Open access peer-reviewed chapter

# Achievable Energy Efficiency and Spectral Efficiency of Large‐ Scale Distributed Antenna Systems

By Wei Feng, Ning Ge and Jianhua Lu

Submitted: April 19th 2016Reviewed: September 28th 2016Published: December 14th 2016

DOI: 10.5772/66049

## Abstract

In the large‐scale distributed antenna system (LS‐DAS), a large number of antenna elements are densely deployed in a distributed way over the coverage area, and all the signals are gathered at the cloud processor (CP) via dedicated fiber links for globally joint processing. Intuitively, the LS‐DAS can inherit the advantage of both large‐scale multiple‐input‐multiple‐output (MIMO) and network densification; thus, it offers enormous gains in terms of both energy efficiency (EE) and spectral efficiency (SE). However, as the number of distributed antenna elements (DAEs) increases, the overhead for acquiring the channel state information (CSI) will increase accordingly. Without perfect CSI at the CP, which is the majority situation in practical applications due to limited overhead, the claimed gain of LS‐DAS cannot be achieved. To solve this problem, this chapter considers a more practical case with only the long‐term CSI including the path loss and shadowing known at the CP. As the long‐term channel fading usually varies much more slowly than the short‐term part, the system overhead can be easily controlled under this framework. Then, the EE‐oriented and SE‐oriented power allocation problems are formulated and solved by fractional programming (FP) and geometric programming (GP) theories, respectively. It is observed that the performance gain with only long‐term CSI is still noticeable and, more importantly, it can be achieved with a practical system cost.

### Keywords

• large‐scale distributed antenna system (LS‐DAS)
• energy efficiency (EE)
• spectral efficiency (SE)
• long‐term channel state information (CSI)
• fractional programming (FP)
• geometric programming (GP)

## 1. Introduction

The large‐scale distributed antenna system (LS‐DAS) is a promising candidate technology for the future 5G wireless network. In a LS‐DAS, as shown in Figure 1, a large number of distributed antenna elements (DAEs) are densely scattered over the coverage area, and the signals from/to all the DAEs are gathered via dedicated fiber links, at the cloud processor (CP), where the globally joint processing is performed [1, 2]. On one hand, the LS‐DAS can be regarded as a special large‐scale multiple‐input‐multiple‐output (MIMO) system, as shown in Figure 2, with distributed deployment of antenna elements. On the other hand, it can be regarded as a special heterogeneous small‐cell network, as shown in Figure 3, with global intercell coordination. As a consequence, the LS‐DAS can inherit the advantage of both large‐scale MIMO and network densification. Notably, existing studies have already shown that it can offer enormous gains in terms of both energy efficiency (EE) [3, 4] and spectral efficiency (SE) [5, 6].

Due to the distributed deployment of antenna elements, the average access distance of all the mobile terminals (MTs) is reduced. Moreover, due to the global coordination among all the DAEs, the multiplexing gain and diversity gain offered by multiple antenna elements can be obtained [79]. These are the main reasons for high EE and SE offered by a LS‐DAS. However, to exploit the benefit of LS‐DASs, the channel state information (CSI) is crucially required at the CP [10, 11]. Without perfect CSI, the interference among different DAEs will become intractable, and accordingly the system performance will be severely degraded.

In practical applications, the acquisition of full CSI would require an overwhelming amount of system overhead, including the training symbols for channel estimation, the system backhaul for CSI exchanging, and so on. Due to this point, in the literature, some researchers have shown that the system cost of CSI is quite an important issue for evaluating and designing multi‐antenna systems. For example, in [12], it has been proved that the optimal number of transmit antennas is equal to the channel coherence interval (CCI). Thus, it will become useless to utilize more antennas than CCI under given channel dynamics. The authors of [13] particularly focused on the cost of CSI for network MIMO systems; they have shown that the optimal number of base stations that can be coordinated exists, which is mainly determined by the CCI in both time and frequency domains. Particularly, for the massive MIMO in frequency division duplex (FDD) mode, it is also very challenging to acquire full CSI at the base station side. In [14], a one‐bit feedback scheme was proposed by using a set of predefined precoding vectors. The scheme only performs well in some specific cases, e.g., the multi‐antenna channel following one‐ring model.

