Abstract: We investigate the application of time and frequency packingtechniques, an extension of the classical faster-than-Nyquist signaling, tolong-haul optical links. These techniques provide a significant spectralefficiency increase and represent a viable alternative to overcome the theo-retical and technological issues related to the use of high-order modulationformats. Adopting these techniques, we successfully demonstrate throughsimulations the transmission of 1 Tbps over 200 GHz bandwidth in arealistic (nonlinear) long-haul optical link.
References and links1. S. Chandrasekhar and X. Liu, “Enabling components for future high-speed coherent communication systems,”
in Proc. Optical Fiber Commun. Conf. (OFC’09) (Los Angeles, CA, USA, 2011), Paper OMU5.2. G. Colavolpe, T. Foggi, E. Forestieri, and G. Prati, “Robust multilevel coherent optical systems with linear pro-
cessing at the receiver,” J. Lightwave Technol. 27, 2357–2369 (2009).3. J. Zhao and A. Ellis, “Electronic impairment mitigation in optically multiplexed multicarrier systems,” J. Light-
wave Technol. 29, 278–290 (2011).4. G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of nyquist-WDM terabit
superchannels based on PM-QPSK, PM-8PSK or PM-16QAM subcarriers,” J. Lightwave Technol. 29, 53–61(2011).
5. J. E. Mazo, “Faster-than-Nyquist signaling,” Bell System Tech. J. 54, 1450–1462 (1975).6. F. Rusek and J. B. Anderson, “The two dimensional Mazo limit,” in Proc. IEEE International Symposium on
Information Theory, (Adelaide, Australia, 2005), pp. 970–974.7. A. Barbieri, D. Fertonani, and G. Colavolpe, “Time-frequency packing for linear modulations: spectral efficiency
and practical detection schemes,” IEEE Trans. Commun. 57, 2951–2959 (2009).8. N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai, “On information rates for mismatched decoders,” IEEE
Trans. Inform. Theory 40, 1953–1967 (1994).9. D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of infor-
mation rates for channels with memory,” IEEE Trans. Inform. Theory 52, 3498–3508 (2006).10. G. D. Forney, Jr., “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol
interference,” IEEE Trans. Inform. Theory 18, 284–287 (1972).11. L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error
rate,” IEEE Trans. Inform. Theory 20, 284–287 (1974).12. G. Colavolpe and A. Barbieri, “On MAP symbol detection for ISI channels using the Ungerboeck observation
model,” IEEE Commun. Lett. 9, 720–722 (2005).13. G. Ungerboeck, “Adaptive maximum likelihood receiver for carrier-modulated data-transmission systems,” IEEE
Trans. Commun. com-22, 624–636 (1974).14. F. Rusek and D. Fertonani, “Lower bounds on the information rate of intersymbol interference channels based
on the ungerboeck observation model,” in Proc. IEEE International Symposium on Information Theory (2009).
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26600
15. G. Colavolpe, D. Fertonani, and A. Piemontese, “SISO detection over linear channels with linear complexity inthe number of interferers,” IEEE J. Sel. Top. Signal Process. (submitted).
16. A. Barbieri, G. Colavolpe, T. Foggi, E. Forestieri, and G. Prati, “OFDM vs. single-carrier transmission for 100Gbps optical communication,” J. Lightwave Technol. 28, 2537–2551 (2010).
1. Introduction
The evolution of long-haul optical communications is actually oriented beyond the current es-tablished 100 Gbps [1]. Hence, there is a widespread interest in algorithms and techniqueswhich can overcome the difficulties concerning a further capacity growth, enabling the trans-mission of data rates up to 1 Tbps. Among the more severe limitations involved in such a systemupgrade, the technological and practical issues of processing high data rates on a single channel,and the optical channel impairments related to the required transmit power (i.e., the nonlineareffects), should be noticed. Thus, effective and feasible solutions for a deeper exploitation ofthe optical channel and available devices and technologies should be envisaged.
