EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS Eur. Trans. Telecomms. 2007; 18:761–768 Published online 12 February 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/ett.1165
Optical Communications 2-Dimensional code design for an optical CDMA system with a parallel interference cancellation receiver Mika¨el Morelle, Claire Goursaud, Anne Julien-Vergonjanne∗ , Christelle Aupetit-Berthelemot, Jean-Pierre Cances and Jean-Michel Dumas University of Limoges/XLIM Dpt—C2S2 UMR CNRS 6172, Ensil, 16 rue d’Atlantis BP 6804, 87068 Limoges, France
SUMMARY The objective of this paper is to design a two-Dimensional Optical Code Division Multiple Access system (2D-OCDMA) for application in access networks, with coding and decoding functions performed by electronic devices. We present a new construction method of Multi-Wavelength Optical Orthogonal Codes (MWOOC), which permits a high flexibility in the code parameters choice. This work evaluates in the noiseless case, the MWOOC potentialities for two receiver structures: a Conventional Correlation Receiver (CCR) and a Parallel Interference Cancellation receiver (PIC). We show that with a PIC receiver, it is possible to design two-Dimensional codes that respect the access specifications. Copyright © 2007 John Wiley & Sons, Ltd.
1. INTRODUCTION Optical Code Division Multiple Access (OCDMA) technique exhibits as an alternative for fibre optic high-speed access networks [1–3]. The multiple access method consists in allocating to each user a specific and distinct code. A number of different schemes and different coding methods have already been proposed. The first solution deals with spreading codes in one dimension (1D): temporal [2, 4] or spectral [5, 6]. Recently, new coding schemes based on both 1D coding methods (temporal and spectral) simultaneously have been performed. These approaches are called two dimensions (2D) coding methods and are the most promising solutions for practical OCDMA networks [7]. Most of the 2D code constructions are issued from the 1D code families such as: prime/prime [8], Optical Orthogonal Codes (OOC)/prime [9], prime/EQC [10], OOC/OOC [11, 12]. In this paper, we focus on Multi-Wavelength Op-
tical Orthogonal Code (MWOOC) based on the OOC/OOC spreading [11, 12]. Either existing methods for MWOOC construction [11, 12] impose a fixed weight for a value of cross-correlation equal to one or a number of wavelengths L equal to the temporal length. To obtain more flexibility in the code parameters choice, we have developed a new 2D code construction based on the MWOOC method described in Reference [11]. In the first part, the 2D-OCDMA system and the new 2D coding method are described. Then, a conventional correlation method is used at the receiver end. We show that this solution is not advantageous in terms of spectral efficiency. Therefore, we investigate in the last part, a more complex receiver structure based on a Parallel Interference Cancellation (PIC) [13, 14], which permits to decrease the Multiple Access Interference (MAI) effect. We show that the new 2D code construction along with a PIC receiver can be a suitable solution for OCDMA systems.
* Correspondence to: Anne Julien-Vergonjanne, University of Limoges/XLIM Dpt—C2S2 UMR CNRS 6172, Ensil, 16 rue d’Atlantis BP 6804, 87068 Limoges, France. E-mail:
[email protected]
Copyright © 2007 John Wiley & Sons, Ltd.
Received 14 November 2005 Revised 5 May 2006 Accepted 26 July 2006
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Figure 1. 2D-OCDMA emission scheme.
represented by a matrix (L × F ):
2. SYSTEM DESCRIPTION
RL,F (t) = r1,F (t),
2.1. The 2D-OCDMA system We consider an incoherent and asynchronous OCDMA system using 2D unipolar codes. Each user employs an On/Off Keying (OOK) modulation to transmit independent and equiprobable binary data upon an optical channel. Before transmission, data are coded by multiplication with a code matrix of dimension (L × F ). 2D codes are defined by: (L × F , W, ha , hc ) where L is the number of wavelengths, F is the temporal code length (the bit period is subdivided in F intervals called chips whose duration is Tc), W is the weight corresponding to the number of chips set to one, ha and hc are the auto and cross-correlation values. The 2D coding scheme for Nu simultaneous users is presented on Figure 1. We employ light sources such as τ Tc (with τ the coherence time), in this case the beat noise is assumed to be negligible compared to MAI [15]. j The jth user 2D code CL,F , is a matrix composed of L row j
vectors dk,F related to the temporal spreading: j j CL,F = d1,F j
j
j
d2,F j
j
...
dL−1,F
j
j
dL,F
T
(1)
j
with dk,F = [ck,1 , ck,2 , . . . , ck,F ] and ck,i ∈ {0, 1} k is related to the emitted wavelength: k ∈ {1, . . . , L} The signals rk,F (t) are carried on the wavelength λk . They are expressed as: rk,F (t) =
Nu
j
j
bi (t)dk,F
(2)
j=1
Copyright © 2007 John Wiley & Sons, Ltd.