In this chapter, we try to liberate the implementation of LS‐DAS from the acquisition of full CSI. We note that the channel of a LS‐DAS usually consists of path loss, shadowing, and Rayleigh fading [79]. Compared with Rayleigh fading, path loss and shadowing vary much more slowly and can be estimated in a much longer interval than CCI. Thus, it requires a controllable system overhead. In some of the existing studies, path loss and shadowing are classified as large‐scale CSI [4, 6]. To distinguish from the largescale in LS‐DAS, for clarity, we here use long‐term CSI to identify path loss and shadowing. With the knowledge of long‐term CSI, the achievable EE and SE will be particularly investigated in the sequel. Different from the reported EE and SE with perfect CSI assumption, which actually cannot be achieved in most practice, our results can be approached with a limited system cost; thus, it is of great significance for the realistic implementation of LS‐DASs.

In order to control the computational complexity at the CP, we first divide the whole system into a number of virtual cells (VCs) [5, 15]. As shown in Figure 4, the VC is established in a user‐centric manner, i.e., each MT chooses only a subset of the surround DAEs for its data transmission. Then, each MT is served by its own VC under the interference from other VCs. To control the interference, the signals of all the VCs are designed in a coordinated fashion at the CP while maximizing the EE or SE of the system. Given VCs, the EE‐oriented and the SE‐oriented power allocation problems are formulated based on long‐term CSI only, both of which are non‐convex problems, and thus are difficult to solve. By adopting the fractional programming (FP) theory and the geometric programming (GP) theory, we propose two iterative power allocation algorithms. These algorithms can derive the locally optimal EE and SE of the system, respectively. It is further observed from the simulation results that the performance gain with only long‐term CSI is still remarkable, while it can be achieved with a practical system cost.

The rest of this chapter is organized as follows. The system model of a multiuser LS‐DAS is described in Section 2. In the subsequent Sections 3 and 4, the achievable EE and SE are discussed, respectively. Then, we show the simulation results to verify the superiority of the proposed schemes in Section 5. Finally, the conclusion of this chapter is drawn in Section 6.

Notations: Indenotes an identity matrix with a dimension of n, and Ois a zero matrix. (.)Hrepresents the conjugate transpose operation. M×Ndenotes the set of complex M×Nmatrices, and CNrepresents a complex Gaussian distribution. E(.)represents the expectation operator, and tr()represents the trace operator.

## 2. System Model

We consider a LS‐DAS with KMTs. Without loss of generality, all the VCs consist of NDAEs, and the number of antenna elements equipped at each MT is M.

For MT k, the received signal is

y(k)=H(k)x(k)+i=1,ikKH(k,i)x(i)+n(k),E1

where H(k)M×N,k=1,2,…,K,represents the channel between the DAEs in VC kand MT k, H(k,i)M×N,k=1,2,…,K,i=1,2,…,K,denotes the channel between the DAEs in VC iand MT k, x(i)N×1,i=1,2,…,K, is the transmitted signal vector for MT i, and n(k)M×1,k=1,2,…,K, denotes the additive white Gaussian noise with distribution CN(0,σ2IM).

E[x(k)x(k)H]=P(k)=[p1(k)pN(k)],k=1,…,K.E2

Assuming a total transmit power constraint Pmax(k)for MT k, we set

n=1Npn(k)Pmax(k).E3

The channel matrix can be modeled as

H(k,i)=S(k,i)L(k,i),E4

where S(k,i)and L(k,i)reflect the short‐term fading and the long‐term fading, respectively. Particularly, the entries of S(k,i)are all independent and identically distributed (i.i.d.) circular symmetric complex Gaussian variables following CN(0,1)distribution.

L(k,i)=[l1(k,i)lN(k,i)],E5

with

ln(k,i)=(Dn(k,i))γSn(k,i),n=1,2,…,N,E6

where Dn(k,i)is the transmission distance between the DAE nin VC iand MT k, and γis the path loss exponent, and Sn(k,i)represents the shadow fading caused by large objects such as tall buildings or walls.

## 3. Achievable Ee

Given perfect CSI, the authors of [16] have proposed an energy‐efficient power allocation scheme for traditional DASs. In [17], further taking the inter‐VC interference into consideration, an iterative power allocation scheme was presented to improve the EE of a LS‐DAS, via applying the successive Taylor expansion method. In contrast, we investigate the achievable EE with the long‐term CSI only in this section.