In optical communications, as in most digital communication systems, orthogonal signalingis usually adopted to ensure the absence of intersymbol interference (ISI) and, in multi-carrierscenarios, also the absence of interference from adjacent channels. In fact, in coherent opticalsystems, possibly employing polarization multiplexing, given a conventional transmitter, withMach-Zehnder (MZ) modulators and return to zero (RZ) or non-return to zero (NRZ) shap-ing pulses, when group velocity dispersion (GVD) and polarization mode dispersion (PMD)are effectively compensated and nonlinear effects are limited, proper filtering and samplingat the receiver ensure that even a symbol-by-symbol detector enables an almost-optimal per-formance [2]. However, if the orthogonality condition must be satisfied and given the presenthardware limitations, even when adopting spectrally efficient orthogonal techniques such asorthogonal frequency-division multiplexing (OFDM) [3] or Nyquist wavelength-division mul-tiplexing (WDM) [4], the only way to increase the spectral efficiency with the aim of reachingthe goal of 1 Tb/s transmissions is to increase the constellation cardinality, thus employingmodulation formats more sensitive to the nonlinear effects. This is clearly illustrated by Fig. 1,where the achievable spectral efficiency η per polarization for three different modulation for-mats, namely quaternary and octal phase shift keying (QPSK of 8-PSK) and 16-ary quadratureamplitude modulation (16-QAM), is shown (the Shannon limit for additive white Gaussiannoise channels is also reported) as a function of Eb/N0, Eb being the energy per bit and N0/2the noise power spectral density (PSD) per polarization. An excess bandwidth of 20% withrespect to the minimum value ensuring orthogonal signaling (i.e., a transmitted bandwidth of1.2 the signaling frequency) is assumed, since, in real systems a penalty must be expected, fortechnological and practical issues, when implementing Nyquist WDM systems [4, Fig. 4]. Thisgives, for 16-QAM, an asymptotic maximum value of 3.2 bit/s/Hz instead of 4 bit/s/Hz.
An alternative way to improve the spectral efficiency of low-order modulations, such asQPSK or 8-PSK, can be obtained by giving up the orthogonality condition. For example, faster-than-Nyquist signaling (FTN, see [5, 6]) is a well known technique consisting of reducing thespacing between two adjacent pulses in the time domain (i.e., the symbol interval) well be-low that corresponding to the Nyquist rate. Controlled ISI is thus introduced but the ISI-freeperformance is still reached provided the optimal detector (whose complexity could, however,become unmanageable) is employed and proper values of the symbol interval selected (suchthat the minimum Euclidean distance of the system is not reduced). Following the same prin-ciple, in [7] both the symbol interval and the frequency separation among adjacent channelsare optimized with the aim of maximizing the achievable spectral efficiency η , which is thusused as a performance measure instead of the minimum distance. In addition, rather than theoptimal receiver, a symbol-by-symbol detector working on the samples at the output of a filter
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26601
0
0.5
1
1.5
2
2.5
3
3.5
4
-2 0 2 4 6 8 10 12 14
η [b
it/s/
Hz]
Eb/N0 [dB]
Shannon limit16-QAM8-PSKQPSK
Fig. 1. Achievable spectral efficiency for different modulation formats with orthogonalsignaling and a bandwidth equal to the signaling frequency.
matched to the transmitted shaping pulse (matched filter, MF) is considered, thus constrainingthe receiver complexity to its minimum value [7].
