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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 3 — Feb. 2, 2009
  • pp: 1385–1403
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Evaluation of the computational effort for chromatic dispersion compensation in coherent optical PM-OFDM and PM-QAM systems

P. Poggiolini, A. Carena, V. Curri, and F. Forghieri  »View Author Affiliations


Optics Express, Vol. 17, Issue 3, pp. 1385-1403 (2009)
http://dx.doi.org/10.1364/OE.17.001385


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Abstract

Recently, coherent-detection (CoD) polarization multiplexed (PM) transmission has attracted considerable interest, specifically as a possible solution for next-generation systems transmitting 100 Gb/s per channel and beyond. In this context, enabled by progress in ultra-fast digital signal processing (DSP) electronics, both multilevel phase/amplitude modulated formats (such as QAM) and orthogonal-frequency-division multiplexed (OFDM) formats have been proposed. One specific feature of DSP-supported CoD is the possibility of dealing with fiber chromatic dispersion (CD) electronically, either by post-filtering (PM-QAM) or by appropriately introducing symbol-duration redundancy (PM-OFDM). In both cases, ultra-long-haul fully uncompensated links seem to be possible. In this paper we estimate the computational effort required by CD compensation, when using the PM-QAM or PM-OFDM formats. Such effort, when expressed as number of operations per received bit, was found to be logarithmic with respect to link length, bit rate and fiber dispersion, for both classes of systems. We also found that PM-OFDM may have some advantage over PM-QAM, depending mostly on the over-sampling needed by the two systems. Asymptotically, for large channel memory and small over-sampling, the two systems tend to require the same CD-compensation computational effort. We also showed that the effort required by the mitigation of polarization-related effects can in principle be made small as compared to that of CD over long uncompensated links.

© 2009 Optical Society of America

1. Introduction

Optical system research is currently targeting 100 Gb/s per channel transmission and higher. At such speeds, all detrimental fiber propagation effects are exacerbated and this has brought up a number of severe challenges for system implementation. To cope with these challenges, coherent detection (CoD) has been advocated.

CoD has only recently become a practical option, thanks to astounding progress in electronic digital signal processing (DSP). CoD allows to use advanced modulation formats such as polarization-multiplexed (PM) quadrature amplitude modulation (PM-QAM) [1

1. R. Noé, “Phase Noise-Tolerant Synchronous QPSK/BPSK Baseband-Type Intradyne Receiver Concept With Feedforward Carrier Recovery,” J. Lightwave Technol. 23, 802–808 (2005). [CrossRef]

]-[11

11. J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver,” J. Lightwave Technol. 26, 36–42 (2008). [CrossRef]

] and coherent orthogonal frequency division multiplexing (OFDM) [12

12. W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express 16, 841–859 (2008). [CrossRef] [PubMed]

]-[17

17. B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the Effect of FWM in Coherent Optical OFDM Systems,” in Proc. OFC 2008, Anaheim (CA), paper JWA58, San Diego (CA), Feb. 24-28, (2008).

]. The latter can be transmitted PM, too [20

20. S. L. Jansen, I. Morita, and H. Tanaka, “16x52.5-Gb/s, 50-GHz Spaced, POLMUX-CO-OFDM Transmission over 4,160 km of SSMF Enabled by MIMO Processing,” in Proc. ECOC 2007, paper PD 1.3, Berlin (D), Sept. 16-20, (2007).

]-[23

23. E. Yamada, et al., “Novel No-Guard-Interval PDM CO-OFDM Transmission in 4.1 Tb/s (50x88.8 Gb/s) DWDM Link over 800 km SMF Including 50-Ghz Spaced ROADM Nodes”, in Proc. OFC 2008, paper PDP8, San Diego (CA), Feb. 24-28, (2008).

]. OFDM is being very actively investigated with direct-detection (DD) as well. See for instance [18

18. A. J. Lowery and J. Armstrong, “Orthogonal Frequency Division Multiplexing for Dispersion Compensation of Long-Haul Optical Systems,” Opt. Express 14, 2079–2084 (2006). [CrossRef] [PubMed]

], [19

19. A. J. Lowery, “Improving Sensitivity and Spectral Efficiency in Direct-Detection Optical OFDM Systems,” in Proc. OFC 2008, paper OMM4, San Diego (CA), Feb. 24-28, (2008).

]. In this paper, we concetrate on CoD PM-OFDM, because the comparison with CoD PM-QAM is more balanced. However, many of the results obtained here would also apply with straightforward adaptations to some DD-OFDM schemes as well.

The most commonly used Tx and Rx structures for the two formats are shown in Figs. 1–2. Concerning transmission DSP, PM-OFDM needs two IFFTs and four DACs, whereas PM-QAM may avoid DSP altogether for simple constellations but may need it for larger ones. Concerning the Rx’s, the electro-optical analog front-ends are essentially identical for both formats. Fig. 2 only shows the CD-compensation main blocks.

As in [25

25. H. Bulow, B. Franz, A. Klekkamp, and F. Buchali, “40 Gb/s Distortion Mitigation and DSP-Based Equalisation,” in Proc. ECOC 2007, Berlin, Germany, Sept. (2007).

], we chose to express the DSP computational effort in terms of arithmetical operations per transmitted information bit (OPb). Another fundamental assumption that we made is that the same DSP technology is used for both classes of systems and, in particular, the same FFT/IFFT technology is available to both PM-QAM and PM-OFDM. This makes it possible to carry out a fair and meaningful comparison. Note that, from an implementation viewpoint, additions and multiplications have different complexities. However, with both formats, by far the bulk of the computational effort consists of FFTs/IFFTs, as we shall see later on. Therefore, for both formats the DSP ratio of additions to multiplications is essentially established by the common FFT/IFFT technology. Consequently, as far as a comparative analysis of the relative computational effort of PM-QAM to PM-OFDM is concerned, discerning additions from multiplications would not change the result.

While estimating OPb for either PM-OFDM and PM-QAM, we take into account various important implementation aspects such as zero-padding, over-sampling and the efficiency of FIR filter implementation.

