Digital-domain Chromatic Dispersion Compensation for Different Pulse Shapes: Practical Considerations Valery N. Rozental, Victor E. Parahyba, Jacklyn D. Reis, Juliano F. de Oliveira CPqD Foundation, Optical Communications Division Campinas, Brazil Email:
[email protected]
Abstract—We infer, by computer simulations, practical values of the overlap length for frequency-domain chromatic dispersion compensation in long and ultra-long-haul coherent optical systems. The overlap length has a direct impact on computational complexity and the receive-side DSP power consumption. The NRZ, RZ, and Nyquist-shaped signals are investigated. The presented results may be used as guidelines for trading off OSNR margin and computational complexity. In a sufficiently high OSNR scenario, these trade-offs may reduce computational complexity by 10-25%, depending on the particular filter implementation.
I.
I NTRODUCTION
Long and ultra-long-haul optical coherent systems should replace the traditional dispersion management schemes, where chromatic dispersion (CD) is compensated for after each fiber span, by fully digital signal processing (DSP)-based compensation. This choice has several advantages: from the management perspective, it is more effective to replace dozens of DCMs by a receiver-integrated DSP block; it avoids the insertion loss of DCMs, which increases network energy consumption and lowers the signal-to-noise ratio (SNR); and finally, it overcomes nonlinearities generated in dispersion compensating fibers. Nevertheless, digital CD equalizers are among the most power hungry DSP blocks, and so their careful design is essential to enhance system power efficiency. In 100G and beyond optical systems, the time-domain (TD) CD equalizer impulse response length typically varies from a few hundreds to over a thousand signal samples [1], according to the length of the optical link. Therefore, it is a general consensus that CD compensation be more efficiently implemented in the frequency domain (FD). In FD, the TDequivalent impulse response is translated into the size of the overlap for the overlap-&-save, or overlap-&-add algorithms [2]. The efficiency of the FD implementation depends on the size of the overlap in a sense that smaller overlaps requires less mathematical operations for processing a fixed-size data block. On the other hand, the overlap is given by the CDinduced temporal spread of the optical pulse, which depends on the signal spectral content [3]. Finding this spectral content in real systems is not trivial, since it is given by the interplay of the pulse shape and the intrinsic system optical and electrical filtering. Thus, the analytically computed CD equalizer impulse response length may diverge from the de facto required values.
Darli A. A. Mello School of Electrical and Computer Engineering, University of Campinas (UNICAMP) Campinas, Brazil Email:
[email protected]
The aim of this work is to provide practical values of the overlap size for the frequency-domain CD compensation for the non-return-to-zero (NRZ), return-to-zero (RZ) with 50% duty cycle, and raised cosine (RC) Nyquist signals. The NRZ and RZ pulses are traditionally used in optical communications, because they are easily generated in the optical domain. Yet, for DSP-enabled transmitters, digitallygenerated Nyquist pulses are gaining attention, especially in the scope of the so-called superchannels [4], due to their superior spectral efficiency. In case of a sufficient optical signal-to-noise ratio (OSNR) margin, the findings of this work may serve as guidelines for trading off bit error rate (BER) and computational complexity. The rest of this paper is structured as follows: Section II reviews the computational complexity issues related to the frequency-domain CD compensation; Section III describes the simulation setup; Section IV presents and discusses the obtained results; and Section V concludes the paper. II.
F REQUENCY- DOMAIN CD
COMPENSATION AND COMPUTATIONAL COMPLEXITY
The CD quadratic phase response is translated into temporal spreading of the pulse energy. Therefore, the CD equalizer impulse response length should be matched to the fiber pulse broadening due to CD. The temporal broadening of a pulse with a spectral width ∆ω is given by [3]: c ∆T = zβ2 ∆ω = |D|z 2 ∆f, (1) fc where z is the fiber length, β2 is the group velocity dispersion parameter, λ is the optical carrier wavelength, c is the speed of light, fc is the carrier frequency, ∆f is the pulse spectral width in Hz, and D is the fiber dispersion parameter, related to β2 by: D = −β2 (2πc)/λ2 . The rightmost form of Eq. (1) frequently appears in the literature, e.g., in [5], [6], [7]. The equivalent finite impulse response (FIR) filter length, and hence the overlap size in FD equalization, are related to the temporal pulse broadening via the sampling time, Tsa : ⌉ ⌈ ∆T , (2) NCD = Tsa where ⌈x⌉ denotes the nearest integer larger than x. In the following, we use the required number of multiplications as a figure of merit for complexity estimation, since
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Fig. 1: Tap weights of an LMS-based dynamic equalizer for different pulse shapes: (a) NRZ; (b) RZ with 50% duty cycle; (c) Nyquist with roll-off factor = 0.2.