...
rL,F (t)
T
(3)
2.2. Construction of MWOOC codes j
The construction of a MWOOC code matrix CL,F based on a 1D-code vector called OOC [4]. In order to generate MWOOC with high flexibility, we have modified the construction method presented in Reference [11]. The new method permits to construct codes such as L ≤ F and for any value of W. The objective is to construct 2D code with the values of L and F as low as possible and the highest number of users. Indeed, for an electrical implementation of coding function, the maximal temporal code length F has to be low. For example, with an electrical bandwidth of 10 GHz and a data rate of 155 Mbit/s, the maximal temporal code length is Fmax = 64. Furthermore, a low number of wavelengths L is easier to implement and limits the amount of beat noise. Let [ai,0 ; . . . ; ai,w−1 ] be the chip one ‘position’ vector of a 1D OOC code where i ∈ [0, NOOC − 1]. NOOC is the number of 1D available sequence given by the following expression [4]: F −1 NOOC ≤ (4) W (W − 1) with F , the temporal length of the 1D OOC code. A MWOOC code matrix [11] is constructed with the dimension (F × F ) from the Equation (5):
(ai,0 + k, j · ai,0 ), . . . , (ai,w−1 + k, j · ai,w−1 ) mod F
j
where bi (t) ∈ {0, 1} is the ith data bit of the jth user. The L signals rk,F (t) are multiplexed and the total signal RL,F (t) is transmitted on the optical fibre. It can be
r2,F (t)
(5) with j, k ∈ [0, F − 1]2 . Eur. Trans. Telecomms. 2007; 18:761–768 DOI: 10.1002/ett
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2-DIMENSIONAL CODE DESIGN FOR OCDMA
For each value of (i, j and k), we obtain matrices composed of W couples (u, v) representing the 2D positions of pulses one; u is related to the transmitted wavelength and v represents the temporal pulse position. Because the 2D coding method allows only one pulse per row, we have for each (i, k) values corresponding to the spectral pulses position, a set of W wavelengths. As i ∈ [0, NOOC − 1] and k ∈ [0, F − 1] there are NOOC × F sets. In our method, we aim at decreasing the number of wavelengths NS . So, NS will be lower than NOOC × F . The algorithm consists in selecting the appropriate wavelengths (L among F ) which leads to the greatest number of sets. The minimal number of wavelengths we can obtain is L = W. In this case, NS is reduced to one. Due to the use of 1D OOC code with a cross-correlation value equal to one, if the number of wavelengths is such as W + 1 ≤ L < 2W − 1, the number of sets NS is always equal to one. We can also add to the code matrices obtained with Equation (5), the matrices composed of 1D temporal OOC codes emitted on each wavelength: [(m, a0,0 ), . . . , (m, a0,w−1 )] with m ∈ [0, L − 1]
(6)
This method leads to a number of users Nutot lower than the one provided by the method described in Reference [11]. Nutot can be expressed as a function of the number of wavelength set NS as: Nutot = F × NS + L
(7)
Let us take a simple construction example with a MWOOC code using L = 6 wavelengths, a temporal length F = 13 and a weight W = 3. The two pulse position vectors define the 1D OOC temporal coding: [1, 3, 9] and [2, 5, 6]. So, we get NOOC × F = 26 wavelengths set Ai (Table 1). We first fix L = W wavelengths, that is one of the 26 sets, for example the set A1 . Then, we identify the sets which have the biggest number of wavelengths among those already selected (1, 3, 9), we select one, for example A6 . We have now five wavelengths used (1, 3, 6, 8, 9). We repeat the same process until we reach the desired number of wavelengths. In the example, the next step indicates that with one wavelength more (the 10th one), we could have two more sets A8 and A18 . We have selected four wavelengths sets using L = 6 wavelengths (1, 3, 6, 8, 9, 10) to construct the MWOOC (6 × 13, 3, 1, 1). Copyright © 2007 John Wiley & Sons, Ltd.