By employing more sophisticated detection algorithms, η can be further improved. Two re-ceiver architectures will be considered in this paper: (i) a proper filtering of the MF outputplus a symbol-by-symbol detector, and (ii) a low-complexity maximum a posteriori (MAP)symbol detector, which takes into account only a limited amount of interference. Improvingη without increasing the constellation order can be considerably convenient since the largerthe constellation size, the higher the decoding complexity. Moreover, it is well known thatlow-order constellations are more robust to channel impairments such as nonlinearities, whoseeffects are already increased by the higher transmitted power needed to obtain higher spectralefficiency values, and phase noise. In the case of frequency packing, a further improvementcould be achieved by adopting, at the receiver side, a multi-user detector, although this caseis not considered here since it would increase the receiver complexity. The remainder of thispaper is organized as follows. The system model is described in Section 2. The spectral effi-ciency computation and optimization is then described in Section 3, considering detectors withdifferent complexity. Numerical results are reported in Section 4 and, finally, some conclusionsare drawn in Section 5.
2. System Model
We consider a frequency-division multiplexed system where adjacent channels (assumed per-fectly synchronized) employ the same linear modulation format and shaping pulse p(t). Thebaseband equivalent of the received signal is expressed as
r(t) =√
2Es ∑n
∑�
xn,�p(t −nT )e j2π�Ft +w(t) (1)
where Es is the symbol energy, T the symbol interval, xn,� the symbol transmitted over the�-th channel during the n-th symbol interval, F the frequency spacing between adjacent chan-nels, and w(t) a circularly symmetric zero-mean white Gaussian noise process with PSD 2N0.When polarization multiplexing is also employed, r(t) is the received signal on one state of
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26602
polarization. In the following, we will avoid to consider the presence of GVD and PMD since,as known, they can be perfectly compensated through a proper two-dimensional equalizer [2].The transmitted symbols {xn,�} are independent and uniformly distributed and belong to a givenzero-mean M-ary complex constellation χ properly normalized such that E{|xn,�|2}= 1. Notethat, in order to avoid boundary effects, the summations in Eq. (1) extend from −∞ to +∞,namely an infinite number of time epochs and carriers are employed. As in [7], we considerthe central user only and in the definition of the spectral efficiency we will use F as a measureof the signal bandwidth. The symbol interval T and frequency spacing F will be optimized tomaximize the spectral efficiency.
The possibility to generate a transmitted signal with expression√
2Es ∑n
∑�
xn,�p(t −nT )e j2π�Ft
is strictly related to the availability of a linear modulator. In other words, let us consider thetransmitted signal associated to the user for � = 0. If pulse p(t) has support larger than T , thissignal cannot be directly generated through a MZ modulator unless it is properly linearized.This is due to the nonlinear transfer function of the MZ modulator between the electrical signalat its input and the optical signal at its output. We could, however, use a MZ modulator togenerate a linearly modulated signal with shaping pulse having support at most T and then“stretch” the transmitted pulses through an optical filter. Hence, in this case, time packing isnot an available option. The only degree of freedom will be the frequency spacing F and thebandwidth of the optical filter used at the MZ output.
3. Spectral Efficiency optimization
3.1. Symbol-by-Symbol detection
We first consider a symbol-by symbol detector, working on the central user, i.e., that for �= 0.The receiver is composed by a filter matched to the shaping pulse p(t), followed by a properdiscrete-time filter and a symbol-by-symbol detector. Although the discrete-time filter couldbe, in general, fractionally-spaced (FS), the detector will operate on one sample per symbolinterval. These samples will be denoted by {yk,0} and can be expressed as
yk,0 =√
2Esxk,0h(0,0,k)+ ∑(n,�) �=(0,0)
xk−n,�h(n, �,k)+ zk (2)
in which h(n, �,k) is the residual interference at time kT due to the �-th user and the (k−n)-thtransmitted symbol, and {zk} is the additive noise term, in general colored unless a whiteningfilter (WF) is employed after the MF. The discrete-time filter is assumed properly normalizedsuch that the noise variance is 2N0. The dependence of coefficients h(n, �,k) on k is through acomplex coefficient of unit amplitude which disappears for � = 0 (hence h(n,0,k) is indepen-dent of k) and is due to the fact that F is not an integer multiple of 1/T .
