# Two- and four-relay selection schemes for application in interference limited legacy networks

- Gaojie Chen
^{1}Email author, - Ousama Alnatouh
^{1}Email author and - Jonathon Chambers
^{1}

**2012**:351

**DOI: **10.1186/1687-1499-2012-351

© Chen et al.; licensee Springer. 2012

**Received: **7 June 2012

**Accepted: **1 November 2012

**Published: **21 November 2012

## Abstract

### Abstract

The limiting effect of multi-user interference from an adjacent cell upon the relays used for cooperative transmission is considered in the context of legacy networks which adopt max(min(·,·)) type relay selection policies. We extend previous work which considered single relay selection to the selection of two or four such relays, as is required in distributed space–time coding. We obtain new analytical expressions for outage probability over Rayleigh frequency flat fading channels for two signal-to-noise ratio regimes. We confirm by simulation that such a relay selection scheme has robustness to relay selection feedback error and outperforms a single relay selection scheme.

## 1 Introduction

Cooperative relaying can be considered as an effective method to combat fading by exploiting spatial diversity [1], and as a way for two users with no or weak direct connection to attain a robust link. One or more relay nodes are generally used in such relaying to forward signals transmitted from the source node to the destination node. In a cooperative communication system, there are two main cooperative methods: decode-and-forward (DF) (regenerative relaying protocol) and amplify-and-forward (AF) (transparent relaying protocol) methods [2]. In the DF method, relay nodes decode the source information and then re-encode and re-transmit it to the destination. In the AF method, relay nodes only amplify and retransmit their received signals, including noise, to the destination. Therefore, compared with DF, AF-type schemes have the advantage of simple implementation and low complexity in practical scenarios. In addition to complexity benefits, it has been shown in [3] that an AF scheme asymptotically, in terms of appropriate power control, approaches a DF one with respect to diversity.

AF has extensively been studied in the literature, but generally in the context of ideal configurations without interference during the cooperation process [4, 5]. However, more practical systems have been studied in [6–8], which consider the presence of interference effects. The effect of cochannel interference on the performance of multihop wireless networks with AF relaying is analyzed in [6]. The performance of a two-hop channel state information-assisted AF system, with co-channel interference at the relay, is analyzed in [8].

In a cooperative relay network, moveover, when many relays can help the source to transmit to the destination, sometimes some relays provide a poor channel quality which can affect the end-to-end transmission quality [9]. Therefore, the use of a relay selection scheme is attracting considerable attention to overcome this problem and preserve the potential diversity gains [10–12], while mitigating the problem in synchronizing a large number of cooperative nodes.

In [10], exact outage and diversity performance expressions for a single relay selection scheme are provided for a wide range of signal-to-noise ratio (SNR) regimes in the context of an AF transmission protocol. The work in [11] relies on using instantaneous end-to-end wireless channel conditions to obtain the best single relay for cooperative diversity. This study was extended in [12] to obtain outage-optimal opportunistic relaying in the context of selecting a single relay from a set of *N* available relays. They show that cooperative diversity gain is achieved even when certain relays remain inactive. However, these relay selection criteria lack the flexibility to deal with the presence or absence of interference effects. In order to improve the practicality, in [13] the effects of multi-user interference are considered for relay nodes and a single relay selection scheme is used to overcome the effects of the interference, in the context of legacy networks. However, using a single best relay is not always sufficient to satisfy the required outage probability at a destination node. Moreover, these works have not considered feedback error for relay selection, which means sometimes the best relay cannot be chosen because the wrong enable feedback information is received from the destination node. We highlight that this is different from recent work which has considered the effect of only delay in the feedback path [14].

Therefore, in this article, in order to overcome these shortcomings, first, the basic AF protocol [15] is considered when external out-of-cell structural/unmanaged interference affects the cooperation process. We also consider maximum ratio combining (MRC) at the destination node and distributed space–time coding (DSTC) to mitigate the associated bandwidth overhead. Furthermore, to facilitate analysis, we just consider interference at the relays and ignore the effect of interference at the destination node, which matches the approach in [13]. Moreover, this study is targeted at legacy systems where max(min(·,·)) type policies are used for relay selection. Second, we focus upon two selection schemes to select two or four relays from a single group of relays. We derive new outage probability expressions for two or four relay selection and compare them with the results for conventional best single relay selection. Finally, we examine by simulation the bit error rate (BER) performance of the best single relay selection scheme and the best two-relay selection scheme, in the presence of errors in the feedback of relay selection information. In practice, this could be as simple as a single permission to transmit bit.