Table 1. Wavelength sets for an OOC (13, 3, 1, 1) code. A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25 A26
1 2 3 4 5 6 7 8 9 10 11 12 0 2 3 4 5 6 7 8 9 10 11 12 0 1
3 4 5 6 7 8 9 10 11 12 0 1 2 5 6 7 8 9 10 11 12 0 1 2 3 4
9 10 11 12 0 1 2 3 4 5 6 7 8 6 7 8 9 10 11 12 0 1 2 3 4 5
Table 2. Correspondence between old and new λi . λi used in Reference [11] Corresponding λi
1 0
3 1
6 2
8 3
9 4
10 5
Now, we modify the subscript value of each selected wavelength as shown on Table 2. The matrices of the MWOOC (6 × 13, 3, 1, 1) can be obtained from the following relations:
[(0, 1.j); (1, 3.j); (4, 9.j)] (mod 13)
(8)
[(2, 1.j); (3, 3.j); (0, 9.j)] (mod 13)
(9)
[(3, 1.j); (5, 3.j); (1, 9.j)] (mod 13)
(10)
[(2, 2.j); (4, 5.j); (5, 6.j)] (mod 13)
(11)
[(m, 1); (m, 3); (m, 9)]
(12)
with j ∈ [0, 12] and m ∈ [0, 5]. Eur. Trans. Telecomms. 2007; 18:761–768 DOI: 10.1002/ett
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For example with j = 3 we obtain from Equation (8) the matrix: 0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
same wavelength is W 2 /L. Furthermore, the probability that two pulses, from different code matrices but on the same wavelength, to be on the same time slot is 1/F . As an overlap occurs only when both wavelengths and time slots coincide, respectively, the probability that an overlap occurs is given by q = w2 /(L × F ). The theoretical error probability in the noiseless case PECCR is then given by the following equation:
PECCR
It can be verified that Nutot = 13 × 4 + 6 = 58.
To study the performances of the 2D codes, we use a Conventional Correlation Receiver (CCR) [1]. The receiver has the knowledge of the desired user code matrix. At the reception end (Figure 2), each wavelength is separated. The electrical signal corresponding to the optical signal rk,F (t) j at wavelength λk is multiplied by the code sequence dk,F of the desired user #j; then, the resulting signal is integrated over the bit duration. The values obtained for each wavelength are summed. So, we get the decision variable value of the desired user which is compared to the threshold level S of the decision device j to provide an estimation of the transmitted data, bˆ i . It has been shown [1] that in the ideal chip synchronous case, this receiver makes errors only when the sent bit is a zero data. Errors are due to the MAI and interference occurs when the code matrix of an undesired user who has sent a data one, has a common pulse with the code matrix of the desired one. Note that the probability to have an overlap is not uniform due to the used 2D construction [11], but its value can be approximated. Since there is up to one chip per row in a 2D code matrix, the probability that two matrices have a pulse one on the
O/E
RL,F(t)
Wavelength DEMUX
rL,F(t)
O/E
Nu − 1 i
i=S
3. 2D CODE DESIGN WITH CONVENTIONAL CORRELATION RECEIVER
Optical to Electrical Converter (OEC) r1,F(t)
Nu −1 1 ∼ = 2
q i
2
1−
q Nu −1−i (13) 2
with Nu : number of active users. We can note that PECCR decreases when the product L × F increases. We have plotted on Figure 3, the theoretical and numerical values of BER for a MWOOC code issued from the new 2D coding method previously described, with characteristics (L × F = 5 × 23, W = 5, ha = hc = 1) and for
Figure 3. Theoretical and simulated BER for a MWOOC (5 × 23, 5, 1,1) code with a CCR receiver. CCR
[1
[1
4 ... 3 6]
3 ... 4 7]
Tb
∫0
d11, F = [0 0 ... 1 0]
Σ
bˆi1 (t ) S
Tb
∫0
d 1L, F = [1 0 ... 0 0]
Figure 2. 2D-OCDMA reception scheme with a Conventional Correlation Receiver (CCR). Copyright © 2007 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. 2007; 18:761–768 DOI: 10.1002/ett
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2-DIMENSIONAL CODE DESIGN FOR OCDMA
Table 3. Coding parameters for a BER ≤ 10−9 and Nu = 30 with a CCR. W