it is the dominant parameter for power consumption [8]. In [9] it was shown that the number of real multiplications per equalized bit in an FD equalizer is given by: Mb
=
κ (2MR + 3NFFT ), (3) (NFFT − NCD + 1) log2 M
assuming implementation of a complex multiplication by three real ones1 . In Eq. (3), κ is the oversampling factor, M is the size of the digital alphabet, and MR is the number of (nontrivial) real multiplications required for the computation of an 1 (e+jf) = (a+jb)·(c+jd) can be computed as e=(a-b)d+a(c-d) and f=(ab)d+b(c+d). This form requires three real multiplications and five real additions [10].
FFT of size NFFT , given by [10]: 3 NFFT (−3 + log2 NFFT ) + 6, (4) MR = 2 provided that NFFT = 2k , k ∈ N. Thus, as indicated by Eq. (3), the (per bit) computational complexity is inversely proportional to the difference between the FFT size and the overlap. Figure 1 shows the filter tap weights of a Ts /2-spaced dynamic equalizer (Ts is the symbol period) based on the least mean squares update criterion for NRZ, RZ 50%, and RCshaped signals for a 28 GBd QPSK transmission, corrupted by accumulated CD of 20 ns/nm (assuming a 3rd -order model), in the otherwise ideal transmission conditions. Additionally, the signal was filtered by a 50-GHz (passband) 4th -order Gaussian filter, and a 19-GHz (baseband) 2nd -order Gaussian filter, to
TABLE I: Simulation parameters Modulation format Pulse format
Simulation rates Phase noise Fiber parameters CD equalizer Dynamic equalizer algorithm AWGN noise
QPSK NRZ, roll-off = 0.2; RZ, roll-off = 1, duty cycle = 50%; Nyquist (RC), roll-off = 0.15, Pulse-shaping filter length = 512 Symbol rate = 32 GBd Samples per symbol = 32 Carrier linewidth = 100 kHz LO linewidth = 100 kHz D = 16.75 ps/(nm.km); Dispersion slope = 0.0656 ps/(nm2 .km) NFFT = 8192 CMA, Number of taps = 7 step-size = 0.001 OSNR = 14 dB
emulate optical filtering and receiver bandwidth limitations, respectively. Clearly, the length of the optimal impulse response depends on the signal pulse shape. As expected, Nyquistshaped signals, having the smallest spectral content, yield the shortest impulse response. Nevertheless, the quasi-rectangular energy spread of the Nyquist-case filter indicates a possibility of more severe signal degradation if under-dimensioned, than, e.g., in the case of RZ pulses. In the later sections we investigate these assumptions. III.
S IMULATION SETUP
The Matlabr -built simulation setup is as follows. Pseudorandom binary strings are mapped onto the digital alphabet. A pulse shaping procedure creates the complex envelopes for X-pol. and Y-pol. signal tributaries, representing each symbol by 32 samples. After adding the carrier phase noise, the signal is filtered by a 50-GHz 4th -order Gaussian filter to emulate channel optical filtering. Additive white Gaussian noise (AWGN) is added to the signal to yield 14 dB OSNR. Accumulated CD is added using the expansion of the propagation constant, β, in Taylor series up to 3rd -order (that is, including dispersion slope [3]), and X-pol. and Y-pol. signals are mixed at a 45◦ polarization rotation angle. At the receiver, local oscillator (LO) phase noise is added, and the signal is filtered by a 19-GHz low-pass 2nd -order Gaussian filter that emulates the receiver photo-detector and transimpedance amplifier bandwidths. The receiver signal processing consists of resampling to 2 samples per symbol (κ = 2) using native Matlab interpolation; frequency-domain CD compensation (FFT size 8192); adaptive equalization by the constant modulus algorithm (CMA) [11]; and blind phase search [12]. After symbol decision, differential decoding is applied. Table I summarizes the simulation parameters. IV.