First of all, the sum rate of the system can be derived according to Eq. (1) as

R=k=1Klog2det(IM+H(k)P(k)H(k)Hσk2),E7

where

σk2=i=1,ikKn=1N[ln(k,i)]2pn(i)+σ2,E8

is the total interference‐plus‐noise power at MT k.

When only the long‐term CSI is known, the average sum rate can be calculated via taking expectation over the short‐term channel fading Ω={S(k)|k=1,,K}as

R¯=k=1KEΩ[log2det(IM+H(k)P(k)H(k)Hσk2)].E9

Then, the EE of the system, denoted as η, can be derived as

η=R¯ρk=1Kn=1Npn(k)+Pc,E10

where

ρ=εγ,E11

with εand γdenoting the peak‐to‐average power ratio and the power amplifier efficiency, respectively, and Pcdenotes the circuit power consumption [4].

In order to investigate the achievable EE under this framework, we formulate the following optimization problem:

maxηE12a
s.t.n=1Npn(k)Pmax(k),E12b
pn(k)0,k=1,,K,n=1,,N.E12c

Because of the non‐convexity of R¯, the problem shown in Eq. (12) is a complicated non‐convex problem [18]. To simplify it, we introduce an upper bound to the objective function as

η^=k=1Klog2det(IN+MP(k)(L(k))2σk2)ρk=1Kn=1Npn(k)+Pc,E13

the numerator of which is an upper bound to R¯[10]. Accordingly, the problem in Eq. (12) can be reformulated as

maxη^E14a
s.t.n=1Npn(k)Pmax(k),E14b
pn(k)0,k=1,,K,n=1,,N.E14c

which is simpler than Eq. (12). However, it is still non‐convex [18]. To further solve the problem in Eq. (14), we express

η^=f1f2ρk=1Kn=1Npn(k)+Pc,E15

where

f1=k=1Klog2det(σk2IN+MP(k)L(k)2),E16a
f2=k=1KNlog2(σk2),E16b

both of which are clearly concave functions.

We find that if the numerator of η^, i.e., f1f2, can be transformed into a concave form, the problem in Eq. (14) can be recast as a quasi‐concave fractional programming problem, further considering the linearity of its denominator [19]. Toward this end, we linearize f2by applying the first‐order Taylor expansion at a given point P¯as

f˜2(P|P¯)=k=1KNlog2(σk2(P¯))+log2(e)k=1KNσk2(P¯)tr(Gk[PP¯]),E17

where P={P(1),,P(k)}and

Gk=diag{G(k,1),,G(k,K)},E18a
G(k,i)=(L(k,i))2,ki,k,i=1,,K,E18b
G(k,k)=O.E18c

By substituting f˜2(P|P¯)for f2(P), the problem in Eq. (14) can be approximated as

maxη¯=f1(P)f˜2(P|P)ρk=1Kn=1Npn(k)+PcE19a
s.t.n=1Npn(k)Pmax(k),E19b
pn(k)0,k=1,,K,n=1,,N,E19c

whose objective function is fortunately fractional with concave numerator and convex denominator [18]. Adopting the FP theory, the problem in Eq. (19) can be optimally solved in an iterative way. In our previous paper [4], we have shown in detail how to solve the problem in Eq. (19). In the following, for brevity, we just present the basic idea and procedure of the iterative algorithm.

We use t1and s1to denote the successive Taylor expansion iteration step and the FP iteration step, respectively. After introducing a positive variable ω, the following concave optimization problem can be formulated

maxv(P|Pt1,s1,ω)E20a
s.t.n=1Npn(k)Pmax(k),E20b
pn(k)0,k=1,,K,n=1,,N,E20c

where

v(P|Pt1,s1,ω)=f1(P)f˜2(P|Pt1,s1)ωρk=1Kn=1Npn(k)ωPc.E21

Further define

V(ω)=maxv(P|Pt1,s1,ω),E22

Algorithm 1 Iterative power allocation for maximizing EE.