Finally, a discussion of the results is proposed and some conclusions are drawn in Section 8. Throughout the paper, the following notation is used:

  • Rb: total channel bit rate [Tb/s];
  • L: link length [km];
  • λ: channel wavelength [nm];
  • D: fiber dispersion [ps/(nm km)];
  • c: speed of light, 299792 [km/s];
  • Rs: symbol rate [TBaud]; in the case of PM-OFDM it is the rate of the so-called ‘OFDM symbols’, for PM-QAM it is the standard symbol rate;
  • Ts: symbol duration [ps], equal to R -1 s;
Fig. 1. Typical Tx structures for PM-QAM and PM-OFDM. Legend: ‘PBS’, polarizing beam splitter; ‘MOD’, electro-optical modulator (typically nested Mach-Zehnder); ‘DES’, data deserializer; ‘SER’, data serializer; ‘DAC’, digital-to-analog converter. Depending on specific implementations and, for PM-QAM, the size of the constellation, certain blocks/functions could be omitted or simplified.
Fig. 2. Typical Rx structures for PM-QAM and PM-OFDM, with only CD-compensation post-processing shown. All other processing is omitted. Legend: ‘PBS’, polarizing beam splitter; ‘Bal’, dual balanced photo-detector; ‘LPF’, low-pass filter; ‘ADC’, analog-to-digital converter. The ‘CD FIR’ block for PM-QAM may contain an FFT and IFFT if implemented in frequency-domain. The ‘CD comp’ block for PM-OFDM multiplies each element of the FFT output array by a suitable complex number.

M: number of bits per subcarrier and per PM-OFDM symbol, or number of bits per PM-QAM symbol.

2. Number of Subcarriers for PM-OFDM

In this section we derive the number of OFDM subcarriers NSC needed to support a given system-design target amount of uncompensated CD.

In the absence of CD we would have:

Ts=Rs1=NSC·MRb.
(1)

However, CD makes the various subcarriers propagate at different group velocities. The absolute value of the group delay difference between the two outermost subcarriers at frequencies f 1 and fNSC [16

16. S. L. Jansen, I. Morita, T. C. W. Schenck, N. Takeda, and H. Tanaka ‘Coherent Optical 25.8-Gb/s OFDM Transmission Over 4160-km SSMF,’ J. Lightwave Technol. 26, 6–15 (2008). [CrossRef]

]:

Δτg=λ2cD(fNSCf1)L.
(2)

Such group delay difference makes the symbols on each subcarrier slip, relative to one another, and a cyclic prefix needs to be added in order to preserve a suitable common sampling window at the Rx, of duration Ts, good for the OFDM symbols on all subcarriers. This avoids symbol discontinuities and prevents loss of subcarrier orthogonality [28

28. L. Hanzo, M. Munster, B.J. Choi, and T. Keller, OFDM and MC-CDMA, John Wiley and Sons, Hoboken (NJ), (2003).

], [12

12. W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express 16, 841–859 (2008). [CrossRef] [PubMed]

]. The cyclic prefix makes the symbol duration increase to a new value:

T′s=Ts+Δτg.
(3)

Even though the duration of the FFT sampling window at the Rx remains Ts, the actual time taken to transmit one symbol is now Ts > Ts and the actual OFDM symbol rate goes down to:

R′s=1T′s=Rs1+ΔτgTs.
(4)

Note that due to the slow-down of the subcarrier symbol rate, the spectral occupancy of each subcarrier decreases. However, from the viewpoint of the Rx FFT, whose input array of signal samples still spans a time-window Ts, the subcarriers are at minimum spacing when they are spaced Rs and trying to pull them closer would generate loss of orthogonality. Therefore, even though Rs is no longer the symbol rate, it still remains the minimum frequency separation between adjacent OFDM subcarriers.

Since the symbol rate has gone down, also the total bit rate carried by the OFDM channel goes down to a lower value:

R′b=Rs1+ΔτgTs.
(5)

However, this is unacceptable because the nominal OFDM channel total bit rate must remain Rb. To restore the original bit rate, it is necessary to add more carriers, i.e., to increase NSC to a greater NSC. Unfortunately, increasing the number of subcarriers in turn increases Δτg, which would require a longer cyclic prefix and eventually an even greater NSC, and so on. The problem must then be solved in a combined way. First, we remark that:

R′s=RbN′SC·M
(6)

Then, we equate the rightmost-hand side of Eq. (4) to the right-hand side of Eq. (6). By rearrangement of the resulting equation, we achieve an expression for N SC, which contains Δτg. We then use the approximated form of Eq. (2) to eliminate Δτg. Remembering that Ts = R -1 s, we further obtain the following intermediate result: NSC = N SC · (1 + 8 |D|LNSC R 2 s. Using Eq. (1) to replace Rs with Rb/(MNSC), we then get:

N′SC=[18D·L·Rb2NSC·M2]1·NSC
(7)

where NSC is the increased number of subcarriers needed to support the cyclic prefix while keeping Rb constant.

Note that in Eq. (7) Nsc is essentially an initial guess of the needed number of subcarriers. Once a value for Nsc has been somehow decided, then Eq. (7) tells us how many subcarriers N′sc are actually needed to cope with dispersion. However, the fact that N′sc depends on an arbitrary initial guess makes Eq. (7) somewhat unsatisfactory. It would be desirable to eliminate Nsc and directly find the actually needed number of subcarriers N′sc.

It turns out that this is not possible, because the problem does have one degree of freedom which cannot be eliminated. However, such degree of freedom could be attributed to a more meaningful quantity than the arbitrary guess NSC.

We therefore define the CD-induced overhead, as

k=N′SCNSC.
(8)

Note that k ≥ 1. This quantity is crucial because the cyclic prefix has two detrimental effects on the system, both of which are directly expressed in terms of k.

One effect is the loss of bandwidth efficiency. We use the symbol ρB for the bandwidth efficiency in the absence of cyclic prefix. When using the cyclic prefix, we have shown that we need more subcarriers to transmit the same bit rate. Since the spacing among subcarriers must remain the same, this means that the use of the cyclic prefix lowers ρB to a new value ρ′B which is given by:

ρ′B=ρB·k1.
(9)

The other detrimental effect of the cyclic prefix is that of impacting the system sensitivity. Transmission of more subcarriers requires more power, because it is easily found that we cannot lower the power per subcarrier without worsening both the per-subcarrier and the global bit error rate (BER). Put it differently, we are wasting power for cyclic prefix transmission. The resulting optical signal-to-noise-ratio (OSNR) penalty is simply:

ΔOSNRdB=10log10(k).
(10)

Therefore k is a fundamental system design parameter, in the sense that fixing k we can a priori set a limit to both the loss of bandwidth efficiency and the system OSNR penalty.

Based on (8), by means of simple substitutions we can re-write Eq. (7) as:

N′SC=k2k18D·L·Rb2M2.
(11)

This important equation clearly shows that there is not a unique solution for N′SC. Rather, there are many possible solutions, depending on the overhead k that we are willing to accept. Note that if we try to minimize the overhead, i.e., make k close to 1, the number of needed subcarriers N′SC diverges. This means that CD has a definite and unavoidable impact on OFDM systems, because some overhead must be accepted for the system to be feasible.

