Performance of MLSE-Based Receivers in Lightwave Systems with Nonlinear Dispersion and Amplified Spontaneous Emission Noise Mario R. Hueda Diego E. Crivelli Hugo S. Carrer Digital Communications Research Laboratory - National University of Cordoba - Argentina Av. V´elez Sarsfield 1611 - C´ordoba (X5016GCA) - Argentina - Email:
[email protected]
Abstract— Maximum likelihood sequence estimation (MLSE) has been proposed in earlier literature to combat the effects of nonlinear dispersion in intensity modulation/direct detection (IM/DD) optical channels. In this paper, we develop a theory of the bit error rate (BER) of MLSE-based IM/DD receivers operating in the presence of nonlinear intersymbol interference (ISI) and amplified spontaneous emission (ASE) noise. Numerical results confirm the predictions of the theory developed in this work. Based on the new analysis, we also investigate the loglikelihood ratio (LLR) of the received signal yielded by a softinput/soft-output (SISO) front-end decoder. Our study shows that traditional channel codes designed for transmissions over AWGN channels, in combination with SISO front-end detectors, achieve asymptotically optimal performance in transmissions over IM/DD optical channels.
I. I NTRODUCTION Long haul high-speed optical fiber transmission systems suffer from impairments such as chromatic dispersion (CD), polarization mode dispersion (PMD), and the amplified spontaneous emission (ASE) noise introduced by optical amplifiers [1]. In intensity modulation/direct detection (IM/DD) schemes, fiber dispersion combined with the square-law response of the photodetector gives rise to nonlinear intersymbol interference (ISI). Additionally, after the photodetector, ASE noise becomes nongaussian and signal-dependent. Recently, there has been a great deal of interest in using digital equalization to compensate the dispersion of optical channels (e.g., [2]-[5]). The feasibility of a very large scale integrated (VLSI) implementation of digital signal processing based equalization for optical channels has been demonstrated by Agazzi et al. in [6]. It has been shown in [2] that feed forward equalization (FFE) and decision feedback equalization (DFE) are severely degraded in the presence of nonlinearity, whereas MLSE equalization is not. MLSE-based receivers for optical channels have already been reported in [4]-[7]. Unlike in traditional dispersive linear channels with additive white Gaussian noise (AWGN), in IM/DD optical channels ISI is nonlinear and noise is nongaussian and signal-dependent [4]. Therefore, new studies are needed to characterize the performance of MLSE-based IM/DD receivers. Weiss [5] reported computer simulations of MLSE receivers in the presence of CD, PMD, and ASE noise. A semi-analytical method to evaluate the bit error rate (BER) of MLSE based IEEE Communications Society Globecom 2004
IM/DD receivers operating in the presence of nonlinearities and generic nongaussian signal-dependent noise, is introduced in [4]. The accuracy of this technique is shown to be excellent in all the cases considered. However, closed-form analytical expressions for the bit error probability similar to those available for AWGN channels [8] have not been reported so far. Analytical expressions would be useful not only to predict system performance, but also to provide information for the design of other aspects of the transmission system such as channel codes. In this paper, we introduce a closed-form analytical expression for the bit error probability of MLSE-based optical receivers in the presence of nonlinear dispersion and ASE noise. Numerical results show a close agreement with the predictions of the theory. Building on the new analysis, we also investigate the log-likelihood ratio (LLR) of the received signal provided by a soft-input/soft-output (SISO) front-end decoder. Our study shows that traditional channel codes designed for transmission over AWGN channels could be used with SISO front-end detectors in IM/DD optical channels achieving asymptotically optimal performance. The rest of this paper is organized as follows. In Section II we present the channel model and a brief description of MLSE receivers. Performance analysis is introduced in Section III. The predictions of the theory are compared with simulation results in Section IV, while concluding remarks are given in Section V. II. C HANNEL M ODEL AND MLSE-BASED R ECEIVERS This work focuses on long-haul or metro links spanning several hundred kilometers of single-mode fibers with optical amplifiers. Fig. 1 shows a simplified model of the system under consideration. The transmitter modulates the intensity of the transmitted signal using a binary alphabet (e.g., On-Off Keying (OOK) modulation). Let {an } = (a0 , a1 , ..., aN −1 ) and N denote, respectively, the bit sequence to be sent on the optical channel (an ∈ A = {0, 1}) and the total number of transmitted symbols. The optical power ratio between the pulses representing a logical 1 and a logical 0, r10 , is called the extinction ratio. We assume that the intensity level for a logical 0 (an = 0) is different from zero, which is usual in practical −1 0.1 [9]). The optical fiber transmitters [1] (e.g., r01 = r10 introduces chromatic and polarization mode dispersion, as well
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Laser
The maximum likelihood sequence detector chooses, among all possible transmitted bit sequences, the sequence {ˆ an } that minimizes the metric [8]
Modulator Optical Amplifiers
1 T
Optical Filter
T 0
dt
A/D
M=
Detected Bits
yn MLSE
Optical channel model.
