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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20273–20281
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Spectral efficiency limits of pre-filtered modulation formats

Yi Cai, Jin-Xing Cai, Alexei Pilipetskii, Georg Mohs, and Neal S. Bergano  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20273-20281 (2010)
http://dx.doi.org/10.1364/OE.18.020273


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Abstract

We investigate the spectral efficiency (SE) limit of pre-filtered modulation in optical fiber communication systems. We show that pre-filtering induced symbol correlation generates a modulation with memory and, thus, a higher SE limit than that of the original memoryless modulation. The SE limits of a series of modulation formats with varying number L of correlated symbols producing 50% of the original bandwidth are evaluated, which approach the SE limit of a modulation format with 50% pre-filtering as L → ∞. We show that the SE limit of a modulation format with L correlated symbols approximates a lower bound for the SE limit of the corresponding pre-filtered format.

© 2010 OSA

1. Introduction

High spectral efficiency (SE) is becoming a key requirement for further advance of optical fiber communication systems to satisfy the increasing capacity demand. Digital coherent detection techniques have enabled high SE transmission techniques such as polarization multiplexing, multi-level modulation, and orthogonal frequency domain multiplexing [1

1. M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009).

4

4. X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010).

]. These techniques are based on memoryless modulation formats with well defined theoretical SE limits. High SE transmission can also be achieved by employing tight pre-filtering and multi-symbol detection, which has been previously studied and demonstrated in intensity-detection systems [5

5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]

7

7. N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).

]. A high SE scheme employing a pre-filtered quaternary phase shift keying (QPSK) format together with maximum a posteriori probability (MAP) based coherent detection that was recently demonstrated [8

8. J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).

,9

9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

] achieved SE exceeding the limit of memoryless QPSK modulation. This is a counter-example for the commonly assumed SE limit for QPSK modulation with coherent detection. It was pointed out [9

9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

] that the memoryless SE limit no longer holds in the pre-filtered case due to modulation memory induced by pre-filtering. A question naturally follows then: what is the SE limit of a pre-filtered modulation format?

In this paper, we attempt to answer this question by taking into account pre-filtering induced symbol correlation in the SE limit evaluation. Inspired by the experimental observations in [8

8. J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).

,9

9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

], we develop a theoretical analysis which shows that the symbol correlation generates a more complex modulation with memory and, thus, a higher SE limit than that of the original memoryless modulation. Efforts have been made to evaluate the capacity limit [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

] of optical fiber channels taking into account some particular system features such as intensity-detection induced non-Gaussian noise statistics and fiber nonlinearities [11

11. J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. 19(8), 1104–1109 (2001). [CrossRef]

18

18. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010). [CrossRef]

]. To focus on the effect of pre-filtering induced symbol correlation, we assume in this paper an optical fiber channel with amplified spontaneous emission (ASE) noise being the dominant impairment. In this case a linear optical fiber channel is well represented by an additive white Gaussian noise (AWGN) channel model when employing coherent detection.

In particular, we apply the general SE limit formula that includes symbol correlations to a 50% pre-filtered (i.e. half of the spectrum is low-pass filtered) binary modulation format. The numerical evaluation to obtain the exact SE limit curve, however, is prohibitively complicated due to the infinite symbol correlation length induced by the pre-filtering. One possible solution is to approximate an SE limit based on Monte Carlo simulations of n channel inputs and outputs [19

19. D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001).

] that converges for large enough n according to the Shannon-McMillan-Breimann theorem [20

20. T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).

]. In this paper, we simplify the SE limit calculation by taking into account a finite number L of correlated symbols when applying the general SE limit formula. The SE limit of a modulation format with a finite number L of correlated symbols approaches that of a pre-filtered modulation format as L goes to infinity. Based on the trend observed in the statistic distribution of the Euclidean distances in signal space as L increases, we show that the SE limit of a modulation format with finite L approximates a lower bound for the SE limit of pre-filtered modulation formats (infinite L). The SE limit result for 50% pre-filtered binary modulation formats can be extended to other pre-filtering ratios and/or other modulation formats by going through a similar analysis.