Fig. 3. Necessary number of OFDM subcarriers N′sc vs. system OSNR penalty ∆OSNRdB, for uncompensated links of 1000, 2000 and 3000 km (dash-dotted, dashed and solid lines, respectively). The system parameters are shown in Table 1.

Incidentally, Eq. (11) has its minimum for k = 2, i.e., for 3 dB OSNR penalty and 50% reduction of the bandwidth efficiency. The plots in Fig. 3 confirm it. On the other hand, the OSNR penalty at such minimum is too large for k = 2 to be a solution of practical interest.

Finally, there have been proposals for OFDM sub-banding [21

21. S. L. Jansen, I. Morita, and H. Tanaka, “10x121.9-Gb/s PDM-ODFM Transmission with 2-b/s/Hz Spectral Efficiency over 1,000 km of SSMF,” in Proc. OFC 2008, paper PDP2, San Diego (CA), Feb. 24-28, (2008).

], [24

24. W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16, 6378–6386 (2008). [CrossRef] [PubMed]

], to help decrease the cyclic overhead. The concept is that of dividing the transmitter signal into Ksub OFDM sub-bands within the same Tx channel, and then demodulating each sub-band as a separate OFDM signal. This way, each sub-band occupies a smaller bandwidth and the group delay difference between a sub-band extreme subcarriers can be made much smaller than the group delay difference between the extreme subcarriers of the overall Tx channel (ideally Ksub -1 times smaller). In addition, the speed of DACs and ADCs can be likewise reduced, since each sub-band approximately carries a fraction Ksub -1 of the payload. This is a very interesting concept but it has also drawbacks: the TX and Rx must use 4Ksub DACs and ADCs (though slower), and must perform 2Ksub IFFTs and FFTs. Moreover, both the TX and RX must make use of perfectly synchronous RF oscillators and RF mixers to perform sub-band upconversions and downconversions. Other techniques for mo/demodulation are also possible but added complexity is always present, in other forms.

In this paper, we elect to restrict the scope of the investigation to the case of a straightforward single-band PM-OFDM signal, leaving more elaborate architectures for future investigation.

Table 1. Reference System Parameters

table-icon
View This Table

3. Operations per bit for OFDM

The OFDM Tx makes use of an inverse FFT (IFFT) to create the modulating signals. Since we assume polarization multiplexing, two IFFTs are needed. The minimum order of the IFFT for the OFDM Tx coincides with the number N′SC of necessary subcarriers. A higher-order IFFT can be used to increase the number of time-samples per OFDM symbol that the DACs use to create the modulating waveforms. This simplifies the removal of aliases off the spectrum and makes the analog electrical modulating waveforms into the electro-optical modulators more ideal. The increase in the IFFT order can be obtained through zero-padding, by imposing zero-amplitude coefficients to ‘ghost subcarriers’, which may reside on either side of the payload subcarrier spectrum. We assume that the Tx IFFT is of order n Tx N′SC, where the ‘oversampling factor’ n Tx ≥ 1 takes zero-padding into account. Note that zero-padding requires faster DACs and the speed of DACs is one of the most critical aspects of OFDM implementation. Therefore a key design goal is to try and keep it as small as possible.

The minimum order for the Rx FFT is again N′SC, like for the Tx IFFT. Oversampling can be operated at the Rx as well. The FFT would then process a larger number of samples than just N′SC. This would allow some spectral margin against aliasing, specifically to protect the subcarriers at the channel band edges. In the calculations that follow, the FFT is assumed to be performed over n Rx N′SC samples, where n Rx ≥ 1 takes oversampling into account. It is interesting to notice that the oversampling factors at the Tx and Rx, n Tx and n Rx, are independent of each other. They can be separately optimized according to the specific Tx and Rx individual optimization constraints.

Note that we are using the symbol n Rx both in the context of PM-QAM (see Section 4) and of PM-OFDM. In both cases it gives an indication of the oversampling that is carried out at the Rx, though the specific definition is somewhat different. Since the two different contexts of use will always be clearly defined, we keep the same notation.

Oversampling requires faster ADCs. Even though somewhat less critical from a technological viewpoint than DACs, ADCs are challenging too. Here as well, the design goal is to try and keep oversampling to a minimum.

Keeping in mind the above caveats regarding n Tx and n Rx, the number of arithmetic operations required to demodulate a single bit of the payload, or OPb, is given by:

OPb=OPs,Tx+OPs,RxN′SC·M
(12)

where OPs,Tx and OPs,Tx are the total number of operations needed to process a full OFDM symbol at the Tx and Rx , respectively. Such processing requires computing two IFFTs and two FFTs (one per polarization) over n Tx N′SC and n Rx N′SC complex numbers, respectively.

We assume that the available FFT technology is such that a FFT or IFFT performed over an array of N complex numbers requires a number of operations OP:

OP=q·Nlog2(N).
(13)

Note that the asymptotic complexity of the split-radix algorithm [39

39. P. Duhamel and H. Hollmann, “Split-radix FFT algorithm,” Electron. Lett. 20, 14–16 (1984). [CrossRef]

] is such that q ≃ 4. The well-known Cooley-Tukey algorithm has a slightly larger OP count, but essentially behaves similarly. In [25

25. H. Bulow, B. Franz, A. Klekkamp, and F. Buchali, “40 Gb/s Distortion Mitigation and DSP-Based Equalisation,” in Proc. ECOC 2007, Berlin, Germany, Sept. (2007).

], it was conservatively assumed q = 5. However, though important from a system design viewpoint, the actual value of q becomes largely irrelevant within a relative comparison of PM-OFDM with PM-QAM, if we assume that the same FFT technology is available to both systems. Actual implementation details may also deeply affect the on-chip performance of a certain FFT algorithm, but by the same reasoning they are unimportant to the effect of a comparison between the two formats, as long as they are using the same technology.

Taking Eq. (13) into account, we have at the Tx:

OPs,Tx=2q·nTxN′SC·log2(nTxN′SC).
(14)

To compute OP s,Rx one only needs to change the subscripts ‘Tx’ with ‘Rx’. Note that the factor ‘2’ in front of the right-hand side is due to the fact that two separate IFFTs are needed, one per polarization.

Using Eqs. (11), (12

12. W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express 16, 841–859 (2008). [CrossRef] [PubMed]

) and (14), the following result is found:

OPb=2qMnTxlog2(nTxk2k18D·L·Rb2M2)+2qMnRxlog2(nRxk2k18D·L·Rb2M2).
(15)

This estimate of operations per bit is not yet complete because CD has the effect of phase-shifting the symbols on each subcarrier through a different phase factor. Such phase factor can be estimated using pilot tones [29

29. Xingwen Yi, W. Shieh, and Yan Tang, “Phase Estimation for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 19, 919–921 (2007). [CrossRef]

]. Irrespective of how estimation is done, the Rx FFT output must be multiplied times a complex correction factor which costs 6 operations per complex multiplication, per polarization. The total is then 12 operations, per subcarrier.