2D code (L × F ) L=W
2D code (L × F ) F = 64
3 4 5 6 7 8 9 10 11 12
3 × 18328 4 × 3703 5 × 1412 6 × 737 7 × 460 8 × 321 9 × 241 10 × 191 11 × 157 12 × 133
860 × 64 232 × 64 111 × 64 70 × 64 51 × 64 41 × 64
different values of active users. We used the optimal threshold value: S = W = 5. We can notice that the performance degradation is correlated with the number of active users increase. Moreover, the theoretical expressions fit with the numerical results, so we can validate the theoretical formula (13). We use from now on the theoretical error probability expression and we search the optimal coding parameters for the access specifications, that is a BER ≤ 10−9 and Nu = 30 users. As system performance evolves as a function of L × F , code parameter choice must respect a tradeoff between the number of available wavelengths L and the temporal code length F . In order to evaluate the most appropriate 2D code (L and F as low as possible), we have first fixed the number of wavelengths L equal to the minimal value, that is L = W (first column of Table 3). We stopped the search at W = 12 because the next steps would lead to a non-existing 1D OOC code regarding to the temporal length. In this case, the best trade-off regarding to L and F corresponds to code parameters (L × F = 12 × 133, W = 12). In the second column of Table 3, we have searched a code which respects the limitations for an electrical implementation, that is a low temporal length F = 64. The values are presented up to W = 8 for the same reasons as previously. We can see that the most appropriate code is a (L × F = 41 × 64, W = 8, 1, 1). So we have designed 2D codes either with a temporal code length F = 133 > Fmax = 64 or with a wavelength number L which is higher than the number of users L = 41 > Nu = 30. We can conclude that the use of a CCR at the reception end along 2D code is not a suitable solution. Therefore, we study in the next part, the use of a specific receiver named PIC. Copyright © 2007 John Wiley & Sons, Ltd.
4. 2D CODE DESIGN WITH A PARALLEL INTERFERENCE CANCELLATION RECEIVER (PIC) The PIC receiver previously studied for 1D OOC code shown to improve the performance and permits to reduce the constraints on the temporal code length (F ) [13, 14]. The aim of such a receiver is to estimate the interference term due to all interfering users and to remove it from the received signal. The PIC principle consists in detecting the Nu − 1 interfering users with the CCR receiver defined in the previous part with a threshold level S = ST . Each p receiver provides the estimation bˆ i of the non-desired user #p. Next, each estimated data is spread by the corresponding code matrix; the interference is built and removed from the received signal. Then, we detect the transmitted data of the desired user with a CCR receiver and a threshold level S = SF . Figure 4 presents the structure of a PIC receiver for the desired user #1. Based on the analysis of PIC error probability described in Reference [13], we express the error probability in the case of 2D codes, in the ideal chip synchronous case:
PEPIC
∼ =
×
Nu 1 2
N u −1
Nu −1−N 1
Nu − 1 N1
N1 =ST −1 N2 =W+1−SF
Nu − 1 − N1 N2
(Q)N2 (1 − Q)Nu −1−N1 −N2 (14)
1 n1 n1 N1 −n1 with Q = q N and Nu : n1 =ST −1 CN1 (q) (1 − q) number of active users. In Figure 5, we present the theoretical and numerical BER values of a PIC receiver for the same MWOOC code as the CCR #2 ST
bˆi2 (t )
RL,F(t)
C2L, F CCR #Nu ST
Σ
bˆiNu (t )
CNuL, F +
-
CCR #1 SF
bˆi1 (t )
Figure 4. Parallel Interference Cancellation structure. Eur. Trans. Telecomms. 2007; 18:761–768 DOI: 10.1002/ett
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Figure 5. Theoretical and simulated BER for a MWOOC (5 × 23,5,1,1) code with a PIC receiver.