The interference due to adjacent symbols and users is here modeled as a zero-mean Gaus-sian process with PSD equal to 2NI , of course independent of the additive thermal noise—anapproximation exploited only by the receiver, while in the actual channel the interference isclearly generated as in Eq. (2). The interference is really Gaussian distributed only if the trans-mitted symbols xk,� are Gaussian distributed as well, which is a good approximation when Tand F are optimized and a large number of interferers arises.
With the above mentioned Gaussian approximation, the channel model assumed by the re-ceiver is
yk,0 =√
2Esxk,0h(0,0,k)+ vk (3)
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26603
where {vk} are independent and identically distributed zero-mean circularly symmetric Gaus-sian random variables, with variance 2(N0 +NI). From Eq. (2) it is
NI = ∑(n,�) �=(0,0)
Es|h(n, �,k)|2 (4)
which does not depend on k. The achievable information rate (AIR), measured in bit per channeluse, for this mismatched receiver (see [8, 9]) is
I(xk,0;yk,0) = Exk,0,yk,0
⎧⎪⎨
⎪⎩log2
⎛
⎜⎝
MpYk,0|Xk,0(yk,0|xk,0)
∑x∈χ
pYk |χ(yk,0|x)
⎞
⎟⎠
⎫⎪⎬
⎪⎭(5)
where pYk,0|Xk,0(yk,0|xk,0) is a Gaussian probability density function (pdf) of mean xk,0 and vari-
ance 2(N0 +NI) (in accordance with the auxiliary channel model of Eq. (3)), while the outerstatistical average, with respect to xk,0 and yk,0, is carried out according to the real channelmodel of Eq. (2) [9]. Equation (5) can be evaluated efficiently by means of a Monte Carloaverage [9]. From a system viewpoint, the spectral efficiency is more significant than the infor-mation rate. Under the assumption of infinite transmission, η is defined as
η =1
FTI(xk,0;yk,0)
[bit
s ·Hz
]. (6)
For a given constellation and shaping pulse, it is possible to show that the optimal spacingsT and F that provide the largest η , depend on the signal-to-noise ratio (SNR). Also, it mustbe noticed that as T and F are reduced, interference increases and thus the information ratedegrades, but η can be improved. This means that, for a given fixed code, the asymptotic per-formance will degrade. Information theory, however, ensures that with a proper code of lowerrate, those values of spectral efficiency can be obtained.
The properties of the function η(T,F,ES/N0) cannot be easily studied in closed form, but it isclear, by physical arguments, that it is bounded, continuous in T and F , and tends to zero whenT,F → 0 or T,F → ∞. Hence, the function η(T,F,Es/N0) has a maximum value—accordingto our findings, in most cases there are no local maxima other than the global maximum. Theproblem can be solved by evaluating η(T,F,ES/N0), for fixed modulation, shaping pulse andEs/N0, on a grid of values of T and F (coarse search), followed by an interpolation of theobtained values (fine search).
A measure of the SNR more significant than Es/N0 is given by Eb/N0, for which the follow-ing Eq. holds Es = I(Es)Eb. The optimization problem becomes
ηM(Eb/N0) = maxT,F>0
η(T,F,Eb/N0) . (7)
In order to solve it, the AIR is first evaluated for some proper values of the couple (T,F),which ensure an accurate sampling of the AIR, and Es/N0. For each couple (Ti,Fj), cubicspline interpolation can be used to obtain a continuous function of Es/N0 (fine search), denotedas I(Ti,Fj,Es/N0). Then, given a value of Eb/N0 the following fixed-point problems are solvedin Es/N0 for different couples (Ti,Fj),
Es
N0= I
(Ti,Fj,
Es
N0
)Eb
N0
and the AIRs corresponding to the solutions are denoted by I(Ti,Fj,Eb/N0). Further improve-ments could be achieved by adding NI as variable in Eq. (7). However, we have found bynumerical results that choosing NI as in Eq. (4) is almost optimal.
