signal is given by

[mathematical expression not reproducible], (16)

where [[delta].sup.d.sub.i,l] is the channel gain of the lth path from [DT.sub.i] to [DR.sub.i] and [[lambda].sup.d.sub.j,l] is the channel gain of the lth path from [CU.sub.j] to [DR.sub.i]. Here, [mathematical expression not reproducible] represent Rayleigh fading and [m.sup.d.sub.i,l], [m.sup.d.sub.j,l] represent multipath fading for each path. In contrast to a flat-fading channel, intersymbol interference (ISI) is easily introduced in the multipath environment. Its SINR is extended to

[[gamma].sub.i,d] = [P.sub.s]/[[sigma].sup.d.sub.i] + [I.sup.c.sub.i] + [I.sup.d.sub.ISI], (17)

where [I.supl.d.sub.ISI] is the intersymbol interference. With a RAKE receiver, multipath components whose delays are less than one chip period can be used to increase the signal energy, thus allowing [P.sub.s] to be written as (18). Let [L.sub.1] be the set of paths whose delays are less than one chip period [T.sub.c]; that is, [L.sub.1] : l [member of] {[[tau].sub.i,l] < [T.sub.c]}; then,

[P.sub.s] = [summation over (l[member of][L.sub.1])] [[absolute value of [[delta].sup.d.sub.i,l]].sup.2] [[rho].sup. -[alpha].sub.i] [q.sub.i]. (18)

Intersymbol interference originates from multipath signals whose delays exceed one chip period. This set can be defined as [L.sub.2] : l [member of] {[[tau].sub.i,l] [greater than or equal to] [T.sub.c]}:

[I.sup.d.sub.ISI] = [summation over (l[member of][L.sub.2])] [[absolute value of [[delta].sup.d.sub.i,l]].sup.2] [[rho].sup.-[alpha].sub.i] [q.sub.i]. (19)

In MODE 1, cellular interference [I.sup.c.sub.i] can be written as

[I.sup.c.sub.i] = [L-1.summation over (l=0)] [[absolute value of [[lambda].sup.d.sub.i,l]].sup.2] [L.sup. -[alpha].sub.i] [p.sub.i]. (20)

Substitute (18), (19), and (20) into (17); the SINR for MODE 1 can be then expressed as

[mathematical expression not reproducible]. (21)

In MODE2, cellular interference [I.sup.c.sub.i] can be written as

[mathematical expression not reproducible]. (22)

Substitute (18), (19), and (22) into (17); the SINR for MODE2 can be then expressed as

[mathematical expression not reproducible]. (23)

Substitute (21) and (23) into (8), respectively; then, the outage probabilities of MODE1 and MODE2 are obtained. In the following session, we will evaluate the system performance in the frequency-selective channel by simulation.

4. Performance Comparison

This section presents a performance comparison between MODE1 and MODE2. Table 1 gives the simulation parameters. In a WCDMA system, the channel spacing is 5 MHz, and the transmission pulse shaping filter is a root-raised cosine (RRC) function. The chip duration [T.sub.c] is 260.42 ns (1/3.84 MHz). Assume the roll-off factor [mu] to be 0.075. Since [B.sub.p] = (1 + [mu])/[T.sub.c], the signal bandwidth [B.sub.p] is 4.12 MHz, making the guard band ([B.sub.g]) 0.88 MHz. Equation (3) allows us to obtain the interference fraction factor [epsilon], which is 0.3847.

4.1. Performance in a Flat-Fading Channel. Figure 5 shows the outage probability performance of D2D communication, indicating how the outage probability changes when the target cellular SNR [xi] is set to 10 dB and 20 dB, respectively. It can be seen that MODE2 provides better outage probability performance than MODE 1. At the same outage probability level, MODE2 provides an SNR gain of 1.2 dB, which accords with the theoretical analysis, verifying that our derivation for outage probability in a flat-fading channel is correct.

Figure 6 gives the cellular network's capacity improvement factor with the target cellular SNR [xi] set to 5 dB, 10 dB, and 20 dB, respectively. Calculated with (26), the capacity improvement factor [eta] is averaged over a range of channel states and cellular user positions. In Figure 6, [eta] > 1 means that the cellular capacity performance of MODE2 is better than that of MODE1. In other words, MODE2 provides better D2D outage probability performance while guaranteeing the cellular capacity:

[mathematical expression not reproducible] (24)

[mathematical expression not reproducible] (25)

[eta] = [R.sup.(2).sub.i,c]/[R.sup.(1).sub.i,c]. (26)

4.2. Performance in a Frequency-Selective Channel. The frequency-selective channel is modelled in accordance with the Ped B model defined by ITU. Table 2 presents the channel parameters of this outdoor to indoor pedestrian model. Relative delay refers to a time difference relative to the first path, whereas average power refers to power fading relative to the first path.

Since WCDMA has a chip period of 260.42 ns, the first three paths (1, 2, and 3) are located in set [L.sub.1] and can be processed by the RAKE receiver. As the other three paths (4, 5, and 6) are located in set [L.sub.2], they are considered as ISI interference.