The paper is organized in the following way. Section 2 describes the theoretical model for the SE limit analysis of pre-filtered modulation formats. Section 3 derives a general SE limit for pre-filtered modulation formats in linear noise-dominated optical fiber channels. Section 4 gives the upper and lower bounds for the SE limits of 50% pre-filtered binary phase shift keying (BPSK) and QPSK modulation formats. Section 5 summarizes the paper and draws conclusions.

2. Theoretical model for SE limit analysis of pre-filtered modulation

A channel model assuming AWGN is commonly used in channel capacity and SE limit studies. According to Shannon’s theory [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

], a T-second transmission (T → ∞) in an AWGN channel of band W can be specified by 2WT numbers, and the SE limit is given by
C/W=log2(1+PN),
(1)
where C is channel capacity and P and N are average signal and noise power, respectively. This is the ultimate SE limit for any modulation and coding schemes that can be applied to an AWGN channel. Figure 1
Fig. 1 (a) modulated spectrum, (b) in-phase and quadrature components in baseband, and (c) time-domain pulses of a T-second W-bandwidth signal that can approach the SE limit of an AWGN channel.
depicts the spectrum (a), in-phase and quadrature components (b), and time-domain pulses (c) of a T-second signal that can approach the SE limit given in Eq. (1) as T → ∞ and the channel input (x 1, x 2, …, xWT, q 1, q 2, …, qWT) are independent and identically distributed (i.i.d.) Gaussian random variables. The limit approaching signal has a rectangular spectrum of bandwidth W in the frequency domain, and 1/W spaced sinc pulses in the time domain, i.e.

sinc(2Wt)=sin(2πWt)2πWt.
(2)

The in-phase and quadrature components shown in Fig. 1 construct two orthogonal signal dimensions and each can achieve half of the SE limit given in Eq. (1). In our following discussions, we will first focus on 1-dimensional (1D) signals, e.g. the in-phase dimension, and then extend the results to 2-dimensional (2D) signals. If we denote the capacity of 1D signals as C 1D, then the 1D SE limit is half of the limit given in Eq. (1), i.e.,

C1D/W=12log2(1+PN),
(3)

As mentioned above, to approach the ultimate SE limit, we need to have independent noise-like signals as channel inputs. For practical advantages, however, we need to put certain constraints on the signal modulation. For example, a widely used binary modulation format, BPSK, requires to put a binary constraint on the signal (x 1, x 2, …, xWT), i.e. xi ∈ {–1, 1}, with equal probability of taking –1 or 1. The SE limit of such a binary modulation format is given by [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

,13

13. G. Kramer, A. Ashikhmin, A. J. van Wijngaarden, and X. Wei, “Spectral efficiency of coded phase-shift keying for fiber-optic communication,” J. Lightwave Technol. 21(10), 2438–2445 (2003). [CrossRef]

]
CB/W=fY(yi)log2[fY(yi)]dyilog2(2πeN),
(4)
where C B is the binary modulation capacity, yi is the channel output corresponding to the noise-impaired xi, and fY (yi) is the probability density function (pdf) of yi given by [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

]

fY(yi)=12μ=1112πNexp[(yiμ)22N].
(5)

The SE limits given in Eqs. (3) and (4) are compared in Fig. 2
Fig. 2 SE limits for unconstrained channel inputs and BPSK modulation [1D limit: from Eq. (3), BPSK: from Eq. (4)].
, where SNR/bit is the signal to noise ratio (SNR) normalized by information bit rate.