Taking the above into account, the OFDM total operations per bit becomes:

OPb,PMOFDM=2qMnTxlog2(nTxk2k18D·L·Rb2M2)+2qMnRxlog2(nRxk2k18D·L·Rb2M2)+12M.
(16)

4. Number of FIR filter taps for PM-QAM

The use of FIR filters to implement fractionally-spaced equalizers (FSEs) to compensate for CD in CoD systems was proposed by Winters, back in 1990 [30

30. J. H. Winters, “Equalization in Coherent Transmission Systems Using a Fractionally Spaced Equalizer,” J. Lightwave Technol. 8, 1487–1491 (1990). [CrossRef]

]. More recently, Taylor has reframed the concept in the context of modern DSP [31

31. M. G. Taylor, “Coherent Detection Method Using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. 16, 674676 (2004). [CrossRef]

]. Specifically, CoD PM-QAM needs two complex-coefficient FIR filters (Fig. 2) to compensate for CD (see for instance [32

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

], [33

33. S. J. Savory, “Digital Filters for Coherent Optical Receivers,” Opt. Express 16, 805–817 (2008). [CrossRef]

]).

The duration of the FIR filters impulse response τF depends on the channel memory, that we will call μ. We will express both τF and μ in number of symbol intervals. They are different quantities by definition but, if we constrain the FIR filters to exactly compensate for the channel memory induced by CD, then from well-known results of signal theory and filtering theory it follows τFμ.

The channel memory μ depends both on the accumulated dispersion D · L and on the symbol rate Rs. It also depends on the actual Tx pulse spectral/temporal shape. The smoother the pulse, the smaller μ and consequently τF. Therefore, determining the actual length of the FIR filter is not simple. In fact, it should be done “a posteriori”, based on a penalty constraint. One should gradually reduce the length of the FIR filter till a certain target penalty is incurred. However, different FIR impulse-response synthesis/optimization techniques could be used, which may lead to different results. Also, different Tx pulse shapes intrinsically generate different μ’s and therefore require different FIR lengths.

In [32

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

], the FIR filters were designed based on a minimum mean-square-error (MMSE) algorithm. The results fitted an estimate of the FIR filter impulse response duration which reads (see Eq. (30

30. J. H. Winters, “Equalization in Coherent Transmission Systems Using a Fractionally Spaced Equalizer,” J. Lightwave Technol. 8, 1487–1491 (1990). [CrossRef]

) in [32

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

]):

τF=10·λ23π·cD·LRs2.
(17)

The assumptions were: target penalty 2 dB, NRZ pulses obtained by passing square pulses through a 5-th order Bessel filter with bandwidth 0.8 ∙ Rs. Assuming operation at λ = 1550 nm, Eq. (17) becomes: τF ≃ 8.5 · |D| · LR 2 s (the number ‘8.5’ has dimensions [(nm2 · s)/km]).

Given the many factors that impact the FIR filter length, we decided to compare Eq. (17) with a formula obtained in a quite different way. It was best-fitted based on results from uncompensated systems using an MLSE receiver to mitigate dispersion. The MLSE receiver can be used to estimate the channel memory μ, simply by adding more memory to the Viterbi processor itself till the system-required OSNR closely approaches its asymptotic minimum vs. processor memory. Such processor memory (in number of symbols) gives an accurate estimate of μ.

In various MLSE systems, both through simulations and experiments [34

34. G. Bosco, P. Poggiolini, and M. Visintin, “Performance Analysis of MLSE Receivers Based on the Square-Root Metric”, J. Lightwave Technol. 26, 2098–2109 (2007). [CrossRef]

]-[36

36. P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, “Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver,” Opt. Express 16, 12919–12936 (2008). [CrossRef] [PubMed]

], we found that the following law fitted well the needed processor memory vs. CD, for smooth NRZ pulses:

μ=λ2c1DL·Rs2.
(18)

If we again assume to operate at λ=1550 nm, then: μ ≃ 8 |D| · L · R 2 s. Since, as mentioned, the FIR filter impulse response duration should ideally match the channel memory, (18) also gives us an estimate of τF for a FIR filter capable of compensating for the set amount of CD.

Eqs. (17) and (18) differ by about 6%. Given the completely different approach, this is quite a remarkable result and confirms the general validity of both estimates. The slightly higher value of (17) can be explained by the relatively mild Tx impulse smoothing. More drastic Tx filtering than applied in [32

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

] is possible, as it was done for instance in [36

36. P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, “Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver,” Opt. Express 16, 12919–12936 (2008). [CrossRef] [PubMed]

], likely leading to some reduction of τF. In the following, we will therefore take Eq. (18) as the estimate of μ, and hence of τF.

The actual number of taps of the FIR filter needed to compensate for the channel memory would then ideally be: NF = τF · n Rx, where nRx is the number of samples per symbol taken by the Rx analog-to-digital converter (ADC).

The parameter n Rx is critical for system design and it is currently being debated how low it can practically be made. The value n Rx = 2 guarantees good performance whereas the value n Rx = 1 is the theoretical lower limit. It is also possible to use intermediate values, such as n Rx = 1.5. The lower n Rx, the lower is the FIR computational effort. However, operating close, or at, n Rx = 1 may cause large penalties due to aliasing and other problems [32

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

]. The actual penalty will also depend on the pulse shape and consequent spectral occupation.

Note also that irrespective of the value of n Rx, it is mandatory that there is a digital clock-recovery circuit, possibly using an interpolation/decimation stage, that eventually provides one single sample per symbol to carry out decision. As stated in the introduction, we consider this topic outside of the scope of the paper. On the other hand, we point out that CD compensation can and should occur before clock recovery, also because the clock recovery circuit would not lock on a highly dispersed signal. Current system prototypes, such as Nortel’s PM-QPSK at 43 Gb/s, follow this scheme. As a result, the computational effort of the FIR filters for CD compensation is correctly estimated using n Rx as the number of samples per symbol.

This issue is therefore complex. For now we conclude that the number of FIR filter taps is estimated to be:

NF=8nRxDL·Rs2=8nRxDL·Rb2M2=nRx·μ
(19)

keeping in mind all the the caveats regarding n Rx.

If the reference system parameters of Table 1 are substituted into (18), we find that the predicted channel memory at 3000 km is very large: about 310 symbols. Assuming a conservative n Rx = 2, then Eq. (19) yields NF ≃ 620.

