ECOC 2010, 19-23 September, 2010, Torino, Italy
We.6.A.1
Are multilevel pseudorandom sequences really needed to emulate highly dispersive optical transmission systems? Edouard Grellier, Jean-Christophe Antona, Sébastien Bigo Alcatel-Lucent, Bell-Labs, Route de Villejust, 91620 NOZAY – FRANCE,
[email protected] Abstract We discuss what test sequences should be used to accurately emulate dispersive transmissions with multilevel modulations. Contrary to previous understanding, multilevel pseudorandom sequences are not more suited than binary ones. Neither of them truly represents the behavior of random data. Introduction During the propagation along an optical fibre with chromatic dispersion, a symbol can expand over its neighbours and interact with them through nonlinearities. As a consequence, each symbol is affected by nonlinear distortions that depend on the data value of its neighbours, i.e. on the sequence. To assess the performance of the system, few studies recommend emulating all the possible inter-symbol interactions by using Pseudorandom Binary Sequences (PRBS) 1,2 of sufficient length . To extend this approach to multilevel modulation formats, multilevel 3,4 pseudorandom sequences are recommended . However, when increasing the channel symbol rate or when operating over fibre lines with sparse or no in-line dispersion compensation, the number of overlapping symbols can become 2 extremely large (over 10 ). For such systems it is no longer possible to emulate all the possible combinations. This suggests revisiting the approach relying on pseudorandom sequences. We recently presented an approach choosing the sequence randomly and increasing its length until a bit-error-rate (BER) with satisfying statistical accuracy is obtained5. A 10% relative error was achieved with this method after 100/BER bits, and the strong dependence of the required sequence length on the BER was underlined. Tolerating such amount of inaccuracy greatly reduces the required sequence length in comparison with theories of 1,2 15 5 (this reduction can be larger than 10 times ). 5 In this paper, we extend the work of to multilevel modulation formats such as Polarisation Division Multiplexed Quaternary Phase Shift Keying (PDM-QPSK), and improve the method by repeating the detection process several times as it is usually done in test beds or numerical simulations in order to suppress the fluctuations due to noise. Finally, we estimate the accuracy of the BER estimations realized using pseudorandom sequences, and compare with sequences picked up at random.
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System under study In the following, we perform numerical simulations using the split step Fourier method for either a back-to-back system or a transmission over 7 spans of 100km standard fibre (17ps/nm/km local chromatic dispersion, -1 -1 0.22dB/km attenuation and 1.32W .km nonlinear coefficient) for a single channel modulated at 28 Giga-symbol-per-second with Binary Phase Shift Keying (BPSK), QPSK, or PDM-QPSK format, each using differential encoding. Ideal 50GHz-bandwidth rectangular filters are used at both transmitter and receiver. The dispersion is entirely compensated at the receiver end; the maximum cumulated dispersion in the line is thus 11900ps/nm, this corresponds to a possible overlap of more than 100 symbols1. Neglecting the possible nonlinear interactions between noise and signal, we emulate the propagation of the signal without noise, and then the noise is added before the receiver. For each simulation of the signal propagation with a given sequence of length of N symbols, we repeat the detection process Knoise times, each with a different noise seed. The bit-error-rate is then estimated by error counting and averaging over these Knoise detections. It can be noticed that the computational time mainly depends on the sequence length and not on Knoise since nonlinear propagation is computed only once. At the receiver, each polarization of the signal is mixed with a local oscillator with 200MHz detuning in a 90° hybrid and then the components of the signal are detected by balanced photodiodes; phase recovery is 6 performed using the method described in with 7 taps. Penalties arising from receiver electronics were neglected. Methods using sequences chosen randomly We note Rand-1 (resp. Rand-2) the method estimating the BER using a sequence chosen randomly with Knoise=1 (resp. Knoise=1000). To 4 study these methods, we generate Mmax=10
We.6.A.1
independent blocks m of L=1024 random symbols. For either Rand-1 or Rand-2 method, we derive a BER estimation rm for transmission links simulated with the sequence-block m. We estimate the NRMS error (Root Mean Squared Error Normalized by the BER) obtained with L=1024 random symbols as:
εˆ Rand ( L) =
rm − BER0 BER0 m =1
M max
1 M max
2
∑
where BER0 is our most accurate available BER estimation, i.e. the average value over Mmax blocks obtained with Rand-2 method:
P0 = ∑m=max r / M max , with 1 m M
K noise = 1000
Grouping M blocks together, we emulate a new random sequence of N=L×M symbols, which can be used with either Rand-1 or Rand-2 to estimate the BER. Since the blocks are independent, we have the following relation for the NRMS error of the BER estimations obtained with a sequence of N=L×M symbols:
ε Rand ( N ) = ε Rand ( L)
N/L .