Clearly the achievable SE with a BPSK modulation is less than that of an unconstrained modulation format especially at high SNR/bit. As SNR/bit goes to infinity (i.e. noise power N → 0 for a given signal power P), the well-known SE limits C 1D/W → ∞ and C B/W → 1 bit/s/Hz can be derived from Eqs. (3) and (4). The difference between the two SE limits can be understood in terms of the usage of signal power. For a signal with statistical properties that approach that of white noise, the signal power is fully utilized to generate uncertainty of the signal [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

]. Signals with higher uncertainty carry more information. On the other hand for BPSK modulation, only a portion of the signal power contributes to the signal uncertainty (or carries information). If we loosen the binary constraint on the channel inputs, we would be able to exceed the BPSK SE limit and achieve a SE somewhere in between the two limits shown in Fig. 2.

To determine the SE limit for pre-filtered modulation formats, we first show that pre-filtering changes the BPSK constraint on the channel input and generates a more complex modulation with symbol correlation. For simplicity and clarity in the following discussion we assume 50% pre-filtering without loss of generality. Figure 3
Fig. 3 Spectrum and pulses of a binary modulation (a) before and (b) after pre-filtering, where the dashed lines indicate sampling instants.
depicts the spectrum of a binary modulation format before and after pre-filtering along with 6 example sinc pulses (all taking a value of 1). Figure 3(a) shows that the original BPSK modulation occupies a bandwidth of W and all 6 pulses can be modulated independently in the time domain. Each of the 6 pulses carries 1-bit of information without interfering with each other.

With the 50% pre-filtering as shown in Fig. 3(b), the pulses become two times wider and start to overlap. By sampling at the center of the 3 red pulses, we also sample energy (and thus information) from the blue, green, and black pulses through symbol correlation induced by the pre-filtering. Hence, with pre-filtering we do not necessarily lose the information carried by the original signal but the information-bearing signal energy is redistributed. To receive the information bits distributed among the pre-filtered symbols, however, a single-symbol detection scheme would suffer significant degradation of receiver sensitivity due to pre-filtering induced inter symbol interference (ISI) [9

9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

]. Ideally, it requires waveform detection to achieve the optimal receiver sensitivity in the tightly pre-filtered case. Practically, a multi-symbol detection scheme such as MAP [9

9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

,24

24. Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010).

] or MLSE [5

5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]

7

7. N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).

] based detection can help to recover most of the ISI penalty.

According to Nyquist’s sampling theorem, the pre-filtered signal has half the bandwidth of the original signal, therefore, we only need half of the original samples to completely determine the pre-filtered signal. These samples [indicated by dashed lines in Fig. 3(b)], however, are no longer binary but have multiple levels. This results in a complex multi-level alphabet for the channel input. The possible values of a pre-filtered sample can be recursively calculated by
A0={1, 1},    Ak={Ak1sinc(k12), Ak1, Ak1+sinc(k12)},
(6)
where k is the recursion index, and A k is the resulting alphabet after the k-th recursion with 2k−1 symbols included in the calculation on each side of the symbol under investigation. The corresponding probability mass function (pmf) is given by

P0={12, 12},    Pk={14Pk1, 12Pk1, 14Pk1}.
(7)

To calculate the possible signal value and the corresponding pmf of the center red pulse in Fig. 3(b), e.g., for k = 1 the blue and green pulses are included in the recursive calculations in Eqs. (6) and (7). Figure 4
Fig. 4 The probability mass function of the signal value evaluated with Eq. (7) for k = 4.
plots the pmf of the signal value of the 50% pre-filtered BPSK modulation format for k = 4. It shows a much more complex modulation than the original BPSK modulation, which results in a higher SE limit as shown in the following sections.