The remainder of this article is organized as follows. The system model and a statistical expression for interference-based AF are described in Section 2. In Section 3., the relay selection criteria for interference-limited systems and asymptotic outage probability analysis are presented. Simulation results for outage probability analysis and impact of relay selection feedback errors are provided in Section 4. And conclusions are drawn in Section 5.

**Notations:** The following notations are used in the article. *ε*(·) represents the statistical expectation operator. A complex zero mean additive white Gaussian noise *n*∼*CN*(0,*N*_{0}), where *N*_{0} is the noise variance; and *Pr*(*A*) is the probability of *A*. *Γ*(*n*) is the Gamma function, and $\mathbb{R}(\xb7)$ denotes the real value.

## 2 System model

*i*th relay and the destination node are given by

*x*and

*x*

^{ ′ }are the source signals from the target and neighboring clusters, respectively, typically drawn from a prescribed finite constellation.

*E*

_{ s }is the average energy per symbol; $\left(\right)close="">{h}_{S{R}_{i}}$, $\left(\right)close="">{h}_{{S}^{\prime}{R}_{i}}$, and $\left(\right)close="">{h}_{{R}_{i}D}$ are channel gains, which are the zero mean, independent, circularly symmetric, complex Gaussian random variables with variances $\left(\right)close="">{\sigma}_{\mathrm{SRi}}^{2}$, $\left(\right)close="">{\sigma}_{{S}^{\prime}{R}_{i}}^{2}$, and $\left(\right)close="">{\sigma}_{{R}_{i}D}^{2}$, between the source and the

*i*th relay node, between the neighboring source and the relay node and between the

*i*th relay node and the destination node; and the complex additive white Gaussian noise $\left(\right)close="">{n}_{{R}_{i}}$ and

*n*

_{ D }are modeled as zero-mean mutually independent, circularly-symmetric, complex Gaussian random variables with variance

*N*

_{0}at the

*i*th relay and the destination node, respectively. The

*i*th relay gain denoted by $\sqrt{{P}_{i}}$ is calculated from

In our model, the source powers at the target and the neighboring cluster are assumed to be the same. This model is representative of an ad-hoc network environment where there is no power control between adjacent clusters.

*N*

_{ s }denotes the set of two or four relay indices for the relays chosen in our relay selection scheme, and each term in the summation has the same form as in [13]. Substituting (3) into (4), the end-to-end SINR is

*i*th relay and

*i*th relay to destination links, respectively. And $\left(\right)close="">{\gamma}_{\mathrm{INFi}}={\gamma}_{{S}^{\prime}{R}_{i}}=|{h}_{{S}^{\prime}{R}_{i}}{|}^{2}{E}_{s}/{N}_{0}$ denotes the interference-to-noise ratio (INR) for the

*i*th relay as a result of the neighboring source. It is difficult to use (5) to find a closed form expression for the probability density function (PDF) of

*γ*

_{ D }, therefore, for high SNR, we provide an asymptotic bound as

which is the sum of the ratios between the SNR of the first hop and the INR of the interference, because when SNR→*∞*, then $\left(\right)close="">\epsilon \left({\gamma}_{\mathrm{INFi}}\right)\epsilon \left({\gamma}_{{R}_{i}D}\right)\gg \epsilon \left({\gamma}_{\mathrm{INFi}}\right)+\epsilon \left({\gamma}_{S{R}_{i}}\right)+\epsilon \left({\gamma}_{{R}_{i}D}\right)+1$. In this case, the statistical description of the system is independent of the second hop.

where *f*(·) and *F*(·) denote the PDF and the CDF, respectively. The parameter $\left(\right)close="">L=\frac{\epsilon \left({\gamma}_{S{R}_{i}}\right)}{\epsilon \left({\gamma}_{\mathrm{INFi}}\right)}=\frac{{\sigma}_{S{R}_{i}}^{2}}{{\sigma}_{{S}^{\prime}{R}_{i}}^{2}}$. Note that the parameter *L* controls the level of interference in the target and neighboring clusters.