as attenuation. Optical amplifiers are deployed periodically along the fiber to compensate the attenuation, also introducing ASE noise in the signal. ASE noise is modeled as AWGN in the optical domain [10]. At the receiver, the optical signal is filtered and then converted to a current with a PIN diode or avalanche photodetector. The resulting photocurrent is filtered by an integrate-and-dump electrical filter [10]. The output of the filter is sampled at the symbol rate and applied to the MLSE-based detector. The samples of the received signal after the optical-toelectrical conversion can be expressed as yn = In + zn = f (an ) + zn ,
(1)
where In = f (an ) represents the noise-free electrical received signal, which is in general a nonlinear function of a group of δ consecutive transmitted bits an = (an , an−1 , ..., an−δ+1 ) δ (note that In ∈ I = {I (0) , I (1) , ..., I (2 −1) } with δ I (0) = f (0, 0, ..., 0), I (1) = f (0, 0, ..., 1), . . . , I (2 −1) = f (1, 1, ..., 1)); zn are samples of the nongaussian signaldependent noise originated by the direct detection process of the optical signal and the ASE noise. Thermal noise, shot noise, and noise contributions from any other source are ignored since it is assumed that ASE noise is dominant [1]. The probability density function (pdf) of yn is noncentral chi-square with 2M degrees of freedom [10]: M2−1 √ yn yn In 1 − ynN+In 0 e JM −1 2 fy|a (yn |an ) = , N0 In N0 (2) yn > 0, where N0 is related to the variance of the noise in the electromagnetic field domain, M is the ratio of the optical (Bo ) Bo ), to electrical (Be ) bandwidth of the front-end (i.e., M 2B e and Jm (·) is the m-th modified Bessel function of the first kind. Assuming that N0 is sufficiently small and r01 > 0 [9], in the Appendix we show that (2) is well approximated by ( 1 e− fy|a (yn |an ) ≈ √ 2 πKn N0
3 Kn = In + Isp − N0 , 2
√ √ yn − Kn )2 N0
Kn > 0,
,
(3)
(5)
ˆ n = (ˆ with a an , a ˆn−1 , . . . , a ˆn−δ+1 ). The minimization can be efficiently implemented using the Viterbi algorithm (VA). Using (3), it is simple to show that minimizing (5) is equivalent to minimize the metric 2 N −1 √ ˆ n + N0 ln(K ˆ n ), ˆ = M yn − K (6) 2 n=0 ˆ n = Iˆn + Isp − 3 N0 and Iˆn = f (ˆ an ). where K 2 III. P ERFORMANCE A NALYSIS In this section, we analyze the performance of MLSE in the presence of dispersion and ASE noise in IM/DD optical channels. In our case each detection error causes exactly one bit error, therefore the probability of bit error of the Viterbi decoder [11] is upper bounded by ˆ Pr{Ψ|Ψ} ˆ WH (Ψ, Ψ) Pr{Ψ}, (7) Pb ≤ ˆ Ψ=Ψ