3. SE limit of pre-filtered modulation in linear noise-dominated optical fiber channels

The SE limit of BPSK modulation in Eq. (4) was derived assuming independent BPSK symbols. The independency condition simplifies the SE limit analysis on discrete memoryless modulation formats [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

,20

20. T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).

22

22. J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).

]. The independency condition, however, does not hold in the pre-filtered case as shown in Fig. 3(b). To evaluate the SE limit of the pre-filtered modulation format we need to take into account the pre-filtering induced symbol correlation. The general formula for the SE limit with correlated channel inputs is similar to the one for channel with memory [17

17. R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]

,21

21. R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).

], which needs to take into account a sequence of correlated channel inputs and outputs. With correlated inputs, the alphabet set A is a collection of vector values, rather than the scalar values in the memoryless case. In the 50% pre-filtered BPSK case as modeled in Section 2, a T-second transmission with bandwidth W requires L = WT/2 pulses [i.e., the red pulses in Fig. 3(b)] modulated by a vector element from
A={(xi1,xi2,...,xiL), i=1, 2, ..., 2WT},
(8)
and the SE limit is given by [10

10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

,21

21. R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).

],
CF/WF=LimT2TW...fY(y1,y2, ...,yL)log2[fY(y1,y2, ...,yL)]dy1dy2...dyLlog2(2πeN),
(9)
where C F/W F represents the SE limit of a pre-filtered modulation format with W F = W/2, Y = (y 1, y 2, …, yL) is the channel output vector that is the sum of a channel input vector X = (x 1, x 2, …, xL) and a noise vector (n 1, n 2, …, nL), and f Y (y 1, y 2, …, yL) is the joint pdf of Y.

To evaluate f Y (y 1, y 2, …, yL), we need to know the statistics of the noise vector (n 1, n 2, …, nL). In a linear optical fiber channel dominated by ASE noise, the noise samples (i.e., n 1, n 2, …, nL) are Gaussian random variables. For the SE limit evaluation, we assume an ideal rectangular filter shape as shown in Fig. 3(b). Thus, the power spectral density of the ASE noise after the receiver filter (with a bandwidth of W/2 to match the 50% pre-filter at the transmitter) has a rectangular shape. Correspondingly, the autocorrelation function of the filtered noise is a sinc function with nulls at all the sampling instants [Fig. 3(b)]. Hence, the noise samples n 1, n 2, …, nL are independent Gaussian random variables and, thus, the joint pdf of Y is given by [22

22. J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).

]
fY(y1,y2, ...,yL)=(xi1,xi2,xiL)Ap(xi1,xi2,xiL)j=1L12πNexp[(yixij)22N],
(10)
where p(xi1, xi2, …, xiL) is the probability of the chosen alphabet vector (xi1, xi2, …, xiL). Note that if the noise is not sampled at the Nyquist rate as in some practical implementations, the resulting noise samples are no longer independent and the noise correlation must be taken into account in the joint statistics [23

23. N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. 28(10), 1564–1572 (2010). [CrossRef]

]. In Eq. (10), if the binary inputs are equally probable for the original BPSK modulation, the alphabet vectors of the pre-filtered modulation are also equally probable, thus,

p(xi1,xi2,...,xiL)=2WT, i=1, 2, ..., 2WT.
(11)

For a noiseless channel, i.e. N = 0, the evaluation of Eqs. (9) and (10) results in a SE limit of 2 bit/s/Hz, which is two times that of regular BPSK modulation. In other words, the SE limit is doubled by the 50% pre-filtering as SNR/bit → ∞. In noisy channels, however, the evaluation of Eqs. (9) and (10) becomes prohibitively complicated as WT increases. To simplify the calculation, we approximate the SE limit of the pre-filtered BPSK modulation by taking a limited number of correlated symbols, i.e. a small value of L in Eqs. (9) and (10).