one used in Figure 3. We can observe that the numerical results fit with the theoretical ones and so this validates the theoretical analysis. For the next BER values, we will use the theoretical formula (14). Moreover, if we compare the BER results reported in Figures 3 and 5, we verify that the PIC permits to reduce significantly the BER compared to the CCR for the same code parameters. Therefore, to attempt a given performance, the PIC receiver should allow the use of a 2D code with lower dimensions (L × F ) than the CCR. As for the CCR, we have studied for the PIC receiver, the 2D code parameters allowing to obtain a BER ≤ 10−9 for Nu = 30. In Table 4, we compare the 2D codes obtained with the two receiver’s structures. We can notice that the PIC receiver permits to relax the code parameters. Indeed, for the same number of wavelengths L = 12, the PIC receiver permits to divide by 2 the temporal code length F . Moreover, for a comparable value of F (F ≤ 64), the L value decreases significantly from the value 41 to the value 5 and becomes much lower than the number of users. In Figure 6, we have plotted the BER of the 2D-OCDMA system with a PIC receiver for the MWOOC codes with characteristics (12 × 60, 5, 1, 1) and (5 × 56, 5, 1, 1) as a function of the number of active users. We have considered Table 4. Comparison of 2D codes for a CCR and a PIC receiver. BER ≤ 10−9 Nu = 30
CCR
PIC receiver
(L × F, W)
(12 × 133, 12) (41 × 64, 8)
(12 × 60, 3) (5 × 56, 5)
Copyright © 2007 John Wiley & Sons, Ltd.
L = 12 F ≤ 64
Figure 6. BER for a MWOOC (12 × 60, 3, 1, 1) code and a MWOOC (5 × 56, 5, 1, 1) code using a PIC receiver.
the optimal thresholds [7, 8]: ST = W and SF = 1. First, we verify that the access specifications (30 users and a BER ≤ 10−9 ) are well respected. We can also note that when the network is not totally charged, the BER decreases quickly for the MWOOC (5 × 56, 5, 1, 1). This is due to the PIC receiver efficiency to remove MAI when the code weight increases. Moreover, for this code the number of wavelengths is low (L = 5). This is a major advantage to limit the interference effect due to the use of several wavelengths. To conclude, the MWOOC (5 × 56, 5, 1, 1) code is suitable for OCDMA systems with electronic coding/decoding functions if the Signal to Noise Ratio (SNR) is important (≈30 dB), that is if noise perturbation is negligible. Thus, with the new code construction presented in this paper and a PIC receiver, we manage to design 2D codes that permit to have a BER ≤ 10−9 for 30 active users with a data rate of 155 Mbit/s.
5. CONCLUSION We have presented a 2D-OCDMA transmission scheme and a new construction method to obtain MWOOC codes whose parameters have high flexibility. From numerical calculation and simulation, we have validated the reliability of the theoretical analysis. The performance study has shown that the use of 2D codes with a CCR does not permit an electrical implementation of OCDMA with a number of wavelengths lower than the number of users. However, the use of a PIC receiver allows using 2D codes with a minimal number of wavelengths Lmin = W and a short temporal code length F . Eur. Trans. Telecomms. 2007; 18:761–768 DOI: 10.1002/ett
2-DIMENSIONAL CODE DESIGN FOR OCDMA
Thanks to our new 2D code construction method, we can generate a MWOOC (5 × 56, 5, 1, 1) code, which permits to provide 155 Mbit/s to 30 users with a BER ≤ 10−9 with the PIC receiver. Therefore, in the context of the high flow access networks, the specifications are reached with a PIC receiver and 2D MWOOC codes whose characteristics are workable by electronic devices.
7. 8. 9. 10.
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Copyright © 2007 John Wiley & Sons, Ltd.