Furthermore, considering interference at both the relays and the destination nodes is beyond the scope of this study and is left for analysis in future work. Our two- or four-relay selection scheme assuming interference only at the relays will be implemented as presented in the following sections.

## 3 Two- or four-relay selection with outage probability analysis

In order to introduce our proposed two- or four-relay selection schemes, we need to introduce first the conventional relay selection scheme.

### 3.1 Conventional relay selection

where *N* represents the set of indices of all available relays.

The conventional relay selection policy offers the relay with the “best” end-to-end path between source and destination and provides diversity gain on the order of the number of the relays [12]. However, this relay selection criterion is only considered for environments without interference, and the best relay selection is not always sufficient to achieve the required outage probability at a destination node. Finally, when feedback error is present in the relay selection, the performance of the single relay selection scheme is significantly degraded, further discussion of which will be given in the simulation section. Therefore, to overcome these problems two- and four-relay selection schemes are proposed for use in interference configurations for legacy networks which are restricted to adopt a max(min(·,·)) type policy.

### 3.2 Asymptotic two- and four-relay selection criterion

**b**=(

*i*

*i*

^{ ′ }), i.e., the best pair of relay indices, where

*i*denotes the index of the relay with the best link in

*N*, and

*i*

^{ ′ }is that of the best relay among the remaining

*N*−1. In this approach, we select the best two relay nodes from the

*N*available relays in the group in the cluster, namely, select the relays with the maximum

*γ*

_{max}and the second largest

*γ*

_{max−1}from the

*N*relays instantaneous SNRs. Using the theory of order statistics [20], the selection of the maximum and the second largest is not independent, we therefore can find the joint distribution of the two largest values as

*γ*

_{max}=

*x*and

*γ*

_{max−1}=

*y*. Substituting (7) into (10), we obtain

*γ*

_{up}is the sum of

*x*and

*y*, which are identically distributed and formed as the ratios of exponential random variables. Therefore, the CDF is obtained as

*x*and

*y*are non-negative, with

*x*≥

*y*, then,

*F*

_{2,1}(

*a*

*b*

*c*

*z*) is the first hypergeometric function, which can be calculated by using the Hypergeom Matlab function [21]. Furthermore,

*F*

_{1}(

*a*;

*b*1,

*b*2;

*c*;

*x*

*y*) is a formal extension of the Appell hypergeometric function of two variables, which can also be expressed by the simple integral in [22] as

where *Γ*(*n*)=(*n*−1)! is the Gamma function.

*α*. The outage probability can be expressed as

The outage probability of the best two-relay selection can be expressed by using the CDF expression (14).

**b**=(

*i*,

*i*

^{ ′ },

*i*

^{ ′′ },

*i*

^{ ′′′ }), i.e., the best four relay indices, in which

*i*denotes the index of the relay with the best link in

*N*;

*i*

^{ ′ }is that of the best relay among the remaining

*N*−1, and

*i*

^{ ′′ }is that of the best relay among the remaining

*N*−2, and

*i*

^{ ′′′ }is that of the best relay among the remaining

*N*−3. The joint distribution of the four largest values is

*γ*

_{max}=

*w*,

*γ*

_{max−1}=

*x*,

*γ*

_{max−2}=

*y*and

*γ*

_{max−3}=

*z*. Substituting (7) into (17), we can obtain

*γ*

_{up}is formed as the sum of

*w*,

*x*,

*y*, and

*z*random variables, which are identically distributed and ratios of exponential random variables. Therefore, the CDF is obtained by

*w*,

*x*,

*y*, and

*z*are non-negative, with

*w*≥

*x*≥

*y*≥

*z*, then,

Then, exploiting (21) as in (15), the outage probability can be evaluated, for example for the results in Section 4. we employ the Mathematica software package [23].

In this study, we focus on a two- or four-relay selection approach as it is immediately applicable within a cooperative network, which exploits DSTC [16] to improve the end-to-end performance, such as an Alamouti or Quasi-Orthogonal code, according to the number of selected relays. Furthermore, for our relay selection policy, it requires only the SNR of the links from source to relay nodes and the INR of the interference links which can be obtained by the relay nodes during the early stage of transmission. In terms of the relay selection policy, moreover, the information describing the links between the relay and destination is not required at the destination node, therefore, this policy has a lower complexity than that of [12] and may save feedback set-up time.