Figure 7 shows the outage probability performance in a frequency-selective channel with the target cellular SNR [xi] set to 10 and 20 dB, respectively. At low and moderate SNRs, MODE2 achieves better outage probability performance than MODE1, with an SNR gain of about 1.2 dB. However, at high SNRs, the outage probabilities of the two modes become equivalent, because, in a high SNR region, the outage probability is mainly affected by [I.sup.d.sub.ISI]. The value of [I.sup.d.sub.ISI] is determined by the D2D transmission power, and it changes with the D2D SNR, making the ISI interferences [I.sup.d.sub.ISI] of the two modes identical.

5. Conclusion

This paper investigated resource sharing strategies for D2D communications in a cellular network. In contrast to previous research, we aimed at mitigating cellular interference on the basis of the low energy characteristics of signals in the guard band and proposed a novel resource sharing strategy. Mathematical analysis and simulation results show that, in a flat-fading channel, the proposed strategy could improve the outage probability performance. And in a frequency-selective channel, the advantage of the proposed method was obvious in low and moderate SNR regions.

http://dx.doi.org/10.1155/2016/9547560

Appendix

A. Proof of Lemma 5

Let [[absolute value of [g.sup.d.sub.i]].sup.2] and [[absolute value of [h.sup.d.sub.i]].sup.2] follow independent exponential distribution with unit mean, x = [[absolute value of [g.sup.d.sub.i]].sup.2], and y = a[[absolute value of [h.sup.d.sub.i]].sup.2], (a > 0). The PDF of x and y are expressed as [f.sub.X](x) = exp(-x) and [f.sub.Y](y) = (1/a)exp(-(1/a)y), respectively. According to [15] (6-43), the PDF of z = x/y is

[mathematical expression not reproducible]. (A.1)

Substituting [f.sub.X](x) and [f.sub.Y](y) into (A.1),

[mathematical expression not reproducible]. (A.2)

Using integration by parts, the PDF in (A.2) can be evaluated as

[f.sub.Z] (z) = 1/a [(z+ 1/a).sup.2]. (A.3)

The CDF of variable Z can be expressed as

[mathematical expression not reproducible]. (A.4)

Thus, the desired result (6) is obtained.

B. Proof of Lemma 6

Let [[absolute value of [g.sup.d.sub.i]].sup.2], [[absolute value of [h.sup.d.sub.i]].sup.2], and [[absolute value of [h.sup.d.sub.i+1]].sup.2] follow independent exponential distribution with unit mean, and x = [[absolute value of [g.sup.d.sub.i]].sup.2] and y = a[[absolute value of [h.sup.d.sub.i]].sup.2] + b[[absolute value of [g.sup.d.sub.i+1]].sup.2], (a > 0, b > 0). The PDF of % is expressed as [f.sub.X](x) = exp(-x). Let A = 1/a and B = 1/b. According to [15] (6-41), the PDF of y is

[mathematical expression not reproducible]. (B.1)

Since the cellular position is randomly generated, it is more common that B [not equal to] A, so we consider the first case. According to (A.1),

[mathematical expression not reproducible]. (B.2)

Applying integration by part to the first and second term of (B.2), respectively, we obtain

[f.sub.Z] (z) = AB/B - A (1/[(z + A).sup.2] - 1/[(z + B).sup.2]. (B.3)

The CDF of variable Z can be expressed as

[mathematical expression not reproducible]. (B.4)

By substituting A and B with 1/a and 1/b, respectively, the desired result (7) is obtained.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the 973 Program under Grant no. 2013CB329003 and the National Science and Technology Major Project under Grant no. 2012ZX03003011-004.

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Yue Li, Xuejun Sha, Liang Ye, and Xiaojie Fang

The Department of Electronics and Information Technology, Harbin Institute of Technology (HIT), Harbin 150001, China

Correspondence should be addressed to Xuejun Sha; shaxuejun@hit.edu.cn

Received 16 March 2016; Revised 17 June 2016; Accepted 23 June 2016

Academic Editor: Mohammed Nouari

Caption: Figure 1: D2D links sharing uplink resources with cellular users.

Caption: Figure 2: Conventional resource sharing strategy MODE1.

Caption: Figure 3: Proposed resource sharing strategy MODE2.

Caption: Figure 4: Calculation of [epsilon].

Caption: Figure 5: D2D outage probability in a flat-fading channel.

Caption: Figure 6: Cellular capacity improvement factor in a flat-fading channel.

Caption: Figure 7: D2D outage probability in a frequency-selective channel.

Table 1: Simulation parameters. Parameters Values Cell radius R 500 m Maximum cellular power [P.sub.c max] 20 dBm Distance between D2D TX and RX [[rho].sub.i] ~U [50 m, 100 m] Distance between D2D TX and BS [r.sup.d.sub.i] ~U [250 m, 500 m] Channel spacing 5 MHz Pulse shaping filter RRC Gaussian noise powers [[sigma].sup.d.sub.i] -100 dBm and [[sigma].sup.c.sub.i] D2D SINR threshold [[gamma].sub.0] 0 dB Table 2: Ped B channel parameters. Path number Relative delay Average power 1 0 ns 0 dB 2 100 ns -3.6 dB 3 200 ns -7.2 dB 4 300 ns -10.8dB 5 500 ns -18.0 dB 6 700 ns -25.2 dB

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Title Annotation: | Research Article |
---|---|

Author: | Li, Yue; Sha, Xuejun; Ye, Liang; Fang, Xiaojie |

Publication: | Mathematical Problems in Engineering |

Date: | Jan 1, 2016 |

Words: | 4801 |

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