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26604
The spectral efficiency depends on the employed discrete-time filter. Since the optimizationof this filter with the aim of maximizing the spectral efficiency is a hard task, we restricted ouranalysis to the case of a FS minimum-mean-square-error (MMSE) feedforward filter with atmost 22 coefficients, since it provided, among all considered filters, the best results.
As mentioned before, when a nonlinear MZ modulator is adopted, the optimization problemof Eq. (7) will be reduced to the optimization of the frequency spacing and of the bandwidth ofthe optical filter (for this latter parameter, only a coarse search was performed).
3.2. Single-User Trellis Processing
An improved, still achievable, lower bound can be reached by the adoption of more effectivedetection algorithms, namely a more complex receiver, able to cope with part of the interfer-ence introduced by the time-frequency packing. Interference due to the adjacent users is notconsidered—a single-user receiver is adopted.
For a general channel with finite intersymbol interference, an optimal MAP symbol detectorcan be designed working on the samples at the WF output. These samples, denoted as Forneyobservation model [10], can still be expressed as in Eq. (2) with a proper expression of coeffi-cients h(n, �,k). We assume to adopt the optimal receiver for the following auxiliary channel:
yk,0 =√
2Es ∑0≤n≤L
fnxk−n,0 + vk (8)
where { fn}n≥0 are such that fn = h(n,0,k) and, as mentioned, are independent of k, whereasthe noise samples {vn}, that take into account the white noise and the residual interference,are assumed independent and identically distributed zero-mean circularly symmetric Gaussianrandom variables with variance 2(N0 +NI), with
NI = ∑n>L
Es| fn|2 +∑n
∑��=0
Es|h(n, �,k)|2 . (9)
In other words, the MAP symbol detector, which takes the form of the classical algorithm byBahl, Cocke, Jelinek and Raviv (BCJR) [11] working on a trellis whose state is defined asσk,0 = (xk−1,0, . . . ,xk−L,0) , takes into account L interfering symbols only, according to a givenmaximal allowable receiver complexity. Being the number of trellis states equal to S = ML, wewill consider very limited values of L.
Let us define xn = (x0,0,x1,0, ...,xn,0) and yn = (y0,0,y1,0, ...,yn,0). The simulation-basedmethod described in [9] allows to evaluate the AIR for the mismatched receiver, i.e.,
I(x;y) = limn→+∞
1n
I(xn;yn)
= limn→+∞
1n
E
{log2
p(yn|xn)
p(yn)
} [bit
ch.use
]. (10)
In Eq. (10), the probability densities have the same meaning as in Eq. (5) for what concernsthe actual and the auxiliary channels, and can be evaluated recursively through the forwardrecursion of the BCJR detection algorithm matched to the auxiliary channel model [9]. Oncethe AIR has been computed, the spectral efficiency can be derived and the optimal time andfrequency spacings optimized accordingly, as described in the previous section.
For channels with finite ISI, optimal MAP symbol detection can be equivalently implementedby working directly on the MF output [12], i.e., on the so-called Ungerboeck observationmodel [13]. The equivalence does not hold when reduced-complexity detection is consideredand interference from adjacent channels arises. The spectral efficiency η for the Ungerboeck
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26605
observation model can be computed as described for the Forney model. Since the Forney obser-vation model has shown to be less convenient, in terms of spectral efficiency values [14], thanthe Ungerboeck model, it will not be considered further in this paper.
Notice that tighter lower bounds can be obtained by using a more general auxiliary channelmodel and the corresponding optimal receiver, i.e., a multi-user receiver for the central user(that with � = 0) that takes into account J adjacent signals on each side (an approximationexploited only by the receiver, while in the actual channel the interference is generated as inEq. (1)). The benefit is two-fold. First, these tighter bounds allow to evaluate the performancedegradation due to the use of single-user receivers with respect to a more involved multi-userreceiver, which is more “matched” to the real channel. Second, it gives a practical performanceupper bound when low-complexity approximate multi-user receivers, for example based onlinear equalization or interference cancellation (see [15] and references therein) are employed.Obviously, in this case some (limited) degradation must be expected.