Comparing Eq. (19) with (11), we remark that the factor μ = 8|D|L·R 2 b/M 2 appears in the latter, too. This could be expected, since it is clearly ‘channel memory’ that drives the need for the cyclic prefix in PM-OFDM as well. Even though μ in (18) was defined and estimated in the context of PM-QAM, we will use it both in PM-QAM and PM-OFDM formulae to simplify the equations and ease comparisons. For instance, Eq. (11) can be rewritten in compact form as:

N′SC=k2k1μ.
(20)

As a consequence, also the formula expressing the number of OPb for PM-OFDM can be likewise simplified and becomes:

OPb,PMOFDM=2qMnTxlog2(nTxk2k1μ)+2qMnRxlog2(nRxk2k1μ)+12M.
(21)

5. Operations per bit for PM-QAM

FIR filters for CD compensation can be implemented in ‘time domain’ (TD), but their computational effort scales quite unfavorably as the number of required filter taps NF [27

27. S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Publishing, Chapter 18, San Diego (CA), (1997).

]. A straightforward count of operations per bit of the two FIR filters shown in Fig. 2, assuming TD, leads to:

OPb,PMQAM/TD=4nRx2(4μ1)/M
(22)

By comparing Eq. (22) to Eq. (21), giving the computational effort for PM-OFDM, it is immediately seen that Eq. (22) scales as μ whereas Eq. (21) scales as log2(μ). The difference is striking and it shows that the TD approach would result in PM-OFDM having a far superior performance than PM-QAM.

However, FIR filters can also be implemented in frequency domain, through the use of FFTs and IFFTs. Special algorithms are needed, because the use of straightforward FFT/IFFTs would perform a ‘circular convolution’ rather than a standard convolution. Perhaps the most well-known such algorithm is called ‘overlap-and-add’ [26

26. A. V. Oppenheim and R. V. Schafer, Digital Signal Processing, Prentice-Hall Inc., Englewood Cliffs (NJ), pp. 110–113, (1975).

], [27

27. S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Publishing, Chapter 18, San Diego (CA), (1997).

], and we will assume its use in the following.

An important point to stress is that FIR implementation through the overlap-and-add algorithm requires ‘block processing’. In other words, it is necessary to accumulate a certain number P of samples of the incoming signal and then block-process them together. The minimum value for P is NF, but choosing P > NF improves the algorithm efficiency. We will come back to the choice of P later. The filter output at each iteration will consist of a block of P samples of the dispersion-compensated signal. Note that the duration of an iteration is P·Ts/n Rx.

Internally to the overlap-and-add algorithm, because of the way the algorithm operates, the block length is increased from P to P+NF. As a result, at every iteration, the overlap-and-add algorithm needs to perform:

  1. one FFT over P+NF samples
  2. P+NF complex multiplications of the FFT output times the channel transfer function
  3. one IFFT over P+NF samples
  4. NF complex sums.

Keeping Eq. (13) in mind, the number of operations per iteration referred to the above list is, respectively:

  1. q·(P+NF)·log2(P+NF)
  2. 6·(P+NF)
  3. q·(P+NF)·log2(P+NF)
  4. 2NF.

Two FIR filters must be implemented, one per polarization. The total number of operations per iteration OPi is then:

OPi=4q·(P+NF)·log2(P+NF)+12P+16NF.
(23)

As found, OPi operations are needed to process a whole block of P input samples, over both polarizations. Then, we observe that each of these dual-polarization blocks carries P/n Rx symbols, corresponding to (M · P)/n Rx bits. As a result, the number of operations per bit is:

OPb=4nRxM·P[q·(P+NF)·log2(P+NF)+3P+4NF].
(24)

For convenience, we now express the overlap-and-add block length as P = pNF. Using this equation and then (19) to relate NF to the system parameters and CD, we get:

OPb,PMQAM=4nRxpM[q(1+p)log2([1+P]NF)+3P+4]=
=4nRxpM[q(1+p)plog2(8nRx[1+p]DLRb2M2)+3+4p].
(25)

Finally, resorting to the channel memory μ to simplify the notation:

OPb,PMQAM=4nRxpM[q(1+p)plog2(8nRx[1+p]μ)+3+4p].
(26)

6. Comparison of PM-OFDM and PM-QAM computational effort for CD compensation

In this section we compare PM-OFDM with PM-QAM, first by trying to establish a realistic case-study. Later, we will attempt to carry out a comparison in more idealized ‘asymptotic’ conditions, to identify fundamental trends.

For the first comparison, we operate in the context of the reference system set-up of Table 1. Specifically M = 4 and so, as PM-QAM format, we actually assume PM-QPSK.

The parameters of Table 1 do not address all of the quantities that appear in (21) and (26). We need to make further assumptions, which we try to do in a realistic and reasonable way. Nonetheless, it is clear that the following comparison cannot be viewed as a general result but, rather, a specific case-study.

We assumed the following.

  • The PM-OFDM cyclic-prefix overhead parameter k is 1.122, corresponding to an OSNR penalty of 0.5 dB and a 12.2% bandwidth efficiency loss.
  • The PM-OFDM Tx and Rx oversampling factors are n Tx = n Rx = 1.25.
  • The PM-QPSK Rx oversampling factor is n Rx = 1.5, which appears to be reachable without incurring substantial penalties [32

    32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

    ].
  • The PM-QPSK overlap-and-add block-length parameter p is set to 7.5, i.e., the total length of the sample block that is processed by the Rx FFT/IFFT is P = (1 + p)NF = 8.5·NF. This choice of p makes the FFT block-length identical between PM-OFDM and PM-QPSK. This makes the comparison more fair because the FFTs and IFFTs used by the two formats then have the same complexity. Note that p could be set as low as 1, allowing for a much smaller block-length for PM-QPSK. However, the total number of operations per bit would actually increase, since decreasing p makes the overlap-and-add algorithm less efficient.

    For both systems we choose q = 5, as it was suggested in [25

    25. H. Bulow, B. Franz, A. Klekkamp, and F. Buchali, “40 Gb/s Distortion Mitigation and DSP-Based Equalisation,” in Proc. ECOC 2007, Berlin, Germany, Sept. (2007).

    ]. As mentioned before, q has little impact on a comparison between PM-QPSK and PM-OFDM, but it is necessary to set it in order to achieve an approximate estimate of the actual total number of operations per bit.

    Fig. 4.Operations per transmitted bit for PM-OFDM and PM-QPSK, vs. link length at 111 Gb/s (top) and vs. bit rate at 1000 km (bottom). D=16.7 ps/(nm·km). Other parameters: for PM-OFDM, k = 1.122 and n Tx = n Rx = 1.25; for PM-QPSK, p = 7.5 and n Rx = 1.5. The FFT parameter q was set to 5.