### 3.3 Semi-conventional two- and four-relay selection

where **b**=(*i*,*i*^{
′
}), i.e., the best pair of relay indices, where *i* denotes the index of the relay with the best link in *N*, and *i*^{
′
}is that of the best relay among the remaining *N*−1. Here, we need to consider the outage behavior of the ratio $\left(\right)close="">\frac{{\gamma}_{S{R}_{i}}}{{\gamma}_{\mathrm{INFi}}}$ according to the semi-conventional scheme. In order to simplify the approximation of the corresponding outage bound as in [13], two cases will be considered.

*f*(·) and

*F*(·) denote the PDF and the CDF, respectively. Substituting (23) into (10), we obtain

where $\left(\right)close="">{F}_{{\gamma}_{\text{up}}}^{\text{Asy}}\left(\gamma \right)$ and $\left(\right)close="">{F}_{{\gamma}_{\text{up}}}^{\prime}\left(\gamma \right)$ are denoted by (14) and (25), respectively. And the outage probability can be obtained by using (26).

**b**=(

*i*,

*i*

^{ ′ },

*i*

^{ ′′ },

*i*

^{ ′′′ }), i.e., the best four relay indices, wherein

*i*denotes the index of the relay with the best link in

*N*;

*i*

^{ ′ }is that of the best relay among the remaining

*N*−1, and

*i*

^{ ′′ }is that of the best relay among the remaining

*N*−2, and

*i*

^{ ′′′ }is that of the best relay among the remaining

*N*−3. In the first case, the joint distribution of the four largest values can be obtained by substituting (23) into (17), yielding

where $\left(\right)close="">{F}_{{\gamma}_{\text{up}}}^{\text{Asy}}\left(\gamma \right)$ and $\left(\right)close="">{F}_{{\gamma}_{\text{up}}}^{\prime}\left(\gamma \right)$ are given by (21) and (29), respectively. And the outage probability can be evaluated by using (15) and (30), for example with the Mathematica software package [23].

## 4 Simulation results for outage probability analysis and impact of relay selection feedback errors

In this section, in order to verify the results obtained from the above mathematical expressions, we assumed the target source node and the neighboring source node use the same unity transmission power, and there is no direct link between the source and the destination as path loss or shadowing render it unusable. We show outage probability performance of the two- and four-relay selection schemes.

*L*=5 and 20. It can be seen that increasing the number of relays,

*N*, decreases the outage probability, and hence when the number of relays is large, the outage event (no transmission) becomes less likely, for example, with the total number of available relays increasing from 4 to 6, the outage probability of the best two-relay selection is decreased from approximately 0.308 to 0.162 for the semi-conventional case; and from 0.192 to 0.073 for the asymptotic case when the threshold value

*α*is 15 dB and

*L*=5. The outage performance of the asymptotic case closely matches the simulation results, when SNR = 40 dB. Moreover, with increased source-to-interference power ratio, the performance in terms of outage probability is improved.

*L*=5 and 20. It can be seen that increasing the number of relays, decreases the outage probability, for example, with the total number of available relays increasing from 6 to 8, the outage probability of the best four-relay selection is decreased from approximately 0.013 to 0.0025 for the semi-conventional case; and from 0.0022 to 1.6×10

^{−4}for the asymptotic case when the threshold value

*α*is 7 dB and

*L*=5.With increased source-to-interference power ratio, the performance of outage probability again improves. Moreover, the asymptotic results match very well with the simulation results, when SNR = 40 dB.

*L*=5 or 10. Obviously, with increasing the number of selected relays, the outage probability decreases. For example, for the semi-conventional case, when the total number of available relays is 6,

*L*=5 and the threshold value

*α*is 7 dB, the outage probability of a single relay, the best two-relay and the best four-relay selections are approximately 0.1, 0.036, and 0.013, respectively. Furthermore, for the asymptotic case, when

*N*=6,

*L*=10, and

*α*=7 dB, the outage probability of the best relay, the best two-relay, and the best four-relay selections are approximately 0.0045,5.59×10

^{−4}, and 9.01×10

^{−5}, respectively. These results confirm that two- and four-relay selection schemes provide more robust transmission than single relay selection, because for the single relay selection, it just uses a single relay to help the source to transmit the signal. Therefore, we can choose a different number of relays to communicate with the source and destination nodes, according to the target outage probability. Finally, through using DSTC we have the advantage of avoiding the bandwidth overhead in using the MRC scheme. We next consider how the end-to-end BER performance of the relay selection schemes degrades when there is an error in selecting the particular relay(s) to use in transmission.