4. Approximation of SE limit for 50% pre-filtered BPSK and QPSK modulation

Figure 5
Fig. 5 Approximations of the SE limit of 50% pre-filtered BPSK by taking a small number (L) of correlated symbols in the SE limit evaluation.
compares the approximated SE limits for pre-filtered BPSK with L = 2, 3, 4 to the SE limits of 1D unconstrained modulation and BPSK modulation. All the approximated cases fall between the 1D limit and the BPSK limit. At high SNR/bit, the approximated limits approach 2 bits/s/Hz, which is twice as high as the BPSK limit. At low SNR/bit, the approximated limits are very close to the 1D limit and significantly higher than the BPSK limit. These comparisons indicate that the pre-filtering helps to increase the achievable SE and exceed the SE limit of the original discrete memoryless modulation.

Figure 5 shows that as L increases from 2 to 4, the SE limit becomes higher. The trend of the SE limit when increasing the number (L) of correlated symbols in the limit evaluation can be explained with a distance analysis on the channel input vectors. The Euclidean distance between two vectors, (xi 1, xi 2, …, xiL) and (xk 1, xk 2, …, xkL), in a channel input alphabet set A is defined as

Dik=(xi1xk1)2+(xi2xk2)2+...+(xiLxkL)2.
(12)

The larger the distance between channel inputs, the lower the detection error probability and, thus, the smaller the overhead required for error correction coding. Given a channel input alphabet, we can analyze the statistics, in terms of mean and variance, of the distances between all combinations of the component vector pairs. Intuitively, a larger mean and a smaller variance of the vector distances in an input alphabet indicate a better performance modulation format.

Up to this point, we have evaluated approximations for the SE limit of 50% pre-filtered BPSK modulation by including a limited number (L) of correlated symbols (up to L = 4) in the numerical calculations. We also analyzed the statistics of the distance between the channel input-vectors for L up to 9. To evaluate the exact SE limit for pre-filtered BPSK modulation, we need to let L → ∞, which cannot be evaluated with practical means. However, based on the trend of the signal distance statistics shown in Fig. 6, we predict that the distance mean increases and variance decreases as L → ∞. Rigorous proof of this prediction is an open research topic.

Based on the trend of the SE limit and the signal distance statistics observed in Fig. 5 and Fig. 6, we use the SE limit for L = 4 as a lower bound for the SE limit of the 50% pre-filtered BPSK modulation. The SE limit is also bound by the 1D limit of unconstrained modulation and the 2 bits/s/Hz limit derived from Eqs. (9) and (10) for infinite SNR.

Figure 7
Fig. 7 Upper and lower bounds on the 50% pre-filtered BPSK and QPSK modulation.
shows the SE bounds for 50% pre-filtered BPSK with the area between the lower (green line) and upper (black line) bound being shaded. The 1D results can be extended to 2D by doubling the SE limits. The resulting bounds for 50% pre-filtered QPSK modulation are also shown in Fig. 7. It shows that at high SNR/bit the SE limit of discrete memory-less QPSK modulation can be almost doubled with 50% pre-filtering. At low SNR/bit, pre-filtering can also significantly increase the SE limit and closely approach the ultimate 2D SE limit.

For comparison, Fig. 7 also shows the SE limits of memoryless BPSK (blue line), QPSK (blue line), and 16-QAM (blue circle) modulation. It shows that in medium-to-high SNR/bit regular QPSK modulation and 50% pre-filtered BPSK modulation have similar SE limits. On the other hand, at low SNR/bit the SE limit of QPSK modulation (without pre-filtering) is higher than that of pre-filtered BPSK modulation. It is also evident that the SE limit of 16-QAM modulation is very close to the evaluated lower bound for the SE limit of the 50% pre-filtered QPSK. The practical implementations of a memoryless 16-QAM versus a pre-filtered QPSK system (or memoryless QPSK vs. pre-filtered BPSK), however, can be very different. To increase SE with memoryless multi-level signaling the implementation complexity on the transmitter side must be increased. On the other hand, a pre-filtered format with symbol correlation results in a more complicated receiver due to the required multi-symbol detection.