1. Initialization: P0=diag{P0(1),,P0(K)}with P0(k)=Pmax(k)NIN, k=1,…,K, P0,0(k)=P0(k), k=1,…,K, ω=0, and ξ=1×103, δ=1×103, t=1,s=1;

2. Solve Eq. (20), and denote the obtained power matrix by P0,1(k), k=1,…,K, set P1(k)=P0,1(k), k=1,…,K, and P1=diag{P1(1),,P1(K)};

3. while |η^(Pt)η^(Pt1)|/η^(Pt1)>ξdo

4. t=t+1, s=1, and ω=0;

5. Pt1,0(k)=Pt1(k), k=1,…,K;

6. Solve Eq. (20), derived V(ω)and denote the obtained power matrix by Pt1,1(k), k=1,…,K;

7. while V(ω)>δdo

8. ω=η¯(Pt1,s(k)|Pt1(k));

9. s=s+1;

10. Solve Eq. (20), derived V(ω)and denote the obtained power matrix by Pt1,s(k), k=1,…,K;

11. end while

12. Pt(k)=Pt1,s(k), k=1,…,K, and Pt=diag{Pt(1),,Pt(K)};

13. end while

14. Output: Pt.

we can propose an iterative power allocation algorithm for maximizing EE, as described in Algorithm 1. By adopting Algorithm 1, the achievable EE with long‐term CSI only can be derived with low computational complexity [4].

## 4. Achievable Se

For traditional single‐cell DASs, the achievable SE was studied in [20, 21], which by considering the general DAS with random antenna layout has identified that DAS outperforms colocated multi‐antenna systems. In [22], the authors have taken the inter‐cell interference into consideration, and they have presented a close‐form expression for the achievable EE in a multi‐cell environment. However, this work has not considered interference coordination. The authors of [23] took a step further; they have put forward a coordinated power allocation scheme for dealing with the inter‐cell interference. Nevertheless, the result was derived by approximately treating the inter‐cell interference as Gaussian noise, and thus it is only applicable to the low signal‐to‐noise‐ratio (SNR) situation. In a recent work, the SE of single‐cell multiuser LS‐DAS was studied [24]. It however also has not considered interference coordination, which is in general inevitable in most practical applications. Different from all the above existing studies, in this section, we investigate the achievable SE of a LS‐DAS with long‐term CSI only.

With the target of average system sum rate maximization, the problem of SE‐oriented power allocation can be formulated as

maxR¯E23a
s.t.n=1Npn(k)Pmax(k),E23b
pn(k)0,k=1,…,K,n=1,…,N.E23c

As R¯is non‐convex, this problem is complicatedly non‐convex [18]. Besides, the objective function is actually in an integral form as a result of the expectation operator in R¯, and it cannot be directly expressed in a compact closed form, which renders it even more challenging to obtain the optimal solution of Eq. (23).

We try to simplify the formulated problem. To this end, a closed‐form approximation for the average system sum rate R¯is leveraged as

R¯ap=k=1Kn=1Nlog2(1+[ln(k)]2pn(k)ϒk1Mσk2)+Mk=1Klog2(ϒk)Mk=1Klog2e(1ϒk1),E24

where ϒksatisfies

ϒk=1+n=1N[ln(k)]2pn(k)σk2+[ln(k)]2pn(k)ϒk1M,k=1,..,K.E25

This approximation can be derived through using the random matrix theory [10], and the introduced parameter ϒkcan be calculated in an iterative way as shown in the following Algorithm 2.

According to the existing studies [10], R¯apis quite a precise approximation for R¯. Therefore, we directly use it as the objective function, and the joint power allocation problem can be recast as

maxR¯apE26a
s.t.n=1Npn(k)Pmax(k),E26b
pn(k)0,k=1,…,K,n=1,…,N,E26c

which is much simplified. However, due to the non‐convexity of R¯ap[18], the new problem in Eq. (26) is still non‐convex. In the following, we explore the achievable SE of the system by contriving an iterative algorithm, which can find a locally optimal solution of Eq. (26) efficiently.