Next, we compare the BER performance of the best two-relay selection from a group of *N* available relays, *N*=4, with distributed Alamouti code with the best single relay selection in the presence of relay selection feedback errors, when quadrature phase-shift keying symbols are used in transmission.

*L*=50 is assumed. To simulate errors in the feedback of relay selection information from the destination we introduce an error rate in the feedback. An error rate of 0.5 corresponds to 50% of the selections being made in error; that is, rather than selecting the best relay, one of the other relays is chosen with equal probability of selection. As can be seen in Figure 5, when perfect relay selection is made, i.e., an error rate of 0, the BER performance of the best single relay selection is worse than the best two-relay selection for the three different relay selection schemes, which are denoted by circular, square, and diamond dotted lines for the conventional, asymptotic, and semi-conventional schemes, respectively. Moreover, in the presence of errors in the relay selection, i.e., error rate over the range 0 to 1, all of the different best two-relay selection schemes outperform that of the best single relay selection. These results illustrate clearly the increased robustness of the best two-relay selection scheme over the single relay selection scheme in the presence of moderate to severe relay selection feedback errors. For example, for the conventional best two-relay selection scheme, when the SNR is 20 dB, the BER for the conventional best two-relay selection changes from approximately 1×10

^{−4}only to 4.9×10

^{−3}as the error rate changes from 0 to 1, whereas the BER for the single relay selection is increased from approximately 2.15×10

^{−4}to 5.1×10

^{−2}, confirming the improved robustness.

^{−4}only to 8×10

^{−3}as the feedback error rate changes from 0 to 1, whereas the BER for the single relay selection is increased from approximately 3.5×10

^{−4}to 6×10

^{−2}, confirming the improved robustness. Moreover, because of the increased interference from the neighboring cluster, the BER performance of the three selection schemes is slightly worse than that of the small interference environment given in Figure 5, when

*L*=50. For example, for the conventional, asymptotic, and semi-conventional best two-relay selection schemes, when SNR = 25 dB, the BER for the conventional, asymptotic, and semi-conventional best two-relay selection increase from approximately 5×10

^{−6}to 2×10

^{−5}, from approximately 3.8×10

^{−5}to 4.2×10

^{−5}, and from approximately 1.5×10

^{−5}to 3×10

^{−5}as the

*L*decreases from 50 to 10.

## 5 Conclusion

We have examined two different selection schemes which are asymptotical and semi-conventional policies to select the best two and four relays from a group of available relays in the same cluster by using local measurements of the instantaneous channel conditions in the context of legacy systems which adopt max(min(·,·)) type policies. New analytical expressions for the PDF, and CDF of end-to-end SNR were derived together with closed form expressions for outage probability over Rayleigh fading channels. Numerical results were provided to show the advantage of the outage probability performance of the best two- and four-relay selection in a cooperative communication system. Moreover, through simulation study, we confirmed the robustness of the best two-relay selection scheme in the presence of moderate to severe relay selection feedback errors.

## Declarations

### Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for improving the clarity of this article.