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superimposed FBG. Lightwave Technology 2005; 23(2):724–731. DOI: 10.1109/JLT.2004.839984. Wang X. Keys towards practical OCDMA networks, 7th IEEE International Conference on Optoelectronics, Fiber optics and Photonics(Photonics 2004),Cochin (Inde) 2004. Tancevski L, Andonovic I. Wavelength hopping/time spreading code division multiple access systems. Electronics Letters 1994; 30(17):1388–1390. Wan SP, Hu Y. Two-dimensional optical CDMA differential system with prime/OOC codes. IEEE Photonics Technology Letters 2001; 13(12):1373–1375. DOI:10.1109/68.969912. Tancevski L, Andonovic I, Tur M, Budin J. Massive optical LANs using wavelength hopping/time spreading with increased security. IEEE Photonics Technology Letters 1996; 8(7):935–937. DOI:10.1109/68.552276. Guu-Chang Y, Wing KC. Performance comparison of multiwavelength CDMA and WDMA + CDMA for fiber-optic networks. IEEE Transactions on Communications 1997; 45(11):1426–1434. DOI:10.1109/26.649764. Ssang-Soo L, Seung-Woo S. New construction of multiwavelength optical orthogonal codes. IEEE Transactions on Communications 2002; 50(12):2003–2008. DOI: 10.1109/ TCOMM.2002.806504. Goursaud C, Julien-Vergonjanne A, Zouine Y, Aupetit-Berthelemo C, Cances J-P, Dumas J-M. Improvement of parallel interference cancellation technique with hard limiter for ds-ocdma systems. IEEE GLOBECOM 2005; 4:2004–2008. DOI: 10.1109/GLOCOM.2005.1578017. Goursaud C, Saad NM, Zouine Y, Julien-Vergonjanne A, AupetitBerthelemot C, Cances J-P, Dumas J-M. Parallel multiple access interference cancellation in Optical DS-CDMA systems. Annals of Telecommunications 2004; 9(10):1053–1068. Wang X, Kitayama K. Analysis of beat noise in coherent and incoherent time-spreading OCDMA. Journal of lightwave Technology 2004; 22(10):2226–2235. DOI:10.1109/JLT.2004. 833267.
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AUTHORS’ BIOGRAPHIES Mika¨el Morelle received his Diplˆome d’Ing´enieur degree from the Institut d’Ing´enierie Informatique de Limoges (3IL), University of Limoges, France in 2005. He continued his studies as a doctorate student (Ph.D.) in the research group of the Ecole Nationale Sup´erieure d’Ing´enieurs de Limoges (ENSIL). His research work deals with the study of coding techniques applied to optical communications. Claire Goursaud received her Diplˆome d’Ing´enieur degree from the Ecole Nationale Sup´erieure d’Ing´enieurs de Limoges (ENSIL), University of Limoges, France in 2003. She continued her studies as a doctorate student (Ph.D.) in the research group of ENSIL. Her research work deals with the study of the digital signal processing techniques applied to Optical Code Division Multiple Access. Anne Julien-Vergonjanne received her Ph.D. in Microwave and Optical Communications from the University of Limoges in 1987. She joined the Ecole Nationale Sup´erieure d’Ing´enieurs de Limoges (ENSIL), University of Limoges, as Assistant Professor of Electronics and Telecommunications in 1997. Her current research interests deal with digital signal processing for communications and especially with theory and implementation of signal processing for Optical CDMA systems. Christelle Aupetit-Berthelemot received her Diplˆome d’Ing´enieur degree from the Ecole Nationale Sup´erieure d’Ing´enieurs de Limoges (ENSIL), University of Limoges, France in 1995. She received her Ph.D. degree from University of Limoges and Optical Communications from the University of Limoges in 1998. She is now an assistant professor in Electronics and Telecommunications at ENSIL—University of Limoges. Her current research interests include optoelectronic devices, fibre-optic communication systems and Optical CDMA systems. Jean-Pierre Cances graduated in Electrical Engineering from Ecole Nationale Sup´erieure des T´el´ecommunications (ENST) Bretagne in 1990. He received his Ph.D. degree from ENST Paris in satellite communications engineering in 1993. He is now an assistant professor at the Ecole Nationale Sup´erieure d’Ing´enieurs de Limoges (ENSIL), University of Limoges. His current research interests include satellite communication systems, CDMA multi-user detection, multicarrier modulation and synchronization algorithms. Jean-Michel Dumas received his Diplˆome d’Ing´enieur degree from the Institut National des Sciences Appliqu´ees of Toulouse, University of Toulouse in 1973 and the Doctorat e` s-Sciences Physiques degree from the University of Limoges in 1985. He was a member of the technical staff of France T´el´ecom/Centre National d’Etudes des T´el´ecommunications (FT.CNET) in Lannion until 1994. Then, he joined the Ecole Nationale Sup´erieure d’Ing´enieurs de Limoges (ENSIL), University of Limoges, as Professor of Electronics and Telecommunications. At ENSIL, he leads a research group involved in the insertion of high-speed devices into systems and the simulation of digital communication systems.
Copyright © 2007 John Wiley & Sons, Ltd.
Eur. Trans. Telecomms. 2007; 18:761–768 DOI: 10.1002/ett