We define N0 the required sequence length to achieve a NRMS error of 10%, using an interpolation of the above equation for any integer N we get:
N 0 = L × (ε Rand (L) 0.1)
2
Random sequences in Back-to-Back We now study the accuracy of Rand-1 and Rand-2 methods for back-to-back systems. We note N0,btb, the value of N0 for a back-to-back system. For method Rand-1 in the case of a binary system using direct encoding, Monte Carlo theory predicts that N0,btb ≈ 100/BER symbols5,7. We report on the first two lines of Tab1, the required sequence length for 10% NRMS error obtained for Back-to-back systems after adjusting the Optical-Signal-to-Noise-Ratio -3 (OSNR) so as to observe a BER of 10 (i.e. -3 BER0=10 ). These different results can be explained by the following steps: Neglecting consecutive mismatches on the detected signal, the BER is twice larger with differential encoding than with direct encoding. As a result, extending the previous relation to differential encoding, one obtains N0,btb=E×100/BER with E=1 for direct encoding, and E=2 for differential encoding. This explains the back-to-back results of Rand-1 for BPSK. For multilevel modulation formats using a Gray mapping, it is very unlikely to have more than 1 binary error in the same symbol. With no more than 1 error per symbol, replacing the BER by the symbol-error-rate, the system is equivalent to a binary one. As a result, extending the
Tab. 1: N0: Required symbol length to achieve a 10% NRMS error using random sequence.
BSPK Back-to-back (Rand-1) Back-to-back (Rand-2) After trans. (Rand-1) After trans. (Rand-2)
2×10
QPSK PDM-QPSK 5
200 2×10
5 3
1×10
5
100 1×10
5×10
4
50
5
5.2×10
3
4.8×10
9.3×10 6.3×10
4 3
previous relation to Gray-encoded multilevel modulation formats, one obtains N0,btb=E×100/(BER×BPS), with BPS the number of bits per symbol (BPS=1,2,4 for BPSK, QPSK and PDM-QPSK respectively). This explains the back-to-back results observed with Rand-1. In absence of inter-symbol interaction, the accuracy does not depend on the sequence and what matters is N×Knoise, the number of noisy samples. As a result, extending the above relation to Rand-2 one obtains N0,btb=E×100/(BER×BPS×Knoise). Random sequences with nonlinearities We now consider systems after nonlinear propagation. For each modulation format we adjust the fibre input power and the OSNR so as -3 to get 1.5dB OSNR penalty and a BER of 10 . The values of N0 are reported in Tab 1. We observe that with the Rand-1 method, the required sequence length is almost the same after nonlinear propagation as in back-to-back, in agreement with 5. This suggests that the accuracy of the Rand-1 estimator is mainly determined by the noise and not the intersymbol interactions. On the contrary the required sequence length using the Rand-2 method is greatly affected by nonlinear effects. Despite this dependence on nonlinearities, the required sequence length using the Rand-2 method is still more than 1 order of magnitude shorter than using the Rand-1 method. Methods using pseudorandom sequences Next we study the accuracy of methods using pseudorandom sequences, in the case of a PDM-QPSK system. For this modulation format with 4 bits per symbol, we consider the 3 following types of sequence: • A 16-level pseudorandom sequence (PR16S). • Two (1 for each polarisation) pseudorandom quaternary sequences (PRQS). • Four (1 for each I/Q tributary) pseudorandom binary sequences (PRBS). For the PRBS and PRQS methods, different sequences are used for each polarisation (and n-2 different tributaries), (we can generate 2 n pseudorandom sequences of 2 -1 bits8, n integer) and a random shift is introduced between them. One symbol is added to the pseudorandom sequences in order to obtain De
εˆPRS ( N ) =
1 200 pm ( N ) − BER0 ∑ BER 200 m =1 0
2
1 200 pm ( N ) − BER0 Bˆ PRS ( N ) = ∑ BER 200 m =1 0
where pm(N), is the m-th BER value obtained with the considered method using a pseudorandom sequence of N symbols.