## Authors’ Affiliations

## References

- Laneman JN, Wornell GW: Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks.
*IEEE Trans. Inf. Theory*2003, 49(10):2415-2425. 10.1109/TIT.2003.817829MathSciNetView ArticleMATHGoogle Scholar - Dohler M, Li YH:
*Cooperative Communications: Hardware, Channel and PHY*. Wiley & Sons Ltd., New York; 2010.View ArticleGoogle Scholar - Nabar RU, Bolcskei H, Kneubuhler FW: Fading relay channels: performance limits and space-time signal design.
*IEEE J. Sel. Areas Commun*2004, 22(6):1099-1109. 10.1109/JSAC.2004.830922View ArticleGoogle Scholar - Madsen AH, Zhang J: Capacity bounds and power allocation for wireless relay channels.
*IEEE Trans. Inf. Theory*2005, 51: 2020-2040. 10.1109/TIT.2005.847703View ArticleMathSciNetMATHGoogle Scholar - Lai L, Liu K, Gamal HE: The three-node wireless network: achievable rates and cooperation strategies.
*IEEE Trans. Inf. Theory*2006, 52: 805-828.View ArticleMathSciNetMATHGoogle Scholar - Ikki SS, Aissa S: Multihop wireless relaying systems in the presence of cochannel interferences: performance analysis and design optimization.
*IEEE Trans. Veh. Technol*2012, 61(2):566-573.MathSciNetView ArticleGoogle Scholar - Ikki S, Aissa S: Impact of imperfect channel estimation and co-channel interference on dual-hop relaying systems.
*IEEE Commun. Lett*2012, 16(3):324-327.View ArticleMATHGoogle Scholar - Suraweera HA, Garg HK, Nallanathan A: Performance analysis of two hop amplify-and-forward systems with interference at the relay.
*IEEE Commun. Lett*2010, 14(8):692-694.View ArticleGoogle Scholar - Bali Z, Ajib W, Boujemaa H: Distributed relay selection strategy based on source-relay channel. In
*Proceedings of the IEEE ICT*. Montreal, Canada; April 2010.Google Scholar - Adinoyi A, Fan YJ, Yanikomeroglu H, Poor HV, Al-Shaalan F: Performance of selection relaying and cooperative diversity.
*IEEE Trans. Wirel. Commun*2009, 12(8):5790-5795.View ArticleGoogle Scholar - Bletsas A, Knisti A, Reed DP, Lippman A: A simple cooperative diversity method based on network path selection.
*IEEE J. Sel. Areas Commun*2006, 24(3):659-672.View ArticleGoogle Scholar - Bletsas A, Shin H, Win MZ: Cooperative communications with outage-optimal opportunistic relaying.
*IEEE Trans. Wirel. Commun*2007, 6(9):3450-3460.View ArticleGoogle Scholar - Krikidis I, Thompson JS, McLaughlin S, Goertz N: Max-min relay selection for legacy amplify-and-forward systems with interference.
*IEEE Trans. Wirel. Commun*2009, 8(6):3016-3027.View ArticleGoogle Scholar - Musavian L, Aissa S, Lambotharan S: Effective capacity for interference and delay constrained cognitive radio relay channels.
*IEEE Trans. Wirel. Commun*2010, 9(10):1698-1707.View ArticleGoogle Scholar - Laneman JN, Tse D, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior.
*IEEE Trans. Inf. Theory*2004, 50(12):3063-3080.MathSciNetView ArticleMATHGoogle Scholar - Jing Y, Jafarkhani H: Using orthogonal and quasi-orthogonal designs in wireless relay networks.
*IEEE Trans. Inf. Theory*2007, 53(11):4106-4118.MathSciNetView ArticleMATHGoogle Scholar - Michalopoulos DS, Karagianidis GK: Performance analysis of single relay selection in Rayleigh fading.
*IEEE Trans. Wirel. Commun*2008, 7(11):3718-3724.View ArticleGoogle Scholar - Anghel PA, Kaveh M: Exact symbol error probability of a cooperative network in a Rayleigh-fading environment.
*IEEE Trans. Wirel. Commun*2004, 3(5):1416-1421. 10.1109/TWC.2004.833431View ArticleGoogle Scholar - Nadarajah S, Kotz S: On the product and ratio of Gamma and Weibull random variables.
*Econ. Theory (Cambridge University Press)*2006, 22: 338-344.MathSciNetMATHGoogle Scholar - Balakrishnan N, Cohen AC:
*Order Statistics and Inference: Estimation Methods*. Academic Press, London; 1991.MATHGoogle Scholar - Gilat A, Subramaniam V:
*Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB*. Wiley & Sons Ltd., New York; 2011.Google Scholar - Bailey WN: On the reducibility of Appell’s function.
*Quart. J. Math. (Oxford)*1934, 5(1):291-292. 10.1093/qmath/os-5.1.291View ArticleMATHGoogle Scholar - Wolfram S:
*The Mathematica Book*. Wolfram Media Press, Champaign, Illinois, U.S.A.; 2003.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.