ˆ = where Ψ = {an } represents the transmitted sequence, Ψ ˆ {ˆ an } is an erroneous sequence, Pr{Ψ|Ψ} is the probability of ˆ (i.e., the error event that occurs when the error event Ψ → Ψ ˆ instead of Ψ), and the Viterbi decoder chooses sequence Ψ ˆ ˆ or, in other words, WH (Ψ, Ψ) is the Hamming weight of Ψ∧Ψ the number of bit errors in the error event (∧ is the exclusive OR operator). Pr{Ψ} is the probability that the transmitter sent sequence Ψ. Assuming that the incorrect path through the trellis diverges from the correct path at time k and remerges with the correct path at time k + l (l ≥ δ), from (6) note that the MLSE-based ˆ if receiver will choose the incorrect sequence (Ψ → Ψ) 2 k+l−1 k+l−1 √ √ 2 ˆ n + gˆn < yn − K y n − K n + gn , n=k
n=k
(8) ˆ n ). After binomial where gn = N20 ln(Kn ) and gˆn = N20 ln(K expansion and some manipulation, we obtain k+l−1 k+l−1
1 ˆn < ˆ n + Gn , Kn − K un Kn − K 2 n=k n=k (9) √ where un = yn and Gn = gn − gˆn . The probability density function of un can be obtained [12] replacing (3) in fu|a (un |an ) = 2un fy|a (u2n |an )
(4)
where Isp N0 M . Note that Kn ∈ K = δ {K (0) , K (1) , ..., K (2 −1) } with K (s) = I (s) + Isp − 32 N0 , s = 0, 1, ..., 2δ − 1. IEEE Communications Society Globecom 2004
− ln fy|a (yn |ˆ an ) ,
n=0
LPF
Fig. 1.
N −1
un > 0,
(10)
yielding
300
√
fu|a (un |an ) = √
2
(un − Kn ) un N0 e− . πKn N0
(11)
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In practical situations (i.e., r01 0.1 [9]), when the power of optical noise (N0 ) is sufficiently low, that is, N0 I (s) , s = 0, 1, ..., 2δ − 1, (12)
(a) OSNR = 10 dB
Λ(yn)
it has been shown in [13] that (11) is well approximated by fu|a (un |an ) ≈ √
√ (un − Kn )2 − N0
1 e πN0
.
−50 0
(13)
n=k
Taking into account the independence of the samples un (given the transmitted sequence), from (14) and (15) we conclude that rn is also a Gaussian random variable with mean and variance respectively given by k+l−1 ˆ n, ηr = Kn − Kn K (17)
2
Kn −
ˆn K
2 .
(18)
n=k
Moreover, since the power of the optical noise N0 is assumed sufficiently small, from (4) we verify that Gn ≈ 0,
Kn ≈ In ,
ˆ n ≈ Iˆn . K
and
(19)
Thus, based on (17)-(19) and the fact that rn is Gaussian, it is possible to show that (16) reduces to 2 1 k+l−1 ˆ In − Iˆn , (20) Pr Ψ|Ψ ≈ Q 2N0 n=k
where Q(x) = (0)
√1 2π (1)
∞ x
1 2
e− 2 t dt.
Let I and I be the current generated by each constellation symbol in a nondispersive optical channel. Then, defining −1 • Extinction Ratio: r01 = r10 =
I I
(0) (1)
,
• Optical Signal-to-Noise Ratio: OSNR =
(1)
(22)
• Normalized OSNR [10]: SNRT M OSNR, (23) −2 k+l−1 2 (1) (0) I − I In − Iˆn (24) • d2 =
0.8
1
yn
1.2
1.4
1.6
1.8
2
n
Λ(y )
200 100 0 −100 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
yn
LLR versus yn . (a): OSNR = 10 dB. (b): OSNR = 15 dB.