Fig. 4 shows plots of OP b,PM-OFDM and OP b,PM-QPSK, obtained using the above parameters. The top plot assumes Rb = 111 Gb/s whereas L ranges between 10 and 3000 km. Both systems use an identical FFT/IFFT block length of 4000 samples at 3000 km, linearly decreasing to zero as L goes to zero. For the sake of clarity we refrained from rounding up the FFT/IFFT block lengths to powers of 2.

The total number of operations per bit, for both formats, is rather large. When expressed in number of operations per second, the totals are quite challenging: at 3000 km it is about 8.6 and 11.2 TeraOPs per second, for 111 Gb/s.

In the specific case-study considered above, PM-OFDM has an edge over PM-QPSK, but perhaps the main result is that the difference is relatively small and that the two formats, as it could be predicted by visually inspecting Eqs. (21) and (26), exhibit essentially the same trend of computational effort vs. CD.

This general trend can be brought out with more clarity by making some idealizing assumptions. We first assume that the oversampling parameters may be set to their information-theory minimum, i.e.:

  • for PM-OFDM n Tx = n Rx = 1;
  • for PM-QAM n Rx = 1.

Slightly re-arranging the formulas, we then get:

OPb,PMQAM=4qM[(1+p)plog2(μ)+(1+p)plog2(1+p)+3q+4pq],
(27)
OPb,PMOFDM=4qM[log2(μ)+log2(k2k1)+3q].
(28)

OPb,PMQAM=OPb,PMOFDM4qMlog2(μ).
(29)

7. Impact of polarization-related effects

Birefringence compensation is indispensable for Rx operation, both for PM-OFDM and for PM-QAM. PMD and PDL mitigation are quite important in practice, too. Therefore, in actual systems, countermeasures against such effects must be implemented as well. See for instance [3

3. Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express 13, 7527–7534 (2005). [CrossRef] [PubMed]

], [10

10. C. R. S. Fludger, et al., “Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission,” J. Lightwave Technol. 26, 64–72 (2008). [CrossRef]

], [22

22. Yiran Ma, W. Shieh, and Qi Yang, “Bandwidth-Efficient 21.4 Gb/s Coherent Optical 2x2 MIMO OFDM Transmission,” in Proc. OFC 2008, paper JWA59, San Diego (CA), Feb. 24-28, (2008).

], [32

32. E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

], [33

33. S. J. Savory, “Digital Filters for Coherent Optical Receivers,” Opt. Express 16, 805–817 (2008). [CrossRef]

]. It is then interesting and important to assess the added computational effort necessary to deal with such effects and compare it with the one needed for CD alone.

Fig. 5. Single-stage compensation for combined CD and polarization-effects, using the ‘butterfly’ structure which mixes the ‘x’ and ‘y’ received polarizations. ‘POL’ stands for polarization effects, including birefringence, DGD and PDL. The ‘OAFD’ blocks for PM-QAM perform Overlap-and-Add Frequency-Domain FIR filtering, using 4 suitable complex transfer functions. The ‘COMP’ blocks for PM-OFDM perform an element-by-element complex multiplication between the FFT output arrays and suitable complex compensation arrays.

μμCD+μDGD
(30)

where μCD is given by (18) and μDGD = τDGD · M/Rb. In turn, τDGD is the channel memory due to the maximum differential group delay (DGD) that the system is designed to handle, measured in [ps]. We remind the reader that μ is measured in number of PM-QAM symbols but is also used as a channel memory parameter for PM-OFDM (see end of Sect. 4). In long uncompensated systems, however, μDGD can be expected to be small as compared to μCD, so that in many practical cases μμCD.

Considering PM-OFDM, apart from the above amendment to μ, the number and size of the FTT/IFFTs remains unchanged. After executing the FFTs at the Rx, instead of the simple multiplication of the two output arrays times the CD phase-shift complex correction factor, suitable processing needs to be performed, called either ‘butterfly’ or multiple-input-multiple-output (MIMO) processing [22

22. Yiran Ma, W. Shieh, and Qi Yang, “Bandwidth-Efficient 21.4 Gb/s Coherent Optical 2x2 MIMO OFDM Transmission,” in Proc. OFC 2008, paper JWA59, San Diego (CA), Feb. 24-28, (2008).

], which entails twice the multiplications and two further element-by-element complex sums (i.e., four real sums, see Fig. 5). Once this is factored in, the number of operations per bit for OFDM becomes:

OPb,PMOFDM=2qMnTxlog2(nTxk2k1μ)+2qMnRxlog2(nRxk2k1μ)+28M.
(31)

The only change with respect to Eq. (21) is in the last term, whose coefficient increases from 12 to 28.

Regarding PM-QAM, theoretically a very similar reasoning could be used. At the Rx, FIR filtering must take place in a ‘butterfly’ fashion as well [10

10. C. R. S. Fludger, et al., “Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission,” J. Lightwave Technol. 26, 64–72 (2008). [CrossRef]

], again also called MIMO [3

3. Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express 13, 7527–7534 (2005). [CrossRef] [PubMed]

]. Apart from the correction to μ, the computational effort for the FFTs and IFFTs needed for frequency-domain FIR filtering remains the same as for CD alone. We now have four transfer functions rather than two, plus two more complex sums. Considering the overall butterfly filtering, we easily get:

OPb,PMQAM=4nRxpM[q(1+p)plog2(nRx[1+p]μ)+7+8p].
(32)
Fig. 6. Dual-stage compensation for combined CD and polarization-effects, for PM-QAM. The first stage is implemented in frequency-domain. The second-stage ‘butterfly’ structure could be implemented either in time or frequency-domain.

The only change with respect to Eq. (26) is in the last two terms in the square brackets, that increase from 3 + 4/M to 7 + 8/M.

However, for practical reasons, most proposed PM-QAM Rx implementations keep the CD-compensation stage separate from the polarization-effects mitigation stage. If so, a separate butterfly filtering stage follows two CD-compensating overlap-and-add FIR filters (Fig. 6). We will call this solution ‘dual-stage’ compensation, as opposed to the ‘single-stage’ butterfly processing of Fig. 5, and in the following we will evaluate the impact of this alternative filtering architecture too.

Fig. 7. Operations per transmitted bit for PM-OFDM and PM-QPSK, vs. link length at 111 Gb/s, with fixed DGD τDGD = 120 ps. D=16.7 ps/(nm·km). Other parameters: for PM-OFDM, k = 1.122 and n Tx = n Rx = 1.25; for PM-QPSK, p = 7.5 and n Rx = 1.5. The FFT parameter q was set to 5. Solid and dashed lines: ‘single-stage’ processing; dashed-dotted line: ‘dual-stage’ processing for PM-QPSK.