4. Simulation Results
In this section, we report the optimal spectral efficiency ηM as a function of Eb/N0 for differentmodulation formats. Since we consider the case of a MZ modulator, as mentioned. simulationresults for frequency-packing only are presented. The employed shaping pulses are those re-sulting form the use of RZ pulses with duty cycle 33, 50, and 66%, and a Gaussian or 4th-orderGaussian optical filter. The frequency spacing F and the optical filter bandwidth B, have beenoptimized for each value of Eb/N0. Hence, their values change along the curves. The consid-ered modulation formats are QPSK, 8-PSK, and 16-QAM. Regarding QPSK, we would like tomention that, in case of use of Gray mapping, it can be viewed, with a proper rotation of theconstellation, as two independent BPSK signals transmitted over the in-phase and quadraturecomponents, respectively. Hence, at the receiver side, we may use two identical and indepen-dent detectors, one working on the in-phase and the other one on the quadrature component.This is beneficial in case of adoption of a MAP symbol detector. In fact, when L interferingsymbols are taken into account, we have two detectors working on a trellis with 2L states in-stead of a single detector working on a trellis with 4L states. Hence, for a given complexity, alarger number of interferers can be taken into account.
Fig. 2 shows the achievable spectral efficiency ηM for QPSK, 8-PSK, and 16-QAM modu-lations with RZ pulses with 50% duty cycle. No differences were observed for different valuesof the duty cycle, although the optimal values of B and F may be different. At the output ofthe MZ modulator, a 4-th order Gaussian optical filter is employed, and, at the receive side,after the MF an optimized discrete-time MMSE filter is used followed by a symbol-by-symboldetector. The Shannon Limit is also shown for comparison. By comparing these results withthose in Fig. 1 and related to the case of orthogonal signaling, we may observe a significantimprovement for QPSK.
Fig. 3 shows the achievable spectral efficiency ηM of the same system but with trellis pro-cessing, that, as can be noticed, allows to improve the spectral efficiency, with respect to asymbol-by-symbol detector, of almost 30% for QPSK and 8-PSK, whereas a limited improve-ment is obtained for 16-QAM. The trellis processing is here performed with S = 16 for QPSKand 16-QAM, and with S = 64 for 8-PSK. If we compare the theoretical SE curve for QPSKthat can be obtained with the proposed technique and that corresponding to orthogonal signal-ing, it is clear that an asymptotic SE of 3.3 bit/s/Hz per polarization can be obtained insteadof 2 bit/s/Hz per polarization. Thus the gain is of 65%. In addition, it is known that a realis-tic transmission system should envisage an excess bandwidth (as an example see ref. [4]) and,as shown in Fig. 1, a SE of 1.6 bit/s/Hz per polarization with a 20% excess bandwidth (morerealistic value) should be taken as a reference. In this sense the improvement is around 106%.
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26606
0
0.5
1
1.5
2
2.5
3
3.5
4
-2 0 2 4 6 8 10 12 14
η M [
bit/s
/Hz]
Eb/N0 [dB]
Shannon Limit16-QAM8-PSKQPSK
Fig. 2. Achievable spectral efficiency for different modulation formats with a narrow 4th-order Gaussian optical filtering, frequency packing, and symbol-by-symbol detection.
0
0.5
1
1.5
2
2.5
3
3.5
4
-2 0 2 4 6 8 10 12 14
η M [
bit/s
/Hz]
Eb/N0 [dB]
Shannon Limit
16-QAM
8-PSK
QPSK
Fig. 3. Achievable spectral efficiency for different modulation formats with narrow 4th-order Gaussian optical filtering, frequency packing, and trellis processing.
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26607
O/E
End
Front2−D
FFE
BCJRDECODERDECODER
LDPC
Fig. 4. Receiver architecture.