Normalized bias
Accuracy with pseudorandom sequences In this section, we study the accuracy of BER estimations using pseudorandom sequences for a nonlinear propagation in the same conditions of power and OSNR as when studying Rand-1 and Rand-2 methods. In Fig. 1 we report the normalized bias obtained with methods PRBS-2, PRQS-2, PR16S-2 for different sequence lengths, (these methods have the same bias as the methods PRBS-1, PRQS-1, PR16S-1 respectively). We observe that when the pseudorandom sequence is too short the BER estimation is biased especially for multilevel pseudorandom sequences. This can be explained by the fact that some patterns possibly affecting the BER can never occur in a pseudorandom sequence of finite length. In Fig. 2 we report the NRMS error of the different methods. We observe that each method manages to accurately estimate the BER provided the sequence is long enough. It appears that for a given sequence length, the BER estimation is statistically more accurate if the sequence is generated randomly instead of using a pseudorandom sequence. Indeed, when the detection process is repeated 1000 times, the required sequence length for 10% NRMS
a)
10
RAND-1 PRBS-1 PRQS-1 PR16S-1
1 0.1 256 10
1024 4096 16384 12 14 16 Number of symbols
1
NRMS Error
Bruijn sequences. We note PRBS-1 (resp. PRQS-1, PR16S-1) the method estimating the BER with Knoise=1 using a PRBS (resp. PRQS, PR16S), and PRBS-2 (resp. PRQS-2, PR16S-2) the method estimating the BER with Knoise=1000 PRBS (resp. PRQS, PR16S). For each method 8 and for each sequence length (between 2 and 16 2 symbols) we perform 200 BER estimations, each with a different sequence. We then derive the NRMS error εPRS and normalized bias BPRS of the BER estimation methods with pseudorandom sequences as follows:
NRMS Error
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×3.5
0.1
RAND-2 PRBS-2 PRQS-2 PR16S-2
0.01
b)
256 10
×1.7
1024 4096 16384 12 14 16 Number of symbols
Fig. 2: NRMS error on BER o with different methods of sequence generation for a PDMQPSK transmission. a) Detection process repeated once. 3 b) Detection process repeated 10 times.
error using a randomly chosen sequence is roughly 1.75 shorter than using a PRBS, and 3.5 times shorter than with a PRQS or PR16S. Conclusions We estimated the accuracy of the bit-error-rate estimation obtained with either random or pseudorandom sequences for a dispersive transmission system. We show that the required sequence length for accurate emulation is shorter with sequences generated randomly than with pseudorandom binary sequences, and it is shorter with pseudorandom binary sequences than with multi-level pseudorandom sequences. With differential encoding, when the signal is detected only once, 10% accuracy is achieved if the sequence length is enough to observe on average 200 errors. In presence of nonlinear effects, repeating several times the detection process with different noise seeds can reduce the required sequence length for accurate emulation by one order of magnitude. Acknowledgement: The authors would like to thank the French public authorities for partial support under the 100GRIA project. References 1. L. K. Wickham et al., IEEE Photon. Technol. Lett., vol. 16 (2004), pp. 1591-1593. 2. P. Serena et al, ECOC’07, We.3.P093, 2007. 3. B. Spinnler et al, ECOC’07, Mo.2.3.6., 2007.
0.8 0.6 0.4 0.2 0 -0.2
PRBS-2 PRQS-2 PR16S-2
4. P. Ramantanis et al, OFC’09, OThC5, 2009. 5. J.-C. Antona et al, ECOC’08, We.1.E.3, 2008. 6. A. Viterbi, IEEE Transactions on Information Theory, vol. 29, 1983, pp. 543-551.
256 10
1024 4096 16384 12 14 16 Number of symbols
Fig. 1: Normalized bias of the BER estimation using pseudorandom sequences, for a PDM-QPSK transmission.
7. G. Fishman, “Monte Carlo Concepts, th Algorithms and Applications”, 4 ed., (2003). 8. F. S. Annexstein, IEEE Transaction on Computers, vol. 46, no. 2, Feb. 1997