The comparison between single-stage and dual-stage processing for PM-QPSK shows instead a very substantial overhead for the latter. This is striking in view of the fact that the second stage has to deal with a very small memory (5 taps per filter, given that n Rx = 1.5) as compared to the 310 symbols of channel memory which are dealt with by the CD-compensating first stage. The very different performance is due to the extreme efficiency of FIR filtering in frequency-domain, with the overlap-and-add algorithm, as opposed to the great inefficiency of time-domain FIR filtering.

However, the PM-QPSK dual-stage result of Fig. 7 should be taken with caution. Though accurate regarding the number of operations per bit, the plot gives no indication about circuit complexity. The time-domain second stage typically needs a small number of taps and, as a result, it requires a limited number of electronic elements and circuit floor space. Therefore, whereas the four lower curves of Fig. 7 represent a fair comparison, since they use the same FFT and IFFT technology and also use a similarly (large) data block size, the top curve gives a somewhat pessimistic rendering of the PM-QPSK dual-stage solution.

8. Conclusion

Our appraisal of the computational complexity involved in dispersion compensation for coherent polarization-multiplexed PM-OFDM and PM-QAM systems showed that the two classes of systems have similar behaviors.

As a whole, we believe this study shows that the computational effort for CD and polarization-effects compensation does not seem to be a decisive discriminating factor between PM-OFDM and PM-QAM. A possible competition between the two systems would probably be decided by other aspects.

One such aspect is that PM-QAM systems with simple constellations may not need DACs at the Tx, which are quite challenging to implement at high speed. DACs are instead mandatory for all PM-OFDM systems. Also, PM-OFDM seems to be quite susceptible to non-linear effects and in particular to FWM and non-linear phase-noise [37

37. Xingwen Yi, W. Shieh, and Yiran Ma, “Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission,” J. Lightwave Technol. 26, 1309–1316 (2008). [CrossRef]

], [38

38. H. C. Bao and W. Shieh, “Transmission of Wavelength-Division-Multiplexed Channels With Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 19, 922–924 (2007). [CrossRef]

]. Computational effort due to clock/frame recovery and interpolation, especially for PM-QAM, should be investigated. Both formats need to track birefringence, PMD and PDL, dynamically, but this may be somewhat less difficult for PM-OFDM than for PM-QAM. All of these elements form a quite complex picture and how pros and cons balance out will need further investigation.

Acknowledgments

This work was supported by CISCO Systems within a CARD 2008 contract. It was also supported by the EURO-FOS project, a Network of Excellence (NoE) funded by the European Commission through the 7th ICT-Framework Programme. The optical transmission system simulation software OptSim was supplied by RSoft Design Group Inc.

References and links

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R. Noé, “Phase Noise-Tolerant Synchronous QPSK/BPSK Baseband-Type Intradyne Receiver Concept With Feedforward Carrier Recovery,” J. Lightwave Technol. 23, 802–808 (2005). [CrossRef]

2.

S. Tsukamoto, D. S. Ly-Gagnon, K. Katoh, and K. Kikuchi, “Coherent Demodulation of 40-Gbit/s Polarization-Multiplexed QPSK Signals with 16-GHz Spacing after 200-km Transmission,” in Proc. OFC 2005, PD paper 29, Anaheim (USA), March. 6–11, (2005).

3.

Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express 13, 7527–7534 (2005). [CrossRef] [PubMed]

4.

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent Detection of Optical Quadrature Phase-Shift Keying Signals With Carrier Phase Estimation,” J. Lightwave Technol. 24, 12–21 (2006). [CrossRef]

5.

S. J. Savory et al., “Digital Equalisation of 40 Gbit/s per Wavelength Transmission over 2480km of Standard Fibre without Optical Dispersion Compensation,” in Proc. ECOC 2006, paper Th2.5.5, Cannes (FR), Sept. 24-28, (2006).

6.

C. R. S. Fludger, T. Duthel, T. Wuth, and C. Schulien, “Uncompensated Transmission of 86 Gbit/s Polarization Multiplexed RZ-QPSK over 100km of NDSF Employing Coherent Equalisation,” in Proc. ECOC 2006, PD paper Th4.3.3, Cannes (FR), Sept. 24-28, (2006).

7.

K. Roberts, “Electronic Dispersion Compensation Beyond 10 Gb/s,” in Proc. of IEEE LEOS Summer Topical Meetings, Portland (USA), paper MA2.3, Jul. 23-25, (2007).

8.

G. Charlet et al., “12.8 Tbit/s transmission of 160 PDM-QPSK (160X2X40 Gbit/s) channels with coherent detection over 2550 km,” Proc. ECOC 2007, paper PD 1.6, Berlin (D), Sept. 16-20, (2007).

9.

C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, Han Sun, and M. OSullivan, “WDM Performance and PMD Tolerance of a Coherent 40-Gbit/s Dual-Polarization QPSK Transceiver,” J. Lightwave Technol. 26, 168–175 (2008). [CrossRef]

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C. R. S. Fludger, et al., “Coherent Equalization and POLMUX-RZ-DQPSK for Robust 100-GE Transmission,” J. Lightwave Technol. 26, 64–72 (2008). [CrossRef]

11.

J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear Fiber Impairments Mitigation of 40-Gbit/s Polarization-Multiplexed QPSK by Digital Processing in a Coherent Receiver,” J. Lightwave Technol. 26, 36–42 (2008). [CrossRef]

12.

W. Shieh, H. Bao, and Y. Yang, “Coherent Optical OFDM: Theory and Design,” Opt. Express 16, 841–859 (2008). [CrossRef] [PubMed]

13.

W. Shieh and C. Athaudage, “Coherent Optical Orthogonal Frequency Division Multiplexing,” Electron. Lett. 42, 587–589 (2006). [CrossRef]

14.

W. Shieh, X. Yi, and Y. Tang, “Transmission Experiment of Multi-Gigabit Coherent Optical OFDM Systems over 1000 km SSMF Fibre,” Electron. Lett. 43, 183184 (2007). [CrossRef]

15.

S. L. Jansen, I. Morita, N. Takeda, and H. Tanaka, “20-Gb/s OFDM Transmission over 4160-km SSMF Enabled by RF-pilot Tone Phase Noise Compensation,” Proc. OFC 2007, Anaheim (CA), paper PDP 15, March 25-29, (2007).

16.