In the above discussions, we focused on the SE limit when applying 50% pre-filtering to a discrete memory-less binary modulation format. The results can be extended to other pre-filtering ratios and/or other modulation formats by going through a similar analysis. In the theoretical SE limit evaluation, ideal multi-symbol detection and error correction coding are assumed without giving any specific scheme. Practically, the limit can be approached by employing different combinations of advanced coding, detection, and equalization schemes such as the ones described in [5

5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]

9

9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

,23

23. N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. 28(10), 1564–1572 (2010). [CrossRef]

25

25. I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008). [CrossRef]

].

5. Conclusions

We investigated the spectral efficiency (SE) limit of pre-filtered modulation formats in optical fiber communication systems. A theoretical model was constructed to analyze the SE limit for pre-filtered modulation formats. The pre-filtering induced symbol correlation generates a modulation with memory for which a higher SE limit applies than that for the original memory-less format. We evaluated the SE limits of a series of modulation formats with varying number L of correlated symbols that approach the limit for 50% pre-filtered binary modulation as L goes to infinity. We showed that the SE limit of the correlated L-symbol modulation is a lower bound for the SE limit of the corresponding pre-filtered format. At high SNR/bit, the SE limit of discrete memory-less BPSK and QPSK modulation can be almost doubled with 50% pre-filtering. At low SNR/bit, pre-filtering can also significantly increase the SE limit which closely approaches the ultimate SE limit for unconstrained modulation.

References and links

1.

M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009).

2.

A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “69.1-Tb/s (432x171-Gb/s) C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” Proc. OFC/NFOEC’10, paper PDPB7 (2010).

3.

X. Zhou, J. Yu, M. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, Jr., “64-Tb/s (640x107-Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Proc. OFC/NFOEC’10, paper PDPB9 (2010).

4.

X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010).

5.

N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]

6.

M. Rubsamen, P. J. Winzer, and R.-J. Essiambre, “MLSE receivers for narrow-band optical filtering,” Proc. OFC/NFOEC‘06, paper OWB6 (2006).

7.

N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).

8.

J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).

9.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).

10.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).

11.

J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. 19(8), 1104–1109 (2001). [CrossRef]

12.

Y. Cai, N. Ramanujam, J. M. Morris, T. Adali, G. Lenner, A. B. Puc, and A. Pilipetskii, “Performance limit of forward error correction codes in optical fiber communications,” Proc. OFC/IOOC’01, paper TuF2 (2001).

13.

G. Kramer, A. Ashikhmin, A. J. van Wijngaarden, and X. Wei, “Spectral efficiency of coded phase-shift keying for fiber-optic communication,” J. Lightwave Technol. 21(10), 2438–2445 (2003). [CrossRef]

14.

J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004). [CrossRef]

15.

I. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. 23(11), 3755–3763 (2005). [CrossRef]

16.

Y. Cai, “Performance limits of FEC and modulation formats in optical fiber communications,” Proc. LEOS’06, paper WH1 (2006).

17.

R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]

18.

A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010). [CrossRef]

19.

D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001).

20.

T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).

21.

R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).

22.

J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).

23.

N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. 28(10), 1564–1572 (2010). [CrossRef]

24.

Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010).

25.

I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 28, 2010
Revised Manuscript: August 31, 2010
Manuscript Accepted: September 2, 2010
Published: September 8, 2010

Citation
Yi Cai, Jin-Xing Cai, Alexei Pilipetskii, Georg Mohs, and Neal S. Bergano, "Spectral efficiency limits of pre-filtered modulation formats," Opt. Express 18, 20273-20281 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20273