R ESULTS AND DISCUSSION
Figures 2 (a-c) show the BER as a function of the overlap size for NRZ, RZ, and RC-shaped signals, respectively. The accumulated CD values vary from 0 to 90 ns/nm in steps of 10 ns/nm, roughly corresponding to transmission distances up to 5,300 km, assuming a standard single-mode fiber with dispersion parameters as in Table I. The obtained overlap values may be used as guidelines for practical frequencydomain CD equalizer design.
The RZ-shaped signals require longer overlap segments than the NRZ and Nyquist counterparts, as expected from Fig. 1. Nevertheless, those differences become more significant for high accumulated CD values. The curves in all three figures show a rather smooth descent with the overlap increase, allowing for trades-off between the BER and computational complexity. As an example, assume a transmission of an NRZ signal over 4200 km (roughly corresponding to the accumulated CD of 70 ns/nm). If instead of a hard decision coding scheme, operating at the pre-forward error correction (pre-FEC) BER around 3 × 10−3 [13], a soft decision FEC is used, allowing to operate at the pre-FEC BER of around 2 × 10−2 , the overlap length can be reduced from about 1200 to 300 samples, yielding a reduction in CD-compensationassociated computational complexity of η > 11%, computed as [see Eq. (3)]: η
−1 (L−1 1 − L2 ) × 100. L−1 1
=
(5)
In Eq. (5) Li = NFFT − NCDi + 1, where NCD1 and NCD2 are the initial and the final overlap sizes. For an FFT size of 4096, the complexity reduction in the above example is ∼ 24%. Figure 2 (d) shows the normalized BER versus the overlap size for the three investigated pulse formats and different accumulated CD values. Bit error rate normalization is done with respect to the BER value obtained without any dispersion, for comparison. Here, NRZ curves are shown in solid lines with round markers; RZ curves in dashed lines with square markers; and Nyquist curves in dash-dotted lines with asterisk markers. In the quasi-zero-penalty region, for sufficiently high accumulated CD values, Nyquist-shaped signal shows about 30-40% reduction in the overlap size in comparison with the RZ case, corresponding to the computational complexity reduction of ∼ 5-11% for NFFT = 8192, and ∼ 9-27% for NFFT = 4096. V.
C ONCLUSION
We inferred, by computer simulations, practical values of the overlap size required for the frequency-domain CD compensation for three pulse formats: NRZ, RZ with 50% duty cycle, and Nyquist (RC). The accumulated chromatic dispersion was varied up to 90 ns/nm, covering most practical requirements for long and ultra-long-haul systems. We showed, via simple examples, that for sufficiently high OSNR margins, the BER versus overlap trade-offs may yield about 10-24% reduction in computational complexity, depending on the FFT size of the equalizer block. We also showed that in the quasizero-penalty region Nyquist pulses require about 30-40% less overlap than RZ signals, namely for the accumulated CD values ≥ 40 ns/nm. VI.
ACKNOWLEDGEMENTS
This work was supported by the Fund for Technological Development in Telecommunications (FUNTTEL), Brazil. R EFERENCES [1] S. J. Savory, “Digital filters for coherent optical receivers,” Optics Express, vol. 16, no. 2, pp. 804–817, 2008. [2] J. Proakis and D. Manolakis, Digital signal processing: principles, algorithms, and applications, 3rd ed. Prentice Hall, 1996.
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Fig. 2: BER versus overlap size: (a) NRZ; (b) RZ with 50% duty cycle; (c) Nyquist. (d) Normalized BER with respect to the no CD case for the three pulse formats for selected accumulated CD values (20, 40, 60 and 80 ns/nm). NRZ – solid lines and round markers; RZ – dashed lines and square markers; Nyquist – dash-dotted lines and asterisk markers.
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