expression (20) can be rewritten as
ˆ (25) Pr Ψ|Ψ ≈Q SNRT β 2 d2 , √ where β = 1− r01 . Parameters β and d take into account the reduction of SNRT owing to the extinction ratio and channel dispersion. In particular, for nondispersive optical channels √ (1) (0) I − I (δ = l =1), note that In − Iˆn = (In = Iˆn ), therefore from (24) we obtain d2 = 1. Based on (7) and (25) it is possible to derived an upper bound for Pb [11]. In the following we provide a tight lower bound for Pb . Defining dmin as the minimum distance of an ˆ ≥ 1, the error event, and taking into account that WH (Ψ, Ψ) bit error probability can be lower bounded by 2 2 SNRT β dmin , (26) Pb ≥ JQ where J = Ψ∈D Pr{Ψ} and D is the set of transmitted sequences that have a minimum distance error event [11]. The lower bound (26) with J = 1 will be used in Section IV to evaluate the performance of MLSE-based receivers [11]. Analysis of the Log-Likelihood Ratio (LLR) Expressions (3) and (13) can also be used to provide useful information for the design of other aspects of the transmission system such as channel decoders. Consider a nondispersive optical channel (i.e., an = an ). Then, the LLR of the received signal provided by a SISO front-end decoder is given by Λ(yn ) = ln
fy|a (yn |an = 1) . fy|a (yn |an = 0)
(27)
From (3), note that the LLR (27) can be approximated by √
2 yn 1 (0) (1) (0) (1) K − K K −K Λ(yn ) ≈ + N0 N0
n=k
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0.6
Exact Our Analysis
(21) I , 2Isp
0.4
300
Fig. 2.
Then, from (9) notice that
k+l−1
1 ˆ n + Gn ˆ Kn − K . (16) Pr Ψ|Ψ = Pr rn < 2
0.2
(b) OSNR = 15 dB
n=k
σr2 =
50
0
From (13) note that un can be modeled as a Gaussian random variable with mean and variance respectively given by N0 ηu ≈ Kn and σu2 ≈ . (14) 2 Let rn be the random variable defined by k+l−1 ˆn . rn = un Kn − K (15)
n=k k+l−1 N0
Exact Our Analysis
100
301
(0)
+
1 K ln , 2 K (1)
(28)
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−1
−1 Lower Bound (26) Simulation (Exact) Simulation (Approximate)
−2 −3
−3
−5
log10 (BER)
−4
log10 (BER)
−4
−5
r10 = 5dB
−6
−6
−7
6
8
10
12 14 16 OSNR [dB]
18
−9
20
(s)
where K = I + Isp − 32 N0 , s = 0, 1. Fig. 2 shows Λ(yn ) versus yn for OSNR= 10 and 15 dB with M = 3 and r10 = 10 dB. We present the exact function (27) derived from (2) and values obtained from (28). In both cases, we verify the excellent accuracy of approximation (28). The pdf of Λ(yn ) can be easily derived from (11) and (28). However, if condition (12) is satisfied, from (13) we verify that √ the random variable yn (i.e., un ) is Gaussian with parameters defined by (14), therefore Λ(yn ) is also a Gaussian random variable with mean and variance respectively given by √ √ (0) 2 (1) K − K (0) + 12 ln K (1) an = 1 N0 K √ √ (0) 2 (29) ηΛ ≈ (1) K − K (0) − + 12 ln K (1) an = 0 N0 2 σΛ ≈
2 N0
K
(1)
−
K
(0)
K
2
∀an .
,
(30)
As an application of this analysis, we derive simple closedform analytical expressions for the bit error probability. (0) Based 1 on (19), (29), (30), and neglecting term 2 ln K (1) in (29) K (i.e., high OSNR), it is simple to verify that
(31) SNRTβ 2 , Pb ≈ Q which agrees with (26) when J = 1 and d2min = 1 (i.e., 1 2 1 nondispersive optical channel). Since Q(x) ≈ √2πx e− 2 x [11], from (31) we get Pb ≈
1 2πSNRTβ 2
1
2
e− 2 SNRTβ .
(32)
Note that (32) agrees with [14, Eq. (56)] when (i) it is evaluated at high OSNR and (ii) ASE noise is dominant. IV. S IMULATION R ESULTS AND D ISCUSSION In this section, we confirm the theoretical analysis introduced previously by using computer simulations. Our analysis is valid for both linear and nonlinear channel dispersion (including, for example, nonlinear Kerr effect) and it is not limited to the photodetector nonlinearity. However the pdf of IEEE Communications Society Globecom 2004
r10 = 10dB
−8
Fig. 3. BER versus OSNR for an optical channel with no dispersion. M = 3. (s)
r10 = 5dB
−7
r10 = 10dB
−8 −9
Lower Bound (26) Simulation (Exact) Simulation (Approximate)
−2
Fig. 4.