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
−2 0 2 4 6 8 10 12 14
η M[b
it/s/
Hz]
Eb/N0 [dB]
QPSK
Shannon Limit
(1)
(2)
(3)
Fig. 5. Results obtained with practical modulation and coding formats.
What information theory promises can be approached by using proper coding schemes, evenin the presence of nonlinear effects. We consider a compensated optical link (14 spans of about90 km each, described in [16]) with QPSK modulation and three different combinations ofcodes and subchannel data rates. Similar results are expected in a link without inline dispersioncompensation. Two low-density parity-check codes having codewords of length 64800 bits andrates 4/5 and 8/9, respectively, are employed. As the target bit rate in our simulations is 1Tbps on a bandwidth of 200 GHz and given a polarization-multiplexed QPSK transmission,a resulting spectral efficiency of 2.5 bit/s/Hz per polarization is required. We consider threesetups: (1) seven 180 Gbps 30 GHz-spaced channels using the code with rate 4/5, (2) six 188Gbps 35 GHz-spaced channels with the same code, and (3) eight 140 Gbps 26 GHz-spacedchannels with the code of rate 8/9. Transmit and receive optical filter bandwidths are chosenequal to 0.3/T for (1), and 0.325/T for (2) and (3), whereas the launch powers per channelare −2.5,−2.6, and −4 dBm, respectively. A two-dimensional (2-D) adaptive FFE with 9taps processes the signals received over two orthogonal states of polarization to compensatefor GVD and PMD [2], so that its output is provided to a BCJR detector which iterativelyexchange information with the LDPC decoder for a maximum of 50 iterations. The receiverarchitecture is shown in Fig. 4. The values of Eb/N0, estimated as if the channel were linear,for the three systems able to provide a bit-error rate (BER) of 10−7 are reported in Fig. 5
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26608
-50
-40
-30
-20
-10
0
10
-150 -100 -50 0 50 100 150
norm
aliz
ed P
SD
f [GHz]
Fig. 6. Normalized PSD of the transmitted eigth QPSK 140 Gbps channels, carrying 1 Tbpsover a bandwidth of 200 GHz.
along with the theoretical spectral efficiency curves. It may be observed that, despite the lackof optimization in the code design, we are 2.5÷3 dB far from the theoretical results. This lossis due to the presence of nonlinear effects which require a careful redesign of the codes and theinvestigation of the best combination of coding and modulation. The PSD of the transmittedsignal of system (3) is also shown in Fig. 6. It can be noticed that, since the bandwidth of eachsubchannel is highly reduced by filtering, the required sampling rate is always within state-of-the-art technology, i.e, well below 50 Gsample/s.
We point out that no alternative schemes could be adopted in this scenario with this targetvalue of spectral efficiency. In fact, 8-PSK or 16-QAM modulations with orthogonal signalingand proper coding schemes fail to attain the target BER value, in this specific link, due to thehighly detrimental nonlinear effects. In this case, we considered both the cases of single-carriertransmissions, relaxing all constraints on the sampling rate available today, and multicarriertransmissions with a limited number of subcarriers to satisfy such technological constraints.
5. Conclusions
In this paper, we investigated a possible solution to improve the spectral efficiency of low-orderlinear modulations with different kinds of receivers. The improvement is related to the use ofnarrow optical filtering and frequency packing, in order to giving up the signal orthogonalityin the time and in the frequency domain, and on the adoption of detectors with different com-plexity. We showed preliminary results of combinations of modulation and coding formats thatreach promising performance on a realistic optical link, and, in particular, 1 Tbps over 200 GHzbandwidth has been successfully demonstrated by simulations.
#155776 - $15.00 USD Received 30 Sep 2011; revised 28 Nov 2011; accepted 29 Nov 2011; published 14 Dec 2011(C) 2011 OSA 19 December 2011 / Vol. 19, No. 27 / OPTICS EXPRESS 26609
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