To eliminate the effect of the introduced parameters ϒ1, ϒ2, , ϒK, we first fix ϒ1, ϒ2, , ϒKas constants. Then we can equivalently simplify the objective function in Eq. (26) as

R¯ap'=k=1Kn=1Nlog2(1+[ln(k)]2pn(k)ϒk1Mσk2).E27

As log2()is monotonically increasing, the problem shown in Eq. (26) can be equivalently transformed into

mink=1Kn=1Nσk2σk2+[ln(k)]2pn(k)ϒk1ME28a
s.t.n=1Npn(k)Pmax(k),E28b
pn(k)0,k=1,…,K,n=1,…,N.E28c

Define

fn,k(P)=σk2(P)+[ln(k)]2pn(k)ϒk1M=i=1,ikKj=1Ngj(k,i)(P)+gn(k)(P)+σ2,n=1,…,N,k=1,…,K,E29

where

gj(k,i)(P)=[lj(k,i)]2pj(i),ki,E30
gn(k)(P)=[ln(k)]2pn(k)ϒk1M,E31

and then, given a feasible point P¯, an approximation of fn,k(P)can be obtained as

f˜n,k(P|P¯)=(i=1,ikKj=1N(gj(k,i)(P)αn,j(k,i))αn,j(k,i))×(gn(k)(P)αn,n(k))αn,n(k)×(σ2αn,k0)αn,k0,E32

where

αn,j(k,i)=gj(k,i)(P¯)/fn,k(P¯),E33
αn,n(k)=gn(k)(P)/fn,k(P¯),E34
αn,k0=σ2/fn,k(P¯).E35

By using the inequality of arithmetic and geometric means, it is easy to obtain that

fn,k(P)f˜n,k(P|P¯).E36

The equality holds if and only if

P=P¯.E37

By replacing fn,k(P)with f˜n,k(P|P¯), the problem in Eq. (28) can be recast as

mink=1Kn=1Nσk2f˜n,k(P)E38a
s.t.n=1Npn(k)Pmax(k),E38b
pn(k)0,k=1,…,K,n=1,…,N,E38c

which is a good approximation for the original problem in the neighborhood of P¯. More importantly, it is a standard GP problem [25]; thus, it can be efficiently solved via convex optimization tools, e.g., the interior point algorithm [18].

We use t1and s1to denote the updating iteration step of ϒkand the arithmetic‐to‐geometric approximation iteration step, respectively. Then the following convex optimization problem is derived

mink=1Kn=1Nσk2f˜n,k(P|Ps1,ϒkt)E39a
s.t.n=1Npn(k)Pmax(k),E39b
pn(k)0,k=1,…,K,n=1,…,N.E39c

Accordingly, we propose an iterative power allocation algorithm for maximizing SE as described in Algorithm 2. In the algorithm, ϒk,k=1,…,Kand Pare updated in an alternate way. By adopting the algorithm, the achievable SE with long‐term CSI only can be derived with low computational complexity [6].

## 5. Simulation Results

In this section, we illustrate the EE and SE performance of the proposed schemes by simulations. To be general, we consider a circular coverage area with a radius of 500 m. There are 20 DAEs randomly deployed in the coverage area with a two‐dimension uniform distribution. The number of MTs is set as K=3. The number of antenna elements equipped at each MT is set as M=3. In order to fully exploit the spatial degree of freedom of each MT and, in the meantime, well control the system complexity, we set the size of each VC as N=M=3. As for the channel parameters, we set γ=4(path loss exponent), σ2=107dBm (noise power), and the shadowing standard deviation is 8 dB. Without loss of generality, we consider the same transmit power constraint for all MTs, i.e., Pmax(1)=Pmax(2)=Pmax(3). Particularly, 100 randomly selected system topologies are considered in the simulation, and the averaged results are shown in the following.

First, the achievable EE of different schemes is compared in Figure 5. Both the scheme presented in reference [16] and the simplest equal power allocation scheme are considered. It can be seen from Figure 5 that the proposed scheme outperforms the other ones, especially when the transmit power constraint goes larger. The scheme proposed in [16] has not considered interference coordination; thus, in a multi‐VC setting, its performance is worse than the proposed scheme, although it has assumed the perfect CSI as the CP. In contrast, although using the long‐term CSI only, the proposed scheme can still offer the highest EE performance. We can also observe from Figure 5 that the key point for high EE is to set proper transmit power, i.e., when the transmit power has reached a corresponding point, it should no longer be increased even though the power consumption constraint goes larger. Intuitively, this observation can be explained by the fact that when the transmit power goes larger, the sum rate gain will become smaller and smaller due to the impact of interference; thus, the EE of the scheme will fall instead of rising.

Algorithm 2 Iterative power allocation for maximizing SE.