6
8
10
12
14 16 18 OSNR [dB]
20
22
BER versus OSNR for 150 km fiber span. M = 3.
the noise is assumed to be a chi-square. The photodetector response is the only source of nonlinearity considered in this section. We present results for an optical channel with no dispersion and a typical 150 km single-mode fiber as specified by the ITU G.652 Recommendation [15]. The latter is used in the third telecommunications window (1550 nm), which leads to a dispersion parameter D = 17 ps/km-nm [16]. We consider a data rate of 10 Gb/s, M = 3, OOK RZ modulation, r10 = 5 and 10 dB. The pulse shaping function used in the simulations is an unchirped Gaussian with TFWHM = 60 ps [16]. MLSE is implemented with an eight-state VA where a perfect knowledge of the channel response is assumed. Figs. 3 and 4 show the bit error rate versus the optical signal-to-noise ratio for an optical channel with no dispersion and a typical 150 km of fiber span, respectively. We present theoretical values obtained from (26), and simulation results of the entire system using both (i) the optimum metrics based on the exact pdf of the received signal (2), and (ii) the approximate metrics defined by (6). Parameter dmin in (26) is found from the Viterbi algorithm [11]. In all cases, comparisons between the values derived from the theory and simulation confirm the excellent accuracy of the lower bound (26). Furthermore, from Figs. 3 and 4 it can be noted the high accuracy of the simulation results based on the approximate pdf derived in the Appendix. Fig. 5 presents pdfs of the signal at the output of a SISO/MLSE equalizer (i.e., the LLR), obtained from both computer simulation and the Gaussian approximation. The latter is evaluated using mean and variance values obtained from the simulated samples. We consider no dispersion and two values of OSNR, 10 and 12 dB, with r10 = 10 dB and M = 3. In all cases, the excellent agreement between values obtained from the Gaussian approximation and the simulations confirms the validity of the theoretical study presented in the previous section. Also note that the accuracy of the Gaussian approximation for Λ(yn ) improves as OSNR increases, in agreement with our analysis. In particular, the accuracy of the Gaussian approximation is slightly better √ (0) for an = 1 than for an = 0. This is because the ratio NI0 is smaller than
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(a) OSNR = 10 dB
0
(33) in (2) and rearranging terms, it is simple to obtain M2−1 √ √ ( yn − In )2 yn 1 N0 √ e− . fy|a (yn |an ) ≈ In 2 π yn In N0 (34) √ Since r01 > 0 and Isp I (s) ∀s, we can verify that Pr{|yn − In | < ξ} → 1 with ξ > 0 and arbitrary small. Then, using a similar analysis to that reported in the Appendix of [13], it can be shown that (34) reduces to
10
Simulation Gaussian Approx. −2
PDF
10
−4
10
−6
10
−60
−40
−20
0
20
40
Λ(y ) n (b) OSNR = 12 dB
0
60
10
Simulation Gaussian Approx.
√ √ yn − In )2 N0
( 1 e− fy|a (yn |an ) ≈ √ 2 πIn N0
−2
PDF
10
−4
10
−6
10
−80
−60
−40
−20
0
Λ(y )
20
40
60
80
n
√
Fig. 5.
PDFs of Λ(yn ) in optical channels with no dispersion.
(1)
I N0 , therefore the an = 0. From Fig.
accuracy of assumption (12) decreases for 5 we infer that channel coding schemes designed for transmission over AWGN channels could be used with SISO/MLSE equalizers in IM/DD optical channels with negligible performance degradation [13]. This result is of great interest for turbo decoding techniques, which have been considered for optical channel transmission in earlier literature (e.g., see [13] and references thereof). V. C ONCLUSIONS In this paper, we have developed a new theoretical performance analysis of lightwave systems in the presence of nonlinear dispersion and ASE noise. Unlike previous contributions, we have derived a closed-form analytical expression for the bit error probability. Its accuracy has been confirmed by comparison with values derived from computer simulations. As a second important contribution of this paper, we have analyzed the LLR of the received signal yielded by a SISO front-end decoder. Our study has concluded that traditional channel codes designed for transmission over AWGN channels in combination with SISO/MLSE-based front-end detectors, could achieve asymptotically optimal performance in IM/DD optical channels. Finally, although the analysis presented here considers ideal optical/electrical filters (e.g., integrate-anddump electrical filters), simulation results not presented in this work have shown that the accuracy of the Gaussian approximation for the LLR is also satisfactory for more realistic filters such as, for example, five-pole Bessel electrical filters [17]. A PPENDIX In this Appendix we derive approximation (3). Since the m-th modified Bessel function of the first kind can be ap1 ex for x m [18], we obtain proximated by Jm (x) ≈ √2πx √ √ √ yn In yn In N0 JM −1 2 e2 N0 , In > 0, (33) ≈ √ N0 2 π yn In with yn > 0. In practical situations, approximation (33) is √ verified when r01 > 0 and Isp = N0 M I (s) ∀s. Using IEEE Communications Society Globecom 2004
.