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References

  1. M. Salsi, H. Mardoyan, P. Tran, C. Koebele, E. Dutisseuil, G. Charlet, and S. Bigo, “155x100Gbit/s coherent PDM-QPSK transmission over 7,200km,” Proc. ECOC’09, paper PD2.5 (2009).
  2. A. Sano, H. Masuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “69.1-Tb/s (432x171-Gb/s) C- and extended L-band transmission over 240 km using PDM-16-QAM modulation and digital coherent detection,” Proc. OFC/NFOEC’10, paper PDPB7 (2010).
  3. X. Zhou, J. Yu, M. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, and R. Lingle, Jr., “64-Tb/s (640x107-Gb/s) PDM-36QAM transmission over 320km using both pre- and post-transmission digital equalization,” Proc. OFC/NFOEC’10, paper PDPB9 (2010).
  4. X. Liu, S. Chandrasekhar, B. Chu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “Transmission of a 448-Gb/s reduced-guard-interval CO-OFDM signal with a 60-GHz optical bandwidth over 2000 km of ULAF and five 80-GHz-Grid ROADMs,” Proc. OFC/NFOEC’10, paper PDPC2 (2010).
  5. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical pre-amplifier,” Opt. Express 13(12), 4568–4579 (2005). [CrossRef] [PubMed]
  6. M. Rubsamen, P. J. Winzer, and R.-J. Essiambre, “MLSE receivers for narrow-band optical filtering,” Proc. OFC/NFOEC‘06, paper OWB6 (2006).
  7. N. Alic, E. Myslivets, and S. Radic, “1.0 bit/s/Hz spectral efficiency in single polarization at 2000km with narrowly filtered intensity modulated signals,” Proc. IEEE Summer Topical Meeting 2008, paper TuD3.4 (2008).
  8. J. X. Cai, Y. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “Transmission of 96x100G pre-filtered PDM-RZ-QPSK channels with 300% spectral efficiency over 10,608km and 400% spectral efficiency over 4,368km,” Proc. OFC/NFOEC’10, paper PDPB10 (2010).
  9. Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and S. Neal Bergano, “High spectral efficiency long-Haul transmission with pre-filtering and maximum a posteriori probability detection,” Proc. ECOC’2010, paper We.7.C.4 (2010).
  10. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
  11. J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. 19(8), 1104–1109 (2001). [CrossRef]
  12. Y. Cai, N. Ramanujam, J. M. Morris, T. Adali, G. Lenner, A. B. Puc, and A. Pilipetskii, “Performance limit of forward error correction codes in optical fiber communications,” Proc. OFC/IOOC’01, paper TuF2 (2001).
  13. G. Kramer, A. Ashikhmin, A. J. van Wijngaarden, and X. Wei, “Spectral efficiency of coded phase-shift keying for fiber-optic communication,” J. Lightwave Technol. 21(10), 2438–2445 (2003). [CrossRef]
  14. J. Kahn and K. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” J. Sel. Top. Quantum Electron. 10(2), 259–272 (2004). [CrossRef]
  15. I. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightwave Technol. 23(11), 3755–3763 (2005). [CrossRef]
  16. Y. Cai, “Performance limits of FEC and modulation formats in optical fiber communications,” Proc. LEOS’06, paper WH1 (2006).
  17. R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]
  18. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28(4), 423–433 (2010). [CrossRef]
  19. D. Arnold, and H. Loeliger, “On the information rate of binary-input channels with memory,” Proc. ICC’2001, pp. 2692–2695 (2001).
  20. T. Cover, and J. Thomas, Elements of Information Theory, John Wiley & Sons, New York (1991).
  21. R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, New York (1968).
  22. J. G. Proakis, Digital Communications, 4th Edition, McGraw-Hill, New York (2001).
  23. N. Alic, M. Karlsson, M. Sköld, O. Milenkovic, P. A. Andrekson, and S. Radic, “Joint statistics and MLSD in filtered incoherent high-speed fiber optic communications,” J. Lightwave Technol. 28(10), 1564–1572 (2010). [CrossRef]
  24. Y. Cai, D. Foursa, C. R. Davidson, J.-X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” Proc. OFC/NFOEC’10, paper OTuE1 (2010).
  25. I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm. 26(6), 73–83 (2008). [CrossRef]

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