1. Initialization: Set P0={[p1(1)]0,[p2(1)]0,…,[pN(K)]0}, where [pn(k)]0=Pmax(k)N, k=1,…,K, n=1,…,N, and ε=1×104, δ=1×103, s=1;

2. for k=1to Kdo

3. t=1;

4. ϒk0=1;

5. ϒk1=1+n=1N[ln(k)]2[pn(k)]0σk2(P0)+[ln(k)]2[pn(k)]0[ϒk0]1M;

6. while |ϒktϒkt1|>εdo

7. t=t+1;

8. ϒkt=1+n=1N[ln(k)]2[pn(k)]0σk2(P0)+[ln(k)]2[pn(k)]0[ϒkt1]1M;

9. end while

10. Output ϒk'=ϒkt,k=1,…,K.

11. end for

12. Solve Eq. (39) with ϒk=ϒk',k=1,…,K, and denote the obtained power matrix by P1;

13. while |R¯ap(Ps)R¯ap(Ps1)|/R¯ap(Ps1)>δdo

14. for k=1to Kdo

15. t=1;

16. ϒk0=1;

17. ϒk1=1+n=1N[ln(k)]2[pn(k)]sσk2(Ps)+[ln(k)]2[pn(k)]s[ϒk0]1M;

18. while |ϒktϒkt1|>ϵdo

19. t=t+1;

20. ϒkt=1+n=1N[ln(k)]2[pn(k)]sσk2(Ps)+[ln(k)]2[pn(k)]s[ϒkt1]1M;

21. end while

22. Output ϒk'=ϒkt,k=1,…,K.

23. end for

24. s=s+1;

25. Solve Eq. (39) with ϒk=ϒk',k=1,…,K, and denote the obtained power matrix by26: Ps;

26. end while

27. Output: Ps.

Then, we evaluate the performance of the proposed scheme in terms of achievable SE. The scheme presented in reference [23] and equal power allocation scheme are taken into comparison. The results are shown in Figure 6. We can find that the proposed scheme performs the best among the three schemes. The scheme presented in [23] is only applicable to the low SNR condition; thus, the performance gas between it and the proposed scheme goes larger when the transmit power constraint increases, which implies that the impact of inter‐VC interference becomes bigger. The results identify that it is still effective for enhancing the SE of the system when only the long‐term CSI is available.

According to the discussion in [4, 6], the proposed Algorithms 1 and 2 are assured to converge to a local optimum. The histogram of the number of iteration steps is illustrated in Figures 7 and 8, for Algorithms 1 and 2, respectively. We can observe from the figures that 15 iteration steps are enough for the convergence of Algorithm 1 and that for Algorithm 2 is 11.

## 6. Conclusions

The LS‐DAS is a promising candidate technology for the future 5G wireless network, due to its remarkable gains in terms of both EE and SE. In this chapter, we try to liberate the implementation of LS‐DAS from the acquisition of full CSI. With the knowledge of long‐term CSI, including the path loss and shadow fading, the achievable EE and SE have been investigated. Different from the reported EE and SE with perfect CSI condition, which actually cannot be achieved in most practice, our results can be achieved with a limited system cost; thus, it is of great significance for the realistic implementation of LS‐DASs. We also use the concept of VC to control the computational complexity at the CP. Accordingly, we design the transmit power of all the VCs in a coordinated fashion, to control the interference and finally maximize EE or SE of the system. Particularly, the EE‐oriented and the SE‐oriented power allocation problems are formulated based on long‐term CSI only, both of which are non‐convex problems, and thus are difficult to solve. By adopting the FP theory and the GP theory, we propose two iterative power allocation algorithms. These algorithms can derive the locally optimal EE and SE of the system, respectively. It is further observed from the simulation results that the performance gain with only long‐term CSI is still remarkable, while it can be achieved with a practical system overhead.

## Acknowledgments

This work was supported in part by the National Science Foundation of China for Young Scholars under grant no. 61201186 and the National Basic Research Program of China under grant no. 2013CB329001 and the National Science Foundation of China under grant no. 61132002.

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Wei Feng, Ning Ge and Jianhua Lu (December 14th 2016). Achievable Energy Efficiency and Spectral Efficiency of Large‐ Scale Distributed Antenna Systems, Towards 5G Wireless Networks - A Physical Layer Perspective, Hossein Khaleghi Bizaki, IntechOpen, DOI: 10.5772/66049. Available from:

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