S. L. Jansen, I. Morita, T. C. W. Schenck, N. Takeda, and H. Tanaka ‘Coherent Optical 25.8-Gb/s OFDM Transmission Over 4160-km SSMF,’ J. Lightwave Technol. 26, 6–15 (2008). [CrossRef]

17.

B. Goebel, B. Fesl, L. D. Coelho, and N. Hanik, “On the Effect of FWM in Coherent Optical OFDM Systems,” in Proc. OFC 2008, Anaheim (CA), paper JWA58, San Diego (CA), Feb. 24-28, (2008).

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A. J. Lowery and J. Armstrong, “Orthogonal Frequency Division Multiplexing for Dispersion Compensation of Long-Haul Optical Systems,” Opt. Express 14, 2079–2084 (2006). [CrossRef] [PubMed]

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A. J. Lowery, “Improving Sensitivity and Spectral Efficiency in Direct-Detection Optical OFDM Systems,” in Proc. OFC 2008, paper OMM4, San Diego (CA), Feb. 24-28, (2008).

20.

S. L. Jansen, I. Morita, and H. Tanaka, “16x52.5-Gb/s, 50-GHz Spaced, POLMUX-CO-OFDM Transmission over 4,160 km of SSMF Enabled by MIMO Processing,” in Proc. ECOC 2007, paper PD 1.3, Berlin (D), Sept. 16-20, (2007).

21.

S. L. Jansen, I. Morita, and H. Tanaka, “10x121.9-Gb/s PDM-ODFM Transmission with 2-b/s/Hz Spectral Efficiency over 1,000 km of SSMF,” in Proc. OFC 2008, paper PDP2, San Diego (CA), Feb. 24-28, (2008).

22.

Yiran Ma, W. Shieh, and Qi Yang, “Bandwidth-Efficient 21.4 Gb/s Coherent Optical 2x2 MIMO OFDM Transmission,” in Proc. OFC 2008, paper JWA59, San Diego (CA), Feb. 24-28, (2008).

23.

E. Yamada, et al., “Novel No-Guard-Interval PDM CO-OFDM Transmission in 4.1 Tb/s (50x88.8 Gb/s) DWDM Link over 800 km SMF Including 50-Ghz Spaced ROADM Nodes”, in Proc. OFC 2008, paper PDP8, San Diego (CA), Feb. 24-28, (2008).

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W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express 16, 6378–6386 (2008). [CrossRef] [PubMed]

25.

H. Bulow, B. Franz, A. Klekkamp, and F. Buchali, “40 Gb/s Distortion Mitigation and DSP-Based Equalisation,” in Proc. ECOC 2007, Berlin, Germany, Sept. (2007).

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27.

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29.

Xingwen Yi, W. Shieh, and Yan Tang, “Phase Estimation for Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 19, 919–921 (2007). [CrossRef]

30.

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31.

M. G. Taylor, “Coherent Detection Method Using DSP for Demodulation of Signal and Subsequent Equalization of Propagation Impairments,” IEEE Photon. Technol. Lett. 16, 674676 (2004). [CrossRef]

32.

E. Ip and J. M. Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). [CrossRef]

33.

S. J. Savory, “Digital Filters for Coherent Optical Receivers,” Opt. Express 16, 805–817 (2008). [CrossRef]

34.

G. Bosco, P. Poggiolini, and M. Visintin, “Performance Analysis of MLSE Receivers Based on the Square-Root Metric”, J. Lightwave Technol. 26, 2098–2109 (2007). [CrossRef]

35.

P. Poggiolini, G. Bosco, and M. Visintin, “MLSE Receivers and Their Applications in Optical Transmission Systems”, in Proc. of The 20th Annual Meeting of the IEEE LEOS, Lake Buena Vista, Florida (U.S.A.), 21-25 Oct., pp. 216–217, (2007).

36.

P. Poggiolini, G. Bosco, Y. Benlachtar, S. J. Savory, P. Bayvel, R. I. Killey, and J. Prat, “Long-Haul 10 Gbit/s Linear and Non-Linear IMDD Transmission over Uncompensated Standard Fiber Using a SQRT-Metric MLSE Receiver,” Opt. Express 16, 12919–12936 (2008). [CrossRef] [PubMed]

37.

Xingwen Yi, W. Shieh, and Yiran Ma, “Phase Noise Effects on High Spectral Efficiency Coherent Optical OFDM Transmission,” J. Lightwave Technol. 26, 1309–1316 (2008). [CrossRef]

38.

H. C. Bao and W. Shieh, “Transmission of Wavelength-Division-Multiplexed Channels With Coherent Optical OFDM,” IEEE Photon. Technol. Lett. 19, 922–924 (2007). [CrossRef]

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OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
(060.4080) Fiber optics and optical communications : Modulation
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: September 4, 2008
Revised Manuscript: January 2, 2009
Manuscript Accepted: January 14, 2009
Published: January 22, 2009

Citation
P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, "Evaluation of the computational effort for chromatic dispersion compensation in coherent optical PM-OFDM and PM-QAM systems," Opt. Express 17, 1385-1403 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1385


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References

  1. R. Noé, "Phase Noise-Tolerant Synchronous QPSK/BPSK Baseband-Type Intradyne Receiver Concept with Feedforward Carrier Recovery," J. Lightwave Technol. 23, 802-808 (2005). [CrossRef]
  2. S. Tsukamoto, D. S. Ly-Gagnon, K. Katoh, K. Kikuchi, "Coherent Demodulation of 40-Gbit/s Polarization-Multiplexed QPSK Signals with 16-GHz Spacing after 200-km Transmission," in Proc. OFC 2005, PD paper 29, Anaheim (USA), March. 6-11, (2005).
  3. Y. Han and G. Li, "Coherent optical communication using polarization multiple-input-multiple-output," Opt. Express 13, 7527-7534 (2005). [CrossRef] [PubMed]
  4. D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, "Coherent Detection of Optical Quadrature Phase-Shift Keying Signals With Carrier Phase Estimation," J. Lightwave Technol. 24, 12-21 (2006). [CrossRef]
  5. S. J. Savory et al., "Digital Equalisation of 40 Gbit/s per Wavelength Transmission over 2480km of Standard Fibre without Optical Dispersion Compensation," in Proc. ECOC 2006, paper Th2.5.5, Cannes (FR), Sept. 24-28, (2006).
  6. C. R. S. Fludger, T. Duthel, T. Wuth, and C. Schulien, "Uncompensated Transmission of 86 Gbit/s Polarization Multiplexed RZ-QPSK over 100km of NDSF Employing Coherent Equalisation," in Proc. ECOC 2006, PD paper Th4.3.3, Cannes (FR), Sept. 24-28, (2006).
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