(35)
The mean value obtained from (35) (∼ In + 32 N0 ) is different from the exact value (In +Isp [10]). This difference (negligible at high OSNR), can be overcome by replacing In with In + Isp − 32 N0 in (35), yielding thus expression (3). The accuracy of (3) has been found to be highly satisfactory in most cases of practical interest (e.g., r01 0.1 [9], OSNR>3 dB). R EFERENCES [1] R. Ramaswami and K. Sivarajan, Optical Networks: a Practical Perspective. Morgan Kaufmann, 2002. [2] O. E. Agazzi and V. Gopinathan, “The impact of nonlinearity on electronic dispersion compensation of optical channels,” in Proc. of OFC, Feb. 2004. [3] W. Sauer-Greff, A. Dittrich, R. Urbansky, and H. Haunstein, “Maximumlikelihood sequence estimation in nonlinear optical transmission systems,” in Proc. of LEOS 2003, vol. 1, Oct. 2003, pp. 167–168. [4] O. E. Agazzi, M. R. Hueda, H. S. Carrer, and D. E. Crivelli, “Maximum likelihood sequence estimation in dispersive optical channels,” Accepted at J. Ligthwave Technol., Aug. 2004. [5] A. J. Weiss, “On the performance of electrical equalization in optical fiber transmission systems,” IEEE Photon. Technol. Lett., vol. 15, no. 9, pp. 1225–1227, Sept. 2003. [6] O. E. Agazzi et al., “DSP based equalization for optical channels - Feasibility of a VLSI implementation,” IEEE 802.3ae Task Force, New Orleans., Sept. 2000. [Online]. Available: http://grouper.ieee.org/groups/802/3/ae/public/sep00/agazzi 1 0900.pdf [7] H. F. Haunstein, K. Sticht, A. Dittrich, W. Sauer-Greff, and R. Urbansky, “Design of near optimum electrical equalizers for optical transmission in the presence of PMD,” in Proc. of OFC, vol. 3, 2001, pp. 558–560. [8] J. G. Proakis, Digital Communications, 3rd ed. McGraw-Hill, 1995. [9] W. van Etten and J. van der Plaats, Fundamentals of Optical Fiber Communications. Prentice Hall, 1991. [10] D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol., vol. 8, no. 12, pp. 1816–1823, Dec. 1990. [11] E. A. Lee and D. G. Messerschmitt, Digital Communications, 2nd ed. KAP, 1988. [12] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. McGraw-Hill, 1991. [13] M. R. Hueda, D. E. Crivelli, and H. S. Carrer, “Analysis of SISO frontend decoders in IM/DD optical channels with application to turbo code decoding,” in Proc. of ICCS, Singapore, Sept. 2004. [14] D. Marcuse, “Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise,” J. Lightwave Technol., vol. 9, no. 4, pp. 505–513, Apr. 1991. [15] Characteristics of Single-Mode Optical Fibre and Cable, International Telecommunications Union ITU-T Recommendation G.652, Mar. 2003. [16] G. P. Agrawal, Fiber-Optic Communication Systems. WileyInterscience, 1997. [17] E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering,” J. Lightwave Technol., vol. 18, pp. 1493–1503, Nov. 2000. [18] W. H. Press et al., Numerical Recipies in C: The Art of Scientific Computing, 2nd ed. Cambridge